ON THE STABILITY OF VISCOUS FLUID MOTION 271
where T is a function of t. On substitution in (6) he finds
dT
whence T = Ce-^{&2+»a-«*^+i&Wj ........................ (g)
and comes ultimately to zero. Equations (7) and (8) determine £ and so suffice for the heat and salinity problems in an infinitely extended fluid. As an example, if we suppose n = 0 and take the real part of (7),
&t.y), ........................... (9)
reducing to f= G cos /car simply when t~Q. At this stage the lines of constant £ are parallel to y.' As time advances, T diminishes with increasing rapidity, and the lines of constant £ tend to become parallel to x. If x be constant, £ varies more and more rapidly with y. This solution gives a good idea of the course of events when a liquid of unequal salinity is stirred.
In the hydrodynamical problem we have further to deduce the small velocities u, v corresponding to £ From (2) and (3), if u and v are proportional to eikx,
Thus, corresponding to (9),
v —
No complementary terms satisfying d?v/dy2 — k*v = 0 are admissible, on account of the assumed periodicity with as. It should be mentioned that in Kelvin's treatment the disturbance is not limited to be two-dimensional.
Another remarkable solution for an unlimited fluid of Kelvin's equation (6) with U~fty has been given by Oseen*. In this case the initial value of £ is concentrated at one point (£, 17), and the problem may naturally be regarded as an extension of one of Fourier relating to the conduction of heat. Oseen finds
~
where 0 = Jj £(£ *i, 0)^; .........................(13)
and the result may be verified by substitution.
* ArUvf'dr Hatematik,Astronomi och Fysik, Upsala, Ed. vn. No. 15 (1911).lvin'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