TY - JOUR

T1 - A solution of the Boltzmann equation in the presence of inelastic collisions

AU - Shishkova, Irina

AU - Sazhin, Sergei

AU - Xie, Jianfei

N1 - © 2012 Elsevier Inc.

PY - 2012/7/20

Y1 - 2012/7/20

N2 - The effect of inelastic collisions between two molecules on the solution of the Boltzmannequation is taken into account by presenting the change of state of molecules after collisionsas a random (with uniform probability distribution) movement along a surface ofan N-dimensional sphere, the squared radius of which is equal to the total energy of themolecules before and after the collision in the reference system of the centre of mass.The projection of a point on the surface of this sphere in each of N directions gives the rootsquare of the kinetic energy in one of three directions in the physical space, or the internalenergy of one of degrees of freedom, of one of two molecules. The kinetic energies of twomolecules are described by the first six dimensions of the system, and the remainingðN 6Þ dimensions describe the internal energies. This approach is applied to three testproblems: shock wave structure in nitrogen, one-dimensional heat transfer through a mixtureof n-dodecane and nitrogen and one-dimensional evaporation of n-dodecane intonitrogen. In the first problem, the predictions of the model are shown to be close to experimentaldata and also to the predictions of the earlier developed model, based on a differentapproach to taking into account the effects of inelastic collisions. The predicted heatflux for the second problem and mass flux for the third problem are shown to be very weakfunctions of the number of internal degrees of freedom when this number exceeds about15. These results open the way for considering systems with arbitrarily large numbers ofinternal degrees of freedom by reducing the analysis of these systems to the analysis ofsystems with relatively small numbers of internal degrees of freedom.

AB - The effect of inelastic collisions between two molecules on the solution of the Boltzmannequation is taken into account by presenting the change of state of molecules after collisionsas a random (with uniform probability distribution) movement along a surface ofan N-dimensional sphere, the squared radius of which is equal to the total energy of themolecules before and after the collision in the reference system of the centre of mass.The projection of a point on the surface of this sphere in each of N directions gives the rootsquare of the kinetic energy in one of three directions in the physical space, or the internalenergy of one of degrees of freedom, of one of two molecules. The kinetic energies of twomolecules are described by the first six dimensions of the system, and the remainingðN 6Þ dimensions describe the internal energies. This approach is applied to three testproblems: shock wave structure in nitrogen, one-dimensional heat transfer through a mixtureof n-dodecane and nitrogen and one-dimensional evaporation of n-dodecane intonitrogen. In the first problem, the predictions of the model are shown to be close to experimentaldata and also to the predictions of the earlier developed model, based on a differentapproach to taking into account the effects of inelastic collisions. The predicted heatflux for the second problem and mass flux for the third problem are shown to be very weakfunctions of the number of internal degrees of freedom when this number exceeds about15. These results open the way for considering systems with arbitrarily large numbers ofinternal degrees of freedom by reducing the analysis of these systems to the analysis ofsystems with relatively small numbers of internal degrees of freedom.

U2 - 10.1016/j.jcp.2012.07.007

DO - 10.1016/j.jcp.2012.07.007

M3 - Article

SN - 0021-9991

VL - 232

SP - 89

EP - 99

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 1

ER -