A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives

The performing spectral method, developed by Haldenwang et al. [J. Comput. Phys. 55 (1984) 115], to solve multi-dimensional Helmholtz equations, associated to mixed boundary conditions with constant coefficients, is extended to boundary conditions mixing a first order normal derivative with a second order tangential derivative. The accuracy of the proposed algorithm is evaluated on two test cases for which analytical solutions exist: an academic problem and a physical configuration including an interface with shear viscosity. The procedure is also applied to the research of the Rayleigh–Bénard instability thresholds in closed cavities with thin diffusive walls.