192 ON THE MOTION OF A VISCOUS FLUID [376
Of the remaining 18 terms of the second degree, 9 vanish as before when integrated, in virtue of the equation of continuity satisfied by u,, v0, w0. Finally we have*
<®L = -F'-
dt
,dv0 ,dv<> ,dvn dx dy dz
(22)
—^---- T" M/ j I I ww *.v y «-v~- • ........' \ /
dy dz}]
If the motion u0, va, w0 be in two dimensions, so that «;„ = (), while ?t and fl0 are independent of z, (22) reduces to
dF_
dt
Under this head comes the case of uniform rotation expressed in (16), for which
dun _ n dv0 _ - dw0 , ^o __ o
dx~~ ' dy dy ~dx
Here then dT'jdt~-F' simply, that is T' continually diminishes until it becomes insensible. Any motion superposed upon that of uniform rotation gradually dies out.
When the motion u0, va, w0 has a velocity-potential $, (22) may be, written
,...... , + 2i)V , r + 2wV -r-T- dx dii dz.......(24)
dxdy dydz dzdx\
So far as I am aware, no case of complete stability for all values of //. is known, other than the motion possible to a solid body above considered. It may be doubted whether such cases exist. Under the head of (24) a simple example occurs when 0 = tan"1 (y/x), the irrotational motion taking place in concentric circles. Here if r2 = of- + y*,
fJT' r r <> o -i
_ = _ j" _ 2p J |j| (ut - v't) + L^ My J da; dydz.......(25)
* Compare 0. Eeynolds, Phil. Trans. 1895, Part i. p. 146. In Lorentz's deduction of a similar equation (Abhandlwigen, Vol. i. p. 46) the additional motion is assumed to be small. This memoir, as well as that of Orr referred to below, should be consulted by those interested. See also Lamb's Hydrodynamics, § 346. we see from (3) that