The Stokes flow problem for a class of axially symmetric bodies

Abstract

The Stokes flow problem is concerned with fluid motion about an obstacle when the motion is such that inertial effects can be neglected. This problem is considered here for the case in which the obstacle (or configuration of obstacles) has an axis of symmetry, and the flow at distant points is uniform and parallel to this axis. The differential equation for the stream function ψ then assumes the form L2−1ψ = 0, where L−1 is the operator which occurs in axiallysymmetric flows of the incompressible ideal fluid. This is a particular case of the fundamental operator of A. Weinstein's generalized axially symmetric potential theory. Using the results of this theory and theorems regarding representations of the solutions of repeated operator equations, the authors (1) give a general expression for the drag of an axially symmetric configuration in Stokes flow, and (2) indicate a procedure for the determination of the stream function. The stream function is found for the particular case of the lens-shaped body.Explicit calculation of the drag is difficult for the general lens, without recourse to numerical procedures, but is relatively easy in the case of the hemispherical cup. As examples illustrative of their procedures, the authors briefly consider three Stokes flow problems whose solutions have been given previously.

The iterated equation of generalized axially symmetric potential theory. V. Generalized weinstein correspondence principle

“…If the profile in the meridian plane is of the first type considered in section 2, the function W is then given in terms of <f> by (13). If the profile were of type (ii) however, a similar calculation for f in the form This stream function was given by Payne and Pell [5] in their equation (8.7) which, however, contains a slight misprint.…”

“…The problem of determining the flow about a given symmetrical body when the flow is uniform at infinity can be simplified by the use of this principle (5) which, in Weinstein's words [1], 'reduces the determination of ^j, to the determination of the electrostatic potential <f> v+2 of the body with the same meridian profile but in a space of two more dimensions. ' The procedure is set out by Weinstein [1] and by Payne [4] who applies the method to find the stream function for bodies such as a spindle, lens, spheroid set in a stream which is uniform at infinity.…”

“…The method will be illustrated in some problems in the Stokes flow of a viscous fluid which are analogous to the problems in in viscid flow mentioned above; in these problems the stream function xp satisfies the equation L'ij,(y)) = 0. Payne and Pell [5,6,7] have discussed these Stokes flow problems and have carried out the difficult analysis required to produce the solutions. As would be expected, essentially the same detailed calculations are required however the problems are approached but the new procedure proposed here sets these calculations in a unified context and provides a natural approach to them.…”

Stokes flow of an incompressible micropolar fluid past a porous spheroidal shell

“…We assume that the flow is Stokesian as in the classical investigation of the problem by Payne and Pell in the case of classical viscous fluid [23] and Lakshmana Rao and Iyengar in the case of micropolar fluid [16]). This enables us to drop the inertial terms in the momentum equation and bilinear terms in the balance of first stress moments.…”

Hilbert formulas for $r$-analytic functions and the Stokes flow about a biconvex lens

“…However, in contrast to the stream function in [31], the one in this paper includes an additional term to provide proper representations of boundary conditions in the form of Mehler-Fock integrals. This term corresponds to the solution for the problem of axially symmetric steady motion of a rigid sphere in the Stokes fluid and is different from those suggested in [16,24]. Using the Hilbert formulas derived in the first part of the paper, we obtain an analytic expression for the pressure in the fluid, based on which we calculaté epures of the pressure at the contour of the body and isobars about the body.…”