Improved box models for Newtonian and power-law viscous gravity currents

Consider the flow of gravity currents of Newtonian and power-law non-Newtonian viscous fluid, injected over a horizontal boundary in rectangular and cylindrical (axisymmetric) systems. We present some novel box-model (BM) predictions. The previously published theoretical studies are concerned with a power-law volume V = qt^α (influx rate Θ = αqt^α−1 ) where q > 0 and α ≥ 0 are constants and t is time. The lubrication simplification equations predict a self-similar flow: the propagation is K_Lt^β , and the height (thickness) profile is determined by a second-order ODE in the reduced length ξ ∈ [0, 1]. The predicted β and K_L are in good agreement with laboratory data. Previous studies reported that a basic BM predicts K₁t^β propagation with the same β as the lubrication model, but the discrepancy between K₁ and K_L is in general not small. We point out two inconsistencies of the basic BM with the physical system, and present an improved BM prediction, K₂t^β . We show that K₂ is in general more accurate than K₁ (including in comparison with experimental data). Next, we show that the BM provides an effective simple predictor also for a general influx Θ(t) (not a power law), while for such systems the lubrication theory lacks analytical reduction and requires numerical solution of a non-linear PDE (in time and length).