Model for polygonal hydraulic jumps

We propose a phenomenological model for the polygonal hydraulic jumps discovered by Ellegaard and co-workers [Nature (London) 392, 767 (1998); Nonlinearity 12, 1 (1999); Physica B 228, 1 (1996)], based on the known flow structure for the type-II hydraulic jumps with a "roller" (separation eddy) near the free surface in the jump region.

The model consists of mass conservation and radial force balance between hydrostatic pressure and viscous stresses on the roller surface. In addition, we consider the azimuthal force balance, primarily between pressure and viscosity, but also including nonhydrostatic pressure contributions from surface tension in light of recent observations by Bush and co-workers [J.

Fluid Mech. 558, 33 (2006); Phys. Fluids 16, S4 (2004)]. The model can be analyzed by linearization around the circular state, resulting in a parameter relationship for nearly circular polygonal states. A truncated but fully nonlinear version of the model can be solved analytically. This simpler model gives rise to polygonal shapes that are very similar to those observed in experiments, even though surface tension is neglected, and the condition for the existence of a polygon with N corners depends only on a single dimensionless number phi.

Finally, we include time-dependent terms in the model and study linear stability of the circular state. Instability occurs for sufficiently small Bond number and the most unstable wavelength is expected to be roughly proportional to the width of the roller as in the Rayleigh-Plateau instability.