ORIGINAL

Simultaneously developing laminar flow in an isothermalmicro-tube with slip flow models

Nizar Loussif • Jamel Orfi

Received: 15 November 2012 / Accepted: 17 October 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract In this study, a two-dimensional steady state

simultaneously developing laminar flow inside a micro-tube

is investigated numerically under slip flow conditions. The

first and second-order slip flow models have been imple-

mented for the case where the viscous dissipation and axial

conduction are included and a constant wall temperature

boundary condition is specified. The results are obtained for

several combinations of the Knudsen number Kn, the coef-

ficient b and the Brinkman number Br. The study reveals a

significant impact of slip flow and temperature jump on the

hydrodynamic and thermal fields. A comparison of the first

and second-order slip flow shows a considerable variation of

the hydrodynamic flow and a weak impact on the thermal

field particularly when the flow is fully developed.

List of symbols

Br Brinkman number,U2

inlTin�Twð Þk

cp Specific heat at constant pressure (kJ kg-1 K-1)

D Tube diameter (m)

ft Thermal accommodation coefficient

fv Tangential momentum accommodation

fapp Apparent friction coefficient, DPrh

qU2in

z=2

h Convective heat transfer coefficient (W m-2 K-1)

k Thermal conductivity (W m-1 K-1)

Kn Knudsen number k/D

L Micro-tube length (m)

Nux Local Nusselt number hxD/k

P Pressure (Pa)

Pr Prandtl number lcp/k

r Radial coordinate (m)

�r Nondimensional radial coordinate r/R

R Micro-tube radius (m)

Re Reynolds number Uin qR/lT Temperature (�C)�T Nondimensional temperature

U Velocity component in the z-direction (ms-1)�U Nondimensional velocity component in the z-direction

V Velocity component in the r-direction (ms-1)�V Nondimensional velocity component in the r-

direction

z Axial coordinate (m)

�z Dimensionless axial coordinate, z/R

z* Dimensionless axial coordinate, z/2DRePr

z? Dimensionless axial coordinate, z/2DRe

Greek symbols

a Thermal diffusivity (m2 s-1)

bv ð2� fvÞ=fv

b 2�ftft

2c1þc

1Pr

c Ratio of specific heats

k Gas mean free path

l Dynamic viscosity (Pa s)

q Fluid density (kg m-3)

Subscripts

in Inlet

w Wall

N. Loussif

Ecole Nationale d’Ingenieur de Monastir, Universite de

Monastir, Monastir, Tunisie

N. Loussif (&)

Unite de Recherche Materiaux, Energie et Energies

Renouvelables, faculte des sciences Gafsa, Universite de Gafsa,

2100 Gafsa, Tunisie

e-mail: loussif_nizare@voila.fr

J. Orfi

Department of Mechanical Engineering, College of Engineering,

King Saud University, Riyadh, KSA

123

Heat Mass Transfer

DOI 10.1007/s00231-013-1251-7

1 Introduction

Flow and heat transfer at micro-scale have attracted an

extensive research interest in recent years due to the rapid

development of micro-electromechanical systems, bio-

medical applications and innovative cooling techniques for

integrated circuits.

Modeling the heat and fluid flow for such micro devices

is different from that of the macro scale systems. The ratio

of the molecular mean free path of gas to a characteristic

dimension of the duct is known as the Knudsen number,

Kn = k/L. It defines the region where the continuum

assumption is valid and where it becomes no longer valid.

For small values of Kn, the fluid behavior can be analyzed

using the Navier–Stokes equations with no-slip flow

boundary conditions. For values of Kn varying between

0.001 and 0.1, the regime is called slip flow regime. Na-

vier–Stokes equations with slip flow conditions should be

applied. For higher Knudsen numbers, the flow is either in

the transition regime (0.1 \ Kn B 10) or in the free

molecular regime (Kn [ 10) [1, 2].

Many investigations have been done to show the effect

of slip velocity and temperature jump conditions on the

hydrodynamic and thermal profiles inside various config-

urations of micro-ducts. The majority of the studies have

been interested to solve the energy equation when the

hydrodynamic flow is fully developed for both cases of

constant wall temperature and constant wall heat flux [3–

5]. Some other studies included the viscous dissipation and

the axial conduction for fully developed flow and devel-

oping thermal field inside micro-tubes and micro-channels.

Cetin et al. [2] solved the energy equation numerically by

first solving the velocity profile analytically for parallel

plates and micro-tubes for both cases of constant wall

temperature and constant wall heat flux. The viscous dis-

sipation has been considered with first-order slip condition

and temperature jump. Larrode et al. [3] proposed an

analytical solution for the developing temperature profiles

in an isothermal circular tube. A fully developed slip

velocity profile and a temperature jump boundary condition

were used. Yu and Ameel [5, 6] studied analytically the

laminar slip-flow forced convection in rectangular micro-

channels assuming that the hydrodynamic field is fully

developed. Developing and fully developed flow Nusselt

numbers were obtained.

Various studies concluded that slip flow and tempera-

ture-jump conditions have significant influence on heat

transfer with a decrease of the Nusselt number when

increasing Knudsen number [7–9].

Jeong and Jeong [10] investigated the heat transfer in

micro-channels for both cases of constant wall temperature

and constant wall heat flux when the hydrodynamic flow

is fully developed, including first-order slip velocity,

temperature jump and viscous dissipation effects. Their

results show significant impact of Knudsen number, Kn and

Brinkman number, Br on the temperature profiles and the

Nusselt number. Barron et al. [11, 12] extended analyti-

cally the original Graetz problem of thermally developing

heat transfer in laminar flow through a circular tube to slip

flow. Relationships for Knudsen numbers ranging from 0 to

0.12 were developed to describe the effect of slip flow on

heat transfer coefficient.

Zhang et al. [13] reported that the rarefaction, viscous

heating, surface roughness and compressibility have sig-

nificant effects on the flow and heat transfer patterns in the

slip flow region. Using the superposition principle, Zhang

et al. [13] developed an analytical solution for steady heat

transfer in a two dimensional micro-channel including the

effects of the velocity slip and temperature jump at the wall

and the viscous heating. The authors assumed the fluid is

incompressible and the hydrodynamic fully developed.

Xiao et al. [14] solved the momentum and the energy

equation under slip condition and temperature jump when

the flow is assumed to be hydro-dynamically fully devel-

oped. It was shown that increasing the Knudsen number

decreases the maximum velocity inside the tube. Second

order velocity slip boundary condition effects have been

also presented.

Renksizbulut et al. [15] investigated the rarefaction

effect on the flow in rectangular micro-channel and par-

ticularly on the friction coefficient; an important decrease

of the fully developed friction coefficient happens when

increasing the Knudsen number.

El-Genk and Yang [16] studied numerically the influ-

ence of slip and viscous heat dissipation on the friction

number of thermally developing laminar water flow in

micro-tubes. The effects of the fluid viscosity, the Rey-

nolds number and the micro-tube diameter on the friction

coefficient were presented and discussed.

It is of interest to mention that the majority of the pre-

vious studies on convective heat transfer in micro-ducts

adopted the first order slip flow model. The second order

formulation has been considered only in few works.

Barber and Emerson [17] presented a review of avail-

able boundary conditions for modeling gas-phase micro-

flows in the slip flow regime. They highlighted some of the

challenges when extending the Navier–stokes equations

into the transition regime. They noticed also the urgent

need for more studies in this field, in particular, the

experimental ones. Cetin and Bayer [18] determined ana-

lytically the fully developed temperature profile and the

Nusselt number for a gas flow in a micro tube submitted to

uniform heat flux boundary conditions. Second order slip

flow model and temperature jump boundary conditions are

used for the implementation of the rarefaction effect. Some

results comparing the fully developed Nusslet number

Heat Mass Transfer

123

using first order and second order models are presented for

different values of Knudsen and Brinkman numbers.

The recent work of Colin [19] is of great importance

since it presented a state of the art review on convective

heat transfer in micro-channels focusing on rarefaction

effects in the slip flow regime. Several analytical and

numerical models are compared and commented for vari-

ous geometries and various types of boundary conditions.

Besides, some comments on the first and second order

formulations are presented. Rij et al. [20] analyzed

numerically the convective heat transfer and pressure los-

ses in micro-channels under slip flow regime. They

reviewed first the most commonly applied second order

boundary condition models. They observed that the effects

of second order slip boundary conditions, creep flow, vis-

cous dissipation and axial conduction are all significant.

On the other hand, Colin et al. [21] conducted an exper-

imental work on a gaseous flow in rectangular micro-chan-

nels to confirm the domain of accuracy of first and second

order slip models. It was found that the second-order model is

valid for Knudsen numbers up to 0.25, whereas the first-order

model is no longer accurate for values higher than 0.05.

In the present study, we focused on the resolution of the

continuity, momentum and energy equations for micro flow

in circular ducts under slip velocity and temperature jump

for a simultaneously developing flow with a constant wall

temperature condition. The viscous dissipation and the

axial conduction are included. The first order and second

order formulations are applied for the slip flow boundary

conditions. Their effects on the hydrodynamic and thermal

profiles as well as on the Nusselt number and friction factor

distributions are presented and analyzed.

2 Formulation

A simultaneously developing flow in a micro-tube for a

constant wall temperature boundary condition is investi-

gated under slip flow and temperature jump conditions

including viscous dissipation. The coordinate system and

geometry for micro-tube are shown in Fig. 1.

The following assumptions were made: steady state and

axisymmetric flow, constant fluid properties and negligible

gravitational force.

The following dimensionless quantities are introduced:

r ¼ r

R; z ¼ z

R; V ¼ V

Uin

; U ¼ U

Uin

; P ¼ P

qU2in

;

T ¼ T � Tw

Tin � Tw

ð1Þ

where R is the radius of the micro-tube, Uin and Tin are

respectively the inlet velocity and the inlet temperature of

the fluid, Tw is the wall temperature.

For a flow inside an isothermal micro-tube including

viscous dissipation, the governing equations and their

boundary conditions in dimensionless form are as follows:

1

r

o

orrV� �

þ o

ozU� �¼ 0 ð2Þ

VoV

orþ U

oV

oz

� �¼ � oP

orþ 1

Re

o

or

1

r

o

orrV� �� �

þ o2V

oz2

� �

ð3Þ

VoU

orþ U

oU

oz

� �¼ � oP

ozþ 1

Re

1

r

o

orroU

or

� �þ o2U

oz2

� �

ð4Þ

VoT

orþ U

oT

oz

� �¼ 1

Re Pr

1

r

o

orroT

or

� �þ o2T

oz2

� �

þ Br

Re Pr2

oV

or

� �2

þ V

r

� �2

þ oU

oz

� �2" #(

þ oV

ozþ oU

or

� �2)

ð5Þ

where Br, Re and Pr refer to the Brinckman, Reynolds and

Prandtl numbers respectively.

At the inlet z ¼ 0

U ¼ 1; V ¼ 0; T ¼ 1 ð6Þ

Symmetry conditions at r ¼ 0

V ¼ 0;oU

or¼ 0;

oT

or¼ 0 ð7Þ

At the tube wall, r ¼ 1

V ¼ 0 ð8Þ

U ¼ �2bvKnoU

or

����r¼1

� 9

2Kn2o

2U

or2

����r¼1

ð9Þ

Fig. 1 Geometry and coordinate system of flow domain

Heat Mass Transfer

123

T ¼ �2bKnoT

or

����r¼1

ð10Þ

Equation (9) represents the general form of the boundary

condition for the velocity as described by Deissler [22], the

first term is known as first-order boundary condition while

the second term refers to second-order boundary condition.

First-order temperature jump boundary condition is pre-

sented by Eq. (10). The case of no temperature jump is

obtained when b and/or Kn equal to zero.

At the tube outlet z ¼ L

oU

oz¼ 0;

oV

oz¼ 0;

oT

oz¼ 0 ð11Þ

For the remainder of the study, bv will be taken equal to

unity as it was considered in the previous studies [1, 2, 5].

For a fully developed flow, the analytic solution for the

axial velocity and the friction coefficient in non-dimen-

sional form are for the first (Eq. 12) and second (Eq. 13)

order slip conditions as following:

Table 1 Grid dependance study

Fully developed

velocity

Fully developed Nusselt

Br = 0 and b = 1.667

Kn = 0 Kn = 0.12 Kn = 0 Kn = 0.12

300 9 50 1.9990 1.5099 9.5969 3.3838

250 9 40 1.9989 1.5097 9.5971 3.3841

200 9 30 1.9985 1.5095 9.5973 3.3844

150 9 20 1.9983 1.5094 9.5974 3.3845

Fig. 2 Fully developed

centerline velocity and friction

coefficient for different

rarefaction values

Table 2 Fully developed Nu numbers (first-order slip velocity)

Kn Br = 0 Br = 0

b = 0 b = 1.667 b = 0 b = 1.667

Cetin et al. [2] This study Cetin et al. [2] This study Cetin et al. [2] This study Cetin et al. [2] This study

0 3.656 3.6573 3.656 3.6573 9.598 9.5971 9.598 9.5971

0.02 3.855 3.8558 3.488 3.4884 9.871 9.8692 7.427 7.4248

0.04 4.020 4.0207 3.292 3.2919 10.088 10.0839 6.031 6.0274

0.06 4.160 4.1599 3.087 3.0873 10.264 10.2629 5.065 5.0609

0.08 4.279 4.2787 2.887 2.8867 10.441 10.4400 4.359 4.3555

0.10 4.382 4.3813 2.697 2.6968 10.535 10.5312 3.822 3.8197

0.12 – 4.4710 – 2.5210 – 10.6614 – 3.3841

Heat Mass Transfer

123

U ¼ 2ð1� r2Þ þ 8Kn

1þ 8Kn; f Re ¼ 16

1þ 8Kn; ð12Þ

U ¼ 2ð1� vr2Þ2� v

; f Re ¼ 16v2� v

; ð13Þ

where

v ¼ 1þ 8

3Knþ 6Kn2

� ��1

3 Numerical method and validation

The governing equations subjected to the associated

boundary conditions are solved numerically using the

control volume method and the SIMPLER Algorithm

[23]. A grid dependence analysis was performed as

mentioned in Table 1. The number of nodes in the axial

and radial directions was chosen for all computations to

be 250 9 40.

Fig. 3 Effect of Knudsen number and slip flow models on the axial velocity profiles

Heat Mass Transfer

123

The proposed numerical model was validated using

previous results and data and the various comparisons

between the present results and those from literature show

very good agreements.

In fact, as it is shown in Fig. 2, the fully developed

friction coefficient and the centerline velocity are in

excellent agreement with the exact solutions for first and

second-order slip flow models.

Besides, Fig. 2 reveals a significant effect of the slip

flow conditions on the fully developed centerline velocity

and friction coefficient. When increasing Kn, U? and fRe

decrease for both cases first and second-order slip flow

models. Such decreases are more important when the first

order model is used. This is very clear in particular when

Kn takes higher values.

Table 2 presents and compares the computed fully

developed Nusselt number with the results of Cetin al. [2]

for the cases of slip flows using the first order model with

and without viscous dissipation.

4 Results

4.1 Hydrodynamic flow

Figure 3 depicts the axial development of normalized axial

velocity profiles and presents the impact of introducing the

slip conditions using the first and second formulations. The

profiles concern three different axial positions along the

micro-tube namely z/R = 3, 10 and 100. When the slip

flow condition is applied, the fluid particles adjacent to the

solid surface of the tube wall no longer attain the velocity

of the solid surface. In the core region of the tube, the fluid

decelerates and its maximum velocity occurring at the

centerline of the tube decreases significantly reaching a

value of 1.51 for Kn = 0.12 when using the first order slip

flow model. On the other side, when considering the sec-

ond-order slip condition, the maximum velocity decreases

from 2 (Kn = 0) to 1.55 for Kn = 0.12 which represents a

decrease of about 22.50 %.

Besides, for each location z/R along the micro-duct,

there is a particular radius rz/R where the first-order

velocities are lower than the second-order ones when

0 \ r/R \ rz/R. For r/R [ rz/R, the trend reverses until

reaching the wall. Moreover, the first-order slip model

gives lower centerline velocity and higher wall slip

velocity which would affect the shear stress distribution.

Therefore, a more pronounced flattening of the velocity

profiles is observed with the first order model rather than

the second order one. For the locations z/R = 3, 10 and

100, rz/R is equal to 0.738, 0.720 and 0.707 respectively.

These results are consistent with those of Xiao et al. [14]

who considered the fully developed velocity profiles.

Figure 4 shows the axial evolution of the centerline

velocity across the micro-tube for different rarefaction

degrees. Results for the non-slip condition are also pre-

sented and used as reference case. They are in a very good

agreement with those provided by Hornbek [24]. Figure 4

shows that increasing the rarefaction degree reduces the

centerline velocity. In the other side, it is obvious that

Fig. 4 Evolution of the centerline velocity for different values of Kn

and slip flow models Fig. 5 Axial variation of the apparent friction coefficient

Heat Mass Transfer

123

adopting a first-order slip condition induces a more sig-

nificant reduction in the velocity values compared to the

case of the second-order formulation.

Figure 5 presents the axial distribution of the apparent

friction coefficient for different values of Kn. For the no-

slip case, the evolution of the apparent friction coefficient

is in good agreement with that provided by Hornbeck [24].

It reaches the usual value of 16 when the flow becomes

fully developed. When a tangential slip-velocity boundary

condition is taken into consideration, a decrease of the

apparent friction coefficient occurs along the tube. This

decrease is significant at the tube entrance explained by a

decrease of the pressure drop and leads to almost uniform

distribution of this coefficient along the micro-tube length.

In the fully developed region, a specific value of the

friction coefficient is reached for each value of Kn.

Using the second order model leads to higher values for

the friction coefficient than when considering the first order

slip flow model. This is clear in particular in the entrance

region of the micro-tube and for small values of Kn. The

differences become not significant as Kn number increases.

4.2 Thermal flow

The Brinkman number, Br, is introduced to account for the

influence of viscous dissipation such as heating or cooling

of the viscous fluid due to internal friction. Brinkman

number approaches zero when the viscous dissipation is

negligible. It takes negative values when the fluid is heated

at the wall (wall temperature is higher than the inlet fluid

temperature) while for the cooling case, Br is positive.

Figure 6 depicts the local Nusselt number distribution

along the micro-tube for the continuum flow (Kn = 0) with

viscous dissipation. When there is no viscous dissipation

(Br = 0), the local Nusselt distribution is in good agree-

ment with the results of Jensen [25]. Fully developed

Nusselt number reaches two different values namely 3.66

and 9.59 for the cases of Br = 0 and Br = 0 respectively.

Fig. 6 Axial variation of the local Nusselt as a function of z* for

different values of Br (Kn = 0)

Fig. 7 Axial variation of local Nusselt number for different b, Kn and

positive Br numbers

Fig. 8 Axial variation of local Nusselt number for different values of

b, Kn and negative Br numbers

Heat Mass Transfer

123

Therefore, including viscous dissipation results in a sig-

nificant increase in the Nusselt number regardless the value

of the Brinkman number. Furthermore, one can distinguish

two behaviors of the local Nusselt number depending on

the sign of Br. A negative Br for this constant wall tem-

perature boundary condition refers to the fluid being heated

as it flows along the tube while the viscous dissipation

tends to reinforce the external heating effect. Therefore, the

bulk temperature can reach the wall temperature along the

tube length. The particular location where the bulk tem-

perature is equal to the wall temperature corresponds to a

singular point where the local Nusselt number goes to the

infinity. This particular location in the tube length depends

on the value of Br. It moves downstream when the absolute

value of Br decreases. For the cooling case (positive Br),

the local Nusselt number decreases in the entrance region

of the tube, increases at an intermediate axial position and

approaches the constant value of 9.59 near the tube exit.

Similar behavior has been observed by Kakac et al. [26].

Figure 7 shows the axial variation of the local Nusselt

number along the micro-tube for positive Br when

Kn = 0.12. The standard case of continuum model

(Kn = 0) is also given. The effect of the temperature jump

at the wall is presented. Therefore, two values of b (b = 0;

1.667) are considered. One can observe first that Fig. 7

depicts three different behaviors corresponding to the cases

of no slip flow and no temperature jump (Kn = 0), slip

flow and no temperature jump (Kn = 0 and b = 0) and

slip flow with temperature jump. When b = 0 (no tem-

perature jump), the behavior of the local heat transfer

Fig. 9 Axial variation of Nusselt number for first and second-order slip flow models for different Br, Kn and b combinations

Heat Mass Transfer

123

coefficient is similar to that of continuum model (Kn = 0).

The fully developed Nusselt number for b = 0 takes the

values of 4.47 and 10.66 respectively when Br = 0 and

Br = 0. Such Nux values are slightly higher than those of

continuum model.

When increasing the value of b to 1.667 (temperature

jump for air flow), the heat transfer rates are reduced

drastically not only in the fully developed region but also in

the tube entrance one. The effect of viscous dissipation is

noticeable only in the vicinity of the tube exit region. The

fully developed Nusselt number reaches 3.38 and 2.52

respectively with and without viscous dissipation. There-

fore, viscous dissipation enhances the heat transfer mech-

anism by about 34 %.

It is also important to note that when considering the

viscous dissipation effects (positive Br numbers), the type

of boundary condition at the micro-tube wall defines the

value of the asymptotic Nusselt number.

The axial distribution of the local Nusselt number for

negative values of Brinkman number is presented in

Fig. 8. Kn is set equal to 0.12 while b has two values: 0

and 1.667. For negative Br, the fluid is being heated along

the micro-tube. As observed in Fig. 6, the Nusselt number

exhibits a particular behavior when the bulk temperature

reaches the wall temperature. The thermal field develop-

ment continues after the location of the singular point until

reaching the fully developed regime. Similar to positive Br

cases, applying the temperature jump condition reduces

drastically the heat transfer rates along the whole tube

length.

Besides, for fixed values of Kn and b, the fully devel-

oped Nusselt number converges to the same value as it is

the case for positive Br.

Figure 9 present the behavior of the Nusselt number for

both cases of first and second-order slip flow models as

well as for different Br, Kn and b combinations. It is

important to notice that when the temperature jump con-

dition is not considered (b = 0), the local Nusselt number

increases with increasing Kn. However, when b = 0 the

reverse case is observed i.e. Nux is reduced with increasing

Kn. Therefore, one can conclude that the velocity slip and

temperature jump conditions have reverse effects on the

Nusselt number behavior.

As Kn increases and when the temperature jump is

considered the heat transfer coefficient values are consid-

erably reduced except in the discontinuity region for neg-

ative Br numbers.

The effect of using either the first or the second order

slip flow model is not significant in particular when

Br = 0.1 where curves of Nux are for both models almost

identical. Small differences are however observed for the

case of Br = -0.1 where the second order model under-

estimates the heat transfer rates.

5 Conclusion

Simultaneously developing laminar flow in a micro-tube is

modeled using first and second-order slip boundary con-

ditions applied to the continuity, momentum and energy

equations. The constant wall temperature boundary con-

dition is used and the viscous dissipation is accounted in

the energy equation.

The obtained results were presented in terms of devel-

opment of axial velocity profiles and axial distributions of

the heat transfer coefficient and the friction factor for dif-

ferent combinations of Knudsen number, coefficient b and

Brinkman number. The main results of this study can be

summarized as follows:

• The hydrodynamic field is found to be dependent on the

degree of rarefaction represented by Kundsen number

and significant variations of the velocity profiles and

the friction coefficient have been observed. Increasing

Kn reduces the axial velocity of the tube center and

induces a flattening of the velocity profiles.

• Temperature jump condition has significant effect on

heat transfer mechanism and high values of the

coefficient b induce important reductions in the heat

transfer rates.

• The influence of viscous dissipation depends if the fluid

is heated (negative Br) or it is cooled (positive Br). For

the heating case, the Nusselt number distribution

exhibits a special behavior characterized by the exis-

tence of a singular point where Nusselt number goes to

the infinity.

• The viscous dissipation effect is important on the

Nusselt number distribution along the micro-tube. This

effect is accentuated when introducing slip flow and

temperature jump conditions.

• The flattening of the axial velocity profiles is more

pronounced when using the first order slip flow model

rather than the second one. For the heat transfer,

implementing the first order or the second-order model

leads almost to the same results.

Acknowledgments The second author (Dr J Orfi) extends his

appreciation to the Deanship of Scientific Research at King Saud

University (Research group project No: RGP-VPP-091).

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