By A. B. Basset
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Extra resources for A Treatise on Hydrodynamics, Vol. 1
FpT(lu energy which flows into $ + mv + nw) St dS + (pwT) Taking account of the equation of continuity increase in the kinetic energy r r r ^ '7^ flip oi Stdxdydz. 9 (10) the total EQUATIONS OF MOTION 32 fff ftT p h~- Whence ]jy \dt dQ + v dQ u-jdx dQ\ dy w-j-] dz] jn JO -j which requires that ST If we substitute the values of u, dT _ last d dx f dt The jn \ d\ d dy (23). , and the operator is therefore equal to qd/ds. Integrating along a stream line, and restoring the values of Q and T, we obtain Now if dt p F(f) dt .
Let us therefore as functions of u, between v, u', w ; suppose that u This is (w'y - v',) + v' (u' + w' (vx - u'v ) = 0. u'dx + vdy 4- wdz should ' w'x ) a the condition that have an we may therefore put this quantity equal to which proves the proposition. It therefore follows that, integrating factor, = dx = + X dx The components of = +X dy dz dy +X molecular rotation are ..... (13). dz given by the equations (14). , = const. Therefore du 1 = d fd(f> Crelle, vol. LVI. p. 1. Phil. Soc. vol. xiv.
Soc. EQUATIONS OF MOTION. 24 after Since solidification. the impulsive forces are internal, all we have u= v u, = w = w. v, of angular moby the principle of conservation &c. z 0, 2m [y (w w') (^ -')}= mentum, m denoting an element of the mass of the element considered. We have also, l But u t - j]z - y, and u du dx ultimately equal to is du , du , ** t dz dy and similar expressions hold good for the other quantities. and observing that stituting in the above equation, v'y' we have We f = = 0, and 2ra#' dv\ & ---, fdw .