The Stokes flow about a spindle

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

“…The aim is to find a function W to satisfy equation (6), boundary conditions (7) (with % = 0) on the circle r = a and condition (9) at infinity.…”

“…The profile in the meridian plane in this problem (see [6]) is the arc of a circle with centre on the «/-axis joining the points (±c, 0) on the rc-axis. Coordinates |, r\ are chosen so that…”

“…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.…”

“…In finding ip to satisfy (6), it is convenient to use the generalized correspondence principle (2) to write ip as y p+1 $ where $, like <f>, satisfies equation (11). $ is then found in the same way as <j>, the only change being to increase by p-\-l the power of y which is a common factor of the terms making up the general solution.…”

“…The result of this is to change the parameter X which occurs in the solution for <j> to n = A-p -1 = -^p+l-n. The rest of the procedure remains the same and when <j> and $> have both been obtained, (14) gives The method is illustrated by showing how to find the stream function for axially symmetric Stokes flow of a viscous fluid in two cases previously considered by Pell and Payne [6,7] who have solved the problems in detail. The account given here will be limited to demonstrating the systematic approach provided by the present method, the analysis being carried only far enough to link it to the original authors' presentation.…”