(296b) Electrokinetic Fingering: A Problem in Vector Laplacian Growth

Pattern formation is ubiquitous in various physical and chemical processes and has fascinated many scientists over the past couple of decades. In many instances, interfacial instabilities play a crucial role in creating these patterns and controlling their spatiotemporal evolution. Perhaps one of the most well-known examples is the striking figures generated when a high-viscosity fluid is displaced by a low-viscosity fluid. This “viscous fingering” phenomenon was originally described by Saffman and Taylor and is one of the limiting factors in determining the extraction efficiency in Enhanced Oil Recovery (EOR). More generally, viscous fingering belongs to the class of “Laplacian Growth” phenomena which includes many other notable examples such as crystal growth, solidification, and dendritic growth in metal electrodepostion. In this framework, the interfacial velocity is related to the gradient of a harmonic function (e.g. pressure, temperature, or concentration) and the instability is attributed to the curvature-dependence of the gradient.

More generally, however, the interface could be driven by multiple competing forces. In this talk, we will consider a class of problems which we term “Vector Laplacian Growth”. In these problems, fluxes are linearly proportional to the gradient of a vector-valued potential function through a generalized mobility tensor. Specifically, we will study the linear electrokinetic response of two immiscible electrolytes in a Hele-Shaw cell where the interface motion is set by both the pressure-driven and electro-osmotic flows. By using linear stability analysis, as well as nonlinear numerical simulations, we will show that the coupling between pressure and potential fields strongly affects the interfacial stability and, under certain conditions, can suppress viscous fingering. Remarkably, our theory suggests that for a given pair of electrolytes, the ratio of electric current to the flowrate controls the onset of instability, suggesting new directions for further experimental investigations. We believe that our findings could have implications for electrically enhanced oil recovery application and might help with understanding other similar multi-field driven interfacial instabilities.