190 ON THE MOTION OF A VISCOUS FLUID [376
incompressible fluid there exists at a certain moment a mode of motion M which does not satisfy equation (4), then, the velocities along the boundary being maintained constant, the change which must occur in the mode of motion will be such (neglecting squares and products of velocities) that the dissipation of energy by internal friction is constantly decreasing till it reaches the value F0 and the mode of motion becomes identical with M0."
This theorem admits of instantaneous proof. If the terms of the second order are omitted, the equations of motion, such as (7), are linear, and any two solutions may be superposed. Consider two solutions, both giving the same velocities at the boundary. Then the difference of these is also a solution representing a possible motion with zero velocities at the boundary. But such a motion necessarily comes to rest. Hence with flux of time the two original motions tend to become and to remain identical. If one of these is the steady motion, the other must tend to become coincident with it.
The stability of the slow steady motion of a viscous fluid, or (as we may put it) the steady motion of a very viscous fluid, is thus ensured. When the circumstances are such that the terms of the second order must be retained, there is but little definite knowledge as to the character of the motion in respect of stability. Viscous fluid, contained in a vessel which rotates with uniform velocity, would be expected to acquire the same rotation and ultimately to revolve as a solid body, but the expectation is perhaps founded rather upon observation than upon theory. We might, however, argue that any other event would involve perpetual dissipation which could only be met by a driving force applied to the vessel, since the kinetic energy of the motion could not for ever diminish. And such a maintained driving couple would generate-angular momentum without limit—a conclusion which could not be admitted. But it may be worth while to examine this case more closely.
We suppose as before that «„, v0, w0 are the velocities in the steady motion M6 and u, v, w those of the motion M, both motions satisfying the dynamical equations, and giving the prescribed boundary velocities; and we consider the expression for the kinetic energy T' of the motion (1) which is the difference of these two, and so makes the velocities vanish at the boundary. The motion M' with velocities u', v', w' does not in general satisfy the dynamical equations. We have
1 dT \ ( , du' , dt)' , dw'\ . 7 7
----7r = iu —rr + v -r- + w -T-4 dxdy dz.............(14)
p dt j { dt dt dt} J ^ '
In equations (7) which are satisfied by the motion M we substitute u = M0 + u', &c.; and since the solution M0 is steady we have
du» _ dvo _ dwn _
dt ~'dt ~ dt .............................^ 'ith this anomaly; but