1913] ON THE MOTION OF A VISCOUS FLUID 191
We further suppose that V2w0) V2u0, V2?.t;0 are derivatives of a function H, as in (9). This includes the case of uniform rotation expressed by
MO = 2/i v0 = — as, «/„ = (), ........................ (16)
as well as those where there is a velocity-potential. Thus (7) becomes du' _, , dm . fdu0 du'\
-j-=V W - -y- - (W0 + U') ~-° + —
at ax ' \dx dxJ
, ,N /du0 du'\ , , fdtif, du'\ ,_,_.
-(v0 + v) -7- + -T- ~(w0 + w') --,- + 1- > ...... (17)
\dy dyj ^ ' \dz dz ) ^ '
with two analogous equations, where
™=V+p/p-vH, v = ^lp ...................... (18)
These values of du'/dt, &c., are to be substituted in (14).
In virtue of the equation of continuity to which u', v'} w' are subject, the terms in in- contribute nothing to dT'/dt, as appears at once on integration by parts. The remaining terms in dT'/dt are of the first, second, and third degree in u ', v', w' . Those of the first degree contribute nothing, since ua, v0> w0 satisfy equations such as
The terms of the third degree are
, ( , du' , du' , du' u u' -J- + V -T-- + W '-=-dx dy dz
, ( ,dv' , dv' , dvf u-j- + v'-j- + w -r dx dy dz
, ( , dw' , dw' , dw'} "1 7 7 7 + w \u -j- + v -j— + w -y-4 dxdiidz, { dx dy dz) ] J
which may be written
d (w/2 + v'2 + w's) ,d(u'z +
> \
dx dy
iz\
and this vanishes for the same reason as the terms in •or.
We are left with the terms of the second degree in u1, v', w'. Of these the part involving v is
' (20)
So far as this part is concerned, we see from (3) that
dT'fdt^-F', ............. .. .......... .....(21)
F' being the dissipation-function calculated from u', v, w'.t limit—a conclusion which could not be admitted. But it may be worth while to examine this case more closely.