We study the linear stability of some Marangoni flows (thin films) on an inclined plane. The Orr-Sommerfeld eigenproblem contains two non standard boundary conditions of second and third orders. Both depend on the eigenparameter as well as on some non dimensional physical parameters, i.e., Reynolds and capillary numbers and dimensionless surface stress. The basic state is a parabolic profile with the linear part depending on the surface stress. Using a long wave approximation we find a critical value of Reynolds number. The eigenproblem is comparatively solved by a modified Chebyshev-tau approximation. We construct bases in both trial and test spaces such that the differentiation matrices involved are sparse and better conditioned than those involved in the classical tau. Our method is more stable than the classical tau approach and the spurious eigenvalues are completely removed.
Babeș-Bolyai University, Faculty of Mathematics and Informatics
Tiberiu Popoviciu Institute of Numerical Analysis
Chebyshev-Galerkin spectral method; differential eigenvalue problem
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I.S. Pop, C.I. Gheorghiu, A Chebyshev-Galerkin method for fourth order problems, Proceedings of ICAOR (International Conference on Approximation and Optimization – Romania), Cluj-Napoca, July-August 1996, vol. II, pp. 217-220.
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Approximation and Optimization
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