Cent. Eur. J. Phys. • 8(3) • 2010 • 498-508DOI: 10.2478/s11534-009-0105-x

Central European Journal of Physics

Effects of partial slip on axisymmetric flow of anelectrically conducting viscoelastic fluid past astretching sheet

Research Article

Bikash Sahoo∗

Department of Mathematics, National Institute of Technology, Rourkela, Rourkela-769008

Received 12 February 2009; accepted 4 June 2009

Abstract: The entrained flow of an electrically conducting non-Newtonian, viscoelastic second grade fluid due to anaxisymmetric stretching surface with partial slip is considered. The partial slip is controlled by a dimen-sionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suitable similaritytransformations are used to reduce the resulting highly nonlinear partial differential equation into an or-dinary differential equation. The issue of paucity of boundary conditions is addressed, and an effectivenumerical scheme has been adopted to solve the obtained differential equation even without augmentingthe boundary conditions. The important findings in this communication are the combined effects of thepartial slip, magnetic interaction parameter and the second grade fluid parameter on the velocity and skinfriction coefficient. It is observed that in presence of slip, the velocity decreases with an increase in themagnetic parameter. That is, the Lorentz force which opposes the flow leads to enhanced deceleration ofthe flow. Moreover, it is interesting to find that as slip increases in magnitude, permitting more fluid to slippast the sheet, the skin friction coefficient decreases in magnitude and approaches zero for higher valuesof the slip parameter, i.e., the fluid behaves as though it were inviscid.

PACS (2008): 47.11.Bc, 47.15.Cb, 47.50.Cd, 47.65.Cb

Keywords: second grade fluid • MHD flow • partial slip • finite difference method • Broyden’s method© Versita Sp. z o.o.

1. Introduction

The study of laminar boundary layer flow over a stretchingsheet has received considerable attention in the past dueto its applications in the industries, for example, materialsmanufactured by extrusion processes, the boundary layeralong a liquid film in condensation processes and heat-∗E-mail: bikashsahoo@nitrkl.ac.in

treated materials traveling between a feed roll and thewind-up roll or on a conveyor belt pose the features of amoving continuous surface. In view of these applicationsSakiadis [1] initiated the study of boundary layer flow overa continuous solid surface moving with constant speed.Due to the entrainment of the ambient fluid, this boundarylayer is quite different from that in the Blasius flow [2, 3]past a flat plate. It is often tacitly assumed that the sheetis inextensible, but situations may arise in the polymerindustries in which it is necessary to deal with a stretch-ing plastic sheet. This is bound to alter significantly the498

Bikash Sahoo

boundary layer characteristics of the flow considered bySakiadis [1]. The steady two-dimensional laminar flow ofan incompressible, viscous fluid past a stretching sheet hasbecome a classical problem in fluid dynamics as it admitsan unusual simple closed form solution, first discoveredby Crane [4]. In fact, realistic stretching of the sheet maynot necessarily be linear. Many authors [5, 6] have dealtwith this situation. Moreover, the flow past a stretchingsheet need not be necessarily two-dimensional, becausethe stretching of the sheet can take place in a variety ofways. If the flow is axisymmetric, an exact closed form so-lution does not seem to exist [7–11]. The analytic solutionfor the axisymmetric flow of a viscous fluid over a non-linearly stretching sheet has been considered by Sajid etal. [12]. The unsteady axisymmetric flow and heat trans-fer of a viscous fluid has been investigated by Sajid etal. [13]. All of the above investigations were restrictedto the flows of Newtonian fluids. Many materials suchas polymer solutions or melts, drilling mud, clastomers,certain oils, greases and many other emulsions are clas-sified as non-Newtonian fluids. There are many modelsdescribing the properties of non-Newtonian fluids. Thesemodels or constitutive equations, however, cannot describeall the behavior of theses non-Newtonian fluids, for ex-ample, normal stress differences, shear thinning or shearthickening, stress relaxation, elastic effects and memoryeffects etc. A rigorous study of the boundary layer flowof different non-Newtonian fluids past a stretching sheetwas required due to its immense industrial applications.Rajagopal et al. [14] considered the flow of a viscoelasticsecond order fluid past a stretching sheet and obtainedthe numerical solution of the fourth order nonlinear dif-ferential equation. Andersson [15] and Ariel [16] reportedthe analytical closed form solutions of the fourth ordernonlinear differential equations arising due to the MHDflow of viscoelastic Walters’ B’ fluid and the second gradefluid, respectively. Further, one can refer to the work ofLiu [17] and all the references therein regarding the flowand heat transfer of viscoelastic second grade fluids withdiverse physical effects. Recently, Sahoo and Sharma [18]carried out an analysis studying the existence, unique-ness and behavior of the fourth order nonlinear coupledordinary differential equations arising due to the flow andheat transfer of an electrically conducting second gradefluid past a stretching sheet. Ariel [19] adopted an effec-tive numerical method to study the flow of a viscoelasticsecond grade fluid past an axisymmetric stretching sheet.Subsequently, Hayat and Sajid [20] extended the prob-lem considered by Ariel [19] by incorporating the energyequation. Recently, Hayat et al. [21] obtained the analyti-cal solution of the three-dimensional flow of a viscoelasticfluid over a stretching surface.

A common feature of the above investigations and all theassociated references is the assumption that the flow fieldobeys the conventional no-slip condition at the sheet, i.e.,the velocity component u(x, y) parallel to the sheet be-comes equal to the sheet velocity U = Cx at the sheet. Incertain situations, however, the assumption of no-slip nolonger applies and should be replaced by the partial slipboundary condition [22]. The inadequacy of the no-slipcondition is evident for most of the non-Newtonian flu-ids. For example, polymer melts often exhibit macroscopicwall slip, in general governed by a nonlinear and mono-tone relation between the slip velocity and the traction.This may be an important factor in shear skin, spurt andhysteresis effects. Also, the fluids that exhibit boundaryslip have important technological applications such as inthe polishing of artificial heart valves and internal cavities.Wang [23] considered the influence of partial slip on theflow of a viscous fluid over a stretching sheet. He ob-tained the solution numerically by adopting the shootingmethod along with the fifth order Runge-Kutta method.Andersson [24] discussed the partial slip effects on the flowcharacteristics of a viscous fluid by finding an exact ana-lytical solution for the problem considered by Wang [23].It is noteworthy to mention that the solution obtained byAndersson [24] is an exact solution of the Navier-Stokesequations, and as such formally valid for any Reynoldsnumber. Ariel [8] studied the steady, laminar, axisymmet-ric flow of a Newtonian fluid due to a stretching sheetwith partial slip boundary condition. Unlike the case oftwo-dimensional flow (see Ref. [24]), the axisymmetric flowproblem with partial slip does not admit an exact solution.This affords a number of interesting techniques and algo-rithms for obtaining the solution. Very recently, Wang [25]revived an interest in the viscous flow due to a stretchingsheet with slip and suction. In this study, he consideredboth the two-dimensional and the axisymmetric cases. Fora two-dimensional stretching sheet, a closed form solutionis found and uniqueness is proved. For an axisymmet-ric stretching sheet, both existence and uniqueness areshown, and since there are no closed form solutions, nu-merical solution of the axisymmetric case is obtained.In all the above mentioned studies, no attention has beengiven to the effects of partial slip on the flow of any non-Newtonian fluid over a stretching sheet. Ariel et al. [26]tried to fill this gap, and have studied the effects of partialslip on the flow of an elastico-viscous Walters’ B’ liquidover a stretching sheet. They reported an analytical solu-tion of the governing boundary layer problem. Recently,Hayat et al. [27] obtained analytical solution of the slipflow and heat transfer of a second grade fluid past a pla-nar stretching sheet. It seems that there has been rela-tively little information regarding the joint effects of the499

Effects of partial slip on axisymmetric flow of an electrically conducting viscoelastic fluid past a stretching sheet

Figure 1. Schematic diagram of the flow domain.

partial slip and the non-Newtonian flow parameters onthe flow due to a stretching sheet. The axisymmetric flowof a non-Newtonian fluid due to a stretching sheet whenthere is partial slip at the boundary is an equally impor-tant and interesting problem that remains unexplored sofar. The objective of the present study is to investigatethe effects of partial slip on the MHD flow of an electri-cally conducting second grade fluid past an axisymmetricstretching sheet. To the best of our knowledge, no atten-tion has been given to the MHD boundary layer flow of asecond grade fluid over an axisymmetric stretching sheetwith partial slip boundary condition.2. Formulation of the problemWe consider the steady, laminar flow of an electricallyconducting second grade fluid over a stretching sheetwhich is placed in the plane z = 0 and is stretched,keeping the origin fixed, along the radial direction with avelocity proportional to the distance from the origin (seeFig. 1). The flow takes place in the upper half plane z > 0.A uniform magnetic field B = (0, 0, B0) is imposed alongthe z axis. The fluid adheres to the sheet partially, andthus motion of the fluid exhibits the slip condition.2.1. Flow analysisThe equations of motion and continuity are

ρdVdt =∇ · T + J× B, (1)

and

∇ · V = 0, (2)where ρ is the density of the fluid assumed to be constant,T is the Cauchy stress [28] for an incompressible homo-geneous fluid of grade two and J is the electric current.The term due to the Lorentz force (J× B) is simplified bymaking use of the following relevant assumptions [29]:

• all physical quantities are constant,• the magnetic field is perpendicular to the velocityfield,• the induced magnetic field is small compared to theapplied magnetic field,• the electric field is assumed to be zero.

These assumptions are valid when the magnetic Reynoldsnumber is small and there is no displacement current. It isnatural to work with cylindrical polar coordinate system(r, φ, z). In view of the symmetry, ∂∂φ ≡ 0. Also v , thetransverse component of the velocity V = (u, v, w) van-ishes identically. Hence, Eq. (1) has the following com-ponents:

ρ(u∂u∂r + w ∂u∂z

) = ∂Trr∂r + ∂Trz

∂z + Trr − Tφφr − σB20u,(3)

ρ(u∂w∂r + w ∂w∂z

) = 1r∂∂r (rTrz) + ∂Tzz

∂z , (4)where σ is the electrical conductivity of the fluid, and thenon-vanishing physical components Trr , Tφφ, Tzz and Trzof T are given by

Trr = −p+ 2µ∂u∂r + 2α1[u∂

2u∂r2 + w ∂2u

∂r∂z + 2(∂u∂r)2

+∂w∂r

(∂u∂z + ∂w

∂r

)]+α2[4(∂u∂r )2 + (∂u∂z + ∂w

∂r

)2],

(5)

Tφφ = −p+ 2µur + 2α1(ur∂u∂r + w

r∂u∂z + u2

r2)+ 4α2 u2

r2 ,(6)

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Bikash Sahoo

Tzz = −p+ 2µ∂w∂z + 2α1[u ∂

2w∂r∂z + w ∂

2w∂z2 + ∂u

∂z

(∂u∂z + ∂w

∂r

)+ 2(∂w∂z)2]+ α2

[(∂u∂z + ∂w

∂r

)2 + 4(∂w∂z)2]

, (7)

Trz = µ(∂u∂z + ∂w

∂r

)+ α1[(u ∂∂r + w ∂

∂z

)(∂u∂z + ∂w

∂r

)+ ∂u∂r∂w∂r + ∂u

∂z∂w∂z

+3(∂u∂r ∂u∂z + ∂w∂r

∂w∂z

)]+ 2α2(∂u∂r + ∂w

∂z

)(∂u∂z + ∂w

∂r

). (8)

The continuity Eq. (2) takes the form∂u∂r + u

r + ∂w∂z = 0. (9)

The appropriate Navier’s boundary conditions (see Ref. [22]) of the velocity field are,u− Cr

∣∣∣z=0 = λ1ν

[(∂u∂z + ∂w

∂r

)+ α1{(

u ∂∂r + w ∂∂z

)(∂u∂z + ∂w

∂r

)+ ∂u∂r∂w∂r + ∂u

∂z∂w∂z

+3(∂u∂r ∂u∂z + ∂w∂r

∂w∂z

)}+ 2α2(∂u∂r + ∂w

∂z

)(∂u∂z + ∂w

∂r

)]z=0 , w(0) = 0,

u→ 0 as z →∞, (10)where C > 0 is the constant of proportionality relating to the stretching of the sheet, λ1 is the slip coefficient havingdimension of length and ν is the kinematic viscosity.Problems of the kind under consideration admit similarity solutions. We find that the transformations

u = Crφ′(ζ), w = −2√Cµρ φ(ζ), where ζ = √Cρ

µ z, (11)reduce the Navier-Stokes equations to ordinary differential equations.From Eq. (3), we obtain

∂p∂r = ρC 2r

[φ′′′ − φ′2 + 2φφ′′ − 2α1C

µ(φ′′2 − φφiv)+ α2C

µ(φ′′2 − 2φ′φ′′′)]− σB20Crφ′, (12)

and Eq. (4) simplifies to∂p∂ζ = −4Cµφφ′ − 2Cµφ′′ + 4α1C 2 [φφ′′′ + 11φ′φ′′ + ρCr2

µ φ′′φ′′′]+ 2α2C 2 [14φ′φ′′ + ρCr2

µ φ′′φ′′′], (13)

where a prime denotes the derivative with respect to ζ. Integrating Eq. (13) with respect to ζ, we obtain,p = −2Cµφ2 − 2Cµφ′ + 2α1C 2 [2φφ′′ + 10φ′2 + ρCr2

µ φ′′2]+ α2C 2 [14φ′2 + ρCr2

µ φ′′2]+ f(r), (14)

where f(r) is an arbitrary function of r.501

Effects of partial slip on axisymmetric flow of an electrically conducting viscoelastic fluid past a stretching sheet

Substituting p from Eq. (14) into Eq. (12), we obtain,f ′(r)ρC 2r = [φ′′′ − φ′2 + 2φφ′′ − 2K (φ′′2 + φφiv

)− L

(φ′′2 + 2φ′φ′′′)]−Mnφ′, (15)

where K = α1Cµ , L = α2C

µ and Mn = σB20ρC are respectively the dimensionless non-Newtonian fluid parameters and themagnetic parameter (Hartmann number). It is clear from Eq. (15) that the left hand side is a function of r and the righthand side is a function of ζ only. Hence, for consistency, each side must be equated to a constant, say A. Thus, wehave,

f ′(r) = ρC 2rA. (16)Integrating the above equation with respect to r, we obtain,f(r) = p0 + 12ρC 2r2A, (17)

where p0 is the constant of integration.Substituting the expression for f(r) from Eq. (17) into Eq. (14), we get an explicit expression for the pressure at anypoint in the fluid, which is given byp = p0 + 12ρC 2r2A− 2Cµφ2 − 2Cµφ′ + 4α1C 2 (φφ′′ + 5φ′2)+ 14α2C 2φ′2 + (2α1 + α2) ρr2C 3

µ φ′′2. (18)Since, the entire motion of the fluid is due to the stretching of the sheet, the pressure far away from the sheet must begiven by Bernoulli’s equation, i.e., p+ 12ρw2 = constant, and this implies A = 0. Hence, from Eq. (15), we obtain,

φ′′′ + 2φφ′′ − φ′2 − 2K (φ′′2 + φφiv)− L

(φ′′2 + 2φ′φ′′′)−Mnφ′ = 0. (19)

But for the second grade fluid model, we haveK ≥ 0 and K + L = 0. (20)

With the help of conditions (20), we obtain the following fourth order highly nonlinear differential equation for φ:φ′′′ + 2φφ′′ − φ′2 − K (2φφiv − 2φ′φ′′′ + φ′′2)−Mnφ′ = 0, (21)

which is to be solved under the following partial slip boundary conditions:φ(0) = 0, φ′(0)− 1 = λφ′′(0) [1 + 4Kφ′(0)] ,φ′ → 0 as ζ →∞, (22)

where λ = λ1√ Cν is the non-dimensional parameter indicating the relative importance of partial slip.Another quantity of interest in the boundary layer flow is the local skin friction coefficient or frictional drag coefficient,which is related to the wall shear stress Trz |z=0, and is given by

Cf (r) = Trz |z=012ρ(Cr)2 , (23)which in terms of the dimensionless quantities is

Cf (r) = 2√Rr

[φ′′(ζ) + 2K {φ′(ζ)φ′′(ζ)− φ(ζ)φ′′′(ζ)}]ζ=0 , (24)where Rr = Cr2

ν is the local Reynolds number based on the length scale r.502

Bikash Sahoo

3. Numerical solution of the prob-lemAt this point one can see that the order of the Eq. (21) isfour, but there are only three available adherence bound-ary conditions, out of which the known boundary condi-tions are two, in contrast to the no-slip case [19]. In fact,the equations governing the flow of fluids of second gradeare of higher order than the Navier-Stokes equations be-cause of the presence of the term dA1

dt in the expression forthe stress and since only the adherence boundary condi-tion pertains, we do not have enough boundary conditionsto make the problem determinate. The above nonlinearequation under the relevant boundary conditions is solvedby a similar numerical scheme as described in [19, 30–34].The semi-infinite integration domain ζ ∈ [0,∞) is re-placed by a finite domain ζ ∈ [0, ζ∞), where ζ∞ is suffi-ciently large. In this case, the extra boundary conditionto be used implicitly is φ′′′(0). Using the boundary condi-tions (22) in (21), we obtain,φ′′′(0) = φ′2(0) + Kφ′′2(0) +Mnφ′(0)1 + 2Kφ′(0) . (25)

However, since φ′(0) and φ′′(0) are not known a priori,in contrast to the no-slip condition (where φ′(0) = 1),one has to rely on the shooting method to get the correctvalue of φ′′′(0) for comparatively small values of the flowparameters. In practice, we have made initial guesses onφ′(0) and φ′′(0). Then φ′′′(0) can be obtained from φ′(0)and φ′′(0) by using Eq. (25). Thus, now we have fourboundary conditions at ζ = 0, which are sufficient to solveEq. (21) as an initial value problem using any standardroutine. In this work, the fourth order Runge-Kutta methodhas been adopted to solve the initial value problem. Theguesses on φ′(0) and φ′′(0) are improved using Broyden’smethod [35, 36] so as to satisfy the boundary conditions

at infinity, namely φ′(∞) = 0. Broyden’s method is quiteefficient in the sense that it avoids the calculation of theJacobian matrix (contrary to Newton’s method) by obtain-ing approximations to them involving only function values.The use of Broyden’s method has helped us in achievingthe greatest accuracy with the least computational cost.To explain the solution scheme developed in this work, westart by introducing the following variables:y1 = φ, y2 = φ′, y3 = φ′′. (26)

The governing Eq. (21) can then be written asy′3 + 2y1y3 − y22 − K (2y1y′′3 − 2y2y′3 + y23)−Mny2 = 0.(27)The boundary conditions become

ζ = 0 : y1 = 0, y2 − 1 = λy3(1 + 4Ky2),ζ → ζ∞ : y2 → 0. (28)

We now introduce a mesh defined byζi = ih; i = 1, 2, 3, . . . n, (29)

where n is a sufficiently large number. Using the firstorder approximations,F ′ = F j − F j−1

h +O(h), andF ′′ = F j − 2F j−1 + F j−2

h2 +O(h), (30)we rewrite Eq. (27) as

yj3 − yj−13h + 2yj1yj3 − (yj2)2

− K[2yj1(yj3 − 2yj−13 + yj−23

h2)− 2yj2(yj3 − yj−13

h

)+ (yj3)2]−Mnyj2 = 0, (31)yj+12 − yj2 = h2 yj3, (32)yj+11 − yj1 = h2 yj2. (33)

Note that in Eqs. (32) and (33), the first order approximations for the derivatives have been used, but centered at thepoint (j + 12 ). Thus, the discretization scheme has an accuracy of O(h).To construct the rest of the solution, we proceed as follows. We obtain the values of y11 (= φ(0 + h)), y12 (= φ′(0 + h))and y13 (= φ′′(0 + h)) by Taylor series expansion near ζ = 0. During these expansions, the values of φ′(0) and φ′′(0) are503

Effects of partial slip on axisymmetric flow of an electrically conducting viscoelastic fluid past a stretching sheet

required, which are already found by the shooting method. The value of φ′′′(0) can be found from (25). Having found thevalues of y11, y12, and y13, one can obtain the values of y22 and y21 from(32) and (33), respectively. These values are thenused in (31) to calculate y23. At the next cycle, the values of y32 and y31 are calculated from (32) and (33) and subsequentlyused in (31) to get y33. The order indicated above is followed for the subsequent cycles, straightforwardly up to j = n.At that point, y(n)2 , or equivalently φ′(ζ∞), is compared with the given boundary conditions at ζ∞. The initial guesses onφ′(0) and φ′′(0) are corrected using Broyden’s method until the convergence criterion is satisfied.It is noteworthy to point out that a serious problem is encountered if one discretizes Eq. (27) by a second order centraldifference approximation. In fact, with the central difference approximation, Eq. (27) can be discretized as

yj+13 − yj−132h + 2yj1yj3 − (yj2)2 − K[2yj1(yj+13 − 2yj3 + yj−13

h2)+ 2yj2(yj+13 − yj−132h

)+ (yj3)2]−Mnyj2 = 0, (34)which can be readily solved for yj+13 to yield,yj+13 = [1− 2K (2yj1

h − yj2)]−1{[1 + 2K (2yj1

h + yj2)]

yj−13 −[4hyj1 + 2K (4yj1

h − hyj3)]

yj3 + 2h [(yj2)2 +Mnyj2]}

.

(35)The trouble in using Eq. (35) is that, because K > 0,as the integration proceeds, the denominator changes itssign, even for the partial slip case. At the point wherethis transition occurs, the integration process breaks down.Ariel [19] has pointed out this drawback in the central dif-ference scheme while solving the axisymmetric flow prob-lem with no-slip boundary conditions. For the no-slipcase, a remedy for this problem is to solve Eq. (34) foryj−13 instead of yj+13 , or in other words, the Eq. (27) mustbe integrated backwards rather than forwards. But for thepresent investigation with partial slip boundary condition,such a backward journey is not preferable due to the ac-cumulation of so many unknown boundary conditions [seeEq. (28)] at ζ = 0.The fact that the algorithm has an accuracy of only O(h)need not concern us unduely as we can easily hike the ac-curacy to order O(h2) and O(h4) respectively by invokingRichardson’s extrapolation. With reasonably close trialvalues to start the iterations, the convergence to the actualvalues within an accuracy of O(10−6) could be obtained in7− 9 iterations. No difficulties were encountered in com-puting the flow for the entire range of values of K , Mn andλ.4. Results and discussionThe method described above was translated into a FOR-TRAN 90 program and run on a Pentium IV personal com-puter. The value of ζ∞, the numerical infinity, has beentaken large enough and kept invariant throughout the runof the program. If ζ∞ is not large enough, the numerical

solution will not only depend on the physical flow param-eters, but also on ζ∞. Although the results are shown onlyfrom the surface ζ = 0 to ζ = 10.0, for higher values of theflow parameters, the numerical integrations are performedover a substantially larger domain in order to assure thatthe numerical solution closely approximates the terminalboundary condition φ′ → 0. To see if the program runscorrectly, the results of the missing initial guess −φ′′(0)are compared with the exact numerical solution, approxi-mate solution and homotopy perturbation method [37–39](see Tab. 1) for a second grade fluid without slip and fora Newtonian fluid with partial slip, respectively, reportedby Ariel [8, 19], and the comparison is found to be in goodagreement.In order to have insight into the flow characteristics, re-sults are plotted graphically in Figs. 2-9 for differentchoices of the flow parameters.In Figs. 2 and 3, we plot the non-dimensional velocitycomponent φ′(ζ), the mainstream velocity, against ζ forvarious values of K , keeping the values of the other flowparameters constant. It can be observed that φ′(ζ) de-creases with ζ for K keeping constant. Fig. 3 depictsa cross over in the velocity profile, indicating the sim-ilarity curves are not similar to each other. In fact, inpresence of slip (λ = 1.0), as the value of K increases inmagnitude, the flow slows down for distances close to thesheet and for distances away from the sheet, the oppo-site is true, contrary to the results obtained analyticallyby Ariel et al. [26] for the viscoelastic Walters’ B’ modelpast a planar stretching sheet. Our result seems to bephysically more plausible. Since, the viscoelasticity ofthe fluid thickens the width of the boundary layer within504

Bikash Sahoo

Figure 2. Variation of φ′ with K at Mn = 1.0 & λ = 0.

Figure 3. Variation of φ′ with K at Mn = 1.0 & λ = 1.0.

which the effects of viscosity are confined and it takeslonger distance for φ′(ζ) to approach its asymptotic valuezero as the value of K is increased. The reason behindsuch a discrepancy of results may be due to the sign ofthe non-Newtonian fluid parameter1 K . However, fromFig. 2 it is clear that for the no-slip boundary conditions(λ = 0), φ′(ζ) increases with K throughout the domainof integration. A comparison with the corresponding plotof Ariel [19] for λ = 0 indicates good agreement betweenthe two results. Thus, the viscoelasticity increases themomentum boundary layer thickness.In Figs. 4 and 5, the value of φ′(ζ) is plotted for vari-ous values of Mn and λ, respectively. It is observed thatthe mainstream velocity decreases in both cases. It isclear from Fig. 5 that as the slip parameter increases in1 For Walters’ B’ fluid K < 0.

Figure 4. Variation of φ′ with Mn at K = 2.0 & λ = 1.0.

Figure 5. Variation of φ′ with λ at K = 2.0 & Mn = 1.0.

magnitude, permitting more fluid to slip past the sheet,the normalized fluid velocity on the boundary φ′(0) re-mains less than the normalized stretching surface velocityof unity, or in other words, the flow slows down for dis-tances close to the sheet, as was expected. In the limitingcase, as λ→∞, the resistance between the viscous fluidand the surface is eliminated and the stretching of thesheet no longer imposes any motion on the fluid, i.e., theflow behaves as though it were inviscid.Figs. 6-9 elucidate the variation of the skin friction coef-ficient Cf (r) with different flow parameters with Rr = 1.0.Fig. 6 shows that in absence of slip (λ = 0), the skinfriction coefficient is increased in magnitude with an in-crease in the viscoelastic parameter K . This predictionis of course undesirable from an industrial standpoint be-cause it translates into a large driving force (or torque).Fig. 7 depicts the variation of Cf (r) with K in presence ofslip. One of the novel findings of the present investigationis that in the presence of the slip factor, the skin friction505

Effects of partial slip on axisymmetric flow of an electrically conducting viscoelastic fluid past a stretching sheet

Figure 6. Variation of Cf (r) with K at Mn = 1.0 & λ = 0.

Figure 7. Variation of Cf (r) with K at Mn = 1.0 & λ = 0.5.

Figure 8. Variation of Cf (r) with Mn at K = 2.0 & λ = 1.0.

Figure 9. Variation of Cf (r) with λ at K = 2.0 & Mn = 1.0.

coefficient initially decreases in magnitude, reaches theminimum value (critical value not precisely determined)and then starts increasing rapidly as the value of K in-creases. Such a turning point is absent in the no-slipcase (see Fig. 6). The skin friction coefficient increasesin magnitude with an increase in the magnetic interactionparameter Mn, as is clear from Fig. 8. It is interesting tofind that (see Fig. 9) the skin friction coefficient decreasesin magnitude with an increase in slip. As λ→∞, Cf (r) ap-proaches zero. This implies that the frictional resistancebetween the fluid and the surface of the sheet is elimi-nated and the stretching of the sheet no longer imposesany motion of the fluid. This observation is in agreementwith that reported by Andersson [24] for a nonmagneticviscous fluid.5. ConclusionsThe present work is a worthwhile attempt to study the ef-fects of partial slip on the axisymmetric flow of an electri-cally conducting non-Newtonian second grade fluid pasta stretching sheet. The new set of slip-flow boundaryconditions is aimed to accommodate the partial slip ef-fect. The applicability of these conditions in boundarylayer theory is explored, and numerical similarity solu-tions are presented. An effective numerical method hasbeen adopted to solve the resulting highly nonlinear dif-ferential equation subject to the slip boundary conditions.The use of Broyden’s method has indeed enhanced theefficiency of the present algorithm by reducing the com-putational (CPU time) time. The combined effects of theslip, magnetic parameter and the viscoelastic parameteron the velocity field and the skin friction coefficient arestudied in detail. In presence of slip, it is interesting to

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Bikash Sahoo

Table 1. Variations of −φ′′(0) with K and λ at Mn = 0.

K λ Current result Ariel [19] Ariel [8]Numerical Approximate Numerical HPM

2.0 0.615210 0.613547 0.6048584.0 0.470976 0.463684 0.4597636.0 0.402141 0.387808 0. 3854798.0 0.361422 0.340047 0.33846510.0 0.334430 0.306462 0.30529810.5 0.0 0.329042 - -11.0 0.324071 - -11.5 0.319470 - -12.0 0.315199 - -15.0 0.294867 - -20.0 0.253321 0.220079 0.219645

0.0 1.173193 1.173721 1.1785110.1 1.000984 1.001834 1.0003080.2 0.877495 0.878425 0.8744530.5 0.649697 0.650528 0.6453041.0 0.461918 0.462510 0.458333

0.0 2.0 0.298714 0.299050 0.2965343.0 0.222909 - -4.0 0.178559 - -5.0 0.149277 0.149393 0.14845410.0 0.082840 0.082912 0.082532

find a turning point in the curve representing the variationof the skin friction coefficient with non-Newtonian param-eter. Moreover, it is observed that as the slip parameterλ increases, permitting more fluid to slip past the sheet,the skin friction coefficient decreases in magnitude andapproaches zero for higher values of the slip parameter,i.e., the fluid behaves as though it were inviscid.Acknowledgements

The suggestions made by the referees to improve the qual-ity of the paper are highly acknowledged.

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