Rheol Acta (2012) 51497ndash510DOI 101007s00397-012-0618-x

ORIGINAL CONTRIBUTION

Perturbation solutions of weakly compressible NewtonianPoiseuille flows with Navier slip at the wall

Stella Poyiadji middot Georgios C Georgiou middotKaterina Kaouri middot Kostas D Housiadas

Received 7 October 2011 Revised 13 January 2012 Accepted 17 January 2012 Published online 8 February 2012copy Springer-Verlag 2012

Abstract We consider both the planar and axisym-metric steady laminar Poiseuille flows of a weaklycompressible Newtonian fluid assuming that slip occursalong the wall following Navierrsquos slip equation and thatthe density obeys a linear equation of state A perturba-tion analysis is performed in terms of the primary flowvariables using the dimensionless isothermal compress-ibility as the perturbation parameter Solutions up tothe second order are derived and compared with avail-able analytical results The combined effects of slipcompressibility and inertia are discussed with emphasison the required pressure drop and the average Darcyfriction factor

Keywords Newtonian flow middot Poiseuille flow middotCompressibility middot Slip middot Perturbation solution middotDarcy friction factor

S Poyiadji middot G C Georgiou (B)Department of Mathematics and StatisticsUniversity of Cyprus PO Box 205371678 Nicosia Cypruse-mail georgiosucyaccy

S Poyiadjie-mail map4sp1ucyaccy

K KaouriDepartment of Computer ScienceIntercollege Limassol Ag Fylaxeos 923507 Limassol Cypruse-mail katerinakaourigmailcom

K D HousiadasDepartment of MathematicsUniversity of the Aegean Karlovassi Samos 83200 Greecee-mail housiadaaegeangr

Introduction

In a recent paper (Taliadorou et al 2009) we derivedsecond-order perturbation solutions of both the planarand axisymmetric Poiseuille flows of weakly compress-ible Newtonian fluids using a methodology in whichthe primary flow variables ie the velocity componentsand pressure are perturbed a linear equation of stateis employed and compressibility serves as the pertur-bation parameter The same solutions were derivedby Venerus (2006) and Venerus and Bugajsky (2010)respectively for the axisymmetric and planar flow prob-lems using a streamfunctionvorticity formulation Wehave recently extended the primary-variable methodol-ogy to derive perturbation solutions of the planar andaxisymmetric Poiseuille flows of a weakly compress-ible Oldroyd-B fluid (Housiadas and Georgiou 2011Housiadas et al 2012) The aforementioned referencesprovide useful reviews of previous perturbation andother approximate solutions of the flow problems un-der consideration

The objective of the present paper is to extend ourprevious work for a Newtonian liquid allowing linearslip at the wall in order to study the combined effectsof weak compressibility slip and inertia The impor-tance of slip in a variety of macroscopic flows andprocesses has been emphasized in numerous studies inthe past few decades (Denn 2001 Hatzikiriakos andMigler 2005 and references therein) Strong interesthas also been recently generated due to the effectsof slip in microfluidic applications (Stone et al 2004)Slip of molten polymers has been recently reviewed byHatzikiriakos (2012)

In flows of liquids such as polymer melts andwaxy crude oils compressibility may become important

498 Rheol Acta (2012) 51497ndash510

when the liquids are processed at high pressures whichis the case with polymer extrusion (Hatzikiriakos andDealy 1992 Piau and El 1994) or with flow throughlong tubes (Vinay et al 2006 Wachs et al 2009)The stick-slip polymer extrusion instability referringto the sustained pressure and flow rate oscillationsobserved under constant throughput is attributed tothe combination of compressibility with nonmonotonicslip laws relating the wall shear stress to the slip veloc-ity (Hatzikiriakos and Dealy 1992 Georgiou 2005) asconfirmed by one-dimensional phenomenological mod-els (Dubbeldam and Molenaar 2003) as well numericalsimulations (Taliadorou et al 2007) Tang and Kalyon(2008a b) also developed a mathematical model de-scribing the time-dependent pressure-driven flow ofcompressible polymeric liquids subject to pressure-dependent slip and reported that undamped periodicpressure oscillations in pressure and mean velocityare observed when the boundary condition changesfrom weak to strong slip Taliadorou et al (2008) re-ported extrusion simulations showing that severe com-pressibility combined with inertia may lead to stablesteady-state free surface oscillations similar to thoseobserved experimentally with liquid foams Mitsoulisand Hatzikiriakos (2009) carried out steady flow simu-lations of polytetrafluoroethylene (PTFE) paste extru-sion under severe slip taking into account the significantcompressibility of these pastes

The above material flows are weakly compressiblewhich means that the Mach number Ma is low ieMa ltlt 1 The latter number is defined as the ratioof the characteristic speed of the flow to the speedof sound in the fluid Georgiou and Crochet (1994)pointed out that taking into account the weak com-pressibility of the fluid may not have an effect onthe steady flow solution but changes dramatically theflow dynamics Similarly Felderhof and Ooms (2011)studied the flow of a viscous compressible fluid in a cir-cular tube generated by an impulsive point source andreported that compressibility has a significant effect onthe flow dynamics in confined geometries

The combination of slip with compressibility isalso very important in rarefied gas flows through mi-crochannels and need to be taken into account in themicro-electro-mechanical systems technology (Beskokand Karniadakis 1999 Zhang et al 2009) There areof course some important differences from the liquidflow problem under consideration (a) the continuumassumption may not be valid and slip velocity is ex-pressed in terms of the Knudsen number Kn (the ratioof the mean free path of the gas to the characteristicdimension of the tube) (b) the ideal gas law is usedinstead of the linear equation of state and (c) the flow is

non-isothermal Arkilic and Schmidt (1997) and morerecently Qin et al (2007) derived perturbation approx-imations for compressible gas flow in microchannelswith slip at the wall using the aspect ratio as the pertur-bation parameter According to conventional theorycontinuum-based models for channels apply as long asthe Knudsen number is lower than 001 (Kohl et al2005) On the other hand according to Venerus andBugajsky (2010) effects of slip in microchannels canbe neglected for Knudsen numbers less than 0001Therefore the present analysis concerns not only flowsof compressible liquids with slip at the wall but also gasflows for 0001 lt Kn lt 001

The paper is organized as follows first the govern-ing equations and boundary conditions for the steadycompressible plane Poiseuille flow with slip at the wallare presented the results for the axisymmetric floware provided in the Appendix Both the state and slipequations are assumed to be linear Then the per-turbation method in terms of the primary variableswith the isothermal compressibility as the perturbationparameter is outlined and a solution is derived up to thesecond order Explicit analytical solutions for the twonon-zero velocity components the pressure and thedensity are obtained Finally the results are analyzedand discussed with the emphasis given on the combinedeffects of slip and compressibility on the pressure dropand the Darcy friction factor

Governing equations

We consider the steady laminar plane Poiseuille flowof a Newtonian fluid in a slit of length L and width2H in Cartesian coordinates (x y) as shown inFig 1 Note that throughout the text dimensionalquantities are denoted by a star symbols without astar denote dimensionless variables and numbers It is

xu

wu

w wuτ β =

L

Hx

y

Fig 1 Geometry and symbols for plane Poiseuille flow with slipalong the wall

Rheol Acta (2012) 51497ndash510 499

assumed that slip occurs along the wall according to alinear slip equation

τ lowastw = βlowastulowast

w (1)

where τ lowastw is the wall shear stress β is the constant slip

coefficient and ulowastw is the slip velocity The limiting case

βrarr infin corresponds to the no-slip boundary condition(ulowast

w rarr 0) whereas β = 0 corresponds to the theoreti-cal case of full slip in which the velocity profile is plug

Let us consider first the incompressible one-dimensional flow under constant pressure gradient(minuspartplowastpartxlowast) The velocity ulowast

x (ylowast) is given by

ulowastx

(ylowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)+ 1

2ηlowast

(minuspartplowast

partxlowast

)(Hlowast2 minus ylowast2)

(2)

where η is the constant viscosity Obviously the slipvelocity is given by

ulowastw = ulowast

x

(Hlowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)(3)

If the fluid is compressible the flow becomes bidirec-tional and the two velocity components ulowast

x and ulowasty are

in general functions of both x and y The isothermalcompressibility is a measure of the ability of the ma-terial to change its volume under applied pressure atconstant temperature This is defined by

κlowast equiv minus 1Vlowast

0

(partVlowast

partplowast

)

plowast0T

lowast0

(4)

where V is the specific volume and Vlowast0 is the specific

volume at the reference pressure plowast0 and temperature

Tlowast0 Assuming that κ is constant the above equation

can be integrated yielding an exponential equation ofstate In the present work however we employ a linearequation of state

ρlowast = ρlowast0

[1 + κlowast (

plowast minus plowast0

)] (5)

where ρ is the density and ρlowast0 is the density at the

reference pressure and temperature Equation 5 ap-proximates well the exponential equation of state forsmall values of κ and for small pressures The value ofκ is of the order of 0001 MPaminus1 for molten polymers(Hatzikiriakos and Dealy 1994) and increases by anorder of magnitude (00178ndash00247 MPaminus1) in the caseof PTFE pastes (Mitsoulis and Hatzikiriakos 2009)Mitsoulis and Hatzikiriakos (2009) suggest that forweakly compressible flows the values of κ rangebetween 0 (incompressible fluids) and 002 MPaminus1

(slightly to moderately compressible materials) Thelinear equation of state can also be viewed as a specialcase of the well-established Tait equation and its vari-ants for liquids and polymer melts (Guaily et al 2011)

In order to dedimensionalize the governing equa-tions and the boundary conditions of the flow we scalex by the length of the channel L y by the channelhalf-width H the density ρ by the reference densityρlowast

0 the horizontal velocity ulowastx by the mean velocity at

the channel exit U

Ulowast equiv Mlowast

ρlowast0 HlowastWlowast

where Mlowast is the mass flow rate and W is the unit lengthin the z-direction and the transversal velocity ulowast

y byUlowast HlowastLlowast The Mach number is defined by

Ma equiv Ulowast

σ lowast (6)

where

σ lowast equiv[γ

(partplowast

partρlowast

)

Tlowast

]12

=(

γ

κlowast ρlowast0

)12

(7)

is the speed of sound in the fluid γ being the heatcapacity ratio or adiabatic index (γ equiv clowast

pclowastv)

With the above scalings the dimensionless slip equa-tion becomes

τw = B uw (8)

where all variables are now dimensionless and B is theslip number defined by

B equiv βlowast Hlowast

ηlowast (9)

The dimensionless velocity profile in the case of incom-pressible flow becomes

ux(y) = 3B + 3

+ 3B2 (B + 3)

(1 minus y2) (10)

or

ux (y) = BB

+ B2

(1 minus y2) (11)

where

B equiv 3BB + 3

(12)

is an auxiliary slip number In the no-slip limit B rarr infinand B rarr 3 Therefore

u(0)x (y) = 3

2

(1 minus y2) (13)

500 Rheol Acta (2012) 51497ndash510

which is the standard velocity profile for incompressibleflow with no slip at the wall

By demanding that the dimensionless pressure dropin the case of incompressible flow with no slip atthe wall be equal to 1 the pressure scale should be3ηlowastLlowastUlowastHlowast2 The dimensionless form of the equationof state (Eq 5) is then

ρ = 1 + εp (14)

where

ε equiv 3κlowastηlowastLlowastUlowast

Hlowast2 (15)

is the compressibility number The Mach number takesthe form

Ma =radic

εα Re3γ

hArr ε =(

3γ

αRe

)Ma2 (16)

The present work deals with weakly compressibleflows eg Ma lt 03 Assuming that γ is of the orderof unity there must be εαRe lt 027

The dimensionless forms of the continuity and thex-and y-momentum equations in the case of compress-ible Poiseuille flow under the assumptions of zerobulk velocity and zero gravity (Taliadorou et al 2009)are

part (ρux)

partx+ part

(ρuy

)

party= 0 (17)

αRe ρ

(ux

partux

partx+ uy

partux

party

)= minus3

partppartx

+ α2 part2ux

partx2 + part2ux

party2

+α2

3

(part2uy

partxparty+ part2ux

partx2

)(18)

α3 Reρ(

uxpartuy

partx+uy

partuy

party

)= minus3

partpparty

+α4 part2uy

partx2 + α2 part2uy

party2

+α2

3

(part2ux

partxparty+ part2uy

party2

)(19)

where

Re equiv ρlowast0 HlowastUlowast

ηlowast (20)

is the Reynolds number and

α equiv Hlowast

Llowast (21)

is the aspect ratio of the channel

As for the boundary conditions the usual symmetryconditions are applied along the symmetry plane alongthe wall ux obeys the slip equation (Eq 8) while uy van-ishes Moreover the pressure at the upper right cornerof the flow domain is set to zero and the mass flowrate at the exit plane should be equal to 1 Thereforethe conditions that close the system of the governingequations are the following

partux

party(x 0) = uy(x 0) = 0 (22)

minuspartux

party(x 1) = B ux(x 1) and uy(x 1) = 0 (23)

p(1 1) = 0 (24)

int 1

0ρuxdy

∣∣∣∣x=1

= 1 (25)

As in Venerus (2006) and Taliadorou et al (2009) noboundary conditions for the velocity are imposed atthe entrance and exit planes (x = 0 and 1) The flowproblem defined by Eqs 14 17ndash19 and 22ndash25 involvesfour dependent variables ux uy p and ρ and fourdimensionless numbers ε B Re and α Even thoughthe density ρ can be eliminated by means of Eq 14it is kept in order to facilitate the derivation of theperturbation solution

Perturbation solution

The present work deals with weakly compressibleflows that is the Mach number is small typically Malt 03 From Eq 16 it is deduced that as long as Ma issmall and γ (αRe) is of the order of unity or smallerthe compressibility number ε is also a small numberthat can be used as the perturbation parameter Wethus perturb all primary variables ux uy p and ρ asfollows

ux = u(0)x + εu(1)

x + ε2u(2)x + O

(ε3

)

uy = u(0)y + εu(1)

y + ε2u(2)y + O

(ε3

)

p = p(0) + εp(1) + ε2 p(2) + O(ε3

)

ρ = ρ(0) + ερ(1) + ε2ρ(2) + O(ε3

)(26)

By substituting expansions (Eq 26) into the governingEqs 14 17ndash19 and also in the boundary conditions22ndash25 and by collecting the terms of a given order inε the corresponding perturbation equations and the

Rheol Acta (2012) 51497ndash510 501

boundary conditions are obtained These can be foundin Taliadorou et al (2009) who present the more gen-eral case with non-zero bulk viscosity As for the slipequation it can easily be shown that

minuspartu(k)x

party(x 1) = B u(k)

x (x 1) k = 0 1 2 (27)

where k is the order of the perturbation The deriva-tion of the leading-order solutions which is basedon the assumption that the transverse velocity uy

is zero is straightforward and the methodology isdescribed in detail by Poyiadji (2012) The pertur-bation solution of the flow problem up to secondorder is

ux = B2B

(B+2minusBy2)

+ ε

[

minus B2

6B

(B+2minusBy2)(1minusx)minusαReB

4

7560B2

[5B2+45B+98minus3

(11B2+77B+140

)y2+35

(B2+5B+6

)y4minus7

(B2+3B

)y6]

]

+ ε2

[B

3

12B

(B + 2 minus By2) (1 minus x)2 minus αReB

5

7560B4

times [19B4+171B3+658B2+1260B+840+3

(3B4+21B3minus140B

)y2minus35

(B4+5B3+6B2)y4+7

(B4+3B3)y6]

times (1minusx)minus α2 B4

648B2

[B2+3B+8 + 2

(B2+7B

)y2 minus 3

(B2 + 3B

)y4] minus α2 Re2 B

7

314344800B5

times [2193B5 + 35088B4 + 221641B3 + 731346B2 + 1409100B + 1358280

minus 4(2839B5 + 39746B4 + 239316B3 + 803880B2 + 1576575B + 1455300

)y2

+ 2310(B5+12B4+111B3+602B2+1470B+1260

)y4+924

(13B5+130B4+448B3+560B2+105B

)y6

minus5775(B5 + 8B4 + 21B3 + 18B2) y8 + 616

(B5 + 6B4 + 9B3) y10]

]

+ O(ε3) (28)

uy = ε2 αReB5

11340B2 y(1 minus y2) [

5B2 + 45B + 98 minus 2(3B2 + 16B + 21

)y2 + (

B2 + 3B)

y4] + O(ε3) (29)

p = B3

(1 minus x) + ε

[

minus B2

18(1 minus x)2 + αReB

4

315B3

(2B3 + 14B2 + 35B + 35

)(1 minus x) + α2 B

2

54

(1 minus y2)

]

+ ε2

[B

3

54(1minusx)3minus αReB

5

945B3

(4B3+28B2+70B+70

)(1minusx)2minus α2 B

4

486B2

[11B2+53B+72minus3

(B2 + 3B

)y2] (1 minus x)

+ α2 Re2 B7

29469825B6

(3044B6+42616B5+267036B4+951720B3+1964655B2+2182950B+1091475

)(1 minus x)

+ α3 ReB5

204120B3

[97B3 + 889B2 + 2310B + 1260 minus 3

(79B3 + 553B2 + 1120B + 420

)y2

+175(B3 + 5B2 + 6B

)y4 minus 35

(B3 + 3B2) y6]

]

+ O(ε3) (30)

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

498 Rheol Acta (2012) 51497ndash510

when the liquids are processed at high pressures whichis the case with polymer extrusion (Hatzikiriakos andDealy 1992 Piau and El 1994) or with flow throughlong tubes (Vinay et al 2006 Wachs et al 2009)The stick-slip polymer extrusion instability referringto the sustained pressure and flow rate oscillationsobserved under constant throughput is attributed tothe combination of compressibility with nonmonotonicslip laws relating the wall shear stress to the slip veloc-ity (Hatzikiriakos and Dealy 1992 Georgiou 2005) asconfirmed by one-dimensional phenomenological mod-els (Dubbeldam and Molenaar 2003) as well numericalsimulations (Taliadorou et al 2007) Tang and Kalyon(2008a b) also developed a mathematical model de-scribing the time-dependent pressure-driven flow ofcompressible polymeric liquids subject to pressure-dependent slip and reported that undamped periodicpressure oscillations in pressure and mean velocityare observed when the boundary condition changesfrom weak to strong slip Taliadorou et al (2008) re-ported extrusion simulations showing that severe com-pressibility combined with inertia may lead to stablesteady-state free surface oscillations similar to thoseobserved experimentally with liquid foams Mitsoulisand Hatzikiriakos (2009) carried out steady flow simu-lations of polytetrafluoroethylene (PTFE) paste extru-sion under severe slip taking into account the significantcompressibility of these pastes

The above material flows are weakly compressiblewhich means that the Mach number Ma is low ieMa ltlt 1 The latter number is defined as the ratioof the characteristic speed of the flow to the speedof sound in the fluid Georgiou and Crochet (1994)pointed out that taking into account the weak com-pressibility of the fluid may not have an effect onthe steady flow solution but changes dramatically theflow dynamics Similarly Felderhof and Ooms (2011)studied the flow of a viscous compressible fluid in a cir-cular tube generated by an impulsive point source andreported that compressibility has a significant effect onthe flow dynamics in confined geometries

The combination of slip with compressibility isalso very important in rarefied gas flows through mi-crochannels and need to be taken into account in themicro-electro-mechanical systems technology (Beskokand Karniadakis 1999 Zhang et al 2009) There areof course some important differences from the liquidflow problem under consideration (a) the continuumassumption may not be valid and slip velocity is ex-pressed in terms of the Knudsen number Kn (the ratioof the mean free path of the gas to the characteristicdimension of the tube) (b) the ideal gas law is usedinstead of the linear equation of state and (c) the flow is

non-isothermal Arkilic and Schmidt (1997) and morerecently Qin et al (2007) derived perturbation approx-imations for compressible gas flow in microchannelswith slip at the wall using the aspect ratio as the pertur-bation parameter According to conventional theorycontinuum-based models for channels apply as long asthe Knudsen number is lower than 001 (Kohl et al2005) On the other hand according to Venerus andBugajsky (2010) effects of slip in microchannels canbe neglected for Knudsen numbers less than 0001Therefore the present analysis concerns not only flowsof compressible liquids with slip at the wall but also gasflows for 0001 lt Kn lt 001

The paper is organized as follows first the govern-ing equations and boundary conditions for the steadycompressible plane Poiseuille flow with slip at the wallare presented the results for the axisymmetric floware provided in the Appendix Both the state and slipequations are assumed to be linear Then the per-turbation method in terms of the primary variableswith the isothermal compressibility as the perturbationparameter is outlined and a solution is derived up to thesecond order Explicit analytical solutions for the twonon-zero velocity components the pressure and thedensity are obtained Finally the results are analyzedand discussed with the emphasis given on the combinedeffects of slip and compressibility on the pressure dropand the Darcy friction factor

Governing equations

We consider the steady laminar plane Poiseuille flowof a Newtonian fluid in a slit of length L and width2H in Cartesian coordinates (x y) as shown inFig 1 Note that throughout the text dimensionalquantities are denoted by a star symbols without astar denote dimensionless variables and numbers It is

xu

wu

w wuτ β =

L

Hx

y

Fig 1 Geometry and symbols for plane Poiseuille flow with slipalong the wall

Rheol Acta (2012) 51497ndash510 499

assumed that slip occurs along the wall according to alinear slip equation

τ lowastw = βlowastulowast

w (1)

where τ lowastw is the wall shear stress β is the constant slip

coefficient and ulowastw is the slip velocity The limiting case

βrarr infin corresponds to the no-slip boundary condition(ulowast

w rarr 0) whereas β = 0 corresponds to the theoreti-cal case of full slip in which the velocity profile is plug

Let us consider first the incompressible one-dimensional flow under constant pressure gradient(minuspartplowastpartxlowast) The velocity ulowast

x (ylowast) is given by

ulowastx

(ylowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)+ 1

2ηlowast

(minuspartplowast

partxlowast

)(Hlowast2 minus ylowast2)

(2)

where η is the constant viscosity Obviously the slipvelocity is given by

ulowastw = ulowast

x

(Hlowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)(3)

If the fluid is compressible the flow becomes bidirec-tional and the two velocity components ulowast

x and ulowasty are

in general functions of both x and y The isothermalcompressibility is a measure of the ability of the ma-terial to change its volume under applied pressure atconstant temperature This is defined by

κlowast equiv minus 1Vlowast

0

(partVlowast

partplowast

)

plowast0T

lowast0

(4)

where V is the specific volume and Vlowast0 is the specific

volume at the reference pressure plowast0 and temperature

Tlowast0 Assuming that κ is constant the above equation

can be integrated yielding an exponential equation ofstate In the present work however we employ a linearequation of state

ρlowast = ρlowast0

[1 + κlowast (

plowast minus plowast0

)] (5)

where ρ is the density and ρlowast0 is the density at the

reference pressure and temperature Equation 5 ap-proximates well the exponential equation of state forsmall values of κ and for small pressures The value ofκ is of the order of 0001 MPaminus1 for molten polymers(Hatzikiriakos and Dealy 1994) and increases by anorder of magnitude (00178ndash00247 MPaminus1) in the caseof PTFE pastes (Mitsoulis and Hatzikiriakos 2009)Mitsoulis and Hatzikiriakos (2009) suggest that forweakly compressible flows the values of κ rangebetween 0 (incompressible fluids) and 002 MPaminus1

(slightly to moderately compressible materials) Thelinear equation of state can also be viewed as a specialcase of the well-established Tait equation and its vari-ants for liquids and polymer melts (Guaily et al 2011)

In order to dedimensionalize the governing equa-tions and the boundary conditions of the flow we scalex by the length of the channel L y by the channelhalf-width H the density ρ by the reference densityρlowast

0 the horizontal velocity ulowastx by the mean velocity at

the channel exit U

Ulowast equiv Mlowast

ρlowast0 HlowastWlowast

where Mlowast is the mass flow rate and W is the unit lengthin the z-direction and the transversal velocity ulowast

y byUlowast HlowastLlowast The Mach number is defined by

Ma equiv Ulowast

σ lowast (6)

where

σ lowast equiv[γ

(partplowast

partρlowast

)

Tlowast

]12

=(

γ

κlowast ρlowast0

)12

(7)

is the speed of sound in the fluid γ being the heatcapacity ratio or adiabatic index (γ equiv clowast

pclowastv)

With the above scalings the dimensionless slip equa-tion becomes

τw = B uw (8)

where all variables are now dimensionless and B is theslip number defined by

B equiv βlowast Hlowast

ηlowast (9)

The dimensionless velocity profile in the case of incom-pressible flow becomes

ux(y) = 3B + 3

+ 3B2 (B + 3)

(1 minus y2) (10)

or

ux (y) = BB

+ B2

(1 minus y2) (11)

where

B equiv 3BB + 3

(12)

is an auxiliary slip number In the no-slip limit B rarr infinand B rarr 3 Therefore

u(0)x (y) = 3

2

(1 minus y2) (13)

500 Rheol Acta (2012) 51497ndash510

which is the standard velocity profile for incompressibleflow with no slip at the wall

By demanding that the dimensionless pressure dropin the case of incompressible flow with no slip atthe wall be equal to 1 the pressure scale should be3ηlowastLlowastUlowastHlowast2 The dimensionless form of the equationof state (Eq 5) is then

ρ = 1 + εp (14)

where

ε equiv 3κlowastηlowastLlowastUlowast

Hlowast2 (15)

is the compressibility number The Mach number takesthe form

Ma =radic

εα Re3γ

hArr ε =(

3γ

αRe

)Ma2 (16)

The present work deals with weakly compressibleflows eg Ma lt 03 Assuming that γ is of the orderof unity there must be εαRe lt 027

The dimensionless forms of the continuity and thex-and y-momentum equations in the case of compress-ible Poiseuille flow under the assumptions of zerobulk velocity and zero gravity (Taliadorou et al 2009)are

part (ρux)

partx+ part

(ρuy

)

party= 0 (17)

αRe ρ

(ux

partux

partx+ uy

partux

party

)= minus3

partppartx

+ α2 part2ux

partx2 + part2ux

party2

+α2

3

(part2uy

partxparty+ part2ux

partx2

)(18)

α3 Reρ(

uxpartuy

partx+uy

partuy

party

)= minus3

partpparty

+α4 part2uy

partx2 + α2 part2uy

party2

+α2

3

(part2ux

partxparty+ part2uy

party2

)(19)

where

Re equiv ρlowast0 HlowastUlowast

ηlowast (20)

is the Reynolds number and

α equiv Hlowast

Llowast (21)

is the aspect ratio of the channel

As for the boundary conditions the usual symmetryconditions are applied along the symmetry plane alongthe wall ux obeys the slip equation (Eq 8) while uy van-ishes Moreover the pressure at the upper right cornerof the flow domain is set to zero and the mass flowrate at the exit plane should be equal to 1 Thereforethe conditions that close the system of the governingequations are the following

partux

party(x 0) = uy(x 0) = 0 (22)

minuspartux

party(x 1) = B ux(x 1) and uy(x 1) = 0 (23)

p(1 1) = 0 (24)

int 1

0ρuxdy

∣∣∣∣x=1

= 1 (25)

As in Venerus (2006) and Taliadorou et al (2009) noboundary conditions for the velocity are imposed atthe entrance and exit planes (x = 0 and 1) The flowproblem defined by Eqs 14 17ndash19 and 22ndash25 involvesfour dependent variables ux uy p and ρ and fourdimensionless numbers ε B Re and α Even thoughthe density ρ can be eliminated by means of Eq 14it is kept in order to facilitate the derivation of theperturbation solution

Perturbation solution

The present work deals with weakly compressibleflows that is the Mach number is small typically Malt 03 From Eq 16 it is deduced that as long as Ma issmall and γ (αRe) is of the order of unity or smallerthe compressibility number ε is also a small numberthat can be used as the perturbation parameter Wethus perturb all primary variables ux uy p and ρ asfollows

ux = u(0)x + εu(1)

x + ε2u(2)x + O

(ε3

)

uy = u(0)y + εu(1)

y + ε2u(2)y + O

(ε3

)

p = p(0) + εp(1) + ε2 p(2) + O(ε3

)

ρ = ρ(0) + ερ(1) + ε2ρ(2) + O(ε3

)(26)

By substituting expansions (Eq 26) into the governingEqs 14 17ndash19 and also in the boundary conditions22ndash25 and by collecting the terms of a given order inε the corresponding perturbation equations and the

Rheol Acta (2012) 51497ndash510 501

boundary conditions are obtained These can be foundin Taliadorou et al (2009) who present the more gen-eral case with non-zero bulk viscosity As for the slipequation it can easily be shown that

minuspartu(k)x

party(x 1) = B u(k)

x (x 1) k = 0 1 2 (27)

where k is the order of the perturbation The deriva-tion of the leading-order solutions which is basedon the assumption that the transverse velocity uy

is zero is straightforward and the methodology isdescribed in detail by Poyiadji (2012) The pertur-bation solution of the flow problem up to secondorder is

ux = B2B

(B+2minusBy2)

+ ε

[

minus B2

6B

(B+2minusBy2)(1minusx)minusαReB

4

7560B2

[5B2+45B+98minus3

(11B2+77B+140

)y2+35

(B2+5B+6

)y4minus7

(B2+3B

)y6]

]

+ ε2

[B

3

12B

(B + 2 minus By2) (1 minus x)2 minus αReB

5

7560B4

times [19B4+171B3+658B2+1260B+840+3

(3B4+21B3minus140B

)y2minus35

(B4+5B3+6B2)y4+7

(B4+3B3)y6]

times (1minusx)minus α2 B4

648B2

[B2+3B+8 + 2

(B2+7B

)y2 minus 3

(B2 + 3B

)y4] minus α2 Re2 B

7

314344800B5

times [2193B5 + 35088B4 + 221641B3 + 731346B2 + 1409100B + 1358280

minus 4(2839B5 + 39746B4 + 239316B3 + 803880B2 + 1576575B + 1455300

)y2

+ 2310(B5+12B4+111B3+602B2+1470B+1260

)y4+924

(13B5+130B4+448B3+560B2+105B

)y6

minus5775(B5 + 8B4 + 21B3 + 18B2) y8 + 616

(B5 + 6B4 + 9B3) y10]

]

+ O(ε3) (28)

uy = ε2 αReB5

11340B2 y(1 minus y2) [

5B2 + 45B + 98 minus 2(3B2 + 16B + 21

)y2 + (

B2 + 3B)

y4] + O(ε3) (29)

p = B3

(1 minus x) + ε

[

minus B2

18(1 minus x)2 + αReB

4

315B3

(2B3 + 14B2 + 35B + 35

)(1 minus x) + α2 B

2

54

(1 minus y2)

]

+ ε2

[B

3

54(1minusx)3minus αReB

5

945B3

(4B3+28B2+70B+70

)(1minusx)2minus α2 B

4

486B2

[11B2+53B+72minus3

(B2 + 3B

)y2] (1 minus x)

+ α2 Re2 B7

29469825B6

(3044B6+42616B5+267036B4+951720B3+1964655B2+2182950B+1091475

)(1 minus x)

+ α3 ReB5

204120B3

[97B3 + 889B2 + 2310B + 1260 minus 3

(79B3 + 553B2 + 1120B + 420

)y2

+175(B3 + 5B2 + 6B

)y4 minus 35

(B3 + 3B2) y6]

]

+ O(ε3) (30)

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 499

assumed that slip occurs along the wall according to alinear slip equation

τ lowastw = βlowastulowast

w (1)

where τ lowastw is the wall shear stress β is the constant slip

coefficient and ulowastw is the slip velocity The limiting case

βrarr infin corresponds to the no-slip boundary condition(ulowast

w rarr 0) whereas β = 0 corresponds to the theoreti-cal case of full slip in which the velocity profile is plug

Let us consider first the incompressible one-dimensional flow under constant pressure gradient(minuspartplowastpartxlowast) The velocity ulowast

x (ylowast) is given by

ulowastx

(ylowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)+ 1

2ηlowast

(minuspartplowast

partxlowast

)(Hlowast2 minus ylowast2)

(2)

where η is the constant viscosity Obviously the slipvelocity is given by

ulowastw = ulowast

x

(Hlowast) = Hlowast

βlowast

(minuspartplowast

partxlowast

)(3)

If the fluid is compressible the flow becomes bidirec-tional and the two velocity components ulowast

x and ulowasty are

in general functions of both x and y The isothermalcompressibility is a measure of the ability of the ma-terial to change its volume under applied pressure atconstant temperature This is defined by

κlowast equiv minus 1Vlowast

0

(partVlowast

partplowast

)

plowast0T

lowast0

(4)

where V is the specific volume and Vlowast0 is the specific

volume at the reference pressure plowast0 and temperature

Tlowast0 Assuming that κ is constant the above equation

can be integrated yielding an exponential equation ofstate In the present work however we employ a linearequation of state

ρlowast = ρlowast0

[1 + κlowast (

plowast minus plowast0

)] (5)

where ρ is the density and ρlowast0 is the density at the

reference pressure and temperature Equation 5 ap-proximates well the exponential equation of state forsmall values of κ and for small pressures The value ofκ is of the order of 0001 MPaminus1 for molten polymers(Hatzikiriakos and Dealy 1994) and increases by anorder of magnitude (00178ndash00247 MPaminus1) in the caseof PTFE pastes (Mitsoulis and Hatzikiriakos 2009)Mitsoulis and Hatzikiriakos (2009) suggest that forweakly compressible flows the values of κ rangebetween 0 (incompressible fluids) and 002 MPaminus1

(slightly to moderately compressible materials) Thelinear equation of state can also be viewed as a specialcase of the well-established Tait equation and its vari-ants for liquids and polymer melts (Guaily et al 2011)

In order to dedimensionalize the governing equa-tions and the boundary conditions of the flow we scalex by the length of the channel L y by the channelhalf-width H the density ρ by the reference densityρlowast

0 the horizontal velocity ulowastx by the mean velocity at

the channel exit U

Ulowast equiv Mlowast

ρlowast0 HlowastWlowast

where Mlowast is the mass flow rate and W is the unit lengthin the z-direction and the transversal velocity ulowast

y byUlowast HlowastLlowast The Mach number is defined by

Ma equiv Ulowast

σ lowast (6)

where

σ lowast equiv[γ

(partplowast

partρlowast

)

Tlowast

]12

=(

γ

κlowast ρlowast0

)12

(7)

is the speed of sound in the fluid γ being the heatcapacity ratio or adiabatic index (γ equiv clowast

pclowastv)

With the above scalings the dimensionless slip equa-tion becomes

τw = B uw (8)

where all variables are now dimensionless and B is theslip number defined by

B equiv βlowast Hlowast

ηlowast (9)

The dimensionless velocity profile in the case of incom-pressible flow becomes

ux(y) = 3B + 3

+ 3B2 (B + 3)

(1 minus y2) (10)

or

ux (y) = BB

+ B2

(1 minus y2) (11)

where

B equiv 3BB + 3

(12)

is an auxiliary slip number In the no-slip limit B rarr infinand B rarr 3 Therefore

u(0)x (y) = 3

2

(1 minus y2) (13)

500 Rheol Acta (2012) 51497ndash510

which is the standard velocity profile for incompressibleflow with no slip at the wall

By demanding that the dimensionless pressure dropin the case of incompressible flow with no slip atthe wall be equal to 1 the pressure scale should be3ηlowastLlowastUlowastHlowast2 The dimensionless form of the equationof state (Eq 5) is then

ρ = 1 + εp (14)

where

ε equiv 3κlowastηlowastLlowastUlowast

Hlowast2 (15)

is the compressibility number The Mach number takesthe form

Ma =radic

εα Re3γ

hArr ε =(

3γ

αRe

)Ma2 (16)

The present work deals with weakly compressibleflows eg Ma lt 03 Assuming that γ is of the orderof unity there must be εαRe lt 027

The dimensionless forms of the continuity and thex-and y-momentum equations in the case of compress-ible Poiseuille flow under the assumptions of zerobulk velocity and zero gravity (Taliadorou et al 2009)are

part (ρux)

partx+ part

(ρuy

)

party= 0 (17)

αRe ρ

(ux

partux

partx+ uy

partux

party

)= minus3

partppartx

+ α2 part2ux

partx2 + part2ux

party2

+α2

3

(part2uy

partxparty+ part2ux

partx2

)(18)

α3 Reρ(

uxpartuy

partx+uy

partuy

party

)= minus3

partpparty

+α4 part2uy

partx2 + α2 part2uy

party2

+α2

3

(part2ux

partxparty+ part2uy

party2

)(19)

where

Re equiv ρlowast0 HlowastUlowast

ηlowast (20)

is the Reynolds number and

α equiv Hlowast

Llowast (21)

is the aspect ratio of the channel

As for the boundary conditions the usual symmetryconditions are applied along the symmetry plane alongthe wall ux obeys the slip equation (Eq 8) while uy van-ishes Moreover the pressure at the upper right cornerof the flow domain is set to zero and the mass flowrate at the exit plane should be equal to 1 Thereforethe conditions that close the system of the governingequations are the following

partux

party(x 0) = uy(x 0) = 0 (22)

minuspartux

party(x 1) = B ux(x 1) and uy(x 1) = 0 (23)

p(1 1) = 0 (24)

int 1

0ρuxdy

∣∣∣∣x=1

= 1 (25)

As in Venerus (2006) and Taliadorou et al (2009) noboundary conditions for the velocity are imposed atthe entrance and exit planes (x = 0 and 1) The flowproblem defined by Eqs 14 17ndash19 and 22ndash25 involvesfour dependent variables ux uy p and ρ and fourdimensionless numbers ε B Re and α Even thoughthe density ρ can be eliminated by means of Eq 14it is kept in order to facilitate the derivation of theperturbation solution

Perturbation solution

The present work deals with weakly compressibleflows that is the Mach number is small typically Malt 03 From Eq 16 it is deduced that as long as Ma issmall and γ (αRe) is of the order of unity or smallerthe compressibility number ε is also a small numberthat can be used as the perturbation parameter Wethus perturb all primary variables ux uy p and ρ asfollows

ux = u(0)x + εu(1)

x + ε2u(2)x + O

(ε3

)

uy = u(0)y + εu(1)

y + ε2u(2)y + O

(ε3

)

p = p(0) + εp(1) + ε2 p(2) + O(ε3

)

ρ = ρ(0) + ερ(1) + ε2ρ(2) + O(ε3

)(26)

By substituting expansions (Eq 26) into the governingEqs 14 17ndash19 and also in the boundary conditions22ndash25 and by collecting the terms of a given order inε the corresponding perturbation equations and the

Rheol Acta (2012) 51497ndash510 501

boundary conditions are obtained These can be foundin Taliadorou et al (2009) who present the more gen-eral case with non-zero bulk viscosity As for the slipequation it can easily be shown that

minuspartu(k)x

party(x 1) = B u(k)

x (x 1) k = 0 1 2 (27)

where k is the order of the perturbation The deriva-tion of the leading-order solutions which is basedon the assumption that the transverse velocity uy

is zero is straightforward and the methodology isdescribed in detail by Poyiadji (2012) The pertur-bation solution of the flow problem up to secondorder is

ux = B2B

(B+2minusBy2)

+ ε

[

minus B2

6B

(B+2minusBy2)(1minusx)minusαReB

4

7560B2

[5B2+45B+98minus3

(11B2+77B+140

)y2+35

(B2+5B+6

)y4minus7

(B2+3B

)y6]

]

+ ε2

[B

3

12B

(B + 2 minus By2) (1 minus x)2 minus αReB

5

7560B4

times [19B4+171B3+658B2+1260B+840+3

(3B4+21B3minus140B

)y2minus35

(B4+5B3+6B2)y4+7

(B4+3B3)y6]

times (1minusx)minus α2 B4

648B2

[B2+3B+8 + 2

(B2+7B

)y2 minus 3

(B2 + 3B

)y4] minus α2 Re2 B

7

314344800B5

times [2193B5 + 35088B4 + 221641B3 + 731346B2 + 1409100B + 1358280

minus 4(2839B5 + 39746B4 + 239316B3 + 803880B2 + 1576575B + 1455300

)y2

+ 2310(B5+12B4+111B3+602B2+1470B+1260

)y4+924

(13B5+130B4+448B3+560B2+105B

)y6

minus5775(B5 + 8B4 + 21B3 + 18B2) y8 + 616

(B5 + 6B4 + 9B3) y10]

]

+ O(ε3) (28)

uy = ε2 αReB5

11340B2 y(1 minus y2) [

5B2 + 45B + 98 minus 2(3B2 + 16B + 21

)y2 + (

B2 + 3B)

y4] + O(ε3) (29)

p = B3

(1 minus x) + ε

[

minus B2

18(1 minus x)2 + αReB

4

315B3

(2B3 + 14B2 + 35B + 35

)(1 minus x) + α2 B

2

54

(1 minus y2)

]

+ ε2

[B

3

54(1minusx)3minus αReB

5

945B3

(4B3+28B2+70B+70

)(1minusx)2minus α2 B

4

486B2

[11B2+53B+72minus3

(B2 + 3B

)y2] (1 minus x)

+ α2 Re2 B7

29469825B6

(3044B6+42616B5+267036B4+951720B3+1964655B2+2182950B+1091475

)(1 minus x)

+ α3 ReB5

204120B3

[97B3 + 889B2 + 2310B + 1260 minus 3

(79B3 + 553B2 + 1120B + 420

)y2

+175(B3 + 5B2 + 6B

)y4 minus 35

(B3 + 3B2) y6]

]

+ O(ε3) (30)

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

500 Rheol Acta (2012) 51497ndash510

which is the standard velocity profile for incompressibleflow with no slip at the wall

By demanding that the dimensionless pressure dropin the case of incompressible flow with no slip atthe wall be equal to 1 the pressure scale should be3ηlowastLlowastUlowastHlowast2 The dimensionless form of the equationof state (Eq 5) is then

ρ = 1 + εp (14)

where

ε equiv 3κlowastηlowastLlowastUlowast

Hlowast2 (15)

is the compressibility number The Mach number takesthe form

Ma =radic

εα Re3γ

hArr ε =(

3γ

αRe

)Ma2 (16)

The present work deals with weakly compressibleflows eg Ma lt 03 Assuming that γ is of the orderof unity there must be εαRe lt 027

The dimensionless forms of the continuity and thex-and y-momentum equations in the case of compress-ible Poiseuille flow under the assumptions of zerobulk velocity and zero gravity (Taliadorou et al 2009)are

part (ρux)

partx+ part

(ρuy

)

party= 0 (17)

αRe ρ

(ux

partux

partx+ uy

partux

party

)= minus3

partppartx

+ α2 part2ux

partx2 + part2ux

party2

+α2

3

(part2uy

partxparty+ part2ux

partx2

)(18)

α3 Reρ(

uxpartuy

partx+uy

partuy

party

)= minus3

partpparty

+α4 part2uy

partx2 + α2 part2uy

party2

+α2

3

(part2ux

partxparty+ part2uy

party2

)(19)

where

Re equiv ρlowast0 HlowastUlowast

ηlowast (20)

is the Reynolds number and

α equiv Hlowast

Llowast (21)

is the aspect ratio of the channel

As for the boundary conditions the usual symmetryconditions are applied along the symmetry plane alongthe wall ux obeys the slip equation (Eq 8) while uy van-ishes Moreover the pressure at the upper right cornerof the flow domain is set to zero and the mass flowrate at the exit plane should be equal to 1 Thereforethe conditions that close the system of the governingequations are the following

partux

party(x 0) = uy(x 0) = 0 (22)

minuspartux

party(x 1) = B ux(x 1) and uy(x 1) = 0 (23)

p(1 1) = 0 (24)

int 1

0ρuxdy

∣∣∣∣x=1

= 1 (25)

As in Venerus (2006) and Taliadorou et al (2009) noboundary conditions for the velocity are imposed atthe entrance and exit planes (x = 0 and 1) The flowproblem defined by Eqs 14 17ndash19 and 22ndash25 involvesfour dependent variables ux uy p and ρ and fourdimensionless numbers ε B Re and α Even thoughthe density ρ can be eliminated by means of Eq 14it is kept in order to facilitate the derivation of theperturbation solution

Perturbation solution

The present work deals with weakly compressibleflows that is the Mach number is small typically Malt 03 From Eq 16 it is deduced that as long as Ma issmall and γ (αRe) is of the order of unity or smallerthe compressibility number ε is also a small numberthat can be used as the perturbation parameter Wethus perturb all primary variables ux uy p and ρ asfollows

ux = u(0)x + εu(1)

x + ε2u(2)x + O

(ε3

)

uy = u(0)y + εu(1)

y + ε2u(2)y + O

(ε3

)

p = p(0) + εp(1) + ε2 p(2) + O(ε3

)

ρ = ρ(0) + ερ(1) + ε2ρ(2) + O(ε3

)(26)

By substituting expansions (Eq 26) into the governingEqs 14 17ndash19 and also in the boundary conditions22ndash25 and by collecting the terms of a given order inε the corresponding perturbation equations and the

Rheol Acta (2012) 51497ndash510 501

boundary conditions are obtained These can be foundin Taliadorou et al (2009) who present the more gen-eral case with non-zero bulk viscosity As for the slipequation it can easily be shown that

minuspartu(k)x

party(x 1) = B u(k)

x (x 1) k = 0 1 2 (27)

where k is the order of the perturbation The deriva-tion of the leading-order solutions which is basedon the assumption that the transverse velocity uy

is zero is straightforward and the methodology isdescribed in detail by Poyiadji (2012) The pertur-bation solution of the flow problem up to secondorder is

ux = B2B

(B+2minusBy2)

+ ε

[

minus B2

6B

(B+2minusBy2)(1minusx)minusαReB

4

7560B2

[5B2+45B+98minus3

(11B2+77B+140

)y2+35

(B2+5B+6

)y4minus7

(B2+3B

)y6]

]

+ ε2

[B

3

12B

(B + 2 minus By2) (1 minus x)2 minus αReB

5

7560B4

times [19B4+171B3+658B2+1260B+840+3

(3B4+21B3minus140B

)y2minus35

(B4+5B3+6B2)y4+7

(B4+3B3)y6]

times (1minusx)minus α2 B4

648B2

[B2+3B+8 + 2

(B2+7B

)y2 minus 3

(B2 + 3B

)y4] minus α2 Re2 B

7

314344800B5

times [2193B5 + 35088B4 + 221641B3 + 731346B2 + 1409100B + 1358280

minus 4(2839B5 + 39746B4 + 239316B3 + 803880B2 + 1576575B + 1455300

)y2

+ 2310(B5+12B4+111B3+602B2+1470B+1260

)y4+924

(13B5+130B4+448B3+560B2+105B

)y6

minus5775(B5 + 8B4 + 21B3 + 18B2) y8 + 616

(B5 + 6B4 + 9B3) y10]

]

+ O(ε3) (28)

uy = ε2 αReB5

11340B2 y(1 minus y2) [

5B2 + 45B + 98 minus 2(3B2 + 16B + 21

)y2 + (

B2 + 3B)

y4] + O(ε3) (29)

p = B3

(1 minus x) + ε

[

minus B2

18(1 minus x)2 + αReB

4

315B3

(2B3 + 14B2 + 35B + 35

)(1 minus x) + α2 B

2

54

(1 minus y2)

]

+ ε2

[B

3

54(1minusx)3minus αReB

5

945B3

(4B3+28B2+70B+70

)(1minusx)2minus α2 B

4

486B2

[11B2+53B+72minus3

(B2 + 3B

)y2] (1 minus x)

+ α2 Re2 B7

29469825B6

(3044B6+42616B5+267036B4+951720B3+1964655B2+2182950B+1091475

)(1 minus x)

+ α3 ReB5

204120B3

[97B3 + 889B2 + 2310B + 1260 minus 3

(79B3 + 553B2 + 1120B + 420

)y2

+175(B3 + 5B2 + 6B

)y4 minus 35

(B3 + 3B2) y6]

]

+ O(ε3) (30)

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 501

boundary conditions are obtained These can be foundin Taliadorou et al (2009) who present the more gen-eral case with non-zero bulk viscosity As for the slipequation it can easily be shown that

minuspartu(k)x

party(x 1) = B u(k)

x (x 1) k = 0 1 2 (27)

where k is the order of the perturbation The deriva-tion of the leading-order solutions which is basedon the assumption that the transverse velocity uy

is zero is straightforward and the methodology isdescribed in detail by Poyiadji (2012) The pertur-bation solution of the flow problem up to secondorder is

ux = B2B

(B+2minusBy2)

+ ε

[

minus B2

6B

(B+2minusBy2)(1minusx)minusαReB

4

7560B2

[5B2+45B+98minus3

(11B2+77B+140

)y2+35

(B2+5B+6

)y4minus7

(B2+3B

)y6]

]

+ ε2

[B

3

12B

(B + 2 minus By2) (1 minus x)2 minus αReB

5

7560B4

times [19B4+171B3+658B2+1260B+840+3

(3B4+21B3minus140B

)y2minus35

(B4+5B3+6B2)y4+7

(B4+3B3)y6]

times (1minusx)minus α2 B4

648B2

[B2+3B+8 + 2

(B2+7B

)y2 minus 3

(B2 + 3B

)y4] minus α2 Re2 B

7

314344800B5

times [2193B5 + 35088B4 + 221641B3 + 731346B2 + 1409100B + 1358280

minus 4(2839B5 + 39746B4 + 239316B3 + 803880B2 + 1576575B + 1455300

)y2

+ 2310(B5+12B4+111B3+602B2+1470B+1260

)y4+924

(13B5+130B4+448B3+560B2+105B

)y6

minus5775(B5 + 8B4 + 21B3 + 18B2) y8 + 616

(B5 + 6B4 + 9B3) y10]

]

+ O(ε3) (28)

uy = ε2 αReB5

11340B2 y(1 minus y2) [

5B2 + 45B + 98 minus 2(3B2 + 16B + 21

)y2 + (

B2 + 3B)

y4] + O(ε3) (29)

p = B3

(1 minus x) + ε

[

minus B2

18(1 minus x)2 + αReB

4

315B3

(2B3 + 14B2 + 35B + 35

)(1 minus x) + α2 B

2

54

(1 minus y2)

]

+ ε2

[B

3

54(1minusx)3minus αReB

5

945B3

(4B3+28B2+70B+70

)(1minusx)2minus α2 B

4

486B2

[11B2+53B+72minus3

(B2 + 3B

)y2] (1 minus x)

+ α2 Re2 B7

29469825B6

(3044B6+42616B5+267036B4+951720B3+1964655B2+2182950B+1091475

)(1 minus x)

+ α3 ReB5

204120B3

[97B3 + 889B2 + 2310B + 1260 minus 3

(79B3 + 553B2 + 1120B + 420

)y2

+175(B3 + 5B2 + 6B

)y4 minus 35

(B3 + 3B2) y6]

]

+ O(ε3) (30)

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

502 Rheol Acta (2012) 51497ndash510

ρ =1+εB3

(1 minus x)

+ ε2

[

minus B2

18(1minusx)2+ αReB

4

315B3 (2B3+14B2

+35B+35)(1minusx)+ α2 B2

54

(1minusy2)

]

+O(ε3) (31)

Letting B rarr infin we get the solution obtained byTaliadorou et al (2009) and Venerus and Bugajsky(2010) for flow with no-slip at the wall The perturba-tion solution of the axisymmetric flow is given in theAppendix

The volumetric flow rate

Q (x) equivint 1

0ux (x y) dy (32)

is given by

Q(x)=1minusεB3

(1 minus x)

+ ε2

[B

2

6(1minusx)2minus αReB

4

315B3

(2B3+14B2+35B+35

)

times (1minusx)minus α2 B3

405B(2B+5)

]

+O(ε3) (33)

Discussion

Let us first discuss the effect of the slip number onthe two velocity components In creeping flow (Re =0) the transverse velocity component uy is zero atsecond order The effect of the slip number B on thetransverse velocity is shown in Fig 2 The transversevelocity is reduced as the slip number is reduced from

2

yu

Reα ε

0 02 04 06 08 10

0005

001

0015

002

0025

003

Fig 2 Effect of the slip number on the transverse velocitycomponent

infinity (no slip) to zero (full-slip) As slip becomesstronger the velocity in the flow direction tends to be-come more uniform and thus the flow tends to becomeone-dimensional Given that the transverse velocitycomponent is always positive (Eq 29) the streamlinesof the flow under study are either horizontal or have aslight positive slope which reaches its maximum valueroughly in the middle of the y-interval [01] The effectof slip on the transverse velocity component is moreclearly illustrated in Fig 3 where the reduced meanvalue

uy

αRe ε2 equiv 1αRe ε2

int 1

0uy(y)dy

= B3

1120 (B+3)5

(19B2+209B+504

)+O(ε3) (34)

is plotted versus the slip number B Appreciable slipoccurs in the range 1 lt B lt 100 and slip may beconsidered as strong for B lt 1 In conclusion theunidirectionality assumption is valid when the flow iscreeping andor slip is strong

In Fig 4 the contours of the velocity in the flowdirection for B = infin (no slip) and 1 (strong slip) with Re= 0 ε = 01 and α = 001 are compared Even thoughthe contour patterns are similar the main difference isthat the range of the velocity values which in the caseof no-slip is the interval [0 15] shrinks with slip Thisis demonstrated in Fig 5 where the velocity profilesat the inlet- mid- and outlet-planes are plotted forB = infin and 1 (moderate slip) In the extreme case offull slip ux is uniform and equal to unity at the channelexit

The effect of the compressibility number ε on thecontours of ux for Re = 0 α = 001 and B = 1 (strongslip) is illustrated in Fig 6 The incompressible flow(ε = 0) is one-dimensional and thus the contour lines

10-2 100 102 1040

0005

001

0015

002

B

2

yu

Reα ε

Full slip

No slip

Fig 3 The mean transverse velocity as a function of the slipnumber

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 503

02

04

06

08

1

12

14

08

09

1

11

Fig 4 Contours of ux for B = infin (no slip) and 1 (strong slip)Re = 0 ε = 01 and α = 001

are horizontal In the case of compressible flow (ε =01) fluid particles accelerate as they are decompresseddownstream due to the conservation of mass As aresult the contours remain parallel only near the exitplane and tend towards the plane of symmetry up-stream

In Fig 7 the velocity contours obtained with Re = 0and 100 and B = 1 ε = 01 and α = 001 are shownThe results are essentially the same since higher-ordercontributions contain the product αRe which is small

0 02 04 06 08 10

05

1

15

0 02 04 06 08 106

08

1

12

x=0

x=1

x=1

x=0

Fig 5 Profiles of the velocity in the flow direction at x = 0 05and 1 for B = infin (no slip) and 1 ε = 01 Re = 0 and α = 001

08

09

1

11

08

09

1

11

Fig 6 Contours of ux for ε = 0 (incompressible flow) and 01Re = 0 B = 1 and α = 001

To magnify the effect of αRe the velocity contours fora shorter channel with aspect ratio α = 01 are plottedin Fig 8 It is observed that the effect of Reynoldsnumber becomes significant The acceleration of thefluid particles within the slit increases with inertiaNote that the Mach number corresponding to Re =100 ε = 01 and α = 01 is equal to 06 (γ is ofunity order) and the flow can no longer be consideredweakly compressible However the asymptotic expan-sions are still valid since the compressibility number isstill small Note that since Re = 3 (γ aε) Ma2 whenthe compressibility number ε and the Mach numberare small (lt03) solutions are admissible only below

08

09

1

11

08

09

1

11

Fig 7 Contours of ux for Re = 0 and 100 α = 001 (long channel)B = 1 and ε = 01

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

504 Rheol Acta (2012) 51497ndash510

08

09

1

11

08

09

1

11

Fig 8 Contours of ux for Re = 0 and 100 α = 01 (shorterchannel) B = 1 and ε = 01

a critical value of the Reynolds number (for examplethe critical value for Re is 270 for the data in Fig 7and is reduced to 27 in Fig 8 where α is increased from001 to 01) Generally as the channel becomes shorter(α increasing) the admissible Reynolds numbers getsmallermdashthe flow tends to creeping flow

Another way to investigate the validity of our so-lution arises by looking into the volumetric flow rategiven by Eq 33 Since the solution is up to second orderQ is a parabolic function of ε for any value of x At theexit plane

Q (1) = 1 minus α2 B2 (2B + 5)

15 (B + 3)3 ε2 + O(ε3) (35)

Obviously Q(1) is slightly below unity given that αε

is small Since the flow is compressible the volumetricflow rate is reduced as we move upstream A solutionis assumed to be admissible if the volumetric flow rateQ(0) at the inlet is a decreasing function of ε andpositive In Fig 9 Q(0) is plotted versus ε for variousReynolds numbers with B = infin (no slip) and α = 001In creeping flow (Re = 0) solutions are admissible forε lt 13 As the Reynolds number is increased Q(0)decreases faster with ε and may become negative foreven smaller compressibility numbers In other wordsgiven the compressibility number the aspect ratio andthe Mach number solutions are admissible only belowa critical value of the Reynolds number which has alsobe noted above As shown in Fig 10 slip weakensthe compressibility effects and reduces the reductionof the volumetric flow rate upstream As a result slipextends the range of admissible solutions by shifting theminimum of Q(0) to the right (Fig 11)

0 01 02 0 3 04 0 5

065

07

075

08

085

09

095

1

Fig 9 The volumetric flow rate at the inlet plane for differentReynolds numbers with α = 001 and no slip at the wall (B = infin)

From Eq 31 we see that the density ρ at the exitplane is 1 at leading order At the inlet plane wherethe density obviously is maximized we have

ρ (x = 0) = 1 + BB + 3

ε + O(ε2) (36)

The maximum value for ρ obtained in the case of noslip (B = infin) is given by

ρmax = 1 + ε + O(ε2) (37)

and is independent of Re and α In creeping flow ε

lt 13 and thus the maximum admissible value of thedensity for any α is ρmax = 43 which restricts the range

0 02 04 06 08 10

05

1

0 02 04 06 08 109

095

1

Fig 10 Variations of the slip velocity and the volumetric flow inthe channel for different slip numbers ε = 01 Re = 0 and α =001

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 505

0 01 02 03 04 0508

085

09

095

1

Fig 11 Effect of slip number on the volumetric flow rate at theentrance plane Re = 0 α = 001

of validity of the solution However more compressionwhich is expected for very small values of α (for verylong channels) can be obtained only if higher valuesof the compressibility number are admissible ie forlower values of the Reynolds number In other wordsmoderately compressible flow is associated with finitemoderate Reynolds numbers Recalling that for weaklycompressible flow we have αεRe lt 027 such a combi-nation of ε and Re is allowed only for smaller values ofthe aspect ratio α

Generally slip reduces the pressure in the channeland the required pressure drop In Fig 12 we show

0 02 04 06 08 10

02

04

06

08

1

Fig 12 Variation of the pressure along the centreline for variousslip numbers ε = 01 Re = 0 α = 001

the distribution of the pressure along the centreline fordifferent slip numbers ε = 01 Re = 0 and α = 001As the slip number tends to zero (full slip) the pressuretends to become zero everywhere Following Venerusand Bugajsky (2010) we calculate the mean pressuredrop as follows

p equiv p(0) minus p(1) equivint 1

0

[p (0 y) minus p (1 y)

]dy (38)

which gives

p= B3

minus[

B2

18minus αReB

4

315B3

(2B3+14B2+35B+35

)]

ε

+[

B3

54minus2αReB

5

945B3

(2B3+14B2+35B+35

)minus α2 B4

243B2

times (5B2 + 25B + 36

) + α2 Re2 B7

29469825B6

times (3044B6+42616B5+267036B4+951720B3

+1964655B2 + 2182950B + 1091475)]

ε2

+O(ε3) (39)

Equation 39 gives the pressure drop for channelflow of a compressible Newtonian fluid with slip atthe wall This is a generalization of the result providedby Venerus and Bugajsky (2010) for the no-slip case(B = infin)

p = 1 minus(

12

minus 1835

αRe)

ε

+(

12

minus 53α2minus 36

35αRe+ 3044

13475α2 Re2

)ε2+O

(ε3)

(40)

(It should be noted that the Reynolds number inVenerus and Bugajsky (2010) is twice the presentReynolds number) It is clear that the required pres-sure drop decreases with compressibility and increaseswith inertia as illustrated in Fig 13 The effect ofslip is illustrated in Fig 14 where the pressure dropsfor various slip numbers are plotted Slip leads to thereduction of the pressure difference required to drivethe flow and consequently alleviates compressibilityeffects This is of course expected and also noted inprevious works For example Zhang et al (2009) intheir analysis of slip flow characteristics of compressiblegases in microchannels reported that ldquoslip effect makesthe flow less compressiblerdquo For the set of values usedto construct Figs 13 and 14 the wall and centerlinepressures are essentially constant ie the pressure is

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

506 Rheol Acta (2012) 51497ndash510

0 005 01 015 0209

092

094

096

098

1

Fig 13 Effect of the Reynolds number on the mean pressuredrop no slip α = 001

essentially a function of x Hence the pressure contoursare practically straight lines parallel to the inlet andexit planes (Fig 15) This is not the case for shortchannels eg when α = 1 since the contributions of thehigher-order terms become more important this effectis illustrated in Fig 16

The mean pressure drop for axisymmetric Poiseuilleflow of a compressible Newtonian fluid with slip at thewall defined by

p equiv p (0) minus p (1) equiv 2int 1

0

[p (0 z) minus p (1 z)

]rdr (41)

0 005 01 015 020

02

04

06

08

1

Fig 14 Effect of the slip number on the mean pressure dropRe = 0 α = 001

02040608

0

05

0

01

02

Fig 15 Pressure contours for different slip numbers ε = 02Re = 0 and α = 001 (long channel)

is

p = B8

minus[

B2

128minus αReB

3

2048B2

(B2 + 4B + 8

)]

ε

+[

B3

1024minus αReB

4

8192B2

(B2 + 4B + 8

) minus α2 B4

294912B2

times(49B2 + 300B + 576

) + α2 Re2 B6

14155776B5

times(2B5+24B4+171B3+648B2+1080B+864

)]

ε2

+ O(ε3) (42)

0

05

0

02

04

0

01

02

Fig 16 Pressure contours for different compressibility numbersε = 02 Re = 0 and α = 1 (short channel)

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 507

The above equation generalizes the result in Venerus(2006) for the no-slip case

p = 1 minus(

12

minus αRe4

)ε

+(

12

minus αRe2

minus 49α2

72+ α2 Re2

27

)ε2+O

(ε3) (43)

We have derived a solution for Eqs 17ndash19 and 22ndash25 which is valid for all values of the channel aspectratio α It is moreover obvious from Eq 19 thatwe recover the lubrication approximation (α2 ltlt 1)with the transverse pressure gradient being zero whenαRe ltlt 1 if all terms of order α2 or higher are neglected(the aspect ratio α cannot be identically zero since inthis limiting case the pressure scale ie the pressurerequired to drive the flow in a channel of infinite lengthwith no slip at the wall is infinite) Therefore oursolution gives the lubrication-theory solution in thepresence of slip if we neglect the terms of order α2

or higher and assume that αRe ltlt 1 The transversevelocity component vanishes the pressure and the den-sity are functions of x only and the pressure drop isgiven by

p = B3

minus B2

18ε + B

3

54ε2 + O

(ε3) (44)

The velocity in the flow direction is simplified to

ux = B2B

(B + 2 minus By2)

times[

1minus B3

(1minusx) ε+ B2

6(1minusx)2 ε2

]

+O(ε3) (45)

As already discussed such a solution is admissible ifQ(0) is a decreasing function of the compressibilitynumber ε This condition is satisfied when

ε ltB + 3

3B= 1

B(46)

If a more refined solution is desired one could con-struct perturbation expansions using α as the perturba-tion parameter (for any compressible flow) or doubleasymptotic expansions where both ε and α are pertur-bation parameters

In the case of the axisymmetric Poiseuille flow theaverage Darcy friction factor defined by

f equiv minus 8Re

int 1

0

partuz

partr(1 z) dz (47)

is of interest (Housiadas et al 2012) Integrating theabove equation yields

Re f32

= BB + 4

times

1minus BB + 4

[12

minus B12 (B+4)

αRe]

ε + B2

(B + 4)2

times[

12

minus 13B + 1272 (B + 4)

α2 minus B2 + 2B + 44B (B + 4)

αRe

+17B3+78B2+360B+14402160 (B+4)3 α2 Re2

]ε2

+O(ε3) (48)

In the no-slip limit (B rarr infin) one finds that

Re f32

= 1minus(

12

minus 112

αRe)

ε

+(

12

minus 1372

α2minus 14αRe+ 17

2160α2 Re2

)ε2+O

(ε3)

(49)

Venerus (2006) compared the pressure drop and thefriction factor for the no-slip case defined respec-tively by Eqs 43 and 49 and noted that the effectof inertia on pressure drop is significantly larger thanon drag force He also pointed out that the one-dimensional models for the no-slip case overpredict the

10-2

10-1

100

101

065

07

075

08

085

09

095

1

105

Fig 17 The average Darcy friction factor for the axisymmetricPoiseuille flow versus αRe for various slip numbers ε = 02 andα = 001

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

508 Rheol Acta (2012) 51497ndash510

10-2

100

102

104

0

01

02

03

04

05

06

07

08

09

1

Fig 18 Average Darcy friction factor for the axisymmetricPoiseuille flow versus the slip number for Re = 0 and 50 ε = 02and α = 01

friction factor by roughly 10 Similarly to the pressuredrop the average Darcy friction factor is reduced dra-matically with slip as shown in Fig 17 For a given slipnumber it is essentially constant for a wide range of theparameter αRe corresponding to the weak compress-ibility regime and then increases rapidly In Fig 18 theaverage Darcy friction factor for Re = 0 and 50 ε =02 and α = 01 is plotted versus the slip number B Itcan be seen that the friction factor is reduced with slip

following a sigmoidal curve and also that the Reynoldsnumber effect becomes weaker by slip

Conclusions

We have derived perturbation solutions of the weaklycompressible plane and axisymmetric Poiseuille flowswith Navierrsquos slip at the wall thus generalizing previ-ous results by Taliadorou et al (2009) and Venerusand Bugajsky (2010) The density is assumed to be alinear function of pressure and the associated isother-mal compressibility number is used as the perturbationparameter In the proposed derivation the primaryflow variables ie the two velocity components thepressure and the density are perturbed Solutions havebeen obtained up to second order The correspondingexpressions of the volumetric flow rate and the pressuredrop are also provided and discussed As expected slipweakens the y-dependence of the solution The unidi-rectionality assumption is valid if the Reynolds numberis very small andor slip along the wall is strong Weare currently studying approximate solutions of weaklycompressible Newtonian Poiseuille flows with pressure-dependent viscosity

Appendix

In the case of compressible axisymmetric NewtonianPoiseuille flow with slip at the wall the perturbationsolution is as follows

uz(r z)= B4B

(B+2minusBr2

)

+ε

[

minus B2

32B

(B+2minusBr2

)(1minusz)+ αReB

4

73728B2

[minus2

(B2+10B+24

)+9

(B2+8B+16

)r2minus9

(B2+6B+8

)r4+2

(B2+4B

)r6

]]

+ ε2

[3B

3

512B

(B+2minusBr2

)(1minusz)2 minus αReB

5

196608B4

times[B4+10B3+72B2+240B+192+6

(B4+8B3+12B2minus16B

)r2minus9

(B4+6B3+8B2

)r4+2

(B4+4B3

)r6

]

times (1minusz) + α2 B4

294912B2

[B2 + 32B minus 48 minus 4

(7B2 + 48B

)r2 + 27

(B4 + 4B3

)r4

]+ α2 Re2 B

7

459848300B5

times[43B5+774B4+1328B3minus42720B2minus268800Bminus460800minus200

(5B5+80B4+360B3minus96B2minus4032Bminus6912

)r2

+100(

33B5+462B4+2112B3+2736B2minus3456Bminus6912)

r4minus1200(

3B5+36B4+148B3+224B2+64B)

r6

+1425(

B5 + 10B4 + 32B3 + 32B2)

r8 minus 168(

B5 + 8B4 + 16B3)

r10]]

+ O(ε3

)(50)

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

Rheol Acta (2012) 51497ndash510 509

ur (r) = ε2 αReB5

1179648B2 r(1 minus r2) [

4(B2 + 10B + 24

) minus (5B2 + 32B + 48

)r2 + (

B2 + 4B)

r4] + O(ε3) (51)

p (r z) = B8

(1 minus z) + ε

[

minus B2

128(1 minus z)2 + αReB

4

16384B3

(B3 + 8B2 + 24B + 32

)(1 minus z) + α2 B

2

768

(1 minus r2)

]

+ ε2

[

minus B3

1024(1 minus z)3 minus αReB

5

65536B3

(B3 + 8B2 + 24B + 32

)(1 minus z)2 minus α2 B

4

147456B2

times [29B2 + 168B + 228 minus 9

(B2 + 4B

)r2] (1 minus z) + α2 Re2 B

7

113246208B6

times (2B6 + 32B5 + 267B4 + 1332B3 + 3672B2 + 5184B + 3456

)(1 minus z) + α3 ReB

5

14155776B3

times [19B3 + 202B2 + 576B + 288 minus 18

(3B3 + 24B2 + 52B + 16

)r2 + 45

(B3 + 6B2 + 8B

)r4

minus10(B3 + 4B2) r6]

]

+ O(ε3) (52)

ρ(r z) = 1+εB8

(1minusz) +ε2

[

minus B2

128(1minusz)2 + αReB

4

16384B3

(B3 +8B2 +24B+32

)(1minusz)

α2 B2

768(1 minus r2)

]

+O(ε3) (53)

In the above solution z is scaled by the tube lengthL r by the tube radius R the axial velocity by Ulowast =Mlowast

(πρlowast

0 Rlowast2) the radial velocity by Ulowast RlowastLlowast and the

pressure by 8ηlowastLlowastUlowastRlowast2 The dimensionless numbersare defined as follows

α equiv Rlowast

Llowast Re equiv ρlowast0 Ulowast Rlowast

ηlowast ε equiv 8κlowastηlowastLlowastUlowast

Rlowast2

B equiv βlowast Rlowast

ηlowast B equiv 8BB + 4

(54)

The volumetric flow rate

Q (z) equiv 2int 1

0uz (r z) rdr (55)

is given by

Q (z)=1minusεB8

(1 minus z)

+ ε2

[3B

2

128(1minusz)2minus αReB

3

2048B2

(B2+4B+8

)(1minusz)

minus α2 B3

9216B(B + 3)

]

+ O(ε3) (56)

For ε = 0 the standard fully-developed Poiseuille flowsolution with slip at the wall is recovered with

uz (r) = B2B

+ B4

(1 minus r2) (57)

References

Arkilic EB Schmidt MA (1997) Gaseous slip flow in long mi-crochannels J MEMS 6167ndash178

Beskok A Karniadakis GE (1999) Simulation of heat and mo-mentum transfer in complex micro-geometries J Thermo-phys Heat Transfer 8355ndash377

Denn MM (2001) Extrusion instabilities and wall slip Ann RevFluid Mech 33265ndash287

Dubbeldam JLA Molenaar J (2003) Dynamics of the spurt in-stability in polymer extrusion J Non-Newton Fluid Mech112217ndash235

Felderhof BU Ooms G (2011) Flow of a viscous compressiblefluid produced in a circular tube by an impulsive pointsource J Fluid Mech 668100ndash112

Georgiou G (2005) Stick-slip instability Chapter 7 In Hatzikiri-akos SG Migler K (eds) Polymer processing instabilitiescontrol and understanding Mercel Dekker Inc pp 161ndash206

Georgiou GC Crochet MJ (1994) Compressible viscous flow inslits with slip at the wall J Rheol 38639ndash654

Guaily A Cheluget E Lee K Epstein M (2011) A new hyper-bolic model and an experimental study for the flow of poly-mer melts in multi-pass rheometer Comp Fluids 44258ndash266

Hatzikiriakos SG (2012) Wall slip of molten polymers ProgrPolym Sci 37624ndash643

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682

510 Rheol Acta (2012) 51497ndash510

Hatzikiriakos SG Dealy JM (1992) Role of slip and fracture inthe oscillating flow of a HDPE in a capillary J Rheol 36845ndash884

Hatzikiriakos SG Dealy JM (1994) Start-up pressure transientsin a capillary rheometer Polym Eng Sci 34493ndash499

Hatzikiriakos SG Migler K (2005) Polymer processing in-stabilities control and understanding Mercel DekkerNew York

Housiadas K Georgiou GC (2011) Perturbation solution ofPoiseuille flow of a weakly compressible Oldroyd-B fluid JNon-Newton Fluid Mech 16673ndash92

Housiadas KD Georgiou GC Mamoutos IG (2012) Laminar ax-isymmetric flow of a weakly compressible viscoelastic fluidRheol Acta doi101007s00397-011-0610-x

Kohl MJ Abdel-Khalik SI Jeter SM Sadowski DL (2005) An ex-perimental investigation of microchannel flow with internalpressure measurements Int J Heat Mass Transfer 481518ndash1533

Mitsoulis E Hatzikiriakos SG (2009) Steady flow simulations ofcompressible PTFE paste extrusion under severe wall slip JNon-Newton Fluid Mech 15726ndash33

Piau JM El Kissi N (1994) Measurement and modelling of fric-tion in polymer melts during macroscopic slip at the wall JNon-Newton Fluid Mech 54121ndash142

Poyiadji S (2012) Flows of fluids with pressure-dependent mater-ial properties PhD Thesis University of Cyprus

Qin FH Sun DJ Yin XY (2007) Perturbation analysis on gas flowin a straight microchannel Phys Fluids 19027103

Stone HA Stroock AD Ajdari A (2004) Engineering flows insmall devices microfluidics toward a lab-on-a-chip AnnuRev Fluid Mech 36381ndash411

Taliadorou E Georgiou E Alexandrou AN (2007) A two-dimensional numerical study of the stick-slip extrusion in-stability J Non-Newton Fluid Mech 14630ndash44

Taliadorou E Georgiou GC Mitsoulis E (2008) Numerical sim-ulation of the extrusion of strongly compressible Newtonianliquids Rheol Acta 4749ndash62

Taliadorou E Neophytou M Georgiou GC (2009) Perturbationsolutions of Poiseuille flows of weakly compressible New-tonian liquids J Non-Newton Fluid Mech 158162ndash169

Tang HS Kalyon DM (2008a) Unsteady circular tube flowof compressible polymeric liquids subject to pressure-dependent wall slip J Rheol 52507ndash525

Tang HS Kalyon DM (2008b) Time-dependent tube flow of com-pressible suspensions subject to pressure-dependent wallslip ramifications on development of flow instabilities JRheol 521069ndash1090

Venerus DC (2006) Laminar capillary flow of compressible vis-cous fluids J Fluid Mech 55559ndash80

Venerus DC Bugajsky DJ 2010 Laminar flow in a channel PhysFluids 22046101

Vinay G Wachs A Frigaard I (2006) Numerical simulation ofweakly compressible Bingham flows the restart of pipelineflows of waxy crude oils J Non-Newton Fluid Mech 13693ndash105

Wachs A Vinay G Frigaard I (2009) A 15D numerical modelfor the start up of weakly compressible flow of a viscoplasticand thixotropic fluid in pipelines J Non-Newton Fluid Mech15981ndash94

Zhang TT Jia L Wang ZC Li CW (2009) Slip flow characteris-tics of compressible gaseous in microchannels Energy ConvManage 501676ndash1682