Vol. 9, No. 1, 2014

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Low Mach number fluctuating hydrodynamics of diffusively mixing fluids

Aleksandar Donev, Andy Nonaka, Yifei Sun, Thomas G. Fai, Alejandro L. Garcia and John B. Bell

Vol. 9 (2014), No. 1, 47–105
Abstract

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations represent a coarse-graining of the microscopic dynamics of the fluid molecules in both space and time and eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatiotemporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge–Kutta temporal integrators that strictly maintain the equation-of-state constraint. The resulting spatiotemporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients and investigate the validity of common simplifications such as neglecting the spatial nonhomogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular-dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.

Keywords
fluctuating hydrodynamics, low Mach expansion, molecular dynamics, giant fluctuations
Mathematical Subject Classification 2010
Primary: 76T99
Secondary: 65M08
Milestones
Received: 26 November 2013
Revised: 14 January 2014
Accepted: 14 January 2014
Published: 13 May 2014
Authors
Aleksandar Donev
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012
United States
Andy Nonaka
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States
Yifei Sun
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012
United States
Leon H. Charney Division of Cardiology
Department of Medicine
New York University School of Medicine
New York, NY 10016
United States
Thomas G. Fai
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012
United States
Alejandro L. Garcia
Department of Physics and Astronomy
San Jose State University
1 Washington Square
San Jose, CA 95192
United States
John B. Bell
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States