Stability analysis of thin viscoelastic liquid film flowing down on avertical wall

Abstract

This paper investigates the stability of thin viscoelastic liquid filmflowing down on a vertical wall using a long-wave perturbation method to findthe solution for generalized nonlinear kinematic equations with a free filminterface. To begin with, a normal mode approach is employed to obtain the linearstability solution for the film flow. The linear growth rate of the amplitudes,the wave speeds and the threshold conditions are obtained subsequently as theby-products of linear solutions. The results of linear analysis indicate thatthe viscoelastic parameter k = k0/(ρh0*2}) destabilizes the film flow asits magnitude increases. To further investigate practical flow stabilityconditions, the weak nonlinear dynamics of a film flow are presented by usingthe method of multiple scales. It is shown that the necessary condition for theexistence of such a solution is governed by the Ginzburg-Landau equation.Modelling results indicate that both the subcritical instability and thesupercritical stability conditions are possible in a viscoelastic film flowsystem. The results of nonlinear modelling further indicate that the thresholdamplitude εa0 in the subcritical instability region becomessmaller as the viscoelastic parameter k increases. If the initial finiteamplitude of disturbance is greater than the value of threshold amplitude, thesystem becomes explosively unstable. It is also interesting to note that boththe the threshold amplitude and the nonlinear wave speed in the supercriticalstability region increase as the value of k increases. Therefore, the flowbecomes unstable when the value of k increases. The viscoelastic parameter kindeed plays a significant role in destabilizing the film flow travelling downalong a vertical plate.