is a function of [psi], say [f]´([psi]), and constant along a stream line;
dH/dv = 2q[zeta], H - [f]([psi]) = constant, (5)
throughout the liquid.
When the motion is irrotational,
d[phi] 1 d[psi] d[phi] 1 d[psi]
[zeta] = 0, u = - ------ = - --- ------, v = - ------ = --- ------, (6)
dx y dy dy y dx
d²[psi] d²[psi] 1 d[psi]
[nabla]²[psi] = 0, or ------- + ------- - --- ------ = 0. (7)
dx² dy² y dy
Changing to polar coordinates, x = r cos[theta], y = r sin[theta], the
equation (2) becomes, with cos[theta] = [mu],
d²[psi] d²[psi]
r²------- + (1 - [mu]²) ------- = 2[zeta]r³ sin [theta], (8)
dr² d[mu]²
of which a solution, when [zeta] = 0, is
/ B \ dPn / B \ dPn
[psi] = ( Ar^(n+1) + --- ) (1 - [mu]²) ----- = ( Ar^(n-1) + ------- ) y²-----, (9)
\ r^n / d[mu] \ r^(n+2) / d[mu]
[phi] = {(n + 1)Ar^n - nBr^(-n-1)} Pn, (10)
where Pn denotes the zonal harmonic of the nth order; also, in the
exceptional case of
[psi] = A0 cos[theta], [phi] = A0/r;
[psi] = B0r, [phi] = -B0 log tan ½[theta]
= -½B0 sh(-1) x/y. (11)
Thus cos[theta] is the Stokes' function of a point source at O, and PA
- PB of a line source AB.
The stream function [psi] of the liquid motion set up by the passage
of a solid of revolution, moving with axial velocity U, is such that
1 d[psi] dy
--- ------ = -U --, [psi] + ½Uy² = constant, (12)
y ds ds
over the surface of the solid; and [psi] must be replaced by [psi]´ =
[psi] + ½Uy² in the general equations of steady motion above to obtain
the steady relative motion of the liquid past the solid.
For instance, with n = 1 in equation (9), the relative stream function
is obtained for a sphere of radius a, by making it
[psi]´ = [psi] + ½Uy² = ½U(r² - a³/r) sin² [theta],
[psi] = -½Ua³ sin² [theta]/r; (13)
and then
[phi]´ = Ux(1 + ½a³/r²), [phi] = ½Ua³ cos [theta]/r², (14)
d[phi] a³ d[phi] a³
- ------ = U -- cos [theta], - --------- = ½U -- sin [theta], (15)
dr r³ rd[theta] r³
so that, if the direction of motion makes an angle [psi] with Ox,
tan ([psi] - [theta]) = ½ tan [theta],
tan [psi] = 3 tan [theta]/(2 - tan² [theta]), (16)
Along the path of a liquid particle [psi]´ is constant, and putting it
equal to ½Uc²,
(r² - a³/r) sin² [theta] = c², sin² [theta] = c²r/(r³ - a³), (17)
the polar equation; or
y² = c²r³/(r³ - a³), r³ = a³y²/(y² - c²), (18)
a curve of the 10th degree (C10).