Stability analysis of thin film flow along a heated porous wall

The time evolution of a thin liquid film flowing down a heated solid porous substrate is investigated. Using the Navier-Stokes and Darcy-Brinkman equations in the film and the porous layer, respectively, the problem is reduced to the study of the evolution equation for the free surface of the liquid film derived through a long-wave approximation. A linear stability analysis of the base flow is performed and the critical Reynolds and Marangoni numbers are obtained. A nonlinear analysis using continuation techniques shows that the base flow yields to stationary surface structures ranging from surface waves to large amplitude structures resembling sliding drops or ridges. It is also shown under what conditions the porous layer can be replaced by an effective slip boundary condition at the liquid-solid interface. Then, the corresponding slip length is calculated from the porous layer characteristics (thickness, porosity, and Darcy number). (C) 2009 American Institute of Physics. [DOI:10.1063/1.3054157]