The effect of surface tension on axisymmetric confined viscous gravity currents

Abstract

We consider the radial spreading of an axisymmetric viscous gravity current, in which fluid released from a point source at a constant flux is confined vertically to a narrow gap between two horizontal plates. A grounding line forms where the free surface of the current intersects with the top plate, creating two regions of flow: an inner, circular contact region near to the source where the fluid fills the entire gap between the two plates; and an outer annular region where the free surface of the gravity current lies below the top plate. Mathematical models of such flows involve solving a partial differential equation for the height of the free surface, subject to appropriate boundary conditions at the grounding line and at the leading edge of the current. In many cases, these systems admit similarity solutions. I will present one such model where the effects of surface tension are included locally at the grounding line and at the leading edge, leading to similarity solutions that depend on two dimensionless parameters, $J$ and $S$, which measure the impact of confinement and the effects of surface tension, respectively. Introducing the surface tension parameter $S$ is shown to provide better agreement between theory and experiment.