Film flow for power-law fluids: Modeling and linear stability

Abstract

The paper deals with modeling of a power-law fluid film flowing down an inclined plane for small to moderate Reynolds numbers. A model, accurate up to second order [first order] for dilatant [pseudoplastic] fluids is proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The resulting model is naturally unique, i.e., independent of the particular form of the trial functions. Reduced models are proposed for the evolution of the local film thickness and flow rate; their linear spectra are compared to that obtained from the full Orr–Sommerfeld numerical solution. To obtain the latter, a new formulation of the eigenvalue problem is proposed to overcome the classical divergence of the apparent viscosity at the free surface. The full model and its reduced forms all have the advantage of the Benney like model close to criticality. Far from the instability threshold the full model continues to follow the Orr–Sommerfeld solution up to sufficiently large Reynolds numbers and gives better predictions than the depth averaging model. An incomplete regularization procedure is performed to cure the rapid divergence of the reduced two-equation model. Due to its relative simplicity the latter might be preferred in practice to the full model, at least at the initial stage of the nonlinear regime. It is also shown that the convective nature of the instability is not affected by the variation of the power law index.