(14)
28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.—A stream-function ψ must be determined to satisfy the conditions
∇2ψ = 0, throughout the liquid;
(1)
ψ = constant, over any fixed boundary;
(2)
dψ/ds = normal velocity reversed over a solid boundary,
(3)
so that, if the solid is moving with velocity U in the direction Ox, dψ/ds = −U dy/ds, or ψ + Uy = constant over the moving cylinder; and ψ + Uy = ψ′ is the stream function of the relative motion of the liquid past the cylinder, and similarly ψ − Vx for the component velocity V along Oy; and generally
ψ′ = ψ + Uy − Vx