In this paper we develop a separation of variables theory, for solving problems of Stokes flow in cone-shaped trenches formed as the intersection of a cone of circular cross-section and a spherical shell centered at the vertex of the cone. The theory leads to a new set of Stokes flow eigenfunctions which describe axisymmetric motions in the vertex of a cone. Asymptotic formulas for the distribution of eigenvalues are derived; an adjoint system is defined and is used to develop an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the spherical boundaries.