A Spherical Particle Moving Slowly in a Fluid with a Radially Varying Viscosity

Abstract

The Stokesian flow field induced by a spherical particle that undergoes a slow rotational and translational motion in an unbounded quiescent fluid with radially varying viscosity is investigated. For a rotating particle, it is demonstrated that only the near viscosity field contributes effectively to the hydrodynamic torque exerted on the particle. A powerful screening effect exists in which the contribution of the distant viscosity field is weighted with the inverse of the distance from the particle to the fourth power. Two specific cases are investigated in which the viscosity field varies either exponentially or periodically. The latter is of particular interest, since it may serve to model the torque exerted on a particle rotating in a suspension. It is shown that a small test particle rotating in a suspension consisting of larger particles is almost unaffected by the large particles and mainly “senses” the fluid viscosity, whereas a large test particle “senses” the suspension viscosity. For a translating particle, a general expression is obtained for the induced velocity, pressure, and drag force exerted on the particle. An approximate result is obtained for the case in which the viscosity field varies slowly with the distance from the particle. An exact solution is obtained for a case in which the viscosity field varies algebraically with the distance from the particle center. An explicit numerical scheme is also suggested, which may assist in obtaining the drag force exerted on a particle translating in a flow field with an arbitrary radially varying viscosity distribution. Based on this numerical scheme and on the approximate solution obtained for a slowly varying viscosity, the drag force exerted on a particle translating inside a fluid with periodically varying viscosity is calculated. We hypothesize that such a periodic distribution can be viewed as a suspension under low Péclet number conditions. Based on this assumption, we obtain that if a test particle is much smaller than the suspended particles, the initial drag force exerted on the test particle is insensitive to the composition of the suspended particles or droplets, and senses the viscosity of only the continuous liquid, provided that the test particle is far from the suspended particles. However, up to first order in suspension concentration, the apparent viscosity of a dilute suspension is indifferent to whether the test particle is forced to move with a constant velocity or is subjected to a constant external force, provided that the test particle is arbitrarily located between the suspended particles.