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Full text of "Scientific Papers - Vi"

ON THE MOTION  OF A VISCOUS FLUID. [Philosophical Magazine, Vol. xxvi. pp. 776786, 1913.]
IT has been proved by Helmholtz* and Kortewegf that when the velocities at the boundary are given, the slow steady motion of an incompressible viscous liquid satisfies the condition of making F, the dissipation, an absolute minimum. If u0) VQ, WQ be the velocities in one motion MQ, and u, v, w those of another motion M satisfying the same boundary conditions, the difference of the two u', v', w', where
u' = -u  u0,    v' = v  v0,    w' = w  WQ, .................. (1)
will constitute a motion M' such that the boundary velocities vanish. If jp0, F, F' denote the dissipation-functions for the three motions M0, M, M' respectively, all being of necessity positive, it is shown that
0) dxdydz, ......... (2)
the integration being over the whole volume.    Also F' = -fj> ((u W + v W + w'W) das dy dz
f Vfdw'    dv'\*    fdu'    dw'\2 , fdv     du'\*~\ ,   ,   ,           ,0,
= P    \[-j --- T-   +   j --- T~   +  T --- 7    \dxdydz ....... (3)
./ \_\dy     dz/     \dz      dx )     \das     dyl J        '7
These equations are purely kinematical, if we include under that head the incom possibility of the fluid. In the application of them by Helmholtz and Korteweg the motion M0 is supposed to be that which would be steady if small enough to allow the neglect of the terms involving the second powers of the velocities in the dynamical equations. We then have
* Collected Works, Vol. i. p. 223 (1869). t Phil. Mag. Vol. xvi. p. 112 (1883). elles peuvent etre en realite1 faibles encore, mais il est peu probable qu'elles soient plus fortes qu'il y a de certain, c'est que la polarisation parallele n'apparait que les fentes les plus fines, et alors que leur largeur est bien moindre q longueur d'une oiylulatioii qui est environ de ^^ de millimetre." I be remembered that the "plane of polarisation" is perpendicular t< electric vector.