Generating a forced vortex
Suppose
that the open-topped tank described above were to be brought up to speed very
quickly. Shearing of the water would take place at the sides and over the
bottom. The water has viscosity and this shearing would cause a shearing force
to act on the water and this force would progressively make the water rotate,
first slowly and then faster and eventually all the water would rotate like a
wheel. The viscous shearing force is acting on a mass of fluid to make it
accelerate. The time taken would depend on the coefficient of viscosity for the
water and on the density of water. If the liquid in the tank were to be changed
to, say, a lubricating oil with viscosity perhaps 2000 times that of water or
to mercury with a high density and low viscosity the process of coming to
equilibrium would be very different.
I thought that I might make a very simple rig to find out what does happen. On the face of it, all that is needed is a cylindrical drum with its axis vertical, partly filled with liquid, and rotated about its vertical axis. I made a very simple rig from odds and ends in my workshop. It is shown in figure 15-8. I had a nicely-made, plastic dome that came with a free, crudely-made clock. It was a short job to make a wooden fitting to attach the dome to my wheel brace and mount the wheel brace in the vice. I found that I needed the white plastic disk to reduce coloured reflections from the red bench-top. The dome turns 3.75 times for each turn of the handle and a typical speed of rotation of the dome is 240 rpm. It is easy to turn the handle at these speeds and there was a need to limit the speed to stop the fluid overtopping the dome. As it is turned by hand there will inevitably be fluctuations in speed.
[I think that it turned out to be more interesting that I expected. I started by wanting to take some pictures of the free surface of a forced vortex but ended up with a device that is very suited to the illustration of the concept of Reynolds number. In the end I tried this little rig with water, lubricating oil for internal combustion engines, with steam oil for steam engines and with a small quantity of mercury.]
At
this point I will just illustrate the shape of the free surface of water
rotating as a forced vortex. I used water and I found that the rig was very
sensitive to having the axis of rotation vertical. The surface became lopsided
very easily. There were waves on the surface and I thought that I might get rid
of these by adding some detergent to the water to reduce surface tension. This
worked and led to the pictures in figures 15-7a to e. But they are not pictures
for untreated water.
It is inevitable that there will be refraction of light in passing through the plastic dome and through the interface between the plastic and the water and this distorts the shape of the surface as seen by the camera. The black “lozenge” is a reflection of the black circle of adhesive foam used to attach the dome to the fitting in the chuck.
Figures 15-7a to d show the shapes at progressively increasing speed. Figures 15-7d and e show the way in which the surface can become lop-sided very easily.
The rig gave me an opportunity to watch the water as the water accelerated to its final state of equilibrium. The water quickly rose up the sides with a fairly flat centre but the rotation spread towards the centre and in less than half a minute the shape was as steady as it ever would be with a hand turned device. When turning was just stopped the water went on rotating and the surface quickly became flat with a rotating pattern of small surface waves like the arms of a spiral galaxy. These died away but the rotation continued until it, too, died away.
One might argue for having a mechanical drive to make the water rotate. One might fit radial blades inside the tank so that the volume is divided into several compartments. This would speed up the process of coming to a steady state but I am not sure whether,
In that case, the original flow pattern of concentric circles is still appropriate.