Nonlinear elastic instability of gravity-driven flow of a thin viscoelastic film down an inclined plane

A nonlinear evolution equation for the thickness of a thin viscoelastic film flowing down an inclined plane is derived for an Oldroyd-B fluid, using a long wave approximation. The evolution equation is valid to the second order in a small parameter which measures the relative thickness of the film to a typical wavelength. For a very thin film, viscoelasticity dominates the stability of the film and it can cause a purely elastic instability. The weakly nonlinear development of a monochromatic wave resulting from this elastic instability is studied using the second-order evolution equation which allows us to investigate the effects of inclination angle on the bifurcation. It is found that although extremely long waves bifurcate subcritically, the linearly most amplified wave does bifurcate supercritically when the surface tension parameter J > 13.8035. It is demonstrated that for a fixed bifurcation parameter δ = Wi - Wi_c, increasing the inclination angle can reduce the equilibrium amplitude for a supercritical bifurcating wave.