On the positivity of an auxiliary function of the BGK model for slow chemical reactions

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Abstract

We show that an auxiliary function of the BGK model for slow chemical reactions that corresponds to the local temperature in the case of the original BGK model for inert gases, can take negative values.

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]: tfi+vxfi=νiτMifi(i=1,2,3,4).The velocity distribution function fi(x,v,t) (i=1,2,3,4) denotes the number density of ith molecule on the position xR3 with the

Violation of positivity of the auxiliary function T3

In this section, we show that the auxiliary function T3 can take a negative value. First we need the following lemma concerning the incomplete gamma function Γ(32,x).

Lemma 2.1

We have 112x<xexΓ(32,x)<1for all 0<x<.

Proof

First, we recall the identity: Γ(s+1,x)=sΓ(s,x)+xsex,s>0to find 1xexΓ(32,x)=Γ(32,x)xexΓ(32,x)=Γ(12,x)2Γ(32,x)>0.Therefore, we clearly have xexΓ(32,x)<1.For the lower bound, we use Γ(12,x)=x1tetdt<xtxetdt=1xΓ(32,x)to get Γ(12,x)Γ(32,x)<1xwhich, 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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