The motion of ellipsoidal particles immersed in a viscous fluid

Abstract

In both physical and biological science, we are often concerned with the properties of a fluid, or plasma, in which small particles or corpuscles are suspended and carried about by the motion of the fluid. The presence of the particles will influence the properties of the suspension in bulk, and, in particular, its viscosity will be increased. The most complete mathematical treatment of the problem, from this point of view, has been that given by Einstein, who considered the case of spherical particles and gave a simple formula for the increase in the viscosity. We have extended this work to the case of particles of ellipsoidal shape. The second section of the paper is occupied with the requisite solution of the equations of motion of the fluid. The problem of the motion of a viscous fluid, due to an ellipsoid moving through it with a small velocity of translation in a direction parallel to one of its axes, has been solved by Oberbeck, and the corresponding problem for an ellipsoid rotating about one of its axes by Edwards. In both cases the equations of motion are approximated by neglecting the terms involving the squares of the velocities. It may be seen, a posteriori, that the condition for the validity of this approximation is that the product of the velocity of the ellipsoid by its linear dimensions shall be small compared with the “kinematic coefficient, of viscosity” of the fluid. In relation to our present problem, it will therefore be satisfied either for sufficiently slow motions, or for sufficiently small particles.