Low Reynolds numbers flow past an ellipsoid of revolution of large aspect ratio

Abstract

The results of Proudman & Pearson (1957) and Kaplun & Lagerstrom (1957) for a sphere and a cylinder are generalized to study an ellipsoid of revolution of large aspect ratio with its axis of revolution perpendicular to the uniform flow at infinity. The limiting case, where the Reynolds number based on the minor axis of the ellipsoid is small while the other Reynolds number based on the major axis is fixed, is studied. The following points are deduced: (1) although the body is three-dimensional the expansion is in inverse power of the logarithm of the Reynolds number as the case of a two-dimensional circular cylinder; (2) the existence of the ends and the variation of the diameter along the axis of revolution have no effect on the drag to the first order; (3) a formula for drag is obtained to higher order.

The Single Particle Motion of Non-Spherical Particles in Low Reynolds Number Flow

“…Here it is argued that measurements in the colloidal range are highly influenced by seismic noise and other effects outlined in the literature. The net effect of seismic noise is a greater velocity than what the particles would otherwise have in quiescent fluid, so if properly accounted for, Equation (11) may well be in better agreement with the experimental data throughout the entire range of particle sizes.…”

“…In this context, the relaxation time will be defined in simple terms as the time lag required to build up the bulk-fluid region with an effective radius of R e f f around the particle. Note that Equation (11) can be used to calculate the velocity u of the particle (velocity at wall) when the bulk fluid region around the particle is growing and the equilibrium condition has not been reached (i.e., at any time during the time lag to achieve equilibrium). Hence,…”

“…u is the velocity at any point in a fully developed velocity profile where the velocity profile exists up to R, whereas u is the velocity on the wall of the sphere at time t within a profile whose maximum extent is defined by e < e emax . In other words, it is R that is changing in Equation (11) to preserve the goal of obtaining the velocity of a particle at time t within the context of the relaxation time.…”

“…The mechanics of the fluid motion around particles of ellipsoidal shape (category D) have later been extended by Breach [10] to include inertial effects. Shi [11] later proposed solutions for spheroids with large aspect ratios. Oseen [12] developed equations to derive a solution for a circular cylinder that is impossible to solve using Stokes' equations (Stokes' paradox).…”

Rectilinear oscillations of a rigid spheroid in an elastic medium

“…In outer variables the prolate spheroid reduces to a needle of zero radius and finite length when Ma is arbitrary. We take the lead from hydrodynamics [4,5] and assume that the end effects are negligible for the leading term of the outer expansion. Accordingly, as a first approximation, we still take q, to be given by (19).…”