Over a concentric cylinder, external or internal, of radius r = b,
| ψ′ = ψ + U1y = [ U ( 1 − | a2 | ) + U1] y, |
| b2 |
(4)
and ψ′ is zero if
U1/U = (a2 − b2)/b2;
(5)
so that the cylinder may swim for an instant in the liquid without distortion, with this velocity U1, and ω in (1) will give the liquid motion in the interspace between the fixed cylinder r = a and the concentric cylinder r = b, moving with velocity U1.
When b = 0, U1 = ∞; and when b = ∞, U1 = −U, so that at infinity the liquid is streaming in the direction xO with velocity U.
If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by ω = −Uz, we are left with
ω = Ua2/z,