13 



Using substantially the same data as von Doenhoff and Tetervin and 

 a somewhat different analysis, Garner 12 obtained a more amenable auxiliary 

 equation for the rate of variation of H 



fl-dH_ f 0.0135(H-1.4) * dtfl 5(h - 1.4) r ~ u1 



dx " f B i/6 " u dx| e KHJ 



R s 



The following flat-plate formula of Palkner 13 for the skin-friction- 

 coefficient is used in its determination 



r w _ 0.00653^ 



pu 2 " R y fi 



[25] 



A noteworthy feature of Garner's equation is the separation of the pressure- 

 gradient parameter -sj- -nr and the Reynolds number R g into different terms. 



Theoretical derivations of an auxiliary equation have proved very 

 difficult. A strenuous but unsuccessful effort was made by Coleman 14 in 1 9^7 • 

 A very promising avenue of approach was introduced, however, more recently by 

 Tetervin and Lin 15 who investigated various integral forms of the Prandtl 

 boundary-layer equation, [1]. By multiplying each term of this equation by 

 an arbitrary function and then integrating, it is possible to derive various 

 families of equations. Using as the multiplying function u * y 2 , where s x 

 and s 2 are arbitrary constants, the general integral form of the boundary- 

 layer equation is written 



r* du s. s„, , r* du. s, s 2 1 r 1 dp s, s, , , 1 r dr s, s 9 , ■._• n 



j u^uiy "dy.+ JT.y s7 u'y My = -- j g u y My + -|-^ u 1 y My [26] 



As before, this equation is accompanied by the equation of continuity, [2]. 

 When both exponents S-l and s 2 are set equal to zero, the von Karman momentum 

 equation is specified. When s-l = 1 and s 2 = 0, an energy equation results; 

 when s x = and s 2 = 1 , a moment- of-momentum equation is indicated. 



The moment -of-momentum equation appears to be the most promising of 

 these derived integrated equations for obtaining a theoretical auxiliary equa- 

 tion for the variation of H. Assuming the general-power-law distribution for 

 the velocity profile, [8], Tetervin and Lin obtained the moment -of -momentum 

 equation in the form * 



;i; >)ii'.i) ||;^ .^Iw . (H+1)(H 2 -MC^-m [27] 



dx 2 U dx pU 2 J o pl]2 \6/ 



