60 



HYDRAULICS AND ITS APPLICATIONS 



/. Total gain 



(3 8 8 



= ~ | g J. (P W) + ^ (P V) + g-J (PI 



But the mass contained at time t = p B jc . 

 and ,- + S = 



B x . B y . B z. 



.-. Gain in time Bt = ^..Bt.Bx. By. B z. 



8 t 



Equating these expressions for the gain, we have 



z.Bt. 



8 p , 8 (p u) _, 8 

 " 1 



v) , 8 (p w) _ 



(5) 

 from which, if p is constant, i.e., if the fluid is incompressible, we have 



8^+ ^y + 87 = (6) 



as the equation of continuity for an incompressible fluid. 



In the case of a gas p is not constant, but we may have ~ = constant, 



in .which case we have regular motion in the gas. 



Now if the stresses in a viscous fluid (which follows the same stress strain 

 law as an elastic solid) be represented by the notation p xx , p xy) p yz , etc., 

 where each of these symbols denotes a stress on the plane perpendicular to 

 the axis of co-ordinates represented by the first suffix, in the direction of 

 the second suffix, so that, for example, the stress p xy is a stress on the 

 plane perpendicular to X, in the direction Y, and is therefore a 

 tangential stress on this plane, the relations holding between the various 

 stresses for equilibrium are given by the following equations. 1 



2 /8 u , 8 v , 8 u-\ 8 u 



8 v 8 ?/' 



8 v 



2 /a . 8 , a w\ . a w 



P - ^- + + + 2 M 



3 



8 v , 8 v 



rx + ry 



8 w ,8 v 

 t~y + 2~ z 



8 u . 8 w 





1 Stokes on " Theories of the internal friction of fluids in motion and of the equilibrium and 

 motion of elastic solids" ; also Lamb's "Motion of Fluids," p. 219, or "Hydrodynamics," 



