The stability of swirling flow

Suppose that somehow we could have an extensive mass of fluid that is rotating steadily about a vertical axis. We could draw a flow pattern for some plane horizontal surface in the flow. It would be just a set of concentric flow lines such as those in figure 15-1. I have spaced these flow lines at equal increments of radius but, to fit in with the common flow patterns, the flow lines might better be pitched so that the lines have equal flows between any two adjacent lines although it is difficult to see how this might be achieved.

 

This is a flow pattern for steady flow but it is not like other flow patterns. It is for a flow where the properties of the fluid, eg pressure, speed, density and so on do not change along a given flow line. This immediately rules out the use of the steady flow energy equation because, even though the properties are constant in time and the flow is steady, the very fact that the properties do not change along the flow line makes it pointless to even attempt to use the steady flow energy equation. It means that, as the properties do not vary along the flow lines, its only value to be derived from considering this model comes from the fact that the properties vary across the flow lines, that is, with radius.

 

For this flow pattern to exist the flow must in some way be stable and we can look to see what the conditions are for this to be the case.

 

Stability comes from there being a balance between the pressure forces and inertia forces that are applied to the fluid that is flowing in circles. I have a flow pattern in which gravitation does not affect the flow but it has no thickness. If we are to consider inertia forces we must have mass somewhere. With this in mind we have to consider not a line but a ring like the one shown in figure 15-2. The ring shown is really just one of many rings that make up a thin disk of fluid.The disk has a small thickness  and the ring has a radial thickness . As the outcome of this analysis will be a differential equation these dimensions are really infinitesimals that I have had to exaggerate in order to make the drawing clear. So the drawing is of a ring of very small cross section that is rotating at uniform speed. There is nothing in this ring of fluid to hold it together other than the forces exerted by the other rings that surround this elemental ring. These act through pressures and ultimately these pressures acting on the ring must create the inward radial forces that gives the ring its centripetal acceleration.

 

We can look at the forces on a small element of the ring contained between two radial surfaces subtending a small angle of  at the centre and shown in figure 15-3. Having let the ring of fluid rotate in a horizontal plane the small element of this elemental ring is in equilibrium under pressure forces acting on the four vertical surfaces, two radial and two tangential, and the inertial force on the mass of fluid in the element.

 

I shall suppose that the tangential force on one side of the element is equal to the area of the element multiplied by the mean pressure on it and that the resultant of the two forces is equal to one force multiplied by . Then the inward radial force on the element is given by :-

              

 

The mass of the element is  where  is the density of the fluid. The centripetal acceleration is  where  is the tangential velocity of the fluid in the ring.

 

It follows that :-

 

This can be simplified if it is expanded and second order quantities are ignored and reduced still more to :-

 

                                  and finally to

 

This is the fundamental expression for swirling flow to be stable. It is is how the tangential velocity varies with radius that determines whether the vortex is “forced” or “free”. I will start with the forced vortex.