Abstract: In this talk, we discuss a multiscale description of fluid dynamics provided by kinetic theory, which describes a fluid by a density distribution of its molecules as a function of time, position and velocity. We consider the evolution of the density distribution given by the Boltzmann equation (BE), which governs the transport and interaction of the fluid molecules. Of particular relevance in our account of multiscale fluid dynamics is the derivation of the Euler and Navier-Stokes-Fourier closure relations as the leading and first-order terms in the so-called Chapman-Enskog expansion of the density distribution in BE [1]. Higher-order closures derived from the Chapman-Enskog expansion, such as the Burnett equations, are unstable and ill-posed [2]. This leaves a gap – a long-standing one in theoretical and computational mathematics [3] – in our understanding of closure relations that follow from the atomistic view of BE to the laws of motion of continua. To address that gap we split this talk into two parts:

In the first part of the talk, we use variational multiscale (VMS) analysis to derive a hierarchy of compressible fluid-dynamic closures from BE that include the Euler, Navier-Stokes-Fourier, and a new alternative to the Burnett equations. We proceed to show that the derived hierarchy of models, including the alternative to the Burnett equations, inherit an entropy inequality that is satisfied by solutions of the Boltzmann equation, rendering all stable and well posed. In the second part of the talk, we propose finite-element methods that leverage the kinetic formulations of the fluid-dynamic models for entropy stability.

We conclude the presentation with numerical results and a discussion of future research directions.

This work is in collaboration with F. Baidoo, L. Caffarelli, I.M. Gamba and T.J.R. Hughes.

[1] Saint-Raymond, L. (2014). A mathematical PDE perspective on the Chapman–Enskog expansion. Bulletin of the American Mathematical Society, 51(2), 247-275.
[2] Bobylev, A. V. (2006). Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations. Journal of statistical physics, 124(2), 371-399.
[3] Gorban, A., & Karlin, I. (2014). Hilbert’s 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. Bulletin of the American Mathematical Society, 51(2), 187-246

Bio: Michael Abdelmalik is a Peter O'Donnell, Jr. Postdoctoral Fellow at the Oden Institute in the University of Texas at Austin, working with Drs. I.M. Gamba and T.J.R. Hughes. Michael received his PhD degree at the Department of Mechanical Engineering as well as two MSc degrees, one in Applied Mathematics and the other in Mechanical Engineering, from the Eindhoven University of Technology where he worked with Dr. H. van Brummelen.

During his PhD, Michael developed state-of-the-art kinetic-theory-based models as well as finite-element methods for rarefied fluid dynamics. Michael’s current research work includes: continuum theory derivation from, and computational methods for, kinetic theory such as the Boltzmann equation for fluids and quasilinear diffusion for plasma; and image-based computational methods for subject-specific bio-continuum mechanics.