Axisymmetric Stokes Flow Past Cones Including the Case of the Needle

Abstract

The Stokes flow past sharp axisymmetric cones of acute semivertex angle has been studied using an axisymmetric spherical coordinate system. The Stokes solution consists of associated Legendre functions, of the first kind, of the first-order and fractional degree related to the eigenvalues of the problem. These Legendre functions as well as the lowest eigenvalues of the Stokes solution have been accurately evaluated using two different approaches. The present results for the eigenvalues appear to be more accurate than those obtained earlier by Schwiderski, Lugt and Ugincius [3]. An important limiting case with semivertex angle δ → 0, i.e., the needle has been correctly analyzed and the results show that, as δ → 0, the Stokes flow is valid in a vanishingly small neighborhood of the needle with the skin friction being infinite at the “surface” of the needle.