Instability of a weakly viscoelastic film flowing down a heated inclined plane

Abstract

The stability of a thin film of Walters-type B″ viscoelastic fluid flowing down a heated inclined plane is investigated. Both the weighted residual method (WRM) and the Benney-type equation (BE) are derived to simplify the original two-dimensional problem. Normal mode analysis is conducted to determine the instability threshold. We also employ the Chebyshev spectral collocation method to solve the eigenvalue problem of the full linearized Navier-Stokes/energy equations, which provides a technique to test the performance of the analytical approximations. The self-similar velocity and temperature profile assumptions made in WRM are validated by a spectral method. The results show that WRM and BE yield the same expression for the critical Reynolds number, which is in agreement with the full equations. The effects of the viscoelastic parameter, Marangoni number, and Biot number are discussed. Both heating and viscoelasticity are found to destabilize the flow, while a critical value of the Biot number is determined at which the flow is the most unstable. Nonlinear simulations are further conducted based on the method of lines, which support the predictions of instability threshold using the linear theory.