Improved Box Models for Newtonian and Power-Law Viscous Gravity Currents in Rectangular and Axisymmetric Geometries

We consider the flow of gravity currents of Newtonian and power-law non-Newtonian viscous fluids, injected over a horizontal boundary in rectangular and cylindrical (axisymmetric) systems. We focus on some novel box model (BM) predictions. Previously published theoretical studies consider a power-law volume V=qtα (influx rate Θ=αqtα−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 KLtβ, and the height (thickness) profile is determined by a second-order ODE in the reduced length ξ∈[0,1]. The predicted β and KL are in good agreement with laboratory data. Previous studies reported that a basic BM predicts K1tβ propagation with the same β as the lubrication model, but the discrepancy between K1 and KL is in general not small. This work points out two inconsistencies of the basic BM with the physical system and presents an improved, more consistent, BM prediction, K2tβ. We show that K2 is in general more accurate than K1 (including in comparison with experimental data). Next, we consider a general influx Θ(t) (not a power law). We demonstrate that the BM provides a simple and flexible framework of initial-value time-dependent ODEs, though for such systems the lubrication theory lacks analytical reduction and requires numerical solution of a non-linear PDE (in time and length).