A mathematical framework for the analysis of particle-driven gravity currents

Recent studies have modelled the flow of particle-driven gravity currents over horizontal boundaries using either shallow-water equations or simple 'box' (integral) models. The shallow-water equations are typically integrated numerically, whereas box models admit analytical solutions. However, the theoretical validity of the latter models has not been fully established. In this paper a novel mathematical technique is developed which permits the derivation of analytical solutions to the shallow-water model for gravity current motion. These solutions, confirmed by comparison with the results of numerical integration, are in good agreement with experimental observations. They also indicate why the simplified box models have been so successful. Moreover, they reveal how the internal dynamics of particle-driven flows are different from gravity currents arising solely due to compositional density differences. While compositionally driven gravity currents, which have a fixed density difference between the intruding and ambient fluids, may be modelled using similarity solutions to the governing equations, particle-driven gravity currents do not possess such solutions because their density is progressively reduced by particle sedimentation. Instead the new analysis determines how their behaviour progressively diverges from the similarity solution. By a change of independent variables, it is possible to develop convergent series expansions for each of the dependent variables which characterize the motion. It is suggested that this approach may find application to a number of other problems in which the dynamics are initially governed by a simple dynamical balance which is progressively lost as extra physical effects begin to influence the motion.