Interdisciplinary Applied Mathematics

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In summary, particulate microflows are difficult to compute directly, but the relatively low particle Reynolds number limit allows some simplifications. Microflows typically correspond to values of volume fraction greater than 10~3, and thus nonlinear hydrodynamic interactions have to be accurately modeled. FCM is suitable for particulate microflows because it models accurately such interactions without the extra expense of special remeshing or solution of stiff algebraic systems as in the arbitrary Lagrangian Eu-lerian (ALE) method, the front tracking method, or the fictitious domain method (DLM). To appreciate the numerical resolution requirements, FCM typically employs approximately 5 grid points per particle compared to 15 to 20 points in the aforementioned approaches. With regard to lubrication effects, simple models derived from Stokesian dynamics can be incorporated, when such corrections are needed as in the case of very large volume fraction. Similarly, different collision strategies can be implemented that are application-specific, as is typically done in particulate macroflows. The main drawback of FCM is that it assumes rigid particles, which may not be valid for some microflow applications. In addition, for dense suspensions the results may not be accurate.


Multiscale Modeling of Gas Flows

In this chapter we discuss theory and numerical methodologies for simulating gas flows at the mesoscopic and atomistic levels. Such a description is necessary for gases in the transition and free-molecular regimes. First, we present the direct simulation Monte Carlo (DSMC) method, a stochastic approach suitable for gases. We discuss limitations and errors in the steady version of DSMC and subsequently present a similar analysis for the unsteady DSMC. In order to bridge scales between the continuum and atomistic scales we present an iterative coupling algorithm used often in elliptic problems. To demonstrate convergence and accuracy of this algorithm we revisit the microfilters (see Section 6.5) and present specific details for several cases. We then give an overview of the Boltzmann equation, describing in some detail gas-surface interactions, and include benchmark solutions for validation of numerical codes and of macromodels. A main result relevant for bridging microdynamics and macrodynamics is the Boltzmann inequality, which we also discuss in the last section on lattice Boltzmann methods (LBM) as the H-theorem. These methods represent a “minimal” discrete form of the Boltzmann equation, and are applicable to both compressible and incompressible flows. An interesting new version is the entropic LBM, which enforces the H-theorem in order to guarantee Galilean invariance and numerical stability even for small viscosity values.

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