On the positivity of an auxiliary function of the BGK model for slow chemical reactions
Section snippets
BGK model for gas mixtures undergoing slow chemical reaction
The understanding of chemically reactive gas mixtures is getting more important as the high energy flows such as the supersonic flows or the combustion flows become ubiquitous. In this paper, we are interested in the positivity of an auxiliary function of the following BGK model for gas mixtures undergoing slow chemical reaction [1]: The velocity distribution function denotes the number density of th molecule on the position with the
Violation of positivity of the auxiliary function
In this section, we show that the auxiliary function can take a negative value. First we need the following lemma concerning the incomplete gamma function .
Lemma 2.1 We have for all .
Proof First, we recall the identity: to find Therefore, we clearly have For the lower bound, we use to get which, in view of (2.1), gives the desired
Acknowledgments
The authors appreciate anonymous reviewers for their fruitful suggestions. The work of D. Kim was supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study. The work of S.-B. Yun was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02.
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