Linear stability of a contaminated shear-imposed viscoelastic liquid flowing down an inclined plane

Abstract

A linear stability analysis is carried out for a contaminated viscoelastic liquid flowing down an inclined plane in the presence of an imposed shear stress, where the elastic behavior of the liquid follows the upper-convected Maxwell model. The earlier work [Wei, “Stability of a viscoelastic falling film with surfactant subjected to an interfacial shear,” Phys. Rev. E 71, 066306 (2005)] conducted analytically in the long-wave regime is revisited again in exploring the results in the arbitrary wavenumber regime. An Orr–Sommerfeld-type eigenvalue problem is formed for the viscoelastic liquid and solved both analytically and numerically by using the long-wave expansion and Chebyshev spectral collocation technique, respectively. It is found that with increase in the value of the Weissenberg number, the critical Reynolds number for the interface mode reduces, but the stable region enhances in the finite wavenumber regime. Furthermore, the unstable domain induced by the interface mode reduces in the presence of insoluble surfactant but enhances in the presence of applied shear stress. If the Reynolds number is high, but the inclination angle is small, the shear mode arises in the numerical simulation, which becomes weaker in the presence of the Weissenberg number and Marangoni number but becomes stronger in the presence of applied shear stress. In a special case, it is demonstrated that the present study recovers the results of Walters's liquid B″ in the limit of low viscoelastic parameter.