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Ritz solution of the Helmholtz equation in a stadium-shaped domain with zero normal derivatives – applications to fluid sloshing and thermo-convective stability

Wang C. Y.

Archives of Mechanics

2017

Vol. 69, nr 6

471--483

The eigenvalues and eigenfunctions of the Helmholtz equation with Neumann conditions are obtained for the stadium-shaped domain. The variational Ritz method is found to be accurate and efficient in determining these eigenvalues and eigenfunctions. The eigenfunctions show the evolution and switching of mode shapes from a long rectangular strip to a circle. These new results are applied to the sloshing of a liquid in a tank, and to the onset of thermo-convective stability in a confined porous layer.