G • COOLING BY PROTECTIVE FLUID FILMS 



formed into the following two ordinary differential equations with / and 

 d as functions of ?? only: 





and 



(I)'- 



= (4-19) 



g + P./l^ = (4-20) 



With the boundary conditions given in Eq. 4-17 the above transformation 

 is based on the assumption that the temperature of the wall is constant 

 and the normal injection velocity at the wall is given by 



-F, — /w = — A — = const (4-21) 



2 Ue \j V 



and v^ '~ l/v^ foi" a constant u^. 



The differential equation (Eq. 4-19) can be solved only numerically 

 and Eq. 4-20 can readily be integrated if the function / is known from 

 the solution of Eq. 4-19. 



Numerical results for a laminar boundary layer flow on a flat plate 

 (m = 0) and for flow near a stagnation point (m = 1) were given in [16]. 

 With the aid of these solutions the heat transfer phenomena in the above 

 two cases for Prandtl number equal to unity were calculated in [15]. The 

 results can be briefly summarized as follows: 



1. In the case of flow on a flat plate, the heat transfer coefficient h de- 

 creases rapidly for an increase in (v^/ue) -y/Rex. It is reduced to one 

 tenth of the value without transpiration cooling for an average ratio of 

 Vw/we = 1 per cent (Rex = 10^). 



2. For flow near the stagnation point (w = 1) it is found that the point 

 of inflection does not appear in the velocity profiles. This is under- 

 standable because the flow in the neighborhood of the stagnation point 

 is under the influence of a favorable pressure gradient and therefore 

 becomes more stable than the case on the flat plate. The heat transfer 

 coefficient h diminishes practically to zero when the coolant injection 

 parameter v^l y/Vc reaches 3.2. 



3. It appears that the required coolant injection to maintain a given wall 

 temperature is much less in the present case than the result obtained 

 in the approximate solution. This result is expected because in the 

 approximate solution a uniform coolant injection was assumed, whereas 

 in the present exact solution the coolant injection is proportional to 

 the reciprocal of the square root of the distance from the leading edge 

 of a flat plate. 



Compressible boundary layer with variable fluid properties. The simul- 

 taneous effects of pressure gradient in the main stream flow over a porous 



( 448 ) 



