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

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

and eq. (B.20)

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

Transcribed Image Text:

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