A modified Chebyshev-tau method for a hydrodynamic stability problem


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.


C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

I. S. Pop
Babes-Bolyai University, Faculty of Mathematics and Informatics


Chebyshev spectral method, differential eigenvalue problem, hydrodynamic stability

Cite this paper as:

C.I. Gheorghiu, I.S. Pop, A modified Chebyshev-tau method for a hydrodynamic stability problem, Proceedings of ICAOR (International Conference on Approximation and Optimization – Romania), 1996, vol. II, pp. 119-126.



About this paper


Proceedings of the International Conference on Approximation and Optimization (Romania) – ICAOR

Publisher Name

Faculty of Mathematics and Informatics, Babes-Bolyai University







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