Instability of gravity-driven flow of a heated power-law fluid with temperature dependent consistency

In this paper we report on our investigation of the instability of a liquid layer flowing along a heated inclined plane. We develop and implement a theoretical model with a power-law constitutive relation which captures the temperature variation in the rheology of the fluid. We carry out a linear stability analysis and obtain Orr-Sommerfeld type equations for the evolution of infinitesimal perturbations imposed on the equilibrium flow. Numerical solutions were obtained, as well as asymptotic approximations based on the assumption of perturbations of long wavelength and small variation in the consistency of the fluid with respect to temperature. We investigate the critical conditions for the onset of instability and determine the effect of a non-Newtonian rheology and the dependence of the fluid properties on temperature. Nonlinear effects were considered by employing a reduced dimensionality model. Calculations of permanent waves arising from unstable uniform flows were made by carrying out numerical simulations of these equations.