150 Physical Properties [OH. vn 



The number of molecules of class A which cross the plane #=- 



Z 

 into the element of volume, in time dt, will be 



, dy ,,dz 

 "+T f+ 



I I *{,*, f , /(**> v > w > %~-y->y> z\dydzdudvdwudt, 



J J 2i ^ ** X 



dy , da 

 *'T f ~~2 



and as far as squares of small quantities this may be written in the form 



/ ^/P \ 



f ( u > v > w > sr> y> j dydzdudvdwudt ...... (346). 



Similarly the number of molecules of class A which cross the plane 



doc 

 x = % + out of the element of volume is given by expression (346) if 



z 



^- is replaced by + -= . By subtraction, the resulting loss to the 



element, of molecules of class A, caused by motion through the two faces 

 perpendicular to the axis of x, is 



a 



fi-[vf(u, v, w)]dx dydzdudvdwudt .................. (347), 



in which the differential coefficient is evaluated at , 77, The net loss of 

 molecules of class A caused by motion through all the faces is therefore 



r\ rl /-) \ 



- + v~- + w^-j [vf(u, v, w)]dudvdwdxdydzdt ...... (348). 



By integration over all values of u, v and w, we obtain the total number 

 of molecules which are lost to the element docdydz in time dt. If we 

 write 



// 



uf(u, v, w)dudvdw = u , etc (349), 



so that M , v , w are the components of mass velocity of the gas at x, y, z, 

 this number is seen to be 



(o o e) \ 

 ~- (1^0)4- ~- (^o) + ?r (^o) I dxdydzdt ...(350). 



cte dt/ d / 



Since, however, the number of molecules in the element at time t is 



vdxdydz, and at time t + dt is (v + -j- dt ) dxdydz, the net loss is 



\ dt J 



--^dtdxdydz (351). 



Cut 



Equating expressions (350) and (351), we obtain 



dv d , ^ . d_ . . i 

 dy ^ VV ' c 



