The Asymptotic Limit, Pr or Sc

A formal solution satisfying (11-19) and (11-20) is obtained easily, namely, 

However, the Blasius function f(rj) is available only as the numerical solution of the Blasius equation, and it is thus inconvenient to evaluate this formula for H(r] ). A simpler alternative is to numerically integrate the Blasius equation and the thermal energy equation (11-19) simultaneously. The function H(r]), obtained in this manner, is plotted in Fig. 11-2 for several different values of the Prandtl number, 0.01 Pr 100. As suggested earlier, it can be seen that the thermal boundary-layer thickness depends strongly on Pr. For Pr 1, the thermal layer is increasingly thin relative to the Blasius layer (recall that / - . 99 for rj 4). The opposite is true for Pr C 1. 

Apart from the special cases discussed in this section, the thermal boundary-layer equation (11-6) can only be solved analytically for the two asymptotic limits, Pr I and Pr 1. This is the subject of the next two sections. 

In the high-T3 case, we obtain the leading-order term in an asymptotic expansion, for the part of the domain where the momentum boundary-layer scaling is applicable, by taking the limit Pr —oo in (11-6). The result is 

We proceed formally. Thus we introduce the rescaled variable 

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