1916]
ON THE DYNAMICS OF REVOLVING FLUIDS
451
for purely theoretical purposes there is no inconsistency in supposing the (u, w) motion resisted .while the v motion is unresisted.
The next supposition to u — 0, w = 0 in order of simplicity is that u is a function of r and t only, and that w = 0, or at most a finite constant. It follows from (8) that P is independent of z, while (6) becomes
_ -dt dr r dr
determining the pressure. In the case of an incompressible fluid u as a function of r is determined by the equation of continuity ur = C, where G is a function of t only ; and when u and the initial circumstances are known, v follows. As the motion is now two-dimensional, it may conveniently be expressed by means of the vorticity £, which moves with the fluid, arid the stream-function -^r, connected with £ by the equation
H
rdr
The solution, appropriate to our purpose, is
B0, .................. (15)
where A and B are arbitrary constants of integration. Accordingly
ety B dty 2 L A nr.
^ = ---33= -» v = -/- — ~ &'dr + — ............. (16)
rd6 r dr r J r
In general, A and B are functions of the time, and £ is a function of the time as well as of r.
A simple particular case is when % is initially, and therefore, permanently, uniform throughout the fluid. Then
*> = £r
Let us further suppose that initially the motion is one of pure rotation, as of a solid body, so that initially A = 0, and that then the outer wall closes in. If the outer radius be initially R0 and at time t equal to R, then at time t
J. = tW-#2)> .......................... .(18)
since vr remains unchanged for a given ring of the fluid ; and correspondingly,
«-£{r + CR.»-#')r-1} ......................... (19)
Thus, in addition to the motion as of a solid body, the fluid acquires that of a simple vortex of intensity increasing as R diminishes.
* It may be remarked that (17) is still applicable under appropriate boundary conditions even when the fluid is viscous.
29—2uid ultimately settles down into stable equilibrium, it may be after many oscillations. In the present problem ordinary viscosity does not meet the requirements, as it would interfere with the constancy of the circulation of given rings of fluid on which our reasoning depends. But