VISCOUS FLOW 67 



i.e., by the rate of change of this force with respect to y, multiplied by the 

 change in y. 



But if b is the width of the stratum, and 8 x its length, the area of each 

 end = b 8 y, while its area A = b 8 x. 



Also, since for equilibrium the difference of traction on the two faces of 

 the stratum is equal to the difference of normal pressures on the two ends, 



Integrating this expression, we get 



u = C + B y + ^ . ?/ 2 . --2L, where C and B are constants. 

 * H ax 



Since the motion is symmetrical about the axis of x, i.e., is the same for 

 equal positive and negative values of y, the term involving the first power 

 of y must vanish, since this would change sign with y, so that B = o. 



Again for uniformity of flow ~ is constant, while, assuming no slip at 



ci x 



e boundaries, we have u = o for y = + h. 



Determining the constant C, so that these boundary conditions are 

 atisfied, we get 



e., the pressure diminishes as x increases. Evidently, too, from the form 

 f the equation the curve of velocities is a parabola. 

 The flux over unit width of the plates is given by 



h 



**/ 



h 



If the measurements are taken in pounds, feet, and seconds, the volume 

 ier second in cubic feet, Q, is given bv 



2 h 5 dp 



^ ~ 3 7 Tx 



1 This may be deduced directly from the general equations of motion, p. 62, for since 

 = o ; iv = o ; -ry = o ; equations (10) reduce to 



d n d u d u dp d* u 



; dt = '> Jic = " ; -j-z = * thisbecomw ji = ^57,- 



F2 



