180 182.] FRICTION IN STRAIGHT PIPE. 223 



182. We proceed to apply our equations to the treatment of 

 a few particular problems. 



Example 1. We take first the flow of a liquid in a straight 

 pipe of uniform circular section, neglecting external impressed 

 forces. If we take the axis of the pipe as the axis of x, and assume 



v = 0, w = 0, the second and third equations of (15) give -^ = 0, 



ay 



-j- 0, and the equation of continuity becomes -r- = 0. If we 



further assume that u is a function of r(= (y* + z 2 fy and t only, 

 the first equation of (15) becomes on transformation of co-ordinates 



du 1 dp , fd?u 1 da 



- 



If a be the radius of the pipe, the surface-condition is, Art. 181, 



-*Tr-t* 



when r = a. 



In steady motion ~ = 0, so that (17) becomes 



&amp;lt;Fu 1 du _ 1 dp 

 dr z r dr fj, p dx 



The left-hand side of this equation does not involve x, and the 

 right-hand side does not involve ?*, so that each must be constant, 

 = A, say. This gives 



- ............ (20). 



The solution of (19) is 



-f- C ..................... (21), 



where S, C are arbitrary constants. Since the velocity at the 

 centre of the pipe is necessarily finite, we must have 5=0; C is 

 then determined by (18), so that 



