On Stokes flow in a semi-infinite wedge

Consider Stokes flow in the semi-infinite wedge bounded by the sidewalls ph i = +/-alpha and the endwall z = 0. Viscous fluid fills the region 0 < r < infinity, 0 < z < infinity bounded by these planes; the motion of the fluid is driven by boundary data given on the endwall z = 0. A consequence of th e linearity of the problem is that one can treat the velocity field q(r, ph i, z) as the sum of a field q(a)(r, phi, z) antisymmetric in phi and one sy mmetric in it, q(s)(r, phi, z). It is shown in each of these cases that the re exists a real vector eigenfunction sequence v(n)(r, phi, z) and a comple x vector eigenfunction sequence u(n)(r, phi, z), each member of which satis fies 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 repres entation for the velocity field as an in finite integral over k of the sums of the real and complex eigenfunctions, each multiplied by unknown real an d complex scalar functions b(n)(k) and a(n)(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 deduce d by this method for given boundary data show interesting vortical structur es. Assuming that the set of eigenfunctions is complete and that the releva nt 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 th e symmetric case. The same conclusion was reached earlier for the infinite wedge by Sane & Hasimoto (1980) and Moffatt & Mak (1999). A difficulty in t he symmetric case when 2 alpha = pi /2, caused by the merger of two real ei genfunctions, has yet to be resolved.