270 ON THE STABILITY OF VISCOUS FLUID MOTION [388
In other applications of (1), e.g. to the diffusion of heat or dissolved matter in a moving fluid, £ is a new dependent variable, not subject to (2), and representing temperature or salinity. We may then regard the motion as known while f remains to be determined. In any case ^D^fDt = v ^V2^. If the fluid move within fixed boundaries, or extend to infinity under suitable conditions, and we integrate over the area included,
f[D%~ d
so that
......... (4)
by Green's theorem. The boundary integral disappears, if either £ or d£/dn there vanishes, and then the integral on the left necessarily diminishes as time progresses*. The same conclusion follows if £ and d£/dn have all along the boundary contrary signs. Under these conditions f tends to zero over the whole of the area concerned. The case where at the boundary £ is required to have a constant finite value Z is virtually included, since if we write Z + £' for £, Z disappears from (1), and £ everywhere tends to the value Z.
In the hydrodynamical problem of the simple shearing motion, £ is a constant, say Z, u is a linear function of y, say U, and v — 0. If in the disturbed motion the vorticity be Z+ £> and the components of velocity be U + u and v, equation (1) becomes
in which £, u, and v relate to the disturbance. If the disturbance be treated as infinitesimal, the terms of the second order are to be omitted and we get simply
In (6) the motion of the fluid, represented by U simply, is given independently of %, and the equation is the same as would apply if £ denoted the temperature, or salinity, of the fluid moving with velocity U. Any conclusions that we may draw have thus a widened interest.
In Kelvin's solution of (6) the disturbance is supposed to be periodic in #, proportional to eikx, arid U is taken equal to J3y. He assumes for trial
Compare Orr, I.e. p. 115.t d?_ - m