520 VISCOSITY. [CHAP, xi 



Also, since there is no motion parallel to y, dpjdy must vanish. 

 These results might of course have been obtained immediately 

 from the general equations of Art. 286. 



It follows that the pressure-gradient dpjdx is an absolute 

 constant. Hence (1) gives 



and determining the constants so as to make u = for y = h, we 

 find 



Zh* dp 

 Hence y= __ ..................... (4). 



289. The investigation of the steady flow of a liquid through 

 a straight pipe of uniform circular section is equally simple, and 

 physically more important. 



If we take the axis of z coincident with the axis of the tube, 

 and assume that the velocity is everywhere parallel to z, and a 

 function of the distance (r) from this axis, the tangential stress 

 across a plane perpendicular to r will be pdw/dr. Hence, con 

 sidering a cylindrical shell of fluid, whose bounding radii are r 

 and r + r, and whose length is I, the difference of the tangential 

 tractions on the two curved surfaces gives a retarding force 



d [ dw _ A ? 



-J- /* -j- - 2 r - 

 dr \ dr J 



On account of the steady character of the motion, this must be 

 balanced by the normal pressures on the ends of the shell. Since 

 dwjdz = 0, the difference of these normal pressures is equal to 



where p r , p 2 are the values of p (the mean pressure) at the two 

 ends. Hence 



dr\dr 



Again, if we resolve along the radius the forces acting on a 

 rectangular element, we find dp/dr = 0, so that the mean pressure 

 is uniform over each section of the pipe. 



