Stability analysis of thin viscoelastic liquid film flowing down on a vertical wall

Abstract

This paper investigates the stability of thin viscoelastic liquid film flowing down on a vertical wall using a long-wave perturbation method to find the solution for generalized nonlinear kinematic equations with a free film interface. To begin with, a normal mode approach is employed to obtain the linear stability solution for the film flow. The linear growth rate of the amplitudes, the wave speeds and the threshold conditions are obtained subsequently as the by-products of linear solutions. The results of linear analysis indicate that the viscoelastic parameter k = k0/(ρh0*2}) destabilizes the film flow as its magnitude increases. To further investigate practical flow stability conditions, the weak nonlinear dynamics of a film flow are presented by using the method of multiple scales. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg-Landau equation. Modelling results indicate that both the subcritical instability and the supercritical stability conditions are possible in a viscoelastic film flow system. The results of nonlinear modelling further indicate that the threshold amplitude εa0 in the subcritical instability region becomes smaller as the viscoelastic parameter k increases. If the initial finite amplitude of disturbance is greater than the value of threshold amplitude, the system becomes explosively unstable. It is also interesting to note that both the the threshold amplitude and the nonlinear wave speed in the supercritical stability region increase as the value of k increases. Therefore, the flow becomes unstable when the value of k increases. The viscoelastic parameter k indeed plays a significant role in destabilizing the film flow travelling down along a vertical plate.