Depth-averaged model for hydraulic jumps on an inclined plate

We examine the dynamics of a layer of viscous liquid on an inclined plate. If the layer's upstream depth h(-) exceeds the downstream depth h(+), a smooth hydraulic jump (bore) forms and starts propagating down the slope. If the ratio η = h(+)/h(-) is sufficiently small and/or the plate's inclination angle is sufficiently large, the bore overturns and no smooth steadily propagating solution exists in this case. In this work, the dynamics of bores is examined using a heuristic depth-averaged model where the vertical structure of the flow is approximated by a polynomial. It turns out that even the simplest version of the model (based on the parabolic approximation) is remarkably accurate, producing results which agree, both qualitatively and quantitatively, with those obtained through the Stokes equations. Furthermore, the depth-averaged model allows one to derive a sufficient criterion of bore overturning, which happens to be valid for the exact model as well. Physically, this criterion reflects the fact that, for small η, a stagnation point appears in the flow, causing wave overturning.