Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples. The Reynolds number of the flow around them is assumed to be small, with the consequence that the hydrodynamic interactions between the spheres can be described by a set of linear relations between, on the one hand, the forces and couples exerted by the spheres on the fluid and, on the other, the translational and rotational velocities of the spheres. These relations may be represented completely by either a set of 10 resistance functions or a set of 10 mobility functions. When non-dimensionalized, each function depends on two variables, the non-dimensionalized centre-to-centre separation s and the ratio of the spheres’ radii λ. Two expressions are given for each function, one a power series in s−1 and the other an asymptotic expression valid when the spheres are close to touching.

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Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow

Two rigid spheres of radii a and b are immersed in infinite fluid whose velocity at infinity is a linear function of position. No external force or couple acts on the spheres, and the effect of inertia forces on the motion of the fluid and the spheres is neglected. The purpose of the paper is to provide a systematic and explicit description of those aspects of the interaction between the two spheres that are relevant in a calculation of the mean stress in a suspension of spherical particles subjected to bulk deformation. The most relevant aspects are the relative velocity of the two sphere centres (V) and the force dipole strengths of the two spheres (S′ij, S″ij), as functions of the vector r separating the two centres.It is shown that V, S′ij and S″ij depend linearly on the rate of strain at infinity and can be represented in terms of several scalar parameters which are functions of r/a and b/a alone. These scalar functions provide a framework for the expression of the many results previously obtained for particular linear ambient flows or for particular values of r/a or of b/a. Some new results are established for the asymptotic forms of the functions both for r/(a + b) [Gt ] 1 and for values of r − (a + b) small compared with a and b. A reasonably complete numerical description of the interaction of two rigid spheres of equal size is assembled, the main deficiency being accurate values of the scalar functions describing the force dipole strength of a sphere in the intermediate range of sphere separations.In the case of steady simple shearing motion at infinity, some of the trajectories of one sphere centre relative to another are closed, a fact which has consequences for the rheological problem. These closed forms are described analytically, and also numerically in the case b/a = 1.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112072002927

The hydrodynamic interaction of two small freely-moving spheres in a linear flow field

Rheology of a dilute suspension of axisymmetric Brownian particles

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 mo...

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

The kinetics of flocculation of heterodisperse suspensions like those in water treatment plants are usually described by the Smoluchowski equation, which incorporates collision frequency functions for particle collisions by Brownian motion, fluid shear, and differential sedimentation. These collision-frequency functions are based on a rectilinear view of collisions, i.e., one that ignores short-range forces and changes in fluid motion as particles approach one another.

Relative) insignificance of G in flocculation

In dieser Arbeit handelt es sich um eine Analyse der asymptotischen Formen fur die Losung von Problemen der langsamen zahen Stromung, die durch die Bewegung von zwei starren spharischen Grenzen verursacht wird, wenn diese sich entweder entlang von Axen bewegen, die senkrecht zu der Verbindungslinie ihrer Zentren liegen, oder sich um diese drehen, Es wird der Grenzfall untersucht, bei dem der den Mindestspielraum zwischen den Spharen definierende Parameter e sich Null nahert Es wird gezeigt, dass fur Kugeln, die ausserhalb einander liegen, die Krafte und die Momente, die auf jede der beiden Kugeln einwirken, singular werden, und zwar wie log e fur e→0, fur alle Werte der Radien Wenn eine Kugel die andere einschliesst, findet man im allgemeinen dasselbe Verhalten, aber wenn eine stationare Kugel eine drehende einschliesst, so verschwinden die Krafte fur e»0, wenn der Radius der ausseren Kugel viermal so gross als der der inneren ist Ein entsprechendes Verhalten zeigen auch die Momente, wenn sich die innere Kugel fortbewegt

Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero

A method is presented for determining the exact solution of the Stokes equation for axisymmetric streaming flow past a torus. By solving directly for the velocity and pressure fields rather than introducing a stream function, the problem is reduced to the solution of a second order difference equation for a coefficient sequence. The force acting on the torus is evaluated for a number of values of the parameter defining the torus geometry.

On axisymmetric stokes flow past a torus

On the Stokes resistance of two equal spheres in contact in a linear shear field

Note on the slow rotation of a concave spherical lens or bowl in two immiscible semi-infinite viscous fluids

Exact solutions of the Stokes equations are given for asymmetric flows produced by a closed torus which steadily translates along or rotates about a direction perpendicular to its axis of rotational symmetry in a quiescent viscous fluid. The force and couple acting on the body are calculated and the general motion of a closed torus is considered.

Samir R. Majumdar

Papers

Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero

On axisymmetric stokes flow past a torus

On the Stokes resistance of two equal spheres in contact in a linear shear field

Note on the slow rotation of a concave spherical lens or bowl in two immiscible semi-infinite viscous fluids

Asymmetric stokes flows generated by the motion of a closed torus