Numerical Investigation of Fluid Flow Stability by Chebyshev Collocation Method

This paper describes a Chebyshev collocation method for solving the eigenvalue problem that governs the linear stability of fluid film flow down an inclined planein presence or absence of surfactant at arbitrary Reynolds numbers. The method is based on the expansion of the eigenfunctions in terms of Chebyshevpolynomials, point collocation, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ-algorithm. This method is developed tosolve Orr-Sommerfeld equation. The results in the absence of surfactant are compared with those presentedby Yih. Finally the effect of inertia and Marangoni modewhich appears when insoluble surfactant is present is investigated. The effect of instability modes, the Yihmode which is associated with perturbation on fluid film surface and Marangoni mode which is associated with variation of surfactant surface concentration, are studiedfor flow instability at arbitrary Reynolds numbers. The agreement between the obtained results at limitedconditions and also results from other researchers shows the effectiveness of Collocation Method and suggested profiles.

Keywords:

Collocation , Orr-Sommerfeld , Linear Stability , Marangoni , Surfactant