Elementary Classical Hydrodynamics by B. H. Chirgwin

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.