Stability of a thin viscoelastic film falling down an inclined plane

The stability of a thin viscoelastic film of Oldroyd-B fluid falling down an incline is investigated. In the weak viscoelasticity limit, a reduced model is derived using the weighted residual techniques, which consists of two coupled equations for the film thickness and local flow rate. Through the normal mode analysis, temporal growth rates and neutral stability curves are calculated to explore linear stability of the film. Results show that the viscoelasticity acts to destabilize the film and decrease the phase speed of linear waves. Good agreement is found between the reduced model and full linearized equations solved by the Chebyshev spectral collocation method when viscoelastic effect is relatively weak. Nonlinear traveling waves are further determined. The speed of fast/slow-wave families is promoted/reduced in the presence of the viscoelasticity, resulting in a dispersion effect on the system; while the wave amplitudes are augmented for both fast and slow waves. Besides, the temporal evolution of surface waves is numerically resolved, which validates the linear prediction of the instability threshold. Steady permanent waves are observed in the final stage; the surface deformation and perturbation energy are enhanced by the viscoelasticity as expected.