A Model for Polygonal Hydraulic Jumps

We present a model for the shape of polygonal hydraulic jumps discovered by Ellegaard et al. 1998 and modeled there by a force balance (an inward push of gravity and an outward pull of viscous stresses) on the ``surface roller" in the jump-region, which is known to be a prerequisite for the occurrence of polygonal jumps. We develop this model, replacing their unexplained ``line tension" by a more detailed modeling of the flow in the roller, including the tangential flow. We solve a simplified model exactly in which each polygon exists in a finite region of parameter space with shapes very similar to those observed in experiments. The number of corners $N$ in the polygon scale as $N \sim (Q \nu)^2 h_o^{-4} h_i^{-3}$, in terms of the volumetric flux $Q$, the inner height $h_i$, the outer height $h_o$ and the viscosity $\nu$. In contrast to recent work by Bush et al. 2006, our model does not include surface tension explicitly, but since it affects the radius of the jump it will also affect $h_i$. A reduction of the surface tension will reduce $h_i$ and therefore increase $N$. If the reduction is strong enough, this could restore the circular symmetry.