Influence of Heterogeneity on Second-Kind Self-Similar Solutions for Viscous Gravity Currents

We report theoretical, experimental and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (textit{a}) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (textit{b}) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviors emerge as a result of horizontal heterogeneity: (textit{a}) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (textit{b}) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared to experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.