Stokes flow past an arbitrary shaped body with slip-stick boundary conditions

Applied Mathematics and Computation 219 (2013) 5367–5375

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Stokes flow past an arbitrary shaped body with slip-stick boundary conditions D. Choudhuri ⇑, R. Radha, B. Sri Padmavati Department of Mathematics & Statistics, University of Hyderabad, Hyderabad 500046, India

a r t i c l e

i n f o

Keywords: Stokes flow Arbitrary shaped body Slip-stick boundary conditions

a b s t r a c t We give an analytical but approximate solution to the Stokes equations governing the motion of an incompressible flow of viscous fluid past a body of arbitrary shape with slip-stick boundary conditions. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The study of low Reynolds number flows plays a vital role in many areas of science and engineering as diverse as lubrication theory, flagellar propulsion of micro-organisms and many areas of rheology, to name a few. Most of the earlier studies were concerned with particles of axisymmetric shapes rather than with particles of arbitrary shapes which would present a considerable degree of complication. However the solutions obtained provided reasonably accurate solutions for relatively small particles which were modelled to be spherical in shape. But a more realistic and general analysis must involve methods for discussing Stokes flows for arbitrary body shapes. Most non-spherical particles in particulate suspensions were modelled as ellipsoidal particles wherein the shapes like disks and needles could also be included as the degenerate cases. But most of these methods either involve the complicated ellipsoidal coordinates or improper integrals or are not simple to use in practice. Very few analytical methods that discuss exact solutions were available like singularity methods and boundary value methods for bodies of arbitrary shapes. Recently Radha et al. [1,2] gave a simple method of solving the problem of an arbitrary axisymmetric Stokes flow past a body of arbitrary shape with no-slip boundary conditions using a solution of Stokes equations due to Palaniappan et al. [3]. They gave an approximate analytical solution using the method of least squares. The solution proposed by Palaniappan et al. [3] had lent itself for effectively solving many boundary value problems in non-axisymmetric Stokes flows involving spherical boundaries in terms of two scalar functions A and B which play the role of a stream function. Moreover, this solution has been proved to be a complete general solution by Padmavathi et al. [4]. Using this solution and the method developed by them for rigid bodies of arbitrary shape, Radha et al. solved the problem of uniform flow past a rigid body made up of two intersecting rigid spheres [1]. They also discussed the corresponding technique for non-axisymmetric flows past rigid bodies of arbitrary shape and illustrated the method for a Stokeslet outside a sphere, for uniform flow past an ellipsoid and the flow generated due to a Stokeslet outside an ellipsoid [2]. However many experiments have indicated that some slipping takes place for many fluids when they are in contact with the surface of a solid. So instead of employing the no-slip boundary conditions, it was suggested that it is more appropriate to use the slip-stick boundary conditions which are the vanishing of the normal velocity and the proportionality of ⇑ Corresponding author. E-mail address: [email protected] (D. Choudhuri). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.117

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the tangential velocities and the stresses on the boundary which are stated in terms of a parameter called the Basset slipcoefficient. These boundary conditions were employed by many researchers like Basset [5], Schmitz and Felderhof [6] to study some important problems of Stokes flows. In fact, it is possible to study the rigid (no-slip) and shear-free boundary conditions as limiting cases of these conditions. Padmavathi et al. [7] also discussed the motion of a viscous, incompressible fluid past a sphere using slip-stick boundary conditions in a non-axisymmetric Stokes flow by employing the complete general solution of Stokes equations proposed by Palaniappan et al. [3]. Several previous results pertaining to rigid and shear-free conditions could be retrieved as limiting cases. In this paper, we adopt the technique of Radha et al. [1,2] to find an analytic but approximate solution for the problem of Stokes flow of a viscous, incompressible fluid past a body of arbitrary shape satisfying the slip-stick boundary conditions. Thus, as we can obtain the rigid and shear-free cases as special cases of the slip-stick boundary conditions, we obtain a more general solution of the problem of an arbitrary Stokes flow past a body of arbitrary shape. We illustrated the method for the example of a singularity driven flow due to a Stokeslet located outside an ellipsoid with different slip parameters. In particular, we discuss the effect of the slip parameter on physical quantities like the drag and the torque. We also discuss the convergence of the method for different values of the slip parameter. 2. Mathematical formulation and method of solution The Stokes equations governing the motion a viscous, incompressible fluid are given as follows.

lr2~ q ¼ rp;

ð2:1Þ

r~ q ¼ 0;

ð2:2Þ

where ~ q is the velocity, p the pressure and mations

~ q0 ¼

~ q ; U

p0 ¼

L

lU

p;

x x0 ¼ ; L

y y0 ¼ ; L

l the co-efficient of dynamic viscosity of the fluid. We consider the transfor z z0 ¼ ; L

where U is a typical velocity and L a typical length to non-dimensionalize the steady Stokes Eqs. (2.1) and (2.2). We can then drop the primes for convenience and write the non-dimensionalized steady Stokes equations as follows.

r2~ q ¼ rp; r~ q ¼ 0:

ð2:3Þ ð2:4Þ

We discuss the problem of an incompressible flow of a viscous fluid past a body X of arbitrary shape satisfying the slip-stick conditions on the boundary @ X. The slip-stick boundary conditions are given in a non-dimensional form as follows.

~ ^ j@ X ¼ 0; qn ~ Tn^  ^t1 j@X ; q  ^t1 j@X ¼ k~ ~ Tn^  ^t2 j ; q  ^t2 j ¼ k~ @X

@X

ð2:5Þ ð2:6Þ ð2:7Þ

^ ; ^t1 ; ^t2  are the local orthowhere k is the slip-parameter which is equal to the reciprocal of the Basset’s slip-coefficient and ½n normal basis vectors on the surface of the particle @ X and ~ Tn^ is defined in Appendix C. The symbol j@ X indicates the evaluation at the boundary. We observe that if k ¼ 0 in (2.5)–(2.7), then we get the no-slip or the rigid boundary conditions which are given as follows.

~ ^ j@ X ¼ 0; qn ~ q  ^t1 j@ X ¼ 0; ~ q  ^t2 j ¼ 0 @X

and if k ! 1, then we get the shear-free boundary conditions which are given as follows.

~ ^ j@ X ¼ 0; qn ^ n ~ T  ^t1 j@X ¼ 0; ~ Tn^  ^t2 j ¼ 0: @X

The motivation for this method is from the work of Radha et al. [1,2] who proposed a method to find an approximate analytical soution of Stokes equations past a rigid body X of arbitrary shape, whose boundary is denoted by @ X satisfying the no-slip conditions on the boundary. Their method can be briefly described as follows. Let ð~ q0 ; p0 Þ denote the

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unperturbed flow in the absence of a rigid body and be expressed in terms of the solution due to Palaniappan et al. [3] as follows

~ rA0 Þ þ r  ð~ rB0 Þ; q0 ¼ r  r  ð~ @ p0 ¼ P 0 þ ðr r2 A0 Þ; @r

ð2:8Þ ð2:9Þ

where P0 is a constant and A0 ; B0 are the corresponding scalar functions satisfying r4 A0 ¼ 0; r2 B0 ¼ 0. Let the disturbance in the presence of a rigid boundary @ X be denoted by ð~ q1 ; p1 Þ. Then the perturbed flow is given by ð~ q; pÞ where

~ q1 ; q¼~ q0 þ ~

ð2:10Þ

p ¼ p0 þ p1 ;

ð2:11Þ

such that ð~ q1 ; p1 Þ ! ð0; 0Þ as r ! 1. Let the scalar functions corresponding to ð~ q; pÞ and ð~ q1 ; p1 Þ be denoted by ðA; BÞ and ðA1 ; B1 Þ respectively in the solution due to Palaniappan et al. [3] as in (2.8) and (2.9), where A1 and A are biharmonic functions and B1 and B are harmonic functions. Thus in the presence of a rigid body, the modified flow can be represented by the scalars A and B where

A ¼ A0 þ A1 ;

ð2:12Þ

B ¼ B0 þ B1 :

ð2:13Þ

In spherical polar coordinates ðr; h; /Þ, the no-slip condition ~ q ¼ 0 on the boundary is the same as satisfying qr ¼ qh ¼ q/ ¼ 0 on the boundary @ X of the rigid body X, where qr ; qh and q/ are the radial, tangential and azimuthal components of velocity of the fluid. Instead of applying the no-slip condition ~ q ¼ 0 on the boundary @ X, Radha et al. [1,2] adopted the condition that

I¼

Z

j~ qj2 dS ¼ 0:

ð2:14Þ

@X

This condition (2.14) can be rewritten as

I¼

Z @X

ðq2r þ q2h þ q2/ ÞdS ¼ 0:

Since ~ q is a continuous function of ðr; h; /Þ and ðq2r þ q2h þ q2/ Þ being a non negative quantity, it can be concluded that R I ¼ @ X ðq2r þ q2h þ q2/ ÞdS ¼ 0 if and only if ðq2r þ q2h þ q2/ Þ  0 and hence qr ¼ qh ¼ q/ ¼ 0, i.e., ~ q ¼ 0 on @ X. The approximate solution is given by ~ qN where

~ rAN Þ þ r  ð~ rBN Þ; qN ¼ r  r  ð~

ð2:15Þ

where

AN ¼ A0 þ AN1 ;

ð2:16Þ

BN ¼ B0 þ BN1 ;

ð2:17Þ

AN1

BN1

where and are the approximations to A1 and B1 respectively when they have been expressed as infinite series in terms of spherical harmonics and the series have been truncated up to N terms. Hence, instead of (2.14), an alternate condition was adopted by them that

IN ¼

Z

j~ qN j2 dS ¼ 0;

ð2:18Þ

@X

and the integral in (2.18) was minimized in order to obtain an approximate analytic solution. This results in a system of linear non-homogeneous equations which were solved for the unknown constants. Using the afore mentioned technique, Radha et al. solved the problem of an axisymmetric flow, namely, a uniform flow past a rigid body made up of two intersecting rigid spheres [1]. They also discussed the corresponding technique for non-axisymmetric flows past rigid bodies of arbitrary shape and illustrated the technique with some examples [2]. We now try to adopt this technique for slip-stick boundary conditions which can be applied to bodies of arbitrary shape ^ þ qt ^t1 þ qt ^t2 , then we can rewrite the slip-stick boundsatisfying more general conditions. Suppose we now write ~ q ¼ qn n 1 2 ary conditions on the boundary @ X given in (2.5)–(2.7) as

qn j@ X ¼ 0; qt1 j@ X ¼ kT t1 j@ X ; qt2 j@ X ¼ kT t2 j@ X ;

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where qn is the normal component of the velocity, qt1 ; qt2 and T t1 ; T t2 are the tangential velocity and stress components k respectively. Since k takes the values from 0 to 1 for slip-stick boundary conditions, we introduce a parameter b ¼ 1þk , so that b ¼ 0 when k ¼ 0 and b ! 1 when k ! 1. In other words, the domain of the slip parameter ½0; 1Þ has been k scaled down to ½0; 1Þ by using the transformation b ¼ 1þk . We use the following form of slip-stick boundary conditions in terms of b as

qn j@X ¼ 0;

ð2:19Þ

ð1  bÞqt1 j@ X ¼ bT t1 j@X ;

ð2:20Þ

ð1  bÞqt2 j@ X ¼ bT t2 j@X :

ð2:21Þ

Let the equation of the surface be given as r ¼ f ðh; /Þ in spherical polar co-ordinates ðr; h; /Þ (refer Appendix B). Motivated by the work of Radha et al. [1,2], we adopt their technique to reformulate the slip-stick boundary conditions (2.19)–(2.21) on the boundary @ X as requiring the fulfillment of the condition that

I¼

Z @X

½q2n þ ðð1  bÞqt1  bT t1 Þ2 þ ðð1  bÞqt2  bT t2 Þ2 dS ¼ 0;

ð2:22Þ

instead of the boundary conditions (2.19)–(2.21). We follow the notation in the work of Radha et al. [2] and assume as in (2.10)–(2.13) that the scalars ðA0 ; B0 Þ represent the given unperturbed flow ð~ q0 ; p0 Þ, the scalars ðA1 ; B1 Þ represent the disturbance ð~ q1 ; p1 Þ due to the presence of the body of arbitrary shape with slip-stick boundary conditions and the scalars ðA; BÞ represent the modified flow ð~ q; pÞ such that the Eqs. (2.10)–(2.13) hold. Then A1 and B1 can be expressed as an infinite series in terms of spherical harmonics as follows. 1 X n  X Anm

    C nm Bnm Dnm cos m/ þ sin m/ Pm þ n ðfÞ; r r n1 rnþ1 r n1 n¼1 m¼0  1 X n  X 1 ðEnm cos m/ þ F nm sin m/Þ nþ1 Pm B1 ¼ n ðfÞ; r n¼1 m¼0

A1 ¼

þ nþ1

ð2:23Þ

where Anm ; Bnm ; C nm ; Dnm ; Enm and F nm are unknown constants to be determined from the boundary conditions. So in order to compute the solution approximately, we truncate the series of A1 and B1 up to N terms and denote the resulting terms as AN1 and BN1 respectively and the approximate solution as ð~ qN ; pN Þ where ~ qN is as given in (2.15)–(2.17) (see Appendix A). Let

    C nm Bnm Dnm cos m/ þ sin m/ Pm þ n ðfÞ; n1 nþ1 n1 r r r r n¼1 m¼0  N X n  X 1 BN1 ¼ ðEnm cos m/ þ F nm sin m/Þ nþ1 Pm n ðfÞ: r n¼1 m¼0

AN1 ¼

N X n  X Anm

þ nþ1

ð2:24Þ

Now instead of (2.22), we impose an alternate condition that

IN ¼

Z @X

fðqNn Þ2 þ ðð1  bÞqNt1  bT Nt1 Þ2 þ ðð1  bÞqNt2  bT Nt2 Þ2 gdS ¼ 0;

ð2:25Þ

where the superscript N indicates that the flow quantities have been truncated up to n ¼ N in the series representation of the solution (refer Appendix A). The number of arbitrary constants occuring in (2.24) which are to be determined using the boundary conditions (2.19)–(2.21) are 3NðN þ 3Þ namely Anm ; Bnm ; C nm ; Dnm ; Enm ; F nm , where n ¼ 1; 2; . . . ; N; m ¼ 0; 1; . . . ; n. Clearly, IN being non-negative, the existence of an infimum of IN is guaranteed. Hence, we find the minimum value of IN by applying the following conditions.

@IN ¼ 0; @Aij @IN ¼ 0; @Dij

@IN ¼ 0; @Bij @IN ¼ 0; @Eij

@IN ¼ 0; @C ij @IN ¼ 0; @F ij

ð2:26Þ ð2:27Þ

i ¼ 1 to N and j ¼ 0 to i. These equations form a linear system of non-homogeneous equations which can be solved for the unknown constants Aij ; Bij ; C ij ; Dij ; Eij and F ij . We now present an example to illustrate this method. 2

2

Example 1. Consider the Stokes flow of a viscous, incompressible fluid past an ellipsoid given by x2 þ y 2 þ z 2 ¼ 1, induced 0:75 0:5 due to a Stokeslet of unit strength located at ð0; 0; 0:75Þ on the z-axis and whose axis lies along the positive x-axis. Clearly this is a non-axisymmetric flow. We assume the slip parameter value b to be equal to 13 in this example. By using the method N suggested for slip-stick boundary conditions, we have calculated ðAN 1 ; B1 Þ for N ¼ 1; 2; . . . ; 9 (refer to Appendices A and B). For N ¼ 9, we obtain

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AN1 ¼ ½fð2:11805r2  3:02799ÞP1 ðfÞ þ ð93:2183r 3  45:0989r1 ÞP2 ðfÞ  ð0:44899r 4  1:28621r 2 ÞP3 ðfÞ  ð17:0513r 5  9:19796r 3 ÞP4 ðfÞ þ ð0:0747035r 6  0:185572r 4 ÞP5 ðfÞ þ ð0:404749r 7  6:13047r5 ÞP6 ðfÞ  ð0:00525081r 8  0:00501518r 6 ÞP7 ðfÞ þ ð0:227785r 9  1:4524r 7 ÞP8 ðfÞ  ð0:000396663r 10  0:00288658r 8 ÞP9 ðfÞg  1012 þ fð1:51287r2  47:2512ÞP11 ðfÞ  ð0:681361r 3  1:99339r 1 ÞP 12 ðfÞ  ð0:224696r 4  1:16619r 2 ÞP 13 ðfÞ  ð0:00745753r 5  0:160097r3 ÞP14 ðfÞ þ ð0:0165318r 6  0:0412963r4 ÞP15 ðfÞ  ð0:00187356r7  0:0109095r5 ÞP16 ðfÞ  ð0:00146645r8  0:00419457r 6 ÞP17 ðfÞ  ð0:000197909r 9  0:00107501r 7 ÞP18 ðfÞ  ð0:0000102052r10  0:000117647r 8 ÞP19 ðfÞg cos /  102 þ fð1:1913r3  0:292837r1 ÞP22 ðfÞ  ð0:000329272r4  0:000749726r 2 ÞP23 ðfÞ  ð0:130568r 5  0:266977r 3 ÞP24 ðfÞ þ ð0:0000229244r6  0:0000468854r4 ÞP25 ðfÞ þ ð0:00494434r 7  0:00545668r 5 ÞP 26 ðfÞ þ ð0:00025619r 9  0:00173157r 7 ÞP28 ðfÞg cos 2/  1010  fð23:8926r 4  163:919r2 ÞP33 ðfÞ  ð6:6041r5  8:53223r 3 ÞP34 ðfÞ  ð2:89022r 6  10:1164r 4 ÞP35 ðfÞ  ð0:272567r 7  1:70785r 5 ÞP36 ðfÞ þ ð0:121787r8  0:317812r 6 ÞP37 ðfÞ þ ð0:0183889r 9  0:0796786r 7 ÞP38 ðfÞ  ð0:000726217r 10  0:00274236r8 ÞP39 ðfÞg  cos 3/  106  fð0:0235396r5  0:0400253r 3 ÞP 44 ðfÞ  ð0:00121001r7  0:00317844r 5 ÞP46 ðfÞ  ð0:0000255744r 9  0:000145017r 7 ÞP 48 ðfÞg cos 4/  1010  fð2:76227r 6 þ 0:528823r 4 ÞP55 ðfÞ þ ð0:0355246r 7 þ 0:576805r5 ÞP56 ðfÞ þ ð0:149749r 8  0:513836r 6 ÞP57 ðfÞ þ ð0:00614185r 9  0:0330422r 7 ÞP58 ðfÞ  ð0:00154886r 10  0:00291031r8 ÞP59 ðfÞg cos 5/  108 þ fð1:02088r 7  0:604942r 5 ÞP 66 ðfÞg cos 6/  1015  fð1:18818r 8  8:08031r 6 ÞP77 ðfÞ þ ð0:0453238r 9  0:171543r 7 ÞP78 ðfÞ  ð0:0205875r10  0:0707192r 8 ÞP79 ðfÞg cos 7/  1011 þ fð7:07611r 10  23:5485r8 ÞP99 ðfÞg cos 9/  1015 þ fð1:37581r 2  0:305558ÞP 11 ðfÞ þ ð7:37886r 3  0:101122r1 ÞP12 ðfÞ  ð0:0470325r 4  0:0419817r 2 ÞP13 ðfÞ  ð0:134944r 5  0:810666r3 ÞP14 ðfÞ þ ð0:0048352r6  0:00716801r4 ÞP15 ðfÞ  ð0:0419146r 7  0:247987r 5 ÞP16 ðfÞ  ð0:000253244r8 ÞP17 ðfÞ  ð0:00138792r 9  0:00866634r 7 ÞP 18 ðfÞ þ ð0:000123917r8 ÞP19 ðfÞg sin /  1012  fð0:660201r4  1:6685r2 ÞP33 ðfÞ þ ð5:09897r5  13:0666r3 ÞP34 ðfÞ  ð0:0164012r6  0:0405661r 4 ÞP35 ðfÞ  ð0:208962r 7  0:531209r 5 ÞP36 ðfÞ þ ð0:00146903r 9  0:00519343r 7 ÞP38 ðfÞg sin 3/  1013 þ fð1:8438r6  3:74806r 4 ÞP 55 ðfÞ þ ð4:03409r 7  7:21168r 5 ÞP56 ðfÞg  sin 5/  1016 ;

BN1 ¼ ½f16:5989r 2 P11 ðfÞ  0:672204r3 P 12 ðfÞ  18:1963r4 P13 ðfÞ þ 0:0883978r 5 P14 ðfÞ þ 3:89238r 6 P15 ðfÞ  0:0103824r 7 P 16 ðfÞ  0:545346r8 P17 ðfÞ þ 0:000623896r9 P 18 ðfÞ þ 0:0383653r 10 P19 ðfÞg cos / þ f1:73625r 3 P22 ðfÞg cos 2/ þ f2:60764r 4 P33 ðfÞ þ 0:0146756r5 P34 ðfÞ þ 0:338524r 6 P35 ðfÞ  0:000941878r 7 P36 ðfÞ  0:0428656r 8 P37 ðfÞ þ 0:00248631r 10 P39 ðfÞg cos 3/  f0:00487845r6 P 55 ðfÞ  0:000312676r 7 P57 ðfÞg cos 5/  1012  f132:538r 2 P11 ðfÞ  5:74295r 3 P 12 ðfÞ  2:58214r4 P 13 ðfÞ  0:236345r 5 P14 ðfÞ  0:0388528r 6 P15 ðfÞ  0:0347912r7 P16 ðfÞ  0:00854601r 8 P17 ðfÞ þ 0:000591397r9 P18 ðfÞ þ 0:0000351253r10 P19 ðfÞg sin /  103  f0:0400477r3 P 22 ðfÞ þ 2:61287r4 P 23 ðfÞ  0:00103875r5 P24 ðfÞ þ 0:661265r 6 P25 ðfÞ þ 0:0000257138r 7 P26 ðfÞ  0:0557074r8 P27 ðfÞ þ 0:00139546r 10 P29 ðfÞg sin 2/  1011 þ f1975:53r 4 P33 ðfÞ þ 143:257r 5 P24 ðfÞ þ 2:61809r6 P25 ðfÞ  1:64987r7 P26 ðfÞ  0:0233622r 8 P27 ðfÞ  0:229851r 9 P28 ðfÞ  0:0539956r 10 P29 ðfÞg sin 3/  107 þ f0:0124402r 5 P44 ðfÞ þ 6:31057r 6 P45 ðfÞ þ 0:583585r 8 P47 ðfÞg sin 4/  1013  f8:34703r6 P 55 ðfÞ þ 3:0247r 5 P56 ðfÞ þ 0:037821r 7 P57 ðfÞ  0:0125032r 9 P 58 ðfÞ  0:000660397r 10 P 59 ðfÞg sin 5/  108  f0:0125032r 8 P67 ðfÞg sin 6/  1016  f2:89462r 8 P77 ðfÞ  0:634302r 9 P78 ðfÞ  0:0114386r 10 P79 ðfÞg sin 7/  1011 þ f4:39749r 10 P99 ðfÞg sin 9/  1015 : Using the approximate analytical solution obtained we have computed the drag experienced by the ellipsoid and the minimum value of IN for different values of N. We have tabulated the corresponding values for b ¼ 13 in Table 1. It is observed

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Table 1 Convergence of the solution and the corresponding drag for b ¼ 13. N

IN

Drag

Torque

N

IN

Drag

Torque

1

3.81746

9:8117^i

6

0.143811

2

3.29739

7

0.129522

11:7351^i 11:7527^i

3:26464^j 3:2632^j

3

1.99579

10:3724^i 11:5728^i

1:19351^j 1:26688^j

0.110037

0.606072

1:28068^j 3:25043^j

8

4

9

0.0945999

11:7533^i 11:8755^i

3:32911^j 3:33104^j

5

0.295513

11:4789^i 11:6886^i

3:27444^j

Fig. 1. Convergence of IN for different values of slip parameter b.

20

18

16

Drag

14

12

10

8

6

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 2. Variation in the magnitude of the drag for different values of slip parameter b.

1

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10 9 8 7

Torque

6 5 4 3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Variation in the magnitude of the torque for different values of slip parameter b.

from Table 1 that IN decreases from 3:81746 when N ¼ 1 to 0:0946 when N ¼ 9. From this observation we can conclude that IN can be made further smaller and closer to zero by increasing N. We also computed the approximate but analytical solution for different values of the slip parameter b and noticed that IN decreases to zero as N increases (refer Fig. 1). Further, we computed the drag experienced by the body and the torque, for different values of slip parameter b in order to understand the effect of b on the drag and the torque (refer Fig. 2 and Fig. 3). Fig. 1 shows that the convergence of the integrals IN to 0 as N ! 1 is slower with increasing values of the slip parameter b. We also observe from Figs. 2 and 3 that the magnitude of the drag and that of the torque decreases with increasing b. 3. Conclusions In this paper, we proposed a method to obtain an analytic but approximate solution by adopting the method proposed by Radha et al. [1,2] for slip-stick boundary conditions, to study the flow due to an arbitrary Stokes flow past an arbitrary shaped body whose surface is given by r ¼ f ðh; /Þ. In particular, we considered the flow generated due to the presence of a Stokeslet outside an ellipsoid in a viscous, incompressible Stokes flow. We observed the convergence of the sequence of approximate solutions with increasing ‘N’ by computing IN at each stage and showed that IN ! 0 as N increases. An advantage of this method over other methods like finite difference methods is that in the latter methods the velocity and pressure are known only at a finite number of points on the surface unlike in this method, where the solution is known in the entire domain, even though approximately. Hence this method enables us to find the physical quantities like drag and torque more accurately as their computation involves calculating the derivatives of velocities. We observed that the integral IN converges to 0 as N ! 1 faster if the slip parameter b is small. We also observed that the magnitude of the drag and that of the torque decreases as b ! 1. In the case b ¼ 1 corresponding to shear-free boundary conditions, the torque is zero as it depends only on the tangential stresses on the boundary which are both zero. This technique can be used for problems pertaining to Stokes flow past bodies of arbitrary shape satisfying slip-stick boundary conditions. As in the case of no-slip boundary conditions, this technique can be used even if the equation of the surface is not given explicitly in the form r ¼ f ðh; /Þ [2]. This can be done by considering a sum of the squares of the velocities j~ qj2 at many points on the boundary, and minimizing this sum instead of minimizing the integral I given in Eq. (2.14). We can improve the accuracy by considering more points on the boundary. Acknowledgments One of the authors (DC) wishes to acknowledge with thanks the financial support received from C.S.I.R., India as J.R.F./ S.R.F. at University of Hyderabad, during the time this work was carried out. The authors thank Prof. T. Amaranath, University of Hyderabad for useful discussions. The authors also thank one of the reviewers for the comments which helped in improving the presentation of the paper.

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D. Choudhuri et al. / Applied Mathematics and Computation 219 (2013) 5367–5375

Appendix A Consider the approximate solution ~ qN given by (2.15)–(2.17). Then

^; qNn ¼ ~ q N :n N ^ ~ q ¼ qN :t1 ; t1

qNt2 ¼ ~ qN :^t2 are the corresponding components of normal and tangential velocities. The corresponding pressure is given by

pN ¼

@ ðr r2 AN Þ; @r

where AN is given as in Eq. (2.16). Similarly, the tangential components of the stress are given by

T Nt1 ¼ ~ TnN^ :^t1 ; T Nt2 ¼ ~ TnN^ :^t2 ; ^ ; TN being the stress tensor of rank 2 whose components are computed with the components of ~ where ~ TnN^ ¼ TN  n qN and pN . Appendix B 2

2

2

Suppose the rigid body is an ellipsoid defined by ax2 þ by2 þ cz2 ¼ 1, and if the equation of the rigid body is expressed in spherical polar coordinates as r ¼ f ðh; /Þ, then we have

f ðh; /Þ ¼

abc 2 ðb c2

2

sin

h cos2

/þ

a2 c2

2

1

2

2

sin h sin / þ a2 b cos2 hÞ2

;

@f abcZ 1 ¼ 2 ; 3 2 2 2 2 @h 2 2 2 ðb c sin h cos / þ a c2 sin h sin / þ a2 b cos2 hÞ2 @f abcZ 2 ¼ 2 ; 3 2 2 2 2 @/ ðb c2 sin h cos2 / þ a2 c2 sin h sin / þ a2 b cos2 hÞ2 where 2

2

2

Z 1 ¼ ðb c2 sin h cos h cos2 / þ a2 c2 sin h cos h sin /  a2 b cos h sin hÞ; 2

2

2

Z 2 ¼ ða2 c2 sin h sin / cos /  b c2 sin h sin / cos /Þ: The element of surface area is

"

(

2 )  2 #12 @f 2 @f þf dS ¼ f sin h f þ : @h @/ 2

2

2



Appendix C ^ . In particular, if the boundary @ X has the form r ¼ f ðh; /Þ, then in Let ~ T be the stress tensor of rank 2. We write ~ Tn^ ¼ ~ Tn 0 1 ^r e ^r T rh e ^r e ^h T r/ e ^r e ^/ T rr e ^ n ~ @ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ the spherical polar coordinates ðr; h; /Þ, we write n ¼ n1 er þ n2 eh þ n3 e/ and T ¼ T    n ¼ T hr eh er T hh eh eh T h/ eh e/ A ^/ e ^r T /h e ^/ e ^h T /h e ^/ e ^/ T /r e 0 1 ^r n1 e @ n2 e ^h A. ^/ n3 e Moreover,

^t1 ¼ 1 ðfh e^r þ f e^h Þ; k2      2   ff/ fh f/ f ^t2 ¼ 1 e^r  e^h þ h þ f e^/ ; k1 k2 k3 r sin h r sin h r        1 T rh fh T r/ f/ T hh fh T h/ f/ T /h fh T // f/ ^¼ T rr   e^r þ T hr   e^h þ T /r   e^/ ; T:n k1 r r sin h r r sin h r r sin h

D. Choudhuri et al. / Applied Mathematics and Computation 219 (2013) 5367–5375

5375

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 fh f/ k1 ¼ 1 þ þ ; r r sin h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ ðfh2 þ f 2 Þ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  2  2 ff/ fh f/ fh f k3 ¼ : þ þ þ rk1 k2 sin h rk1 k2 sin h rk1 k2 k1 k2 ^ ; qt ¼ ~ In addition, suppose qn ¼ ~ qn q  ^t1 ; qt2 ¼ ~ q  ^t2 ; T t1 ¼ ~ T  ^t1 and T t2 ¼ ~ T  ^t2 then 1

  1 fh f/ qr  qh  q/ ; k1 r r sin h 1 qt1 ¼ ðqr fh þ qh f Þ; k2       2  1 ff/ fh f/ fh f  qh þ q/ ; qr þ qt 2 ¼ k3 r sin hk1 k2 r sin hk1 k2 rk1 k2 k1 k2

qn ¼

     1 T rh fh T r/ f/ T hh fh T h/ f/ fh þ T hr  f ; T rr    k2 r r sin h r r sin h          2 1 T rh fh T r/ f/ ff/ T hh fh T h/ f/ fh f/ T /h fh T // f/ fh ¼ T rr    T hr   þ T /r   þr : k3 r r sin h r sin h r r sin h r sin h r r sin h r

T t1 ¼ T t2

The components of stress in spherical polar co-ordinates ðr; h; /Þ are given as follows.

@q T rr ¼ p þ 2l r ; @r   @qh qh 1 @qr  þ ; T rh ¼ l r @h @r r   1 @qr @q/ q/ þ  ; T r/ ¼ l r sin h @/ @r r   1 @qh qr ; þ T hh ¼ p þ 2l r @h r   1 @q/ qr qh cot h ; T // ¼ p þ 2l þ þ r sin h @/ r r   1 @q/ cot hq/ 1 @qh T h/ ¼ l :  þ r @h r sin h @/ r The formula to compute the drag and torque experienced by a body of arbitrary shape is given as follows.

~¼ D ~ T¼

Z Z

~ Tn^ dS;

@X

~ ð~ r  DÞdS;

@X

where dS is the suface element. References [1] R. Radha, B.S. Padmavati, T. Amaranath, New approximate analytical solutions for creeping flow past axisymmetric rigid bodies, Mech. Res. Comm. 37 (2010) 256–260. [2] Radha R, Padmavati BS, Amaranath, T. A new approximate analytical solution for arbitrary Stokes flow past rigid bodies. ZAMP, DOI 10.1007/s00033012-0218-8 (to appear). [3] D. Palaniappan, S.D. Nigam, T. Amaranath, R. Usha, Lamb’s solution of Stokes equations: a sphere theorem, Quart. J. Mech. appl. Math. 45 (1) (1992) 47– 56. [4] B.S. Padmavathi, G.P. Rajasekhar, T. Amaranath, A note on complete general solution of Stokes equations, Quart. J. Mech. appl. Math. 50 (3) (1998) 383– 388. [5] A.B. Basset, A treatise on hydrodynamics with numerous examples, 2, Dover, New York, 1966. [6] R. Schmitz, B.U. Felderhof, Creeping flow about a sphere, Physica A 92 (1978) 423–437. [7] B.S. Padmavathi, T. Amaranath, S.D. Nigam, Stokes flow past a sphere with mixed slip-stick boundary conditions, Fluid Dyn. Res. 11 (1993) 229–234.