The resistance and mobility functions of two equal spheres in low-Reynolds-number flow

Abstract

The resistance and mobility functions which completely characterize the linear relations between the force, torque, and stresslet and the translational and rotational velocities of two spheres in low-Reynolds-number flow have been calculated using a boundary collocation technique. The ambient velocity field is assumed to be a superposition of a uniform stream and a linear (vorticity and rate-of-strain) field. This is the first compilation of accurate expressions for the entire set of functions. The calculations are in agreement with earlier results for all functions for which such results are available. The technique is successful at all sphere–sphere separations except at the almost-touching (gaps of less than 0.005 of the sphere diameter) configuration. New results for the stresslet functions have been used to determine Batchelor and Green’s [J. Fluid Mech. 56, 401 (1972)] order c2 coefficient in the bulk stress (7.1 instead of their 7.6). The two-sphere functions have also been used to determine the motion of a rigid dumbbell in a linear field. We also show that certain functions have extrema.

Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid

“…5.4. Sedimentation of spheroids The motion of two spheroids sedimenting side by side is intriguing since 'periodic' orbits appear for certain initial conditions (Kim 1985 a). Hydrodynamic interactions then cause the particles' directors to rotate past in radians, at which point the trajectory -projected onto a plane perpendicular to gravity -is reversed ( figure 46).…”

“…Their singularity representation, reminiscent of the slender-body theory pioneered by Batchelor (1970), is much more convenient for numerical purposes than the grantedly more compact symbolic operator formalism of Brenner & Haber (1983) (see also Brenner 1966), which, in effect, places an infinite series of disturbances at the centre of the particle, just as in the multipole collocation technique. Kim 1974,1975) in conjunction with the method of reflections to examine the sedimentation of two identical spheroids in an unbounded fluid (Kim 1985~). Since the formulation is exact in the absence of interactions (unlike the multipole collocation approach for non-spherical bodies), better accuracy is expected for the same number of unknown multipoles, even though many iterations may be necessary.…”

“…Note that Mus is defined as being symmetric and traceless in its last two indices, since it is otherwise indeterminate (Kim & Mifflin 1985;Brenner 1964~). To convert M,, into Mus, we thus need to multiply it by g (see (C 1)).…”

Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

“…The functions A and B, as before, characterize the relative translational velocity of two inertialess spheres, while C denotes the corresponding angular velocity correction on account of hydrodynamic interactions; the function E represents the translation-rotation coupling. Explicit expressions for A, B, C, E, G and H may again be obtained from Jeffrey & Onishi (1984) and Kim & Mifflin (1985); for instance, G = x a 11 − x a 12 , H = y a 11 − y a 12 , E = y b 11 − y b 12 , etc. Although we have retained the O(St) denominator term on the right-hand sides of (4.2) and (4.3), the resulting solution is meaningful only to O(St).…”

“…Explicit expressions for A and B may be obtained in terms of the resistance and mobility functions defined in Jeffrey & Onishi (1984) and Kim & Mifflin (1985); for instance, A = x g 11 − x g 12 . For St 1, the inertial velocity St V (1) remains asymptotically small compared to V (0) for large r because…”