Equations of Motion of a Viscous Liquid. Ho 



tion or repnlsion is supposed to be proportional to the veloc- 

 ity with which the molecules, separated l)y a given distance, 

 are approachin*^ or recedin*^ from each other. A second sup- 

 position involves the symmetrical arrangementof the particles. 

 Navier dealt only with an incompressible fluid, and arrived 

 at the following equations: 



l\dx ^ \i\x' ill/ (Izy/ot 

 and the other two can be written by making similar changes 

 in the corresponding equations for a perfect fluid. ^ is a 

 constant, depending on tlie nature of the fluid. The other 

 terms are the same as those used in the following pages. 



Poisson (1781-1840) derived equations not only for an 

 incompressible fluid, but also for an elastic fluid in which 

 the change in density is small. He treated the subject from 

 the standpoint of an elastic solid, supposing the fluid to be 

 continually beginning to be displaced like an elastic solid, 

 and continually rearranging itself so as to make the pressure 

 equal in all directions, as is the case with a fluid at rest. His 

 equations are written thus: 



,^ i- /dp .fdiL^ dii^ du\ j^ d /dii dv dic\\_<'^>u 



r\dx \dx^ dy^ dzy dx\dx dy dzJJ 'H 

 with the other two to correspond. 



For an incompressible fluid these agree with those ob- 

 tained by Navier, since in that case the expression for cubical 

 expansion disappears. It is to be noticed that both the fore- 

 going methods involve a consideration of the ultimate mole- 

 cules of the fluid. 



Barr6 de St. Venant (1797-1886) first succeeded in ob- 

 taining the equations independent of any consideration 

 of the ultimate molecules. He attempted to connect the 

 oblique pressures in different directions about a point with 



the differential coefficents — » — etc., which express the rel- 



dx dy 



ative motion of the fluid particles in the immediate neighbor- 

 hood of the point, by assuming the tangential force on a plane 

 passing through the point to be in the direction of the prin- 

 cipal sliding along that plane. He then employs theorems 



