Wave dynamics of a viscoelastic liquid

Abstract

A falling film of a weakly viscoelastic liquid (Walters’ liquid B′′) is studied in detail for low to moderate values of the Reynolds number. A linear stability analysis within the framework of the Orr–Sommerfeld-type boundary value problem reveals a destabilizing effect of the viscoelastic parameter near the threshold of instability but a stabilizing effect of the viscoelastic parameter on the primary instability far away from the threshold of instability. The surface equation is derived to decipher the existence of nonlinear waves based on the centre manifold approach. However, it renders an unbounded solution in the short-wave regime. In order to avoid singular behaviour found in the surface equation, a regularized equation, a simplified second-order model, and a full second-order model are proposed. In the linear regime, both the regularized equation and the model equations show excellent agreement with the results of the Orr–Sommerfeld-type boundary value problem, i.e., the viscoelastic parameter demonstrates a dual role in the primary instability. In the nonlinear regime, the results of the regularized equation are verified with those of the simplified second-order model. Both the regularized equation and the simplified second-order model show qualitatively similar characteristics of the nonlinear waves. The speed and maximum amplitude corresponding to the steady-state travelling waves attain fixed values in the drag-inertia regime. Moreover, the speed and maximum amplitude acquired from the regularized equation underestimate the results of the simplified second-order model when the Reynolds number lies in the drag-inertia regime. The time-dependent simulation of the regularized equation displays a train of solitary waves downstream to a monochromatic time periodic forcing at the inlet. The separation distance between the first two solitary pulses is computed, and it seems to approach an asymptotic value when the time is very large. This fact assures the possibility of a bound-state formation among the solitary pulses downstream for the weakly viscoelastic liquid.