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 = k 0 /(ρh * 2 0 ) 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 εa 0 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.

Nonlinear stability analysis of the thin pseudoplastic liquid film flowing down along a vertical wall

“…This phenomenon agrees well with the conclusion given by Bird et al 18 By setting nϭ1, the results of a classical Newtonian flow are obtained. As compared to the modeling results given by Cheng et al, 17 it is found that both solutions agree well with each other. …”

“…After deriving the viscoelastic analog of the Orr-Sommerfeld equation with the requisite boundary condition, Gupta pointed out the viscoelastic effect can destabilize the film flow. Cheng et al 17 further studied the nonlinear stability analysis of thin viscoelastic liquid film flowing down on a vertical wall. They also demonstrated that the viscoelastic property has a destabilizing effect on the nonlinear film flow system.…”

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

“…The higher-order terms of OðE 2 Þ can be neglected since they are of little influence on the system [18,19]. The set of zeroth and first order equations can now be expressed as follows: Oð1Þ :…”

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

“…It is worth mentioning here that the investigations on the flow and stability characteristics of non-Newtonian fluids [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] have shown that the deviation from Newtonian rheology has a significant influence on the stability characteristics of non-Newtonian films. The above investigations have analyzed the flow characteristics and the associated time dependent properties and have developed suitable conditions for homogeneous film growth that would be relevant for industrial applications.…”