Effects of Strong Viscosity with Variable Fluid Properties on Falling Film Instability

Abstract

The stability and dynamics of a gravity-driven, viscous, incompressible, Newtonian thin liquid film draining down a uniformly heated inclined plane are examined. We assume the thermophysical properties of the film such as density, surface tension, and thermal diffusivity vary linearly, whereas the viscosity varies exponentially with the small variation of temperature. Employing the classical long-wave expansion technique, a nonlinear evolution equation of Benney type, is derived in terms of film thickness h(x, t). The linear stability analysis is performed using the normal mode approach and a critical Reynolds number is obtained. The linear study reveals that the flow is more stable when the variation of viscosity is exponential as compared to a linear variation. The variation of density also affects the linear stability criteria. The method of multiple scales is used to investigate the weakly nonlinear stability of the flow, and we observe that for the variation of Kμ, Kρ, Kσ both the supercritical stable and subcritical unstable zones are possible together with the unconditional stable and explosive zones. Finally, we perform the numerical simulation in a periodic domain and confirm the results obtained by linear and weakly nonlinear studies.