Biorthogonal Series Solution of Stokes Flow Problems in Sectorial Regions

Abstract

In this paper we develop an eigenfunction expansion method for solving Stokes flow problems in sectorial cavities that arise in fluid dynamics. The method leads to the development of a set of eigenfunctions, adjoint eigenfunctions, biorthogonality conditions, and an algorithm for the computation of the coefficients of the eigenfunction expansion. The resulting infinite system of linear equations is then solved by truncation. These biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions, which are used to compute the coefficients of the eigenfunction expansion solution. The biorthogonality conditions are derived for a class of fourth-order boundary value problems with variable coefficients that arise from separating variables of the governing Stokes equation. The method is applied to the slow, steady flow in a two-dimensional sectorial cavity; the fluid is set into motion by the uniform translation of a covering plate or belt.