On Stokes flow in a semi-infinite wedge

Abstract

Consider Stokes flow in the semi-infinite wedge bounded by the sidewalls ϕ = ±α and the endwall z = 0. Viscous fluid fills the region 0 < r < ∞, 0 < z < ∞ bounded by these planes; the motion of the fluid is driven by boundary data given on the endwall z = 0. A consequence of the linearity of the problem is that one can treat the velocity field q(r, ϕ, z) as the sum of a field qa(r, ϕ, z) antisymmetric in ϕ and one symmetric in it, qs(r, ϕ, z). It is shown in each of these cases that there exists a real vector eigenfunction sequence vn(r, ϕ, z) and a complex vector eigenfunction sequence un(r, ϕ, z), each member of which satisfies the sidewall no-slip condition and has a z-behaviour of the form e−kz. It is then shown that one can, for each case, write down a formal representation for the velocity field as an infinite integral over k of the sums of the real and complex eigenfunctions, each multiplied by unknown real and complex scalar functions bn(k) and an(k), respectively, that have to be determined from the endwall boundary conditions. A method of doing this using Laguerre functions and least squares is developed. Flow fields deduced by this method for given boundary data show interesting vortical structures. Assuming that the set of eigenfunctions is complete and that the relevant series are convergent and that they converge to the boundary data, it is shown that, in general, there is an infinite sequence of corner eddies in the neighbourhood of the edge r = 0 in the antisymmetric case but not in the symmetric case. The same conclusion was reached earlier for the infinite wedge by Sano & Hasimoto (1980) and Moffatt & Mak (1999). A difficulty in the symmetric case when 2α = π/2, caused by the merger of two real eigenfunctions, has yet to be resolved.