Linear stability analysis of gravity-driven viscosity-stratified newtonian coating flow on planar incline

This theoretical investigation motivates, reviews and formulates the linear stability analysis of a gravity-driven and viscosity-stratified coating flow useful in covering a planar surface with one or more liquid layers. The mathematical foundation on which to analyse the stability of this inherently spatial coating flow problem is provided by a Green's impulse function approach which serves as the appropriate mathematical tool to describe (arbitrary) disturbances imposed on the liquid surface. The temporal analysis showed that a two-layered Newtonian coating flow is susceptible to an instability due to viscosity stratification even without the effects of surface tension, density-stratification or, surprisingly, inertia. In addition, waves with the largest growth typically occur at finite wavelengths; the properties of these waves are often of greatest practical interest. However, the temporal growth rate, for the maximally unstable mode, of order 0.0015 was small. The mathematically proper ray-speed approach, originating from a steepest-descent method to determine the perturbed film thickness, removed the interpretational quandary present in the classical spatial approach and corroborated the temporal linear stability results