Two-Fluid-Layer Flow Stability

Abstract

The hydrodynamic stability of the flow of two superposed, immiscible, viscous fluids has attracted considerable theoretical and practical interest over many years. The practical applications for such a configuration are in coating and lubrication technologies, polymer processing and de-/anti-icing of aircraft wings in winter. Many aspects of this generic stability problem were explored intensively via perturbation and asymptotic methods (with their inherent limitations as to regions of validity) ever since late Professor C.-S. Yih tackled it in his JFM (Vol. 27, Part 2, pp. 337-352, 1997) paper and showed the destabilizing effect of viscosity stratification per se. He connected this work to an investigation (JFM, Vol. 212, pp. 41-53, 1990) on the interfacial wave formation on de-icing liquids sprayed on aircraft. In view of the ongoing certification studies on anti-/de-icing fluids being conducted over the years at the Von Karman Institute for Fluid Dynamics and the continuing development of such fluids by industry it was felt appropriate to look at the problem more comprehensively and generate a theoretical data base to complement the experimental data base being built up. In this process extensive parameter studies have been carried out on superposed fluid layers of finite and semi-infinite extent keeping newtonian and non- newtonian flows in view since many of the anti-icing fluids being designed by industry are in fact non-newtonian in character and quit often have been found to satisfy a power-law stress-rate-of-strain relationship. Since the parameters here are rather numerous: viscosity ratio (m), density ratio (r), thickness of the liqiud layer (l), surface tension (S), Froude number (F), and also the power-law index of the liquid (n), sweeping generalizations as to the dominance of the different instability mechanisms and their practical consequences do not seem advisable. However our studies taken together with those of Yih and his followers on 2D Couette, Poiseuille and boundary layer type of flows allow a reasonably well-rounded picture of the linear stability characteristics of this generic two-fluid layer configuration. An accurate and semi-automatic shooting search method for solving the associated eigenvalue problems has been developed and used for elaborate parameter studies. Results of these more comprehensive ongoing studies are presented and discussed against the backdrop of those available in the literature. New aspects treated here are the non-newtonian character of the fluids and effects of a turbulent profile (which is rather the rule in aeronautical practice) in the airflow