In this paper, based on simplified Boltzmann equation, we explore the inversedesign of mesoscopic models for compressible flow using the ChapmanEnskog analysis. Starting from the singlerelaxationtime Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using NavierStokesFourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leadingorder ChapmanEnskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulktoshear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three wellknown mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the ChapmanEnskog expansion at the NSF order. Finally, based on the higherorder ChapmanEnskog expansion of themore »
A Tutorial on Matrix Perturbation Theory (using compact matrix notation)
Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the nonHermitian and degenerate cases, as well as for higher order terms.
 Publication Date:
 NSFPAR ID:
 10322686
 Journal Name:
 ArXivorg
 ISSN:
 23318422
 Sponsoring Org:
 National Science Foundation
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