The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface

Abstract

The viscous gravity current that results when fluid flows along a rigid horizontal surface below fluid of lesser density is analysed using a lubrication-theory approximation. It is shown that the effect on the gravity current of the motion in the upper fluid can be expressed as a condition of zero shear on the unknown upper surface of the gravity current. With the supposition that the volume of heavy fluid increases with time like t α , where α is a constant, a similarity solution to the governing nonlinear partial differential equations is obtained, which describes the shape and rate of propagation of the current. The viscous theory is shown to be valid for t [Gt ] t 1 , when α < α c and for t [Lt ] t 1 when α > α c , where t 1 , is the transition time at which the inertial and viscous forces are equal, with $\alpha_{\rm c} = \frac{7}{4}$ for a two-dimensional current and α c = 3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for α = 1 in the preceding paper by Didden & Maxworthy (1982), as well as evaluating the multiplicative factors appearing in the relationships. The relationships compare very well with experimental measurements of the axisymmetric spreading of silicone oils into air for α = 0 and 1. There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden & Maxworthy and by Britter (1979). The predicted multiplicative constant is within 10% of that measured by Didden & Maxworthy for the spreading of salt water into fresh water in a channel.