The separation of Stokes flows

Abstract

A study is made of the extent to which local boundary geometry can influence separation in a two-dimensional or an axisymmetric Stokes flow. It is shown that a Stokes flow can separate from a point on a smooth body at an arbitrary angle, which can be determined only by reference to the global solution for the flow past the body, and the dominant mode in the stream function near a point of separation is O(r3) in the distance r from the separation point. When the body has a protruding cusped edge it is shown that separation can occur at an arbitrary inclination to the edge which must again be determined from the global solution. In this case the stream function is O(r3/2) near the edge. When the flow is locally within a wedge-shaped region of angle β, where β ≠ π or 2π, and β > 146·3°, it is shown that the dominant modes near the vertex of the wedge are non-separating modes. It follows that, in general, a Stokes flow around such a wedge cannot separate from the vertex. This conclusion is illustrated by reference to the global solution for uniform axisymmetric flow past a spherical lens, in which the structure of the flow near the rim is examined in detail. In the case of a body having a sharp edge of small but non-zero angle protruding into the flow, so that β is very close to 2π, it is shown that separation occurs exceedingly near to the edge. This happens, for example, in the flow past a thin concave-convex lens, for which separation occurs near the rim on the concave side. The analysis also suggests that a similar separation occurs very near the rim on the flatter side of a thin asymmetric biconvex lens. However, for the symmetric biconvex lens, and, as a special case, the circular disk, no separation occurs on either side near the rim. For β < 146·3·, streaming flow into the vertex of a wedge does not occur because of the presence of an infinite set of vortices, and the possibility of separation at the vertex in the sense discussed here does not arise.