Axisymmetric viscous flow between two horizontal plates

Abstract

The flow of viscous fluid injected from a point source into the space between two horizontal plates initially filled with a second fluid of lesser density and different viscosity is studied theoretically and numerically. The volume of the dense input fluid increases with time in proportion to tα. When the fluid has spread far from the source, lubrication theory is used to derive the governing equations for the axisymmetric evolution of the interface between the fluids. The flow is driven by the combination of pressure gradients associated with buoyancy and pressure gradients associated with the input flux. The governing equation is integrated numerically, and we identify that with a constant input flux, the flow is self-similar at all times with the radius growing in proportion to t1/2. In the regimes of injection-dominated and gravity-dominated currents, we obtain asymptotic approximations for the interface shape, which are found to agree well with the numerical computations. For a decreasing input flux (0 < α < 1), at short times, the flow is controlled by injection; the current fills the depth of the channel spreading with radius r ∼tα/2. At long times, buoyancy dominates and the current becomes unconfined with the radius growing in proportion to t(3α+1)/8. The sequence of regimes is reversed in the case of an increasing input flux (α > 1) with buoyancy dominating initially while the pressure associated with the injection dominates at late times. Finally, we consider the release of a fixed volume of fluid (α = 0). The current slumps under gravity and transitions from confined to unconfined, and we obtain asymptotic predictions for the interface shape in both regimes.