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 $\mathcal{V}=q{t}^{\alpha }$ (influx rate $\Theta =\alpha q{t}^{\alpha -1}$) where $q>0$ and $\alpha \ge 0$ are constants and t is time. The lubrication simplification equations predict a self-similar flow: the propagation is $K_L{t}^{\beta }$, and the height (thickness) profile is determined by a second-order ODE in the reduced length $\xi \in [0,1]$. The predicted $\beta$ and $K_L$ are in good agreement with laboratory data. Previous studies reported that a basic BM predicts $K_1{t}^{\beta }$ propagation with the same $\beta$ as the lubrication model, but the discrepancy between $K_1$ and $K_L$ 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, $K_2{t}^{\beta }$. We show that $K_2$ is in general more accurate than $K_1$ (including in comparison with experimental data). Next, we consider a general influx $\Theta \left(t\right)$ (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).