Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow ($x$- and $y$-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number $Re$, Weber number $We$, angle of inclination of the plane $\theta$ and the angle of propagation of the interfacial disturbances $\varphi$. The temporal growth rate ${\omega }_i^{+}$ and second Landau constant $J_2$, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors $B$ and $\beta$ owing to the non-zero steady basic flow along the $y$-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of $Re$, $We$ and $\theta$, any stability that arises in two-dimensional disturbances ($\varphi$ = $0$) must also be present in three-dimensional disturbances. For $\varphi$ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of $Re$ (or $\theta$) due to the involvement of $B$ and $\beta$ in the expression of $J_2$. This explosive unstable zone vanishes after a certain value of $\varphi$ depending upon the values of $Re$, $We$ and $\theta$, which confirms the stabilizing influence of $\varphi$ on the thin film flow dynamics irrespective of the values of $Re$, $We$ and $\theta$.