# Mathematical modeling of rarefied gas mixtures

Gupta, Vinay Kumar; Torrilhon, Manuel (Thesis advisor); Struchtrup, Henning (Thesis advisor)

*Aachen / Publikationsserver der RWTH Aachen University (2015)* [Dissertation / PhD Thesis]

*Page(s): XII, 149 S. : Ill., graph. Darst.*

Abstract

A mathematical framework based on the higher order Grad's moment equations for studying processes in rarefied gas-mixtures is developed. The fully non-linear Grad's N×13-moment (N×G13) and Grad's N×26-moment (N×G26) equations for a gaseous mixture comprised of N monatomic-inert-ideal gases are derived. The strategy for computing the production terms (or Boltzmann collision integrals) associated with the moment equations for gaseous mixtures interacting with any interaction potential is presented. Employing this strategy, the explicit expressions of the non-linear production terms associated with the N×G13 and N×G26 equations are computed and presented for Maxwell as well as hard-sphere interaction potentials.The boundary conditions complementing the N×G13 and N×G26 equations are derived by extending the Maxwell's accommodation model for a single gas to a gaseous mixture. The moment equations and the boundary conditions are then restricted to binary gas mixtures and the linear stability analysis is performed to conclude that the Grad's 2×13-moment (2×G13) and Grad's 2×26-moment (2×G26) equations for a binary gas-mixture are linearly stable for both Maxwell as well as hard-sphere interaction potentials.Next, the 2×G13 and 2×G26 equations, specialized to Maxwell and hard-sphere interaction potentials, along with the boundary conditions are exploited to study some benchmark problems of fluid mechanics in simple geometries. The heat transfer in a binary gas-mixture confined between two infinite plates having different temperatures is analyzed with all four types of moment systems (2×G13 and 2×G26 equations, both with Maxwell and hard-sphere interaction potentials) and the results are compared with those existing in the literature. Furthermore, a one-dimensional problem of binary gas-mixture having one component infinitely diluted is solved analytically with all four types of moment systems in order to study the flow of the diluted component in the mixture.The numerical method based on finite differences for solving the aforementioned moment systems is demonstrated and employed to various problems in order to study the convergence of the numerical method. The convergence is analyzed for the above one-dimensional problems as well as for the standard two-dimensional problems of bottom-heated and lid-driven square cavities. The preliminary results on heat transfer for the latter two problems are also presented. However, the detailed comparison of the results with those obtained with highly accurate (particle based) methods is left for the future.Moreover, for model reduction, the Grad's moment equations for a binary gas-mixture with Maxwell interaction potential are regularized by employing the order of magnitude method and the regularized 17-moment (R17) equations for a binary gas-mixture with Maxwell interaction potential are derived; the R17 equations are third order accurate in the Knudsen number. The linear stability of R17 equations is analyzed and it is concluded empirically that the R17 equations are linearly stable for binary gas-mixtures with small and moderate mass differences while unstable for those with large mass differences.

### Identifier

- URN: urn:nbn:de:hbz:82-rwth-2015-052546
- REPORT NUMBER: RWTH-2015-05254