Self-similar plane solutions for the inertial stage of gravity currents are related to the initial parameters and a coefficient that is determined by the boundary condition at the front. Different relations have been proposed for the boundary condition in terms of a Froude number at the front, none of which have a sound theoretical or experimental basis. This paper focuses on considerations of the appropriate Froude number based on results of lock-exchange experiments in which extended inertial gravity currents are generated in a rectangular cross-section channel. We use 'tophat' vertical density profiles of the currents to obtain an 'equivalent' depth, defined by profiles having the same buoyancy at every position as the real profiles. As in previous work, our experimental results show that in the initial constant-velocity phase the Froude number can be defined in terms of the lock depth. However, as the current enters the similarity phase when the initial release conditions are no longer relevant, we find that the Froude number is more appropriately defined in terms of the maximum height of the head. Strictly speaking, the self-similar solution to the shallow-water equations requires a front condition that uses the height at the rear of the head. We find that this rear Froude number is not constant and is a function of the head Reynolds number over the range 400-4500.

Results of laboratory experiments are presented in which a fixed volume of homogeneous fluid is suddenly released into another fluid of slightly lower density, over a horizontal thin metallic grid placed a given distance above the solid bottom of a rectangular-cross-section channel. Dense liquid develops as a gravity current over the grid at the same time as it partially flows downwards. The results show that the gravity current loses mass at an exponential rate through the porous substrate with a time constant τ; the front velocity and the head of the current also decrease exponentially. The loss of mass dominates the flow and, in contrast to gravity currents running over solid bottoms, no self-similar inertial regime seems to be developed. A simple model is introduced to explain the scaling law of the loss of mass and the evolution of the front position. The flow evolution depends on the characteristic time of the initial (slumping) phase and the time constant τ, related to the initial conditions and the permeability of the porous substrate, respectively. Qualitative comparisons with other gravity currents with loss of mass, such as particle-driven gravity currents, are provided.

This work concerns the spreading of viscous droplets on a smooth rigid horizontal surface, under the condition of complete wetting (spreading parameter S > 0) with the Laplace pressure as the dominant force. Owing to the self-similar character foreseeable for this flow, a self-similar solution is built up by numerical integration from the center of symmetry to the front position to be determined, defined as the point where the free-surface slope becomes zero. Mass and energy conservation are invoked as the only further conditions to determine the flow. The resulting fluid thickness at the front is a small but finite (Z lo-') fraction of the height at the center. By comparison with experimental results the regime is determined in which the spreading can be described by this solution with good accuracy. Moreover, even within this regime, small but systematic deviations from the predictions of the theory were observed, showing the need to add terms modifying the Laplace pressure force.

We study the variation of the Froude number at the front of gravity currents developed in uniform channels whose cross-section shape depends on a parameter usually used in many numerical and theoretical models. The thickness and front velocity of the dense currents running on the bottom are greater for all the cases studied, resulting in a Froude number greater than that corresponding to the rectangular cross-section shape. The light currents developing along the upper boundary show the opposite trend. It is found that the results are not related to the depth and width of the channel. The relationships obtained agree with the results of laboratory experiments in which open and closed channels of different cross-section shapes are used.

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We study the filling of a dry region ͑cavity͒ within a viscous liquid layer on a horizontal plane. In our experiments the cavities are created by removable dams of various shapes surrounded by a silicon oil, and we measure the evolution of the cavity's boundaries after removal of the dams. Experimental runs with circular, equilateral triangular, and square dams result in circular collapse of the cavities. However, dams whose shapes lack these discrete rotational symmetries, for example, ellipses, rectangles, or isosceles triangles, do not lead to circular collapses. Instead, we find that near collapse the cavities have elongated oval shapes. The axes of these ovals shrink according to different power laws, so that while the cavity collapses to a point, the aspect ratio is increasing. The experimental setup is modeled within the lubrication approximation. As long as capillarity is negligible, the evolution of the fluid height is governed by a nonlinear diffusion equation. Numerical simulations of the experiments in this approximation show good agreement up to the time where the cavity is so small that surface tension can no longer be ignored. Nevertheless, the noncircular shape of the collapsing cavity cannot be due to surface tension which would tend to round the contours. These results are supplemented by numerical simulations of the evolution of contours which are initially circles distorted by small sinusoidal perturbations with wave numbers kу2. These nonlinear stability calculations show that the circle is unstable in the presence of the mode kϭ2 and stable in its absence. The same conclusion is obtained from the linearized stability analysis of the front for the known self-similar solution for a circular cavity.

We study the instantaneous Stokes flow near the apex of a corner of angle 2␣ formed by two plane stress free surfaces. The fluid is under the action of gravity with g ជ parallel to the bisecting plane, and surface tension is neglected. For 2␣Ͼ126.28°, the dominant term of the solution as the distance r to the apex tends to zero does not depend on gravity and has the character of a self-similar solution of the second kind; the exponent of r cannot be obtained on dimensional grounds and the scale of the coefficient depends on the far flow field. Within this angular domain, the instantaneous flow is deeply related to the ͑steady͒ flow in a rigid corner known since Moffatt ͓J. Fluid Mech. 18, 1 ͑1964͔͒ and, as in that case, there may be eddies in the flow. The situation is substantially different for 2␣Ͻ126.28°, as the dominant term is related to gravity and not to the far flow. It has the character of a self-similar solution of the first kind, with the exponent of r being given by dimensional analysis. The solution cannot be continued in time since it leads to the curling of the boundaries. Nevertheless, it provides information on how such a cornered contour may evolve. When 2␣Ͻ180°, the corner angle does not vary as the flow develops; on the other hand, if 2␣Ͼ180°the corner must round or tend to a narrow cusp, depending on the far flow. These predictions are supported by simple experiments.