Fluid Dynamics Research 11 (1993) 229-234 North-Holland
FLUID DYNAMICS RESEARCH
Stokes flow past a sphere with mixed slip-stick boundary conditions B.S. Padmavathi, T. A m a r a n a t h a n d S.D. N i g a m School of'Mathematics and Computer b~/brmation Sciences, Unil ersity of ttyderabad P.O. Central Unirersity, 14vderahad-500134, India Received 26 March 1992 Accepted 9 November 1992 Abstract. We give a general solution of Stokes equations for an incompressible, viscous flow past a sphere with mixed slip-stick boundary conditions. The Fax6n's law for drag and torque on the sphere is also given and illustrated with an example.
1. Introduction Recently, a representation for velocity and pressure fields in a general non-axisymmetric Stokes flow was suggested by Palaniappan et al. (1992). This representation which employs two scalar functions A and B (which are biharmonic and harmonic, respectively) was shown to be equivalent to Lamb's (1945) solution. Sphere theorems for Stokes flow with no-slip boundary conditions (Palaniappan et al., 1992) and shear-free boundary conditions (Palaniappan et al., 1990) have been stated and proved, owing to the simplicity of the boundary conditions in terms of A and B. Schmitz and Felderhof (1978) have used the approach of Cartesian coordinates to study the creeping flow past a sphere with mixed slip-stick boundary conditions using a parameter 2. It is more convenient to consider this problem in spherical polar coordinates (r, 0, q~). Hence we use the representation due to Palaniappan et al. (1992) to give a general solution of Stokes equations for an incompressible, viscous flow past a sphere with mixed slip-stick boundary conditions. The advantage of this representation over Lamb's solution is that the solution can be given in a closed form in terms of A and B. We also give the Fax6n's law (1924) for drag and torque on the sphere for all values of 2, for any general non-axisymmetric Stokes flow.
2. Formulation and solution of the problem The equations governing the slow, steady motion of a viscous, incompressible fluid are /W 2 V = Vp,
(1)
v . v = 0,
(2)
where/~ is the coefficient of dynamic viscosity, V is the velocity vector and p is the pressure of the fluid. Correspondence to: Dr. T. Amaranath, School of Mathematics and Computer/Information Sciences, University of Hyderabad, P.O. Central University, Hyderabad-500134, India.
0169-5983/93/$02.50 ~" 1993-The Japan Society of Fluid Mechanics. All rights reserved.
B.S. Padmavathi et al. / Stokes flow past a sphere
230
The Stokes equations (1) and (2) admit the following representation ( P a l a n i a p p a n et al., 1992) for V = (q,, qo, q~) and p in spherical polar coordinates (r, 0, 4)) q,=
--
cotO~+~+cosec
1
A +
qo =
r
qo
rsin0
~r /I
(3)
20~4)2j,
+ cosec 0 - -
A +r
(4)
0 4) '
(5)
80'
0 P = Po + /t Z- (rV2A),
(6)
or
where V4A = 0,
V2B =
0.
(7a, b)
We consider Stokes flow past a stationary sphere of radius 'a' immersed in a viscous, incompressible fluid. O n the surface of the sphere, we assume the following b o u n d a r y conditions
q,(a, O, 4)) = O,
(8)
qo(a, O, 4)) = 2T~o(a, O, 4)),
(9)
q~(a, O, 4)) = 2T~,(a, O, 4)),
(lo)
where Tre, Tro are the tangential stresses. In terms of A and B the b o u n d a r y conditions on r = a are as follows: A =0, 0A Or
(11) 02A 2 -0r 2 '
(12)
,,3, Suppose the basic flow is given as follows ~ n+2 (ct.r" + a.r )S.(O, 4)),
Ao =
(14)
n=l
(15)
Bo = ~ g.r'T.(0,4)). n=l
where
S.(O, 4)) = ~ P~.(()(A.mcosm4) + B.msinm4)), m=O
T.(O, 4)) = Z P~(() (C.,. cos m4) + D.,. sin m4)). m=O
( = cos0,
B.S. Padmavathi et al. / Stokes flow past a sphere
231
In the presence of a sphere of radius 'a', the modified flow is assumed to be A(r, 0,49) and
B(r, O, 49), where A = ~ (cGrn+~'nr"+Z+flnr-"-I
+fl'nr-"+l)S.(0,49),
(16)
n=l
B = ~ (z.r" + a . r
"-1)T.(0,49).
(17)
n=l
The mixed slip-stick b o u n d a r y conditions (1 1) (13) are satisfied if
[S. =
(2n
--
1)a 2n+2
2 [ a + (2n + 1)2]
/~'. -
or. = -
~. +
(2n + 1)(a 2 - 2a2)a 2"+2 2 [ a + (2n + 1)2]
(2n + 1)(a + 22)a 2"-1
[a - (n - 1))~]a z"+l
2 [ a + (2n + 1)2]
c~,,
Z..
[a + (n + 2)2]
(18)
(2n + 3)a 2"+2 ~. -
2 [ a + (2n + 1)2]
ct'.,
(19)
(20)
Suppose 2 ¢ 0, the solution expressed in a closed form then is as follows:
r A(r, O, 49) = Ao(r, O, 49) -- -a Ao(a2/r, O, 49)
+
az
~zr A°(a2/r' O, 49) + 22aa/~_ 1
1 Ia2/rRa/2;~_½(a2 __ R2)VZAod R × 4d° (21)
O B(r,O, 49)=Bo(r,O, 49)+-Bo(ar
2
/r, 0 , 4 9 ) -
(2~
~pa/&+lla2/rRa/2+lBodR" + 3 f ~ ) ° (22)
When 2 ~ zc, eqs. (21) and (22) become
r A(r, O, 49) = Ao(r, O, 49) - -a Ao(a2/r, O, 49),
(23)
a 2 3r I a2/r B(r, O, 49) = Bo(r, O, 49) + -r Bo(a /r, O, 49) -- 75 jo RBo dR,
(24)
232
B.S. Padmavathi et al. ,' Stokes flow past a sphere
which agrees with the result obtained by Palaniappan et al. (1990) corresponding to the shear-free boundary conditions. However, (21) and (22) are not valid for 2 = 0. When 2 = 0 in (18)-(20), the solution (16) and (17) can be expressed in a closed form as follows (Palaniappan et al., 1992):
(r2 2ar + a2) Ao(a2/r, O, 05)
A(r, O, 05) = Ao(r, O, 05) +
(r2 -- (22) OA° 2 a ~r (a /r, O, 05)
r2(r2a 2 ) 2 ~ V 2 ( ! Ao(a2/r,O, 05)),
(25)
a r
(26)
B(r, O, 05) = Bo(r, O, O) - - Bo(a2/r, O, 05).
3. Faxen's laws for drag and torque The drag D and the torque T are found from D = fs(T~,
+ TroOo + T~4,O4~)r=~a2sinOdOd05,
T = f s (rTro#, - rT~,~o),-, a 2 sin 0 d0 d05, where S is the surface of the sphere, Or, ~0, ~,~ are the unit vectors, Trr = - p + 2~ Oqr/Or,
(
1 Oqr Tr° = It r ~0
qo + r Or/1'
g * = l~ r sin O 005
r
The drag F on the sphere is (see Appendix) r =
(. +
12x~aal \ ~ + ~ j [ a + 22"~
+ (a + 32)
<](A,,i+ <,]+
A,o )
~a 4 IV 2 Vo]o, E Volo + - (a + 32)
(27)
where Vo is the velocity corresponding to the basic flow Ao, and [ ]o is the evaluation at the origin r = 0. Similarly, the torque r on the sphere is (see Appendix) T-
8~/~a4 (a + 32)
Z1ECll/"+
Dil~q- Clo/~]
8~/~a4 (a + 3,;.)
(2~Ev × Vo]o).
(28)
B.S. Padmavathi et al. / Stokes ,flow past a sphere
233
When 2 ~ ~c, eqs. (27) and (28) give the drag and torque for shear-free case as follows: T=0,
F = 4nl~a[ Vo]o,
which agrees with the result obtained by Rallison (1978). When 2 = 0, eqs. (27) and (28) give the drag and torque for the case of a rigid sphere which are the well known Fax6n's (1924) laws T=
F = 6nl~a[Vo]o
8TC/Ia3(I[Vx Vo]o).
4. Example- Stokeslet Consider a Stokeslet of strength F1/8nl~ located at (0, 0, c), c > a whose axis is along the positive direction of the x-axis. The corresponding expressions for Ao and Bo due the Stokeslet are (Palaniappan et al., 1990) El R1 Ao(r, 0, qS) = 8 ~ c ( r c o s 0 -- c + R~) cosq~ r sin 0' F1
Bo(r, 0, qS) = 4-~pc (rcos 0 - c + R1) s i n ~ r sin 0'
where R21 = r 2 + C2 - - 2crcosO.
For r < c, A°=~
n=l
B° ~-~4 ~ n = ~
( n + 1)(2n+ 3 ) c ' + 2 - n ( n + - l ~ 2 n
,(F/ -'~ 1)C"+1
- l)c" P.~(~)cos4~,
P~(~)sin(~.
1
Hence, - F ~ ( ( n - 2 ) ) =
8,w
,(,
+
'
~', =- F1/8nl~(n + 1)(2n + 3)c "+2, Z, =- F1/4nlm(n + l)c "+1.
Hence, the drag and torque on the sphere are given by F and T, respectively, where
F = L4 c \ a + 32J and a4 T = c2(a + 3),) F l f
We observe that when 2 ~ F=(a/2c)FIL
co, T=O.
234
B.S. Padmavathi et al. / Stokes flow past a sphere
In the case when 2 = 0,
5. Concluding remarks The general solution of Stokes equations for creeping flow past a sphere with mixed slip-stick b o u n d a r y conditions using a parameter 2 has been obtained. This has enabled us to state the Fax6n's law for drag and torque for all values of 2. The rigid and shear-free b o u n d a r y cases correspond to the values 2 = 0 and 2 --, oc, respectively. The solution in a closed form is also presented.
Acknowledgement The financial assistance received by one of the authors (BSP) from C S I R (India) is gratefully acknowledged.
References Fax~n, H. (1924) Arkiv. Mat. Astron. Fys. 18(29), 3. Lamb, H. (1945) Hydrodynamics (sixth Ed, Dover Publications, New York) 595-596. Palaniappan, D., S.D. Nigam, T. Amaranath and R. Usha (1990) Mech. Res. Comm. 17(6), 429 435. Palaniappan, D., S.D. Nigam, T. Amaranath and R. Usha (1992) Q. J. Mech. Appl. Math, 45(1), 47 56. Rallison, J.M. (1978) J. Fluid. Mech. 88 (3), 529-533. Schmitz, R. and B. Felderhof (1978) Physica 92A, 423 437.
Appendix The representation given in eqs. (3~(5) can be expressed as (Palaniappan et al., 1992): V = c u r l c u r l ( r A ) + curl(rB) =2gradA
+r~rrgradA-rV2A-(rxV)B.
So, [ Vo]o = [ 2 g r a d Ao]o = 2 ~ l [ A l l t +
B,j+
Similarly,
EVe Vo]o = 20~'~[A~/+ B,l./+ Alo/~], [Vx Vo]o = 2z1[C~/+ D , , ] + C~o/~].
Alo/¢3.