Bifurcation analysis of first and second order Benney equations for viscoelastic fluid flowing down a vertical plane

Using the method of multiple scale we have studied the nonlinear stability of a travelling wave solution of an evolution equation for the viscoelastic fluid flowing down a vertical plane. Bifurcation analysis of first and second order Benney equations (BEs) for the viscoelastic fluid shows that the first order BE gives both subcritical unstable and supercritical stable zones depending on the Reynolds number greater or smaller than its critical value and the supercritical stable/subcritical unstable region decreases/increases as the viscoelastic parameter increases. However, the second order BE exhibits only supercritical bifurcation and this stable region increases with the increase in either the Reynolds number or the viscoelastic parameter. The spatially uniform solution of the complex Ginzburg–Landau equation for sideband disturbances is also investigated.