188 ' ON THE MOTION OF A VISCOUS FLUID [376
where V is the potential of impressed forces. In virtue of (4)
I 0'V2ti0 + v'V2^ + w'VX) dxdydz^Q, .................. (5)
if the space occupied by the fluid be simply connected, or in any case if V be single-valued. Hence
F=F« + F', ................................. (6)
or since I" is necessarily positive, the motion M0 makes F an absolute minimum. It should be remarked that F' can vanish only for a motion such as can be assumed by a solid body (Stokes), and that such a motion could not make the boundary velocities vanish. The motion M0 determined by (4) is thus unique.
The conclusion expressed in (6) that MQ makes F an absolute minimum is not limited to the supposition of a slow motion. All that is required to ensure the fulfilment of (5), on which (6) depends, is that V2«0, V2y0, V2w0 should be the derivatives of some single-valued function. Obviously it would suffice that V2M0, V\, V2w0 vanish, as will happen if the motion have a velocity-potential. Stokes* remarked long ago that when there is a velocity-potential, not only are the ordinary equations of fluid motion satisfied, but the equations obtained when friction is taken into account are satisfied likewise. A motion with a velocity-potential can always be found which shall have prescribed normal velocities at the boundary, and the tangential velocities are thereby determined. If these agree with the prescribed tangential velocities of a viscous fluid, all the conditions are satisfied by the motion in question. And since this motion makes F an absolute minimum, it cannot differ from the motion determined by (4) with the same boundary conditions. We may arrive at the same conclusion by considering the general equation of motion
du du du\ _, d(pV+p) ,,_,
If there be a velocity-potential $, so that u — d^/dx, &c.,
du du du 1 d (fdd>\* /dAV fd<b\2\
UJ-+VI- + WT- = »J-\(-^} +(TT +( j~) N ...... (8)
dx~ dy dz 2 dx \\dxJ \dy) \dzj }
and then (7) and its analogues reduce practically to the form (4) if the motion be steady.
Other cases where F is an absolute minimum are worthy of notice. It suffices that
U2 dS ^2 dH _2 dH /m
V"°^~' Vv°=^' VX = U' ............... <9>
* Camb. Trans. Vol. is. (1850) ; Math, and Phys. Papers, Vol. in. p. 73. the aperture ; and let us distinguish by the suffixes TO and p the values applicable upon the negative. (minus), and upon the positive side of the screen. In the present case we have ...es*, to which it would 1