Stability analysis of non-inertial thin film flow over a heterogeneously heated porous substrate

The dynamics and linear stability of a gravity drive thin film flowing over non-uniformly heated porous substrate are studied. A governing equation for the evolution of film-thickness is derived within the lubrication approximation. Darcy-Brinkman equation is used to model flow in the porous medium along with a tangential stress-jump condition at the interface of the porous layer and the fluid film. A temperature profile is imposed at the solid wall to model an embedded heater beneath the porous layer. At the upstream edge of the heater, an opposing thermocapillary stress at the liquid-air interface leads to the formation of a thermocapillary ridge. The ridge becomes unstable beyond a critical Marangoni number leading to the formation of rivulets that are periodic in the spanwise direction. Increase in the values of parameters such as Darcy number, stress jump coefficient, and porosity is shown to have stabilizing effect on the film dynamics. The critical Marangoni number is shown to increase monotonically with Darcy number for various values of porosity. At large values of stress-jump coefficient, a non-monotonic variation in critical Marangoni number versus Darcy number is shown. A correlation is developed numerically for the ratio of critical Marangoni number at large Darcy number to that for a non-porous substrate as a function of porosity and thickness of the porous substrate. A transient growth analysis is carried out followed by non-linear stability analysis. The non-modal growth is found to be negligible thus indicating that the eigenvalues are physically determinant. (C) 2016 AIP Publishing LLC.