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

The stability of a thin film of Walters-type B# viscoelastic fluid flowing down a heated incline plane is investigated. Both the weighted residual method (WRM) and the Benney type equations (BE) are derived to simplify the original two-dimensional problem. The normal mode analysis is conducted to determine the instability threshold. We also employ the Chebyshev spectral collocation method (CSCM) to solve the eigenvalue problem of the linearized Navier-Stokes/energy equations to test the performance of the analytical approximations. The results show that both WRM and BE yield the same expression for the critical Reynolds number, which is also validated by CSCM. The effects of the viscoelastic parameter and the Marangoni number are discussed. Both the heating and the viscoelasticity are found to destabilize the flow.