Equations of Motion of a Viscous Liquid. 45 



Suppose the velocity uniform in the direction of x. Then 

 any parallelopiped, such as AB, would move as a solid; but 

 the element is composed of a great many such parallelepipeds. 

 If the velocity in the direction of .v varies as we pass from 

 R to K, the layers in slipping by each other will produce 

 mutual retardation. The rate of change in ii along the nor- 

 mal to its direction is 



du 



dy 



From the definition of viscosity it follows that the shearing 



force upon the face S B is 



du , , 

 ij. — a.v az, 

 dy 



and that upon the opposite face is 



IJ. — dx dz-\-!J. — dx dz dy, 

 dy dy' 



indicating a loss of force within the element equal to 



fi — dx dy dz. 

 dy- 



In a similar manner we find the retardation upon the 

 face dy dx to be 



du , , 

 fi — dx dy, 



dz 

 and on the opposite face to be 



[i^ dx dy+/i- — dx dy dz, 

 dz dz^ 



showing a loss within the element of 



fjL — dx dy dz. 

 dz' ^ 



All of the loss within the element is now accounted for, 

 provided the velocity in the direction of x is uniform. It 

 remains to examine the effect of an accelerating force along 

 the axis x. To simplify the problem, suppose the motion two 



