Gravity currents are horizontal flows that occur due to density differences between two fluids. In porous media, gravity currents have been extensively studied, typically focusing on two-dimensional rectilinear geometry and assuming a sharp interface between the ambient fluid and the gravity current. In this study, we examine gravity currents in an axisymmetric geometry and develop a theoretical model to account for the mixing effects caused by the dispersive entrainment of ambient fluid into the gravity current. Our analysis reveals that with continuous injection flux, the buoyancy and radius of the gravity current evolve in a self-similar manner, with the radius increasing over time as $t^{1/2}$. However, unlike sharp-interface gravity currents, the height and concentration of the dispersive gravity currents do not display self-similar behavior. Due to the dispersive entrainment, the gravity current mixes with the ambient fluid, resulting in a monotonic dilution. We find that this dilution significantly thickens the current over time and leads to a blunt nose, which contradicts the assumption of a long and thin current at later times. Additionally, we provide estimates of the total volume of entrained fluid over time and the associated mean concentration of the gravity current. We also present results from numerical simulations conducted in comsol to validate some of our assumptions and findings from the theoretical model.

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• Flows in porous media
• Low Reynolds number flows
• Mixing in geophysical flows