Insights on phase speed and the critical Reynolds number of falling films

We revisit the studies of gravity-driven viscous falling films with and without imposed shear stress to provide new perspectives on phase speed and the critical Reynolds number for surface instability. We use the traditional long-wave expansion technique implemented for investigating the linear stability analysis [C. S. Yih, Phys. Fluids 6, 321 (1963)]. The principal purpose is to create a unified relationship between the leading-order phase speed and the critical Reynolds number that will hold for falling films on impermeable substrates with or without shear stress acting at the liquid film surface. The analytical result demonstrates that the critical Reynolds number for the onset of surface instability is $[5/\left(2c_0\right)]cot\theta$, where $c_0$ is the leading-order phase speed of the surface mode and $\theta$ is the angle of inclination with the horizontal. Clearly, the critical Reynolds number of the surface mode is explicitly dependent on the leading-order phase speed. Furthermore, we reveal that the basic parallel flow with or without imposed shear stress is linearly unstable to infinitesimal disturbances if the modified Reynolds number, ${\mathrm{Re}}_M=(\mathrm{Re}{c}_0/cot\theta )[\mathrm{Re}\mathrm{is}\mathrm{the}\mathrm{Reynolds}\mathrm{number},\mathrm{and}\theta \ne \pi /2]$, is greater than its critical value of $5/2$, which is independent of the shear stress applied at the film surface. In addition, it is demonstrated that ${\mathrm{Re}}_M$ controls the surface instability in the long-wave regime for both shear-imposed and non-shear-imposed film flows.