Going down the plughole

By Marcus Wilson 04/07/2014 8

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.


8 Responses to “Going down the plughole”

  • Great post yes vortices are pretty interesting ! Pressure differentials, the weather too. is an interesting system that uses pressure differentials.The energy created by a vortex can be quite phenomenal.
    Like the plug hole and all the water disappearing, so fast. The torque these things can create.!

  • My question is what is the study of Vortices called ? and where would someone go to learn this?
    When Wikked this comes up “Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass. In a stationary vortex, the streamlines and pathlines are closed. In a moving or evolving vortex the streamlines and pathlines are usually spirals.”
    They do have some interesting properties for example, oxygen can be extracted from water with help from a vortex as with that energy state the oxygen bubbles out of the water not too sure exactly how that happens. But in Nature itself fish do this every day with their gills. From Wallace and Grommit ..

  • Your comment that glass is a fluid has now been discredited. Glass is an amorphous solid. Old windows are thicker at the bottom because they were made that way.

  • Funny a real aspect of nature that needs more study as vortices are everywhere.
    But no one can tell me where to study this.
    Is this something being ignored in modern science ?

  • “They do have some interesting properties for example, oxygen can be extracted from water with help from a vortex as with that energy state the oxygen bubbles out of the water not too sure exactly how that happens. But in Nature itself fish do this every day with their gills.”

    Yes, fish extract oxygen from the water with their gills, but no, they don’t do it by means of vortices. The oxygen moves across gill & blood vessel membranes down a concentration gradient (maintained by the fact that blood & water are flowing in opposite directions) from water to blood. It definitely doesn’t come out in the form of bubbles.

  • So how does a concentration gradient extract the oxygen form water ? This doesn’t tell me how the O2 comes out of water. The video does,
    Did you watch the video Alison ? Do you understand what is happening in the video ? Apparently fish do the same thing but on a smaller scale.

  • “Oxygen and carbon dioxide diffuse from areas of high partial pressure to an area with lower partial pressure, so difference in pressure creates a concentration gradient ”
    High pressure to low pressure sounds a lot like a vortex and the fact that the system is never in equilibrium sounds like a sustained vortex.

  • movement down a concentration gradient is simply movement from high to low concentration of whatever is moving (in our bodies that is often across a semipermeable membrane. That doesn’t pre-suppose a vortex.