G • COOLING BY PROTECTIVE FLUID FILMS 



the following stream function is introduced 



^ = 



R' 



U\ X 



m 



(5-10) 



where z = {r/Ry. 



Introducing the expressions of u and v from Eq. 5-9 and 5-10 into 

 Eq. 5-1 and 5-2 results in 



(fO) ^ Re R, 



[RvM"-fn -K^r'+D] (5-11) 



Rv, 



/p _ 2fr\ 



- 2vj" 



= F{z) 



(5-12) 



where X = v^R/v and Re = u^D/v and Ui is the maximum velocity for 

 Vv, = 0. Since the right-hand side of Eq. 5-12 is a function of z only, 

 differentiation of both sides of it with respect to x yields 



d'^p 

 dxdz 



= 



(5-13) 



Hence, differentiating Eq. 5-11 with respect to z gives zero on the left- 

 hand side, and integrating the right-hand side of the equation, one obtains 



for X < 1 and 



zf" + /" - \{P - If) = c 



f' - If" - I i^f" + /") = k 



(5-14) 

 (5-15) 



for X > 1, where c and k are the constants of integration to be determined. 



Eq. 5-14 is an ordinary nonlinear differential equation of the third 

 order which resulted from the Navier-Stokes equations and the continuity 

 equation by the similarity transformation. With the aid of the four given 

 boundary conditions an exact solution can be obtained and the constant 

 of integration c determined. 



It can be seen that the limiting form of Eq. 5-14, by letting v^ ap- 

 proach zero, is the equation describing a flow through a circular pipe with 

 impermeable walls. The solution of this equation which satisfies all the 

 four boundary conditions given in Eq. 5-5 and 5-7 is the well-known 

 Poiseuille law for pipe flow. If small values of X are treated as a pertur- 

 bation parameter, a solution of Eq. 5-14 can be obtained, which is dis- 

 cussed later. 



On the other hand, if large values of X are treated as a perturbation 

 parameter the third order differential equation (Eq. 5-15) is reduced to 

 a second order one. The solution of Eq. 5-15 can also be obtained in the 

 same manner since all four boundary conditions given in Eq. 5-5 and 5-7 

 can be satisfied. 



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