Stability analysis of falling liquid film over a heterogeneously heated slippery substrate

We investigate gravity-driven, Newtonian thin liquid film flow along a heterogeneously heated slippery rigid substrate. Using Benney’s long-wave asymptotic expansion technique (LWE) a free surface evolution equation is constructed. In case of locally heated Gaussian temperature distribution, simulation of the basic flow shows that the increment of thickness of the film for the primary flow in case of variation of the film Marangoni number ($M$) is significant with comparison to the dimensionless slip length ($\beta$). For uniform temperature distribution the linear study confirms that the destabilizing behavior of the slip length ($\beta$) is dominant than that of the destabilizing behavior of the film Marangoni number ($M$); Biot number plays a double role. There exists a critical value of Biot number $\left(B_c\right)$; below this value it exhibits stabilizing effect, but above it, it becomes destabilizing. As the LWE is valid only near the critical point and has the finite time blow up property of the solution, we employed weighted residual method (WRM) for better understanding of the critical condition with the variation of the slip length. For uniform temperature distribution the onset of instability $\left(R{e}_c\right)$ obtained by WRM is exactly same to that obtained by Orr–Sommerfeld/LWE method, in case of small as well as moderate values of the slip length ($\beta$). Further, using Fourier spectral method of the coupled system in terms of film thickness $h(x,t)$ and flow rate $q(x,t),$ the temporal as well as the spatial evolution, in case of locally heated Gaussian temperature distribution, confirms the destabilizing behavior of both $M$ and $\beta$. Numerical simulation of the Benney type evolution equation, for the locally heated Gaussian temperature profile, reveals the destabilizing behavior of M, $\beta$. The dual role of Biot number is missing, it only exhibits stabilizing effect.