Planar non-Newtonian confined laminar impinging jets: Hysteresis, linear stability, and periodic flow

Abstract

This paper considers the linear stability of confined planar impinging jet flow of a non-Newtonian inelastic fluid. The rheology is shear rate dependent with asymptotic Newtonian behavior in the zero shear limit, and the analysis examines both shear thinning and shear thickening behavior. The planar configuration is such that the width of the inlet nozzle is smaller than the distance from the jet exit to the impinging surface, giving an aspect ratio e = 8 for which two-dimensional time dependent flow is readily manifest. For values of the power-law index n in the range 0.4≤n≤1.1, the bi-global linear stability of the laminar flow is analyzed for Newtonian Reynolds numbers Re ≲200. The calculations show that for certain values of n, including the Newtonian value n = 1, the steady flow exhibits multiplicity leading to hysteresis in the primary separation vortex reattachment point and a consequent jump in stability behavior. Even in the absence of hysteresis, relatively small changes in viscosity significantly affect stability characteristics. For Newtonian and mildly shear thinning or shear thickening fluids, an unstable flow shows a decaying perturbation growth rate as Re is increased, and for certain values of n, the flow may be restabilized at a larger Re before eventually becoming unstable again. This decay in the growth rate of the critical antisymmetric mode may be correlated as a function of the reattachment point RP of the primary separation vortex in the underlying steady flow. Representative results are analyzed in detail and discussed in the context of some experimental observations of time-dependent Newtonian impinging flow. The stability results are used to construct the neutral stability curve (n, Re) that displays multiplicity and contains several cusp points associated with flow restabilization and hysteresis. Integration of the full nonlinear equation reveals the structure of the time periodic flow field for both Newtonian and non-Newtonian fluids at Reynolds numbers well beyond the instability threshold.