By B. H. Chirgwin
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Eii (dvi + dv2 , dv3\ = (^ 8^ + A. 8^) = dlv v - (~ 0 . (2 9) - This shows that div v is a measure of the rate at which fluid diverges from a point. e. div v = 0. 2 41 SOME GENERAL THEOREMS In this case co^ is anti-symmetric, in fact (0 \-C2 so that 2(w2)l = C2J3-C3J2, -Cs Ca\ Ci 0/ 2(«2>2 = C3^1~sl>;3? 2(W2)3 = Ciy2~t2yi or 112= K x y . 10) Hence this contribution to the velocity is the same as that of a rigid body rotating about A with angular velocity \ £. ) If we write down the expressions for £1, £2, £3, we obtain c.
MISCELLANEOUS EXERCISES II 1. Incompressible ideal fluid of density Q fills the space between the con centric cylinders r = a and r = b where r2 = x2-\-y2. The velocity of the fluid is vz = -yf(r), vy = xf(r), v2 = 0. Find the pressure throughout the fluid and explain why for general f(r) Bernoulli's theorem (p/Q + v2/2 = constant) does not hold. Prove that if fif) — constant, thenplQ + v2/2 = constant, and state what is special about the motion corresponding to this form of/(r). r
FIG. 15 FIG. 16 Before investigating this apparent contradiction of the uni queness theorem we enumerate some relevant facts concerning multiply connected regions. The volume occupied by the fluid in Fig. 15 is doubly connected, (topologically) equivalent to an anchor-ring (torus); in Fig. 16 we show a triply connected area. ) Any path joining A and B in Fig. 16 can be continuously de formed without leaving the area into one of ACB, ADB or AEB, or a combination of them, but these three paths cannot be deformed into one another.