Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ $B^{\prime }$ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number $k_c$ which include the viscoelastic parameter $\Gamma$, angle of inclination of the plane $\theta$, Reynolds number $Re$ and Weber number $We$. The weakly nonlinear stability analysis that is based on the second Landau constant $J_2$, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is $3(1+3\Gamma )Re-3cot\theta -4ReWe{k}^2$ = 0. It is found that all the four distinct flow zones of this problem exist in $Re$-$k$-, $\theta$-$k$- and $\Gamma$-$k$-plane after the critical value of $R{e}_c,$ ${\theta }_c$ and ${\Gamma }_c$, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of $\Gamma$ irrespective of the values of $Re$ and $\theta$, as for example, a solution of polyisobutylene in cetane, compared with the viscous $(\Gamma =0)$ film flow case. Finally, we scrutinize the effect of $\Gamma$ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.