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45 APPENDIX II 
1. FUNDAMENTAL EQUATIONS 
APPENDIX II 
RADIAL NON-COMPRESSIVE FLOW ABOUT A CENTRAL CAVITY 
I, FUNDAMENTAL EQUATIONS 
When incompressible homogeneous liquid flows with radial symmetry about a 
point 0, its velocity u outward from O can be written 
u=% (1) 
where r denotes radial distance from 0, and u, the velocity at r = 1, which may be a 
function of the time. Suppose the space within a sphere of radius r, about 0 is free 
from liquid; it may be empty or it may contain gas. Let uw, denote the value of u at 
r=yr,. Then u, =u,/7,* and we can also write 
2 
u = Uy (72) {2] 
Because of this simple distribution of the velocity, it is possible to in- 
tegrate the equation of motion of the liquid, 
ee (3) 
ot or s Or 
where p is pressure, ¢t is time, p is density. From [1] 
a Oe Be 
Ho ee [4] 
Hence we can write [3] in the form 
re dt ""* 2 dr p Or 
Taking [e dr of each term in es a take and noting that 
i) [Bar =p eats) 
where p, is the pressure at Laide and p that at distance 7, we obtain 
ed ee 
SS ig 
1 = 
+ p (2) = PP) 
p= p(t fu, — Fw) + py [5] 
This equation can be written in two other useful forms by using [4] or by writing, 
from [1], w,, = 7, °u,: 
p= p (rot — dw) + py [6] 
p=p[iz (r2u,) — 5 w] + po (7] 
Equation [6] expresses the pressure pat any point in terms of the velocity 
near that point. Equation [7] connects p with conditions at the cavity. Another ex- 
pression for p, containing the pressure p, at the cavity, is obtained if we write down 
[5] for r=r,, namely, 
= (ae as 2 
B= Py de 2M) + Po 
