G,5 • TRANSPIRATION-COOLED PIPE FLOW 



average velocities become 



du , du 1 dp 115,, 



dx dr p ax p r dr 



dv , dv Idp , I dr 



u— + v~ = ^+-^ 



dx dr p dr p dx 



where r = —fidu/dr — pu'v'. The continuity equation is 



d{ru) d{rv) 



dx dr 



= 



(5-46) 

 (5-47) 



(5-48) 



The eddy heat transfer equation excluding the dissipation term is 



dT , 



pc,u— + pc,v^^ 



The boundary conditions are 

 r = 0: 



Velocity distribution and skin friction. The first simplification neces- 

 sary in achieving an approximate solution for Eq. 5-46, 5-47, and 5-48 is 

 to assume that v = —v^r/R and that in the region close to the wall 



du u 

 Tr^R 



is valid. With the above assumptions and with the aid of the continuity 

 equation, Eq. 5-46 and 5-47 are reduced to 



and 



v^ d(r^u) _ r dp 1 d(rT) 

 R dr p dx p dr 



dr 



According to momentum transfer theory this gives 



/ 2 



-u'v' = P 



(5-52) 

 (5-53) 



(5-54) 



where I = K{R — r). After neglecting the viscous shearing stress and 

 combining Eq. 5-54 and 5-52, one obtains, after integration with the aid 

 of Eq. 5-50 



K-^{R-rY[^\ =I{ul + v^u) 



(5-55) 



(471 ) 



