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ART. 208 - On the Flow of Viscous Liquids, especially in Two Dimensions

Published online by Cambridge University Press:  05 July 2011

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Summary

The problems in fluid motion of which solutions have hitherto been given relate for the most part to two extreme conditions. In the first class the viscosity is supposed to be sensible, but the motion is assumed to be so slow that the terms involving the squares of the velocities may be omitted; in the second class the motion is not limited, but viscosity is supposed to be absent or negligible.

Special problems of the first class have been solved by Stokes and other mathematicians; and general theorems of importance have been established by v. Helmholtz and by Korteweg, relating to the laws of steady motion. Thus in the steady motion (M0) of an incompressible fluid moving with velocities given at the boundary, less energy is dissipated than in the case of any other motion (M) consistent with the same conditions. And if the motion M be in progress, the rate of dissipation will constantly decrease until it reaches the minimum corresponding to M0. It follows that the motion M0 is always stable.

It is not necessary for our purpose to repeat the investigation of Korteweg; but it may be well to call attention to the fact that problems in viscous motion in which the squares of the velocities are neglected, fall under the general method of Lagrange, at least when this is extended by the introduction of a dissipation function.

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Scientific Papers , pp. 78 - 93
Publisher: Cambridge University Press
Print publication year: 2009
First published in: 1903

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