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Full text of "Scientific Papers - Vi"

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