Stokes flow in a half-filled annulus between rotating coaxial cylinders

Abstract

A model is presented for viscous flow in a cylindrical cavity (a half-filled annulus lying between horizontal, infinitely long concentric cylinders of radii R_i, R_o rotating with peripheral speeds U_i, U_o). Stokes' approximation is used to formulate a boundary value problem which is solved for the streamfunction, ω, as a function of radius ratio R¯= R_i/R_o and speed ratio S=U_i/U_o.

Results show that for S>0 (S<0) the flow domain consists of two (one) large eddies (eddy), each having a stagnation point on the centreline and a potentially rich substructure with separatrices and sub-eddies. The behaviour of the streamfunction solution in the neighbourhood of stagnation points on the centreline is investigated by means of a truncated Taylor expansion. As R¯ and S are varied it is shown that a bifurcation in the flow structure arises in which a centre becomes a saddle stagnation point and vice versa. As R¯→1, a sequence of ‘flow bifurcations’ leads to a flow structure consisting of a set of nested separatrices, and provides the means by which the two-dimensional cavity flow approaches quasi-unidirectional flow in the small gap limit. Control-space diagrams reveal that speed ratio has little effect on the flow structure when S<0 and also when S>0 and aspect ratios are small (except near S=1). For S>0 and moderate to large aspect ratios the bifurcation characteristics of the two large eddies are quite different and depend on both R¯ and S.