A thin film on a porous substrate: A two-sided model, dynamics and stability

Abstract

A two-sided model (TLM) is employed to investigate the dynamics and stability of a thin film of Newtonian fluid overlying a porous substrate; the model consists of a free fluid interfacing a Brinkman-type porous transition layer, which overlies a porous medium described by the Darcy equation. The model explicitly describes the transition flow at the top of the porous medium. A nonlinear evolution equation for the free surface of the film is derived through long-wave approximation. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. It is observed that the stability characteristics of the film are influenced by the permeability, the porosity of the porous medium and the ratio of the porous to liquid layer thickness d^. Further, the conditions under which the two-sided model (TLM) can be replaced by an effective one-sided slip model (SM) is analyzed and the corresponding slip length is computed in terms of the porous layer characteristics. A weakly nonlinear stability analysis is performed and the range of preferred wave numbers for which the disturbances reach finite equilibrium amplitude or an explosive state is obtained. The nonlinear equation is then numerically solved as an initial value problem on a periodic domain and different scenarios of surface structures are captured. The long-time wave forms are shown to agree very well with the corresponding stationary solutions of the evolution equation. The results show that the long-time wave forms are either time-independent waves that propagate or time-dependent modes that oscillate slightly in amplitude. The fundamental modes dominate the stationary solution for shorter periods and the higher modes dominate as the period increases. Further, for certain bands of the period, the steady state is observed to lose its stability to oscillations.