287-288] DISSIPATION OF ENERGY. 519 



If we integrate this over a region such that u, v, w vanish at every point of 

 the boundary, as in the case of a liquid filling a closed vessel, on the hypothesis 

 of no slipping, the terms due to the second line vanish (after a partial integra 

 tion), and we obtain 



............ (iv)* 



In the general case, when no limitation is made as to the boundary 

 conditions, the formula (iii) leads to 



2^=4^ I I ntf + rf + ffdxdydz- n I l-^dtS 



f fffr &amp;gt; w *&amp;gt; 



+ 4^1/1 \u, v, iv, dS (v), 



I &amp;gt; *?&amp;gt; | 



where, in the former of the two surface-integrals, dn denotes an element of the 

 normal, and, in the latter, I, m, n are the direction-cosines of the normal, 

 drawn inwards in each case from the surface-element dS. 



When the motion considered is irrotational, this formula reduces to 



simply. In the particular case of a spherical boundary this expression follows 

 independently from Art. 44 (i). 



Problems of Steady Motion. 



288. The first application which we shall consider is to the 

 steady motion of liquid, under pressure, between two fixed parallel 

 planes, the flow being supposed to take place in parallel lines. 



Let the origin be taken half-way between the planes, and the 

 axis of y perpendicular to them. We assume that u is a function 

 of y only, and that v, w = 0. Since the traction parallel to x on any 

 plane perpendicular to y is equal to pdujdy, the difference of the 

 tractions on the two faces of a stratum of unit area and thickness 

 % gives a resultant fj,d 2 u/dy*. 8y. This must be balanced by the 

 normal pressures, which give a resultant dpjdx per unit volume 



of the stratum. Hence 



d 2 u dp 



* Bobyleff, &quot;Einige Betrachtungen tiber die Gleichungen der Hydrodynamik,&quot; 

 Math. Ann., t. vi. (1873); Forsyth, &quot;On the Motion of a Viscous Incompressible 

 Fluid,&quot; Mess, of Math., t. ix. (1880). 



