dv du d²[psi] d²[psi]
2[zeta] = -- - -- = ------ + ------ = -[Nabla]²[psi], suppose, (3)
dx dy dx² dy²
so that in steady motion
dH d[psi] dH d[psi] dH
-- + [Nabla]²[psi]------ = 0, -- + [Nabla]²[psi]------ = 0, ------ + [Nabla]²[psi] = 0, (4)
dx dx dy dy d[psi]
and [Nabla]²[psi] must be a function of [psi].
If the motion ia irrotational,
d[phi] d[psi] d[phi] d[psi]
u = - ------ = - ------, v = - ----- = ------, (5)
dx dy dy dx´
so that [psi] and [phi] are conjugate functions of x and y,
[phi] + [psi]i = [f](x + yi), [Nabla]²[psi] = 0, [Nabla]²[phi] = 0; (6)
or putting
[phi] + [psi]i = w, x + yi = z, w = [f](z).
The curves [phi] = constant and [psi] = constant form an orthogonal
system; and the interchange of [phi] and [psi] will give a new state
of uniplanar motion, in which the velocity at every point is turned
through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards O, the flow
across any circle of radius r being the same and denoted by 2[pi]m,
the velocity must be m/r, and
[phi] = m log r, [psi] = m[theta],
[phi] + [psi]i = m log re^(i[theta]), w = m log z. (7)