Box Model for Confined Power-Law Viscous Gravity Currents Including Surface Tension Effects

We consider the flow of a viscous fluid (power-law non-Newtonian) injected into a gap of height H between two horizontal plates. When the viscosity of the ambient (displaced) fluid is negligible, the injected fluid forms a tail-slug in contact with both plates connected (at a moving grounding line) to a leading gravity current (GC) whose interface does not touch the top of the gap. Surface tension menisci may appear at the grounding line and nose of the GC. Such systems, of interest in the injection molding industry, have been investigated recently in the framework of the lubrication theory (Hutchinson et al. 2023, Hutchinson 2024, Ungarish 2025) for the volume V = qtα (q and α are positive constants and t is time). Similarity appears for certain values of α. The similarity solution of the lubrication model requires manipulations and numerical calculations which obscure the underlying mechanisms and defy reliable interpretation, because the flow is dependent on four coupled parameters: viscosity exponent n, and J, σ, σN (the height ratio of the unconfined GC, grounding line meniscus and nose meniscus to H, respectively). Here we present a significantly simpler box-model analysis, which provides straightforward insights and facilitates the quantitative predictions. Comparisons with the rigorous lubrication-model solution and with previously-published data demonstrate that the box model provides a reliable physical description of the system, and a fairly accurate prediction of the propagation, for a wide range of parameters.