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A general solution of unsteady Stokes equations
Fluid Dynamics Research 35 (2004) 229 – 236
A general solution of unsteady Stokes equations A. Venkatalaxmi, B.S. Padmavathi∗ , T. Amaranath Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India Received 9 April 2003; received in revised form 29 January 2004; accepted 1 June 2004 Communicated by O. Sano
Abstract A general solution of unsteady Stokes equations is suggested and its completeness is proved. A simple method of solution for the problem of an arbitrary unsteady Stokes 2ow in the presence of a sphere is discussed. Some physical properties like drag and torque experienced by the sphere are given and compared with some earlier known results. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. MSC: 76D07 Keywords: Unsteady Stokes equations; General solution
1. Introduction Steady Stokes 2ows have been a subject of rigorous mathematical study by mathematicians and engineers alike resulting in a rich mathematical theory as well as a wide variety of applications which continue to be exploited even to this day. The unsteady Stokes 2ows have prompted an equally elaborate study resulting in theoretical and experimental investigations in this area. Stokes (1845) studied the force exerted by a viscous, incompressible 2uid on a moving sphere in a steady, axisymmetric motion and also the oscillatory motion of a sphere along a diameter through a viscous, incompressible 2uid which is at rest at in;nity. Basset (1961) gave the force exerted on a sphere in an arbitrary axisymmetric time dependent motion by integrating the result of Stokes (1845). The unsteady, viscous 2ows in the presence of some particles was studied by many researchers, using analytical and numerical methods. Some of the notable researchers in this area in recent times are Lawrence and Weinbaum (1986, 1988), Kim and Karrila (1991), Pozrikidis (1989a, b), Lovalenti and ∗
Corresponding author. E-mail address: [email protected] (B.S. Padmavathi).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.2uiddyn.2004.06.001
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Brady (1993), Sano (1981), Coimbra and Rangel (1998, 2001), Feng and Joseph (1995), Yang and Leal (1991), Maxey and Riley (1983), Ranger (1996), Mazur and Bedeaux (1974), Michaelidis and Feng (1995), Stone and Bush (1996) and Howells (1974). Pozrikidis (1989a, b) used a singularity method for the computation of unsteady Stokes 2ows. He considered linearized oscillatory 2ow in a bounded or unbounded domain using a cartesian tensor notation. In this paper, we present a general solution (also referred to as a complete general solution in literature) of unsteady Stokes equations employing two scalar functions. We then proceed to suggest a method of solution for the problem of an arbitrary unsteady Stokes 2ow in the presence of a sphere. FaxHen’s (1922, 1924) laws are given, compared with previously known results and illustrated by an example. 2. A general solution The equations of motion for the unsteady Stokes 2ow in a viscous, incompressible 2uid are @V (1) = −∇p + ∇2 V; @t ∇ · V = 0;
(2)
where is the density, V the velocity, p the pressure and the coeLcient of dynamic viscosity of the 2uid. We rewrite Eq. (1) as 1 @ ∇2 − V = ∇p; (3) @t where = (=) is the co-eLcient of kinematic viscosity. By taking divergence of (1) and using Eq. (2), we ;nd that pressure is harmonic. We can now also verify that any solution V of (1) and (2) satis;es the equation 1 @ 2 2 V = 0: (4) ∇ ∇ − @t It can be shown that V can also be decomposed as V = V 1 + V2 ; where 2
∇ V1 = 0
(5)
and
1@ ∇ − @t 2
V2 = 0:
(6,7)
From Eq. (2), we know that V is solenoidal and hence we employ the representation due to Padmavathi et al. (1998) V = Curl Curl(rA) + Curl(rB);
(8)
where the scalars A and B are solutions of the following equations: LA = −r · V; LB = −r · CurlV;
(9) (10)
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231
where L is the transverse part of the Laplace operator except for the factor 1=r 2 in spherical polar co-ordinates (r; ; ’). Based on the proof given by Padmavathi et al. (1998), it is easy to show on similar lines that A and B satisfy the equations 1@ 1@ 2 2 2 A = 0; ∇ − B = 0: (11,12) ∇ ∇ − @t @t Using the identity
@ (rA) − r∇2 A + ∇ × (rB) Curl Curl(rA) + Curl(rB) = ∇ @r
(13)
we obtain from Eq. (1), the following relation: @ @ @ 2 2 ∇ (rA) − r∇ A + ∇ × (rB) = −∇p + ∇ ∇ (rA) @t @r @r −r∇2 A + ∇ × (rB) : This may be further simpli;ed as @ 2 ∇ p + (r(At − ∇ A)) − r[∇2 At − ∇4 A] + ∇ × [r(Bt − ∇2 B)] = 0: @r Since A and B satisfy Eqs. (11) and (12), we have @ p = po + (r(∇2 A − At )); @r where po is a constant. So a general solution of unsteady Stokes Eqs. (1) and (2) may be given by Eq. (8) i.e.
(14)
(15)
(16)
V = Curl Curl(rA) + Curl(rB); @ (r(∇2 A − At )); @r where A and B satisfy Eqs. (11) and (12), i.e. 1@ 1@ 2 2 2 A = 0; ∇ − B = 0: ∇ ∇ − @t @t p = po +
(17)
A solution of (11) can be decomposed as follows A = A 1 + A2 ; where 2
∇ A1 = 0
(18)
and
1 @ ∇ − @t 2
A2 = 0:
(19,20)
In fact, if A = A1 and B ≡ 0, then V satis;es the equation ∇2 V = 0. Likewise if A1 ≡ 0, then V satis;es the equation 1@ 2 V = 0: (21) ∇ − @t
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The general solution of Eqs. (11) and (12) can therefore be written as A = A1 + A2 , where ∞ n 2 n A1 = n r + n+1 Sn (; ’)e t ; r n=1 A2 =
∞
(22)
2
(23)
(n fn (r) + n gn (r))Tn (; ’)e t ;
(24)
(n fn (r) + n gn (r))Sn (; ’)e t ;
n=1
B=
∞
2
n=1
and Sn (; ’) = Tn (; ’) =
n m=0 n
Pnm (cos )(Anm cos m’ + Bnm sin m’);
(25)
Pnm (cos )(Cnm cos m’ + Dnm sin m’);
(26)
m=0
n ; ; n ; n ; n ; n ; Anm ; Bnm ; Cnm ; Dnm are constants and Re(2 ) 6 0.
The functions fn (R) = =2RI 1 (R); gn (R) = =2RK 1 (R) are the modi;ed Bessel functions n+
2
n+
2
of fractional order. As an illustration, we can consider the following two examples: (1) Oscillatory 4ow. The expressions for the scalar functions A and B corresponding to an oscillatory 2ow are U A = r sin cos ’ ei!t ; B = 0; U ¿ 0: (27) 2 (2) Unsteady Stokeslet: Consider the velocity ;eld due to an unsteady point force F = F1 ei!t iˆ located at (0; 0; 0). Then the corresponding expressions for A and B are given as 4 2 F1 − g1 (R) + 2 sin cos ’ ei!t ; (28) A= 8 R B = 0;
(29)
where 2 = (i!)=. The velocity components can be written in terms of the scalars A and B in spherical polar co-ordinates (r; ; ’) from (8) as @ 1 @2 @2 2 (30) + csc 2 A; qr = − cot + r @ @ 2 @’ @A @B 1 @ A+r + csc ; (31) q = r @ @r @’ @ @A @B 1 A+r − : (32) q’ = r sin @’ @r @
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
233
Using the expressions for A and B given in (22)–(24) we may rewrite them as ∞ n 2 n− 1 fn (r) gn (r) + n+2 + n Sn e t ; n(n + 1) n r + n qr = r r r n=1 ∞ (n + 1) (n + 1) nn n− 1 q = (n + 1)n r + n fn+1 (r) + fn (r) − n+2 + n gn (r) r r r n=1
−gn+1 (r) q’ =
∞
@Sn 2 t @Tn 2 t e + e ; csc (n fn (r) + n gn (r)) @ @’ n=1
(n + 1)
n (n + 1) n (n + 1)n r n−1 + n fn+1 (r) + fn (r) − n+2 + n gn (r) r r r n=1 ∞
@Sn 2 t @Tn 2 t e − e : −gn+1 (r) csc (n fn (r) + n gn (r)) @’ @ n=1 ∞
Here the pressure is given by ∞
n 2 2 (n + 1)n r n − n+1 n Sn e t : p = p0 − r n=1 Let us consider the unsteady Stokes 2ow in the presence of a sphere of radius a in a viscous, incompressible 2uid with rigid boundary conditions, i.e., qr = 0; q = 0; q’ = 0 on r = a and V → Vo as r → ∞. In terms of A and B, these conditions assume a simple form as A = 0; Ar = 0 and B = 0 on r = a. Suppose the 2uid velocity at in;nity is given by V0 = Curl Curl(rA0 ) + Curl(rB0 );
(33)
where A0 =
∞
2
(n r n + n fn (r))Sn (; ’)e t ;
(34)
n=1
B0 =
∞
2
n fn (r)Tn (; ’)e t ;
(35)
n=1
n ; n and n being known constants. The disturbance caused due to the presence of the sphere of radius a modi;es the 2ow so that the perturbed 2ow is represented by A and B as given in Eqs. (18) and (22)–(24). Using the boundary conditions on r = a we can determine the unknown constants as n [gn+1 (a)fn (a) + fn+1 (a)gn (a)] n gn+1 (a)an n+2 + ; (36) n = a (2n + 1)gn (a) − agn+1 (a) (2n + 1)gn (a) − agn+1 (a) n [(2n + 1)fn (a) + afn+1 (a)] (2n + 1)n an + ; (37) n = − (2n + 1)gn (a) − agn+1 (a) (2n + 1)gn (a) − agn+1 (a) fn (a)n : (38) n = − gn (a)
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The drag on the sphere of radius a is therefore found to be a4 g2 (a)1 3 D = 4 ag2 (a) − 3g1 (a) a3 (f1 (a)g2 (a) + f2 (a)g1 (a))1 ˆ 2 t : (A11 iˆ + B11 jˆ + A10 k)e + ag2 (a) − 3g1 (a) The torque experienced by the sphere is given by (g1 (a)f2 (a) + f1 (a)g2 (a)) 8 ˆ 2 t : (C11 iˆ + D11 jˆ + C10 k)e T = a3 1 3 g1 (a) It can be easily shown that 2 3 a4 g2 (a) D= [V0 ]0 ag2 (a) − 3g1 (a) 2 [3a3 (f1 (a)g2 (a) + f2 (a)g1 (a)) − a4 g2 (a)] 2 [∇ V0 ]0 ; ag2 (a)− 3g1 (a) ea 1 2 a2 1 1 2 2 [V0 ]0 + a [∇ V0 ]0 − = 6 a 1 + a + + + 3 a2 2 3 a 2 a2 = 6 a(B0 [V0 ]0 + a2 B2 [∇2 V0 ]0 );
+
where
2 a2 B0 = 1 + a + 3
;
1 B2 = 2 2 a
2 a2 e − 1 + a + 3 a
(39)
(40)
(41) (42) (43)
;
and
4 a3 (g1 (a)f2 (a) + f1 (a)g2 (a)) [∇ × V0 ]0 ; (44) g1 (a) ea [∇ × V0 ]0 ; (45) = 4 a3 a + 1 where V0 is the velocity of the undisturbed 2ow and [ ]0 is the evaluation at the centre of the sphere r = 0. We observe that the formulae agree with those given by Kim and Karrila (1991), and Pozrikidis (1989a, b). Moreover, they reduce to the well known FaxHen’s (1922, 1924) laws in the steady case when → 0. T=
3. Example Sphere in an oscillatory 2ow. Consider a sphere of radius a in an oscillatory 2ow of a viscous, incompressible 2uid. This amounts to considering the velocity and pressure to be of the form V0 = Uei!t and p = Pei!t , respectively, in the above analysis. Here we seek a solution satisfying the conditions (i) V = 0 on r = a, (ii) V → Uei!t as r → ∞ and p → Pei!t as r → ∞. Here if U = U i,ˆ then U A0 = r sin cos ’ ei!t ; (46) 2 B0 = 0; (47)
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
235
where U is a constant. The modi;ed velocity and pressure owing to the presence of the sphere have the following representation: V = Curl Curl(rA) + Curl(rB); p = p0 +
@ (r(∇2 − 2 )A); @r
(48) 2 =
i! ;
(49)
where ∇2 (∇2 − 2 )A = 0; (∇2 − 2 )B = 0: In this example, Ua4 g2 (a) 3aUg1 (r) 1 Ur + 2 − sin cos ’ ei!t ; A= 2 r (3g1 (a) − ag2 (a)) (3g1 (a) − ag2 (a)) B = 0: We rewrite A as a3 3ea 3 3 U r+ 2 − −1− A= 2 r 2 a2 a 2 a2 2 1 2e−r 3aea 1+ sin cos ’ ei!t : − − 2 2 r 2 r r
(50,51)
(52) (53)
(54)
We can identify the distribution of singularities from Eq. (54) and from expression for A in the case of a Stokeslet situated at the origin as described in Eq. (28). The image system consists of a potential dipole and a Stokeslet due to a point force F = −6 Uaea ei!t iˆ at the origin. The drag is given by the following expression 2 a2 2 3 a4 g2 (a)U ei!t ˆ i = 6 Ua 1 + a + ei!t iˆ D= (55) ag2 (a) − 3g1 (a) 3 which agrees with the well known formula due to Stokes (1845). The torque experienced by the sphere is zero. It is easy to see that in the limit → 0, we recover the expression for A in the steady case from (54) as a3 3a U r+ 2 − sin cos ’; (56) A= 2 2r 2 and the formula given in (55) reduces to the formula for the drag experienced by a sphere of radius ‘a’ in a steady, uniform 2ow. Acknowledgements One of the authors (AVL) wishes to acknowledge gratefully the ;nancial assistance received from C.S.I.R. (India).
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References Basset, A.B., 1961. A Treatise on Hydrodynamics, Vol. 2, Dover Publications Inc., New York. Coimbra, C.F.M., Rangel, R.H., 1998. General solution of the particle momentum equation to unsteady Stokes 2ows. J. Fluid Mech. 370, 53–72. Coimbra, C.F.M., Rangel, R.H., 2001. AIAA J. 39 (9), 1673–1682. FaxHen, H., 1922. Der Widerstand gegen die Bewegung einer starren Kugel in einer zRahen FlRussigkeit, die zwischen zwei parallelen Ebenen WRanden eingeschlossen ist. Ann. Phys. 68, 89–119. FaxHen, H., 1924. Der Widerstand gegen die Bewegung einer starren Kugel in einer zRahen FlRussigkeit, die zwischen zwei parallelen Ebenen WRanden eingeschlossen ist. Arkiv Mat. Astron. Och. Fys. 18 (29), 1–52. Feng, J., Joseph, D.D., 1995. The unsteady motion of solid bodies in creeping 2ows. J. Fluid Mech. 303, 83–102. Howells, I.D., 1974. Drag due to the motion of a Newtonian 2uid through a sparse random array of small ;xed rigid objects. J. Fluid Mech. 64, 449–475. Kim, S., Karrila, S.J., 1991. Microhydrodynamics : Principles and Selected Applications. Butterworth–Heinemann Series in Chemical Engineering, London. Lawrence, C.J., Weinbaum, S., 1986. The force on an axisymmetric body in linearized time dependent motion. J. Fluid Mech. 171, 209–218. Lawrence, C.J., Weinbaum, S., 1988. The unsteady force on a body at low Reynolds number: the axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489. Lovalenti, P.M., Brady, J.F., 1993. The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561–605. Maxey, M.R., Riley, J.J., 1983. Equation of motion for a small rigid sphere in a nonuniform 2ow. Phys. Fluids 26 (4), 883–889. Mazur, P., Bedeaux, D., 1974. A generalization of FaxHen’s theorem to nonsteady motion of a sphere through an incompressible 2uid in an arbitrary 2ow. Physica 76, 235–246. Michaelidis, E.E., Feng, Z.G., 1995. The equation of motion of a small viscous sphere in an unsteady 2ow with interface slip. Int. J. Multiphase Flow 21 (2), 315–321. Padmavathi, B.S., Rajasekhar, G.P., Amaranath, T., 1998. A note on complete general solutions of Stokes equations. Quart. J. Mech. Appl. Math. 51 (3), 383–388. Pozrikidis, C., 1989a. A singularity method for unsteady linearized 2ow. Phys. Fluids A 1 (9), 1508–1520. Pozrikidis, C., 1989b. A study of linearized oscillatory 2ow past particles by the boundary-integral method. J. Fluid Mech. 202, 17–41. Ranger, K.B., 1996. Time dependent decay of the motion of a sphere translating and rotating in a viscous liquid. Quart. J. Mech. Appl. Math. 49 (4), 621–633. Sano, T., 1981. Unsteady 2ow past a sphere at low Reynolds number. J. Fluid Mech. 112, 433–441. Stokes, G.G., 1845. On the theories of internal friction of 2uids in motion. Trans. Camb. Philos. Soc. 8, 287–305. Stone, H.A., Bush, J.W.M., 1996. Time-dependent drop deformation in a rotating high viscosity 2uid. Quart. Appl. Math. 54 (3), 551–556. Yang, Seung-Man, Leal, L.G., 1991. A note on memory-integral contributions to the force on an accelerating spherical drop at low Reynolds number. Phys. Fluids A 3 (7), 1822–1824.
A general solution of unsteady Stokes equations A. Venkatalaxmi, B.S. Padmavathi∗ , T. Amaranath Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India Received 9 April 2003; received in revised form 29 January 2004; accepted 1 June 2004 Communicated by O. Sano
Abstract A general solution of unsteady Stokes equations is suggested and its completeness is proved. A simple method of solution for the problem of an arbitrary unsteady Stokes 2ow in the presence of a sphere is discussed. Some physical properties like drag and torque experienced by the sphere are given and compared with some earlier known results. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. MSC: 76D07 Keywords: Unsteady Stokes equations; General solution
1. Introduction Steady Stokes 2ows have been a subject of rigorous mathematical study by mathematicians and engineers alike resulting in a rich mathematical theory as well as a wide variety of applications which continue to be exploited even to this day. The unsteady Stokes 2ows have prompted an equally elaborate study resulting in theoretical and experimental investigations in this area. Stokes (1845) studied the force exerted by a viscous, incompressible 2uid on a moving sphere in a steady, axisymmetric motion and also the oscillatory motion of a sphere along a diameter through a viscous, incompressible 2uid which is at rest at in;nity. Basset (1961) gave the force exerted on a sphere in an arbitrary axisymmetric time dependent motion by integrating the result of Stokes (1845). The unsteady, viscous 2ows in the presence of some particles was studied by many researchers, using analytical and numerical methods. Some of the notable researchers in this area in recent times are Lawrence and Weinbaum (1986, 1988), Kim and Karrila (1991), Pozrikidis (1989a, b), Lovalenti and ∗
Corresponding author. E-mail address: [email protected] (B.S. Padmavathi).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.2uiddyn.2004.06.001
230
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
Brady (1993), Sano (1981), Coimbra and Rangel (1998, 2001), Feng and Joseph (1995), Yang and Leal (1991), Maxey and Riley (1983), Ranger (1996), Mazur and Bedeaux (1974), Michaelidis and Feng (1995), Stone and Bush (1996) and Howells (1974). Pozrikidis (1989a, b) used a singularity method for the computation of unsteady Stokes 2ows. He considered linearized oscillatory 2ow in a bounded or unbounded domain using a cartesian tensor notation. In this paper, we present a general solution (also referred to as a complete general solution in literature) of unsteady Stokes equations employing two scalar functions. We then proceed to suggest a method of solution for the problem of an arbitrary unsteady Stokes 2ow in the presence of a sphere. FaxHen’s (1922, 1924) laws are given, compared with previously known results and illustrated by an example. 2. A general solution The equations of motion for the unsteady Stokes 2ow in a viscous, incompressible 2uid are @V (1) = −∇p + ∇2 V; @t ∇ · V = 0;
(2)
where is the density, V the velocity, p the pressure and the coeLcient of dynamic viscosity of the 2uid. We rewrite Eq. (1) as 1 @ ∇2 − V = ∇p; (3) @t where = (=) is the co-eLcient of kinematic viscosity. By taking divergence of (1) and using Eq. (2), we ;nd that pressure is harmonic. We can now also verify that any solution V of (1) and (2) satis;es the equation 1 @ 2 2 V = 0: (4) ∇ ∇ − @t It can be shown that V can also be decomposed as V = V 1 + V2 ; where 2
∇ V1 = 0
(5)
and
1@ ∇ − @t 2
V2 = 0:
(6,7)
From Eq. (2), we know that V is solenoidal and hence we employ the representation due to Padmavathi et al. (1998) V = Curl Curl(rA) + Curl(rB);
(8)
where the scalars A and B are solutions of the following equations: LA = −r · V; LB = −r · CurlV;
(9) (10)
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
231
where L is the transverse part of the Laplace operator except for the factor 1=r 2 in spherical polar co-ordinates (r; ; ’). Based on the proof given by Padmavathi et al. (1998), it is easy to show on similar lines that A and B satisfy the equations 1@ 1@ 2 2 2 A = 0; ∇ − B = 0: (11,12) ∇ ∇ − @t @t Using the identity
@ (rA) − r∇2 A + ∇ × (rB) Curl Curl(rA) + Curl(rB) = ∇ @r
(13)
we obtain from Eq. (1), the following relation: @ @ @ 2 2 ∇ (rA) − r∇ A + ∇ × (rB) = −∇p + ∇ ∇ (rA) @t @r @r −r∇2 A + ∇ × (rB) : This may be further simpli;ed as @ 2 ∇ p + (r(At − ∇ A)) − r[∇2 At − ∇4 A] + ∇ × [r(Bt − ∇2 B)] = 0: @r Since A and B satisfy Eqs. (11) and (12), we have @ p = po + (r(∇2 A − At )); @r where po is a constant. So a general solution of unsteady Stokes Eqs. (1) and (2) may be given by Eq. (8) i.e.
(14)
(15)
(16)
V = Curl Curl(rA) + Curl(rB); @ (r(∇2 A − At )); @r where A and B satisfy Eqs. (11) and (12), i.e. 1@ 1@ 2 2 2 A = 0; ∇ − B = 0: ∇ ∇ − @t @t p = po +
(17)
A solution of (11) can be decomposed as follows A = A 1 + A2 ; where 2
∇ A1 = 0
(18)
and
1 @ ∇ − @t 2
A2 = 0:
(19,20)
In fact, if A = A1 and B ≡ 0, then V satis;es the equation ∇2 V = 0. Likewise if A1 ≡ 0, then V satis;es the equation 1@ 2 V = 0: (21) ∇ − @t
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A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
The general solution of Eqs. (11) and (12) can therefore be written as A = A1 + A2 , where ∞ n 2 n A1 = n r + n+1 Sn (; ’)e t ; r n=1 A2 =
∞
(22)
2
(23)
(n fn (r) + n gn (r))Tn (; ’)e t ;
(24)
(n fn (r) + n gn (r))Sn (; ’)e t ;
n=1
B=
∞
2
n=1
and Sn (; ’) = Tn (; ’) =
n m=0 n
Pnm (cos )(Anm cos m’ + Bnm sin m’);
(25)
Pnm (cos )(Cnm cos m’ + Dnm sin m’);
(26)
m=0
n ; ; n ; n ; n ; n ; Anm ; Bnm ; Cnm ; Dnm are constants and Re(2 ) 6 0.
The functions fn (R) = =2RI 1 (R); gn (R) = =2RK 1 (R) are the modi;ed Bessel functions n+
2
n+
2
of fractional order. As an illustration, we can consider the following two examples: (1) Oscillatory 4ow. The expressions for the scalar functions A and B corresponding to an oscillatory 2ow are U A = r sin cos ’ ei!t ; B = 0; U ¿ 0: (27) 2 (2) Unsteady Stokeslet: Consider the velocity ;eld due to an unsteady point force F = F1 ei!t iˆ located at (0; 0; 0). Then the corresponding expressions for A and B are given as 4 2 F1 − g1 (R) + 2 sin cos ’ ei!t ; (28) A= 8 R B = 0;
(29)
where 2 = (i!)=. The velocity components can be written in terms of the scalars A and B in spherical polar co-ordinates (r; ; ’) from (8) as @ 1 @2 @2 2 (30) + csc 2 A; qr = − cot + r @ @ 2 @’ @A @B 1 @ A+r + csc ; (31) q = r @ @r @’ @ @A @B 1 A+r − : (32) q’ = r sin @’ @r @
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
233
Using the expressions for A and B given in (22)–(24) we may rewrite them as ∞ n 2 n− 1 fn (r) gn (r) + n+2 + n Sn e t ; n(n + 1) n r + n qr = r r r n=1 ∞ (n + 1) (n + 1) nn n− 1 q = (n + 1)n r + n fn+1 (r) + fn (r) − n+2 + n gn (r) r r r n=1
−gn+1 (r) q’ =
∞
@Sn 2 t @Tn 2 t e + e ; csc (n fn (r) + n gn (r)) @ @’ n=1
(n + 1)
n (n + 1) n (n + 1)n r n−1 + n fn+1 (r) + fn (r) − n+2 + n gn (r) r r r n=1 ∞
@Sn 2 t @Tn 2 t e − e : −gn+1 (r) csc (n fn (r) + n gn (r)) @’ @ n=1 ∞
Here the pressure is given by ∞
n 2 2 (n + 1)n r n − n+1 n Sn e t : p = p0 − r n=1 Let us consider the unsteady Stokes 2ow in the presence of a sphere of radius a in a viscous, incompressible 2uid with rigid boundary conditions, i.e., qr = 0; q = 0; q’ = 0 on r = a and V → Vo as r → ∞. In terms of A and B, these conditions assume a simple form as A = 0; Ar = 0 and B = 0 on r = a. Suppose the 2uid velocity at in;nity is given by V0 = Curl Curl(rA0 ) + Curl(rB0 );
(33)
where A0 =
∞
2
(n r n + n fn (r))Sn (; ’)e t ;
(34)
n=1
B0 =
∞
2
n fn (r)Tn (; ’)e t ;
(35)
n=1
n ; n and n being known constants. The disturbance caused due to the presence of the sphere of radius a modi;es the 2ow so that the perturbed 2ow is represented by A and B as given in Eqs. (18) and (22)–(24). Using the boundary conditions on r = a we can determine the unknown constants as n [gn+1 (a)fn (a) + fn+1 (a)gn (a)] n gn+1 (a)an n+2 + ; (36) n = a (2n + 1)gn (a) − agn+1 (a) (2n + 1)gn (a) − agn+1 (a) n [(2n + 1)fn (a) + afn+1 (a)] (2n + 1)n an + ; (37) n = − (2n + 1)gn (a) − agn+1 (a) (2n + 1)gn (a) − agn+1 (a) fn (a)n : (38) n = − gn (a)
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The drag on the sphere of radius a is therefore found to be a4 g2 (a)1 3 D = 4 ag2 (a) − 3g1 (a) a3 (f1 (a)g2 (a) + f2 (a)g1 (a))1 ˆ 2 t : (A11 iˆ + B11 jˆ + A10 k)e + ag2 (a) − 3g1 (a) The torque experienced by the sphere is given by (g1 (a)f2 (a) + f1 (a)g2 (a)) 8 ˆ 2 t : (C11 iˆ + D11 jˆ + C10 k)e T = a3 1 3 g1 (a) It can be easily shown that 2 3 a4 g2 (a) D= [V0 ]0 ag2 (a) − 3g1 (a) 2 [3a3 (f1 (a)g2 (a) + f2 (a)g1 (a)) − a4 g2 (a)] 2 [∇ V0 ]0 ; ag2 (a)− 3g1 (a) ea 1 2 a2 1 1 2 2 [V0 ]0 + a [∇ V0 ]0 − = 6 a 1 + a + + + 3 a2 2 3 a 2 a2 = 6 a(B0 [V0 ]0 + a2 B2 [∇2 V0 ]0 );
+
where
2 a2 B0 = 1 + a + 3
;
1 B2 = 2 2 a
2 a2 e − 1 + a + 3 a
(39)
(40)
(41) (42) (43)
;
and
4 a3 (g1 (a)f2 (a) + f1 (a)g2 (a)) [∇ × V0 ]0 ; (44) g1 (a) ea [∇ × V0 ]0 ; (45) = 4 a3 a + 1 where V0 is the velocity of the undisturbed 2ow and [ ]0 is the evaluation at the centre of the sphere r = 0. We observe that the formulae agree with those given by Kim and Karrila (1991), and Pozrikidis (1989a, b). Moreover, they reduce to the well known FaxHen’s (1922, 1924) laws in the steady case when → 0. T=
3. Example Sphere in an oscillatory 2ow. Consider a sphere of radius a in an oscillatory 2ow of a viscous, incompressible 2uid. This amounts to considering the velocity and pressure to be of the form V0 = Uei!t and p = Pei!t , respectively, in the above analysis. Here we seek a solution satisfying the conditions (i) V = 0 on r = a, (ii) V → Uei!t as r → ∞ and p → Pei!t as r → ∞. Here if U = U i,ˆ then U A0 = r sin cos ’ ei!t ; (46) 2 B0 = 0; (47)
A. Venkatalaxmi et al. / Fluid Dynamics Research 35 (2004) 229 – 236
235
where U is a constant. The modi;ed velocity and pressure owing to the presence of the sphere have the following representation: V = Curl Curl(rA) + Curl(rB); p = p0 +
@ (r(∇2 − 2 )A); @r
(48) 2 =
i! ;
(49)
where ∇2 (∇2 − 2 )A = 0; (∇2 − 2 )B = 0: In this example, Ua4 g2 (a) 3aUg1 (r) 1 Ur + 2 − sin cos ’ ei!t ; A= 2 r (3g1 (a) − ag2 (a)) (3g1 (a) − ag2 (a)) B = 0: We rewrite A as a3 3ea 3 3 U r+ 2 − −1− A= 2 r 2 a2 a 2 a2 2 1 2e−r 3aea 1+ sin cos ’ ei!t : − − 2 2 r 2 r r
(50,51)
(52) (53)
(54)
We can identify the distribution of singularities from Eq. (54) and from expression for A in the case of a Stokeslet situated at the origin as described in Eq. (28). The image system consists of a potential dipole and a Stokeslet due to a point force F = −6 Uaea ei!t iˆ at the origin. The drag is given by the following expression 2 a2 2 3 a4 g2 (a)U ei!t ˆ i = 6 Ua 1 + a + ei!t iˆ D= (55) ag2 (a) − 3g1 (a) 3 which agrees with the well known formula due to Stokes (1845). The torque experienced by the sphere is zero. It is easy to see that in the limit → 0, we recover the expression for A in the steady case from (54) as a3 3a U r+ 2 − sin cos ’; (56) A= 2 2r 2 and the formula given in (55) reduces to the formula for the drag experienced by a sphere of radius ‘a’ in a steady, uniform 2ow. Acknowledgements One of the authors (AVL) wishes to acknowledge gratefully the ;nancial assistance received from C.S.I.R. (India).
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