The lift on a small sphere touching a plane in the presence of a simple shear flow

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1 Journal of Applied Mathematics and Physics (ZAMP) $ Vol. 36, January Birkhiuser Verlag Basel, 1985 The lift on a small sphere touching a plane in the presence of a simple shear flow By David Leighton and Andreas Acrivos, Dept. of Chemical Engineering, Stanford University, Stanford, California 94305, USA I. Introduction Gadala-Maria [1] recently reported that, when a settled layer of non-buoyant particles underneath a layer of clear fluid is sheared between two parallel horizontal surfaces, the bed expands and, following a further increase in shear, the particles resuspend. Although such a resuspension, which was subsequently studied in our laboratory over a range of conditions, is normally associated with inertia-dominated flows, it was found to occur in our experiments when the particle Reynolds number was 0 (10-2) or below. Of course, inertia effects are never completely absent and hence it is conceivable that a weak lift acting on a given particle over sufficiently long periods of time could be the cause of the observed net vertical motion of the suspension. This question, then, provided the motivation for the present analysis, the aim of which is to determine the lift on a stationary sphere in contact with a solid plane when placed in a simple shear flow under conditions of small Reynolds numbers. Knowledge of the magnitude of the lift may also be useful in determining the influence of inertia on other flow problems with similar geometries such as sediment packing in ultrafiltration. II. Analysis Consider the situation depicted in Fig. 1. When the fluid inertia is assumed negligible everywhere, the lift on the sphere also vanishes owing to the well-known reversibility property of the creeping flow equations. It becomes necessary therefore to take into account the effects of inertia in order to calculate a non-zero value for the lift. Figure 1 The flow field.

2 Vol. 36, 1985 The lift on a small sphere 175 We begin with the dimensionless form of the incompressible Navier-Stokes equations in the standard notation: ~O'ij ~U i ~!Aj ~xj - R u~ ~x~.' ~xj - 0 (1) a~j = - p 6~j +~ \~x~ + ~x~ (2) where all position vectors x~ have been rendered dimensionless with a, the radius of the sphere, and all velocities u~ with )) a where?) denotes the rate of strain of the undisturbed flow. Also R = p a2v is the shear Reynolds number and a,.~ is the stress tensor. On letting u i = u} ~ + Ru~ 1~ + o(r) (3) and similarly for cr~j we obtain the sequence of equations and ~};~ - O, ~u}~ - 0 ~xj ~xj (4) ~xj -J ~xj ax i - 0 (5) with boundary conditions: u(0) (1) =u~ =0 for x on the sphere and the plane u~ ~ = x36il, u} *) ~ 0(1) for Ixil -, 00. (6) Finally we have for the dimensionless lift L and with n~ being the outer unit normal that: L=R ~ crl))njcsi3da +o(r)=_rljj~ +o(r) (7) sphere since, as mentioned earlier, the contribution to L from the creeping flow solution vanishes. To be sure, as in the large majority of problems at low Reynolds numbers R, the creeping flow solution does not remain uniformly valid over all space but applies only within an "inner" region whose dimensions grow as R ~ 0. In the present case, however, the disturbance velocity u~ ~ x a c~i3 far from the sphere approaches that due to a Stokeslet at x i = c5i3 acting along c$1 which, as shown by Blake [2], decays as 01xl] -2. (We remark, for future reference, that the corresponding velocity disturbance in creeping flow when the Stokeslet acts along c$i3 is 0 (]xil-a). Therefore, to this order of approximation, the lift L can be computed from the solution of the "inner" problem as stated above. Following the lead of past investigations on similar problems, c.f. for example Reference [3], we derive an expression for L(, 1~, equivalent to (7), which it is in terms of known creeping flow solutions. To this end, let v~ and tlj be, respectively, the velocity and the associated stress tensor of the solution to the creeping flow equations satisfying the boundary conditions: vi = O for xl e plane and for I xi[ --+ oo ; vi = - g~i3 for xiesphere.

3 176 D. Leighton and A. Acrivos ZAMP Thus, in view of (7) and the divergence theorem, jl) : sphere (1) nj da 8a~)) a! -1) Ovi dv-- ~ v~ (1) nj da (8) ~ 1)iO'iJ : ~ l)i ~xj ~ ~J OXj S~ (7iJ where S~ is a large hemispherical surface of radius r with its center at the origin (point 0 in Fig. 1) and ~ is the volume for x 3 > 0 between S~ and the solid sphere. But on account of (6) and the divergence theorem: OXj = ~ j dv= s~ nj da (9) which vanishes as r ~ ~ since, as remarked earlier, t u = 0(r -4) whereas u~ 1) remains at most 0 (1). Similarly, the corresponding integral over S~o in (8) also vanishes, and hence, taking account of (5), (o) ~u~~ dv (10) 8xj which, as seen, involves only solutions to the creeping flow equations. We also note that the integral in (10) is uniformly convergent since as mentioned earlier, v~ decays as 0 (r-3) whereas u}o) ~I ~ ~) ~xj becomes 0(r- as r--, oo. Therefore ~ represents the whole space external to the sphere and the plane. Ill. Evaluation of the lift integral An expression for the flow field u~ ~ has been given by O'Neill [4] in polar coordinates: uv=(u + z) coso, Uo=(V- z) sino, Uz= Wcos0 (11) where the functions, U, V, W are independent of 0 and are defined by O'Neill using the tangent sphere coordinate system: 2r 2z r- r2 + Z2, ~ -- r2 + Z2 (12) in which the space occupied by the fluid is mapped into the region 0 < ~ < 1, r> 0 (see Ref. [4], Eqs for details). The corresponding expression for v i can be obtained z I ~<i r<l Figure 2 The tangent sphere coordinate system.

4 Vol. 36, 1985 The lift on a small sphere 177 by a simple extension of Goren's [5] analysis and application of the appropriate boundary conditions. Letting I) r -- V z -- : 1) 0 : 0 r 5z' r 8r we find that ~2)32 c~_ }72 -(r2 2r~ c r~ + [Fa (t) sinh~t- (t) ( osh~t sinh ~ t. )] a~ (rl t) dt where t a e-' sinht + t(t + 1) F2(t ) = F 1 (t) - sinh 2 t - t 2' sinh 2 t - t 2 In cylindrical coordinates the integrand of (10) becomes: ~ (..0 4 aur exj -~\rer + r eo ~ + UzTs since v o = 0. It is convenient to recast (14) in the tangent sphere coordinate system in terms of which (10) becomes Ijl)=i~2~viu(~ 8r 2)3 d0 drd ~ o o o J 8xj (r2+ oo ~ & - ( u + v ) ( V - z ) + W l+~-z (19 or, formally, [( ~W 1 W ~w]~ 8 t~2)3 drd~ +~ V+zl~r-;w(v-~)+ ~jj(~+ 1 oo J~) = ~ ~ ~ f(~, t) atd~. (16) 0 0 Equation (16) has been evaluated numerically to obtain: L ~ 9.22 R (17) or in dimensional terms: (18) IV. Discussion In solving for',i(~ O'Neill [4] derived the leading order term of the lateral force, Fx, on a sphere due to a linear shear field. Combining this result with our calculation of the lift we obtain: LF~ _~ R. (19)

5 178 D. Leighton and A. Acrivos ZAMP Although the lift is large enough to be measured experimentally under appropriate conditions, it is far too small to be of significance relative to the drag at Reynolds numbers of 0 (i0-2). Thus, if, in the more complex system of the resuspension of a layer of settled particles, the inertial forces are of the same order of magnitude as in the simple problem examined here, we may infer that inertia plays only a minor role in bringing about the resuspension of settled particles in low R shear fields. The factors responsible for the observed resuspension remain, therefore, to be identified. Acknowledgement The authors would like to thank the reviewer for recommending that the integral in Eq. (16) be evaluated numerically rather than, as had been done in an earlier draft of the paper, through the use of a far-field approximation which in fact had led to an inaccurate result. This work was supported in part by a contract with the Department of Energy DOE AT03-80-ERI0659. References [1] F. Gadala-Maria in EPRI Project Final Report, A. Acrivos, Investigator, pp (1979). [2] J. R. Blake, Proc. Camb. Phil. Soc. 70, 303 (1971). [3] B. P. Ho and L. G. Leal, J. Fluid Mech. 65, part 2, 365 (1974). [4] M. E. O'Neill, Chem. Eng. Sci. 23, 1293 (1968). [5] S. L. Goren, J. Fluid Mech. 41, part 3, 619 (1970). Abstract The contribution from purely viscous forces to the lift L on a sphere of radius a touching a plane in the presence of a shear flow field of strength?) is zero. An exact integral expression for the lift to leading order in the Reynolds number R- ~a2v is derived using known creeping flow solutions to related problems. The integral is evaluated numerically to obtain the value of the lift L~ Iz a R or LF x " R where F x is the lateral viscous force on the sphere. Zusammenfassung Der Beitrag der Zfihigkeit allein zum Auftrieb L an einer Kugel vom Radius a, welche in einer Scherstr6mung mit dem Gradienten ~ eine unendliche Ebene beriihrt ist null. Ein exakter Integralausdruck ffir den Auftrieb wird in erster Ordnung der Reynoldszahl, R =- ~) a2v, hergeleitet unter Benfitzung bekannter verwandter L6sungen in schleichender Str6mung. Der Wert des Integrals wird numerisch bestimmt und gibt ffir den Auftrieb L~) # a R oder LF x R; dabei ist F x die seitliche Zihigkeitskraft auf die Kugel. (Received: December 30, 1983)

Inertial migration of a sphere in Poiseuille flow

J. Fluid Mech. (1989), vol. 203, pp. 517-524 Printed in Great Britain 517 Inertial migration of a sphere in Poiseuille flow By JEFFREY A. SCHONBERG AND E. J. HINCH Department of Applied Mathematics and