Spatiotemporal evolution of evaporating liquid films sheared by a gas

We study the spatiotemporal evolution of an evaporating liquid film sheared by a gas considering both inertial and thermal instabilities, the latter arising from a combination of evaporation and Marangoni effects. The shearing gas is modeled by imposing a constant shear stress applied along the liquid's interface. Following in the footsteps of Joo et al. [S. W. Joo et al., J. Fluid Mech. 230, 117 (1991)], long-wave theory is used to derive a Benney-like equation governing the temporal evolution of the liquid interface under the effects of inertia, hydrostatic pressure, surface tension, thermocapillarity, evaporation, and gas shear. Linear stability theory is used to investigate the temporal and spatiotemporal characteristics of the flow, where it is found that the evaporation of the film promotes absolute instabilities and can cause convective-absolute transitions of the perturbations. It is also found that a strong enough counterflowing shearing gas can suppress the inertial instability, commonly known as the H mode, affirming similar conclusions found by previous studies for a strongly confined isothermal film. Additionally, our temporal stability analysis indicates that the thinning of the film reduces the phase speed of thermal perturbations, due to the increasing dominance of viscosity. However, our spatiotemporal analysis shows that the thinning of the film actually results in the growth of additional modes with higher group velocities resulting in faster contamination of the flow field. Moreover, the interface evolution equation is solved numerically to (i) simulate the film's interface evolution subject to finite perturbations and (ii) compare to the results of the linear stability analysis. We find qualitative agreement between the temporal dynamics of the linear and nonlinear instabilities. Our subsequent numerical nonlinear spatiotemporal stability analysis demonstrates that for weaker thermal instabilities, the wave-front dynamics are imposed by the nonlinearly saturated wave packet, while for stronger thermal instabilities, the wave-front dynamics are dictated by the linear dispersion relationship. We also study the effects of the dimensionless parameters on the rupture location and the time it takes for the fluid film to rupture. Finally, the shear stress's effect on the rupture mechanics of the film is studied using self-similarity analysis, where we identify the fate of the evolution equation's solutions.