A Uniformly Valid Asymptotic Theory of Rarified Gas Flows Under the Nearly Free
A Uniformly Valid Asymptotic Theory of Rarified Gas Flows Under the Nearly Free
Young Ping Pao
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Therefore, to use expression (4. 17), It Is necessary to make the restriction that p > aT. , where T-, (a) has the property that t, — > oo, aT-, — > 0, and t-, /t — > oo as a — > 0. Since oo (p ) does not depend on a, the validity of each CD (p ) can be extended to cover the Interval i-x^j-x). The n 1 rigorous justification of this restriction will become clear in Appendix VI . Substituting equations (4. 15), (4. L6), and (4. 17) into (4. 15) yields the following result: - 64 - &0 60 (?, oo and... that of cD(p, a) as p — >0. It will be assumed that cr cu is the free-molecule solution and consequently. (jo(r, a j Jirr 5 as 00, a (4. 19) And if CD(p, a) and cD(r, a) are to match each other asymptotically, it is expected that aj(p, a) 2a 5 ^TTP 3 as 0, a 0. (4. 20) With these results the limiting process a — > can be carried out for equation (4. L8). It is now clear from equation (4. L8) that a (a) = a . Indeed, divided by a, equation (4. 18) becomes, in the limit a — > 0, where J o f^o(p) ^ A Jx(p) 3Tr 3 3^^^ (4.
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