Going down the plughole

Being a father of an active, outdoor-loving two-year-old, I am well acquainted with the bath. Almost every night: fill with suitable volume of warm water, check water temperature, place two-year-old in it, retreat to safe distance. He's not the only thing that ends up wet as he carries out various vigorous experiments with fluid flow. 

One that he's just caught on to is how the water spirals down the plug-hole. Often the bath is full of little plastic fish (from a magnetic fishing game), and if one of these gets near the plug hole it gets a life of its own. It typically ends up nose-down over the hole, spinning at a great rate as it gets driven round by the exiting water. 

The physics of rotating water is a little tricky. There are two key concepts thrown in together; first the idea of circular motion  – which involves a rotating piece of water having a force on it towards the centre (centripetal force); second is viscosity – in which a piece of water can have a shear force on it due to a velocity gradient in the water. Although viscosity has quite a technical definition, colloquially, one might think of it as 'gloopiness' [Treacle is more viscous than water. The ultimate in viscosity is glass, which is actually a fluid, not a solid – the windows of very old buildings are thicker at the bottom than the top due to the fluid flow over tens or hundreds of years.] In rotational motion there's a subtle interplay between these two forces which can result in the characteristic water-down-plughole motion. 

In terms of mathematics, we can construct some equations to describe what is going on and solve them. We find, for a sample of rotating fluid, that two steady solutions are possible. 

The first solution is what you'd get if all the fluid rotated at the same angular rate – the velocity of the fluid is proportional to the radius. This is what you'd get if you put a cup of water on a turntable and rotated it – all the water rotates as if it were a solid.

The second solution has the velocity inversely proportional to the radius – so the closer the fluid is to the centre, the faster it is moving. This is like the plughole situation where a long way from the plug hole the fluid circulates slowly, but close in it rotates very quickly. Coupled with this is a steep pressure gradient – low pressure on the inside (because the water is disappearing down the hole) but higher pressure out away from the hole. Obviously this solution can't apply arbitrarily close to the rotation axis because then the velocity would be infinite. This is where vortices often occur. (Actually, Wikipedia has a nice entry and animations on this, showing the two forms of flow I've described above). 

A Couette viscometer expoits these effects as a way of measuring the viscosity of a fluid. Two coaxial cylinders are used, with fluid between them. The outer is rotated while the inner one is kept stationary, and the torque required enables us to calculate the viscosity of the liquid.


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