498 XT - HYDRODYNAMICS. 



If we consider any closed surface fixed in space , expressing the 

 fact that the increase of mass of the fluid contained therein is 

 represented by the mass of fluid which flows into the surface, we 

 shall obtain an additional equation. The velocity being q } the volume 

 of fluid entering through an area dS in unit time is, as in 169, 

 equation 78), qcos(q,n)dS, and the mass, pgcos (q, n) dS. We 

 have therefore for the total amount entering in unit time 



7) I / gq cos (q,ri)dS =1 I p{iecos(w#)-f- vcos(ny)-\-wcos(n0)}dS 



But this is equal to the increase of mass per unit of time, 



for the volume of integration is fixed, that is, independent of the 

 time, consequently we may differentiate under the integral sign. 

 Writing this equal to the volume integral in 7) and transposing, 



Since this holds true for any volume whatever the integrand must 

 vanish, so that we have 



_ A 



^t~ ~^r + ~W~ ~0^~ 



which is known as the Equation of Continuity. 

 Performing the differentiations we have 



00 , 00 , 00 , 00 , (0** , 0^ . 



or in the notation of particle differentiation, 



If now we fix our attention upon a small portion of the fluid 

 of volume V as it moves, its mass will be constant, say, m = $ V= const. 

 By logarithmic differentiation, 



I^ + ^-^-O 



g dt H F dt ~ 



so that the expression 



... du.dv . div _ 1 dQ _ 1 dV 



^^J^^ Tz~ ~^~di~ = T^dt' 



that is, the divergence of the velocity is the time rate of increase 

 of volume per unit volume. This corresponds with the expression 



