# Liquid with corners

A new model explains why a fluid spreading out over a surface can form a polygonal hydraulic jump

April 05, 2012
Complex phenomena can also be observed in the kitchen sink: When a jet of water hits the bottom of a kitchen sink, the water first quickly flows away from the point of impact. At a certain distance, however, the water level rises suddenly and the flow speed decreases abruptly. This so-called hydraulic jump assumes the form of a perfect circle when the bottom of the sink is horizontal and even. A few years ago, researchers discovered that under certain circumstances, complex shapes such as for example triangles, pentagons or octagons, can also emerge from the circular jump. Now scientists at the Max Planck Institute for Dynamics and Self-Organization, together with colleagues from Denmark and Japan, have developed a theoretical model which describes the fundamental mechanisms of this surprising flow phenomenon.

#### Polygons are caused by effects of surface tension

If the liquid level is increased even further, above a certain value, the fluid begins to flow in reverse direction above the vortex. The fluid sloshes inwards, and as a consequence, a second annular vortex is generated but it rotates in the opposite direction compared to the first vortex below. It is in this situation, where the second vortex has appeared, that the jump can become polygonal.

In early approaches to explaining the phenomenon, researchers attempted to account for the formation of the polygonal hydraulic jumps as the result of an interplay between internal friction, i.e. viscosity, and gravitational force of the water towards the inside, i.e. hydraulic pressure, acting on the secondary vortex. However, these forces alone could not explain why polygons are formed. A further piece of the puzzle was provided by the effect of surface tension, i.e. by taking into account the binding forces between the molecules on the fluid surface. The scientists discovered in particular that the mechanism which is responsible for the polygonal form is similar to a phenomenon which is known as the Rayleigh-Plateau instability.

This instability is an everyday phenomenon: when a jet of water flows from a tap, provided it is long enough, the jet breaks up into a chain of drops at a certain distance from the tap. The chain is formed because perturbations, which are not initially visible in the top part of the water jet, are amplified further down the jet due to surface tension effects. This amplification ultimately leads to the breaking up of the jet and thus to the formation of water drops. To explain the formation of the polygons, the researchers consider a similar scenario in their model: small perturbations of the second circular vortex near the surface can be amplified by surface tension, which in the end leads to the formation of the corners.

#### Pressure, viscosity and surface tension together provide the explanation

The new model developed by the team around Erik A. Martens at the Max Planck Institute for Dynamics and Self-Organization combines the aspects of earlier model approaches: hydraulic pressure, viscosity and surface tension. “We can now describe the basic mechanisms that lead to the instability of the circular hydraulic jump and the formation of a polygon,” says Erik A. Martens.

The flow profile in a polygonal hydraulic jump is much more complicated than in a circular jump. Visualizations of the flow structures (see Figure 3) reveal that very strong flow jets emerge from the polygon corners – it looks almost as if the second vortex has broken apart. This is where the researchers come up against one of the limits of their modeling approach. “In order to develop better models for the complicated polygonal flows, we would need more experimental knowledge on the velocity and height profiles of the polygonal patterns,” explains Erik A. Martens. “It is indeed one of the objectives of our work to further stimulate the research in this field.” The researchers envisage that more detailed studies of these flow processes could possibly lead to the observation of novel and unexpected phenomena.

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