Stokes flow in wedge-shaped trenches

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Abstract
In this paper we develop a separation of variables theory for solving problems of Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centred at the vertex of the wedge. The theory leads to a set of Stokes flow eigenfunctions which in the full wedge reduce to the corner eigenfunctions studied by Dean & Montagnon (1949) and Moffatt (1964). Asymptotic formulae 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 circular boundaries. To illustrate the algorithm we find the motion and the shape of the free surface in a wedge-shaped cavity heated from its side.

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