Injection of a non-Newtonian fluid into an otherwise Newtonian boundary layer

We consider the description and subsequent analysis of a natural phenomenon observed in certain species of fish, which have been observed to secrete complex fluids. This thin fluid layer on the skin has been shown to provide beneficial attributes to the fish, in some instances aiding to reduce skin-friction drag. We provide a modified boundary layer analysis to model this secretion.



Firstly, we investigate the relevant flow quantities of a two-phase system, in which a shear-thinning fluid is injected into a larger Newtonian boundary layer. The background fluid is subject to a constant uni-directional flow in the far field, and our non-Newtonian phase is characterised by the Carreau-Yasuda viscosity law. We present the base flow solutions for a variety of parameter space, in which the interfacial location is determined by the flow. These solutions are available both using a self-similar framework, for a specific inclination of flat plate, and in two dimensions, which consider no angle of inclination. Furthermore, we provide a linear perturbation analysis to evaluate the stability of the system to small changes in velocity and pressure.



The fundamental aim of our research is to compare the stability of our shear-thinning injection case study against both the Newtonian equivalent and the one-tier system in the absence of injection. It has been shown in previous studies, that for a specific choice of shear-thinning parameter space, one can delay the onset of instability given no injection. Therefore, we investigate our model to evaluate if the same can be observed with a two-tier boundary layer model. Further key variations, including the change in injection velocity, injection angle and viscosity ratio are considered, and their stability characteristics analysed.