[58] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



At the end of the time t let V be the vis viva of a limited portion of the fluid, occupying 

 the space which lies inside the closed surface 8, and let V + D V be the vis viva of the same 

 mass at the end of the time t + Dt. Then 



V= fffp (u* + »* + w 2 ) dx dy dz, 



rrr ( Du Dv Dw\ 



DV=2DtJJJ P l« — j+v-pj + w j^ ) da)d y d *> • • • (135) 



the triple integrals extending throughout the space bounded by S. Substituting now for 



Du 



— - , &c. their values given by the equations of the system (132), we get 



u z 



DV = 2Dt fffp («Z + vY + wZ) dx dy dz 



- 2Dt ffflu ($& + — * + — ) + v (%& i dTl + 

 •J J •* \ \dx dy dz J \dy dz 



dT 3 \ 

 dx ) 



(dP, dTo dT,\\ 



+ w [^ + "L±l + a ±±)\dxdydz (136) 



\dz dx dy I) 



The first part of this expression is evidently twice the work, during the time Dt, of the 



external forces which act all over the mass. The second part becomes after integration by 



parts 



- 2Dt ff{u P t + vT 3 + w T,) dy dz - 2 Dt ff(v P 2 + w T t + uT 3 ) dz dx 



-zDtffiwPs + uTz + vTi) dxdy 



rrridu dv dw (dv dw\ (dw du\ 



J J J [dx dy dz \dz dy) ' \dx dz) 



(du dv\ 1 

 - + -)T^dxdydz. 



The double integrals in this expression are to be extended over the whole surface S. 

 If dS be an element of this surface, I', m, ri the direction-cosines of the normal drawn outwards 

 at dS, we may write I'dS, tridS, n'dS for dydz, dzdx, dxdy. The second part of DV 

 thus becomes 



-2Dtff{u (tP x + m'T 3 + n'T 2 ) + v (m'P 2 + riT x + ft,) + w (n'P 3 + l'T 2 + m'T,) \ dS. 



The coefficients of u, v, w in this expression are the resolved parts, in the direction of 



x, y, z, of the pressure on a plane in the direction of the elementary surface dS, whence it 



appears that the expression itself denotes twice the work of the pressures applied to the 



surface of the portion of fluid that we are considering 1 . 



On substituting for P 15 &c. their values given by the equations (133), (134), we get for 



the last part of DV 



n. rrr f du dv dw\ , , , 

 + 2Dtfffp( J - + - + - [ -)dxdydz 



dy 



-, rrr f ldu\* (dv\ 2 fdw^ a (du dv dw 

 - 2Dt ftt»Hdx) +2 U) +2 U) "I \dx + Ty + ^ 



(dv dw\* (dw du\ 2 (du dv\ 2 \ , 

 \dx dy) \dx dz) \dy dx) ) 



w\* 



