In 29.4.1 it is asserted that eq. (29.28) describes the steady, circulation-free flow of an incompressible ideal
Question:
In §29.4.1 it is asserted that eq. (29.28)
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
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) = f ∇2g + g∇2 f + 2∇f · ∇g may be useful as well.
R- (& – 2(fcdotf)f) va = vooê + vo m(tcme (1-) R- - ê sine (1+ r2 = vo (î cos 0 ( 1 -2 D = v„ê · r + vR²*," r2
Step by Step Answer:
a Since the second term in the first form of eq 292...View the full answer
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