d / 1 d[psi]´\ d / 1 d[psi]´\ 15 y
2[zeta] = --- ( --- ------- ) + --- ( --- ------- ) = ---U ---, (14)
dx \ y dx / dy \ y dy / 2 a²
so that §34 (4) is satisfied, with
15 15
[f]´([psi]´)= ---Ua^(-2), [f]([psi]´) = ---U[psi]´a^(-2); (15)
2 2
and (10) reduces to
_ _
p 9 | / x² \² / y² \² |
----- + V - ---U | ( --- -1 ) - ( --- - ½ ) | = constant; (16)
[rho] 8 |_ \ a² / \ a² / _|
this gives the state of motion in M. J. M. Hill's spherical vortex,
advancing through the surrounding liquid with uniform velocity.
39. As an application of moving axes, consider the motion of liquid
filling the ellipsoidal case
x² y² z²
--- + --- + --- = 1; (1)
a² b² c²
and first suppose the liquid to be frozen, and the ellipsoid to be
rotating about the centre with components of angular velocity [xi],
[eta], [zeta]; then
u = - y[zeta] + z[eta], v = - z[xi] + x[zeta],
w = - x[eta] + y[xi]. (2)
Now suppose the liquid to be melted, and additional components of
angular velocity [Omega]1, [Omega]2, [Omega]3 communicated to the
ellipsoidal case; the additional velocity communicated to the liquid
will be due to a velocity-function
b² - c² c² - a² a² - b²
[phi] = - [Omega]1 ------- yz - [Omega]2 -------zx - [Omega]3 -------xy, (3)
b² + c² c² + a² a² + b²
as may be verified by considering one term at a time.
If u´, v´, w´ denote the components of the velocity of the liquid
relative to the axes,
2a² 2a²
u´ = u + yR - zQ = ------- [Omega]3 y - ------- [Omega]2 z, (4)
a² + b² c² + a²
2b² 2b²
v´ = v + zP - xR = ------- [Omega]1 z - ------- [Omega]3 x, (5)
b² + c² a² + b²
2c² 2c²
w´ = w + xQ - yP = ------- [Omega]2 x - ------- [Omega]1 y, (6)
c² + a² b² + c²