G • COOLING BY PROTECTIVE FLUID FILMS 



0.01 the wall friction coefficient increases by 85 per cent over the Poiseuille 

 flow case. A comparison of the variation of local wall friction coefficient 

 with fluid injection between the case of flow in a porous-wall pipe and on 

 a flat plate was shown in Fig. G,5c. 



Temperature distribution and heat transfer. For an incompressible 

 fluid, it can be shown that the terms in Eq. 5-4 due to the pressure 

 gradient and the dissipation ^ can be neglected, and furthermore it is 

 assumed that the molecular heat conduction in the axial direction may be 

 neglected in comparison with that in the radial direction. Hence Eq. 5-4 

 can be simplified in the following nondimensional form: 



_v_a0 u^ae _ 1 



Ui dr] Ui d^ PrRe 



1 d 



dd 



7] dr] \ dri/ _ 



(5-28) 



where ? = x/R, r, = Vz = r/R, and d = (T - T^)/{Ti - T^). Upon 

 substitution of the velocity components from Eq. 5-24 and 5-25 into 

 Eq. 5-28, one obtains 



1 - 



^ + 4A^ 

 X ^ Re^ 



18 



dd ^ X 1 

 d^~ Rer{'n)V 



ld_( dd\ 



7] dr] \ dr]/ 



(5-29) 



The energy equation written in the form of Eq. 5-29 yields a solution 

 in the form of an infinite series 





4PrX 



Mjiv, cj) 



where the Mj{r], cy)'s are the particular solutions of the equation. 

 1 



dm 



,2 ">" 



dv' 



Iv 



-\- 2Pr\ 



+ c4(l - 



dM 



dr] 



v') + ^ W - ^v' + 2v') 



(5-30) 



ilf = (5-31) 



and c/s are the eigenvalues of Eq. 5-31 which correspond to the boundary 

 condition 0,=i = 0. The series solution of Eq. 5-31 corresponding to the 

 eigenvalue c,-, which is free from singularities at ?/ = is 



My = 1 - 



+ 



16 



^ 4- C 

 4 ^"^ 



< 466 ) 



PrX - ^ X 4-1 



)]•■ 



+ 



(5-32) 



