In 29.4.1 it is asserted that eq. (29.28) describes the steady, circulation-free flow of an incompressible ideal

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In §29.4.1 it is asserted that eq. (29.28)

R- (& – 2(fcdotf)f) va = vooê + vo m(tcme (1-) R- - ê sine (1+ r2 = vo (î cos 0 ( 1 -2


describes the steady, circulation-free flow of an incompressible ideal fluid around a cylinder. To confirm this, verify 

(a) that va → v x̂ as r → ∞; 

(b) that no fluid flows into or out of the cylinder; 

(c) that the circulation is zero around any circle of radius r > R; 

(d) that ∇ × va = 0 (zero vorticity) and ∇ · v = 0 (incompressibility) everywhere outside the cylinder. For part (d)  va can be written as the gradient of a scalar function, va = ∇Φ, where 

D = v„ê · r + vR²*,


Also make use of the identities eq. (B.18)
 


and eq. (B.20)


, and the facts that ∇(1/rp) = −pr̂/rp+1 and ∇2(1/rp) = p2/rp+2. The identity ∇2( f g) = 2g + g2 f + 2∇f · ∇g may be useful as well.

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The Physics of Energy

ISBN: 978-1107016651

1st edition

Authors: Robert L. Jaffe, Washington Taylor

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