Real swirling flow.

In this chapter I have set out the mechanics of swirling flow for special cases. In doing so I have deliberately ignored some of the inconvenient features of this type of flow. I rather glibly ignored losses when I applied the energy equation. I now want to examine that decision. It implies that the fluid had no viscosity and that is difficult to handle because if a fluid had no viscosity it would be impossible to make a small quantity of fluid rotate if it were to be at rest and impossible to stop rotation if rotation could be established. This follows from the fact that there is no way to apply a tangential force to the small quantity of fluid in the absence of viscosity. We can see what this means from figure 15-12.

 

Here I have drawn the plan view of a disk mounted on a vertical axis so that it can rotate in the horizontal plane. At some point on the disk a small wheel is mounted on a short axle so that the wheel can rotate freely. There is nothing notional about this mechanism, it could be made quite easily. Suppose that the disk is made to rotate at a slow steady speed. The arrowhead drawn on the disk would appear to turn once per revolution of the disk in the clockwise direction. I have marked the small wheel with a black dot and, if there were to be no friction between the small wheel and the axle on which it is mounted, the small wheel would not rotate and, in my diagram, the dot will always be at the lowest point on the small wheel. The wheel appears to be rotating anti-clockwise relative to the disk at the speed of rotation of the disk. This, for obvious reasons, is called counter-rotation. In reality there will be some friction and the small wheel will gradually speed up until it has no rotation relative to the disk. Then the disk and the wheel would appear to rotate as a solid.

 

This has obvious consequences for our notional rotating fluid with the flow pattern in figure 15-4 that we have called a vortex. Every molecule of the fluid will tend to counter-rotate and this may lead to groups of molecules counter-rotating[1]. It is effectively impossible to allow for this complication. All that the engineer can do is ignore counter rotation and then try to take the effect of viscosity into account at a later stage[2]. On top of this, in real fluids there will be a shearing process going on at every radius and we cannot deal with this either so, in order to get some idea of what happens, we ignore viscosity as well. This puts us in the contradictory position of choosing to ignore viscosity knowing that the flow is only possible if there is internal friction. That decision does lead to an insight to the behaviour of fluids when they rotate but leaves us with the need to consider the affect of viscosity.