622 



pro unit of surface acting' at the surface of tiie fluid, F the dissi- 

 pation function. The tirst term on (he right liaud side represents 

 thus (he increase of the kinetic energy l>_v the action of external 

 forces, the second term the inci-ease by the flow inwards of the 

 fluid, the third term the increase by the action of the pressure forces, 

 the fourth the decrease by the transformation of tlie kinetic 

 energy by (he inner resistance. This last integral is always positive, 

 the other ones can be positive or nega(ive. 



When we suppose the surface *S (o be (piite formed by solid 

 bodies, at tlie boundaries of which the friction is neglected except 

 along (he moving body, the second term falls away, while the third 

 term has to be (aken over this body only. When further the state 

 is stationary, equation (1) becomes: 



0=1 i{X,n-{- Y,v\-Z^w)dS - iii Fd.v dy dz . . . (2) 



For (lie moving body we have in this case the equation: 



^(X'u4- y'v-\~Z',r) dt -f i |(-A',u— ]',r-"^i«') dSdt =r U . (3) 



The summation has (o be extended over (he [)oints, where the 

 external forces (A' etc.) act, (he integration over the surface. This 

 notation expresses that the forces acting on the surface layer are 

 equal and opposite to those acting there on the fluid. 



In the case of sliding however the components of veloci(y are 

 not equal as fai- as the}' refer to (he body, on one hand, and to the 

 fluid on the other hand. Calling the components in the fluid ii,- etc. those 

 in the body nt etc. then n,, :=. ni-\- a,- etc. (4), when u,- is written 

 for the componeni of the relative velocity. 



Now equation (2) becomes:- 



I i{X,u,.^ Y,r,-^Z^w,.) dS— i iih'dx dy dz - 

 and equation (3) : 



2{X'u^rv + Z'w) — Cf(X,ui + Y,vi^Z,wi) dS = 0. 

 Summation gives us : 



2{X'u-\-rv+z'io)-\- 



+ fi [X, (u„- u i) -\- Y^ (v, - vi) + Z, {w„—ivi)]dS — CCCf dx dy dz = 

 so, using (4) : 



2:{X'u-{- Y'u-^Zw) = ( ( (f dx dy dz~i \\X,Ur+ Y,Vr^Z,tOr) dS. 



