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Computational fluctuating fluid dynamics

Published online by Cambridge University Press:  26 August 2010

John B. Bell ,
Alejandro L. Garcia  and
Sarah A. Williams
John B. Bell
Affiliation:
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA.
Alejandro L. Garcia
Affiliation:
Department of Physics, San Jose State University, San Jose, California, 95192, USA. [email protected]
Sarah A. Williams
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill NC 27599, USA.
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Abstract

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

Keywords

Fluctuating hydrodynamicsLandau-Lifshitz-Navier-Stokes equationsstochastic partial differential equationsfinite difference methodsbinary gas mixtures
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 1085 - 1105
Copyright
© EDP Sciences, SMAI, 2010

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References

Alexander, F.J. and Garcia, A.L., The direct simulation Monte Carlo method. Comp. Phys. 11 (1997) 588–593. CrossRef
Astumian, R.D. and Hanggi, P., Brownian motors. Phys. Today 55 (2002) 33–39. CrossRef
Baras, F., Nicolis, G., Mansour, M.M. and Turner, J.W., Stochastic theory of adiabatic explosion. J. Statis. Phys. 32 (1983) 1–23. CrossRef
Bell, J.B., Garcia, A.L. and Williams, S.A., Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations. Phys. Rev. E 76 (2007) 016708. CrossRef
Bena, I., Mansour, M.M. and Baras, F., Hydrodynamic fluctuations in the Kolmogorov flow: Linear regime. Phys. Rev. E 59 (1999) 5503–5510. CrossRef
Bena, I., Baras, F. and Mansour, M.M., Hydrodynamic fluctuations in the Kolmogorov flow: Nonlinear regime. Phys. Rev. E 62 (2000) 6560–6570. CrossRef
G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon, Oxford (1994).
Bixon, M. and Zwanzig, R., Boltzmann-Langevin equation and hydrodynamic fluctuations. Phys. Rev. 187 (1969) 267–272. CrossRef
Blömker, D., Maier-Paape, S. and Wanner, T., Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Trans. Amer. Math. Soc. 360 (2008) 449–489. CrossRef
Calzetta, E., Relativistic fluctuating hydrodynamics. Class. Quantum Grav. 15 (1998) 653–667. CrossRef
Ceniceros, H.D. and Mohler, G.O., A practical splitting method for stiff SDEs with application to problems with small noise. Multiscale Model. Simul. 6 (2007) 212–227. CrossRef
Cohen, C., Sutherland, J.W.H. and Deutch, J.M., Hydrodynamic correlation functions for binary mixtures. Phys. Chem. Liquids 2 (1971) 213–235. CrossRef
De Fabritiis, G., Delgado-Buscalioni, R. and Coveney, P.V., Multiscale modeling of liquids with molecular specificity. Phys. Rev. Lett. 97 (2006) 134501. CrossRef
De Fabritiis, G., Serrano, M., Delgado-Buscalioni, R. and Coveney, P.V., Fluctuating hydrodynamic modeling of fluids at the nanoscale. Phys. Rev. E 75 (2007) 026307. CrossRef
J.M.O. de Zarate and J.V. Sengers, Hydrodynamic Fluctuations in Fluids and Fluid Mixtures. Elsevier Science (2007).
Delgado-Buscalioni, R. and De Fabritiis, G., Embedding molecular dynamics within fluctuating hydrodynamics in multiscale simulations of liquids. Phys. Rev. E 76 (2007) 036709. CrossRef
Eggers, J., Dynamics of liquid nanojets. Phys. Rev. Lett. 89 (2002) 084502. CrossRef
Español, P., Stochastic differential equations for non-linear hydrodynamics. Physica A 248 (1998) 77. CrossRef
Fox, R.F. and Uhlenbeck, G.E., Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations. Phys. Fluids 13 (1970) 1893–1902. CrossRef
Garcia, A.L., Nonequilibrium fluctuations studied by a rarefied gas simulation. Phys. Rev. A 34 (1986) 1454–1457. CrossRef
A.L. Garcia, Numerical Methods for Physics. Second edition, Prentice Hall (2000).
Garcia, A.L., Estimating hydrodynamic quantities in the presence of microscopic fluctuations. Commun. Appl. Math. Comput. Sci. 1 (2006) 53–78. CrossRef
Garcia, A.L. and Penland, C., Fluctuating hydrodynamics and principal oscillation pattern analysis. J. Stat. Phys. 64 (1991) 1121–1132. CrossRef
Garcia, A.L., Mansour, M.M., Lie, G. and Clementi, E., Numerical integration of the fluctuating hydrodynamic equations. J. Stat. Phys. 47 (1987) 209–228. CrossRef
Garcia, A.L., Mansour, M.M., Lie, G.C., Mareschal, M. and Clementi, E., Hydrodynamic fluctuations in a dilute gas under shear. Phys. Rev. A 36 (1987) 4348–4355. CrossRef
Giupponi, G., De Fabritiis, G. and Coveney, P.V., Hybrid method coupling fluctuating hydrodynamics and molecular dynamics for the simulation of macromolecules. J. Chem. Phys. 126 (2007) 154903. CrossRef
J.O. Hirshfelder, C.F. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids. John Wiley & Sons (1954).
Horntrop, D.J., Mesoscopic simulation of Ostwald ripening. J. Comp. Phys. 218 (2006) 429–441. CrossRef
Horntrop, D.J., Spectral method study of domain coarsening. Phys. Rev. E 75 (2007) 046703. CrossRef
Ibañes, M., García-Ojalvo, J, Toral, R. and Sancho, J.M., Dynamics and scaling of noise-induced domain growth. Eur. Phys. J. B 18 (2000) 663–673. CrossRef
Kadau, K., Germann, T.C., Hadjiconstantinou, N.G., Lomdahl, P.S., Dimonte, G., Holian, B.L. and Nanohydrodynamics, B.J. Alder simulations: An atomistic view of the Rayleigh-Taylor instability. PNAS 101 (2004) 5851–5855. CrossRef
Kadau, K., Rosenblatt, C., Barber, J., Germann, T., Huang, Z., Carlès, P. and The, B. Alder importance of fluctuations in fluid mixing. PNAS 104 (2007) 7741–7745. CrossRef
Kang, W. and Landman, U., Universality crossover of the pinch-off shape profiles of collapsing liquid nanobridges in vacuum and gaseous environments. Phys. Rev. Lett. 98 (2007) 064504. CrossRef
Kawczynski, A.L. and Nowakowski, B., Stochastic transitions through unstable limit cycles in a model of bistable thermochemical system. Phys. Chem. Chem. Phys. 10 (2008) 289–296. CrossRef
Kelly, G.E. and Lewis, M.B., Hydrodynamic fluctuations. Phys. Fluids 14 (1971) 1925–1931. CrossRef
Lacasta, A.M., Sancho, J.M. and Sagués, F., Phase separation dynamics under stirring. Phys. Rev. Lett. 75 (1995) 1791–1794. CrossRef
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics 6. Pergamon (1959).
L.D. Landau and E.M. Lifshitz, Statistical Physics, Course of Theoretical Physics 5. Pergamon, 3rd edition, part 1st edition (1980).
Law, B.M. and Nieuwoudt, J.C., Noncritical liquid mixtures far from equilibrium: the Rayleigh line. Phys. Rev. A 40 (1989) 3880–3885. CrossRef
Lemarchand, A. and Nowakowski, B., Fluctuation-induced and nonequilibrium-induced bifurcations in a thermochemical system. Mol. Simulat. 30 (2004) 773–780. CrossRef
Mansour, M.M., Garcia, A.L., Lie, G.C. and Clementi, E., Fluctuating hydrodynamics in a dilute gas. Phys. Rev. Lett. 58 (1987) 874–877. CrossRef
Mansour, M.M., Garcia, A.L., Turner, J.W. and Mareschal, M., On the scattering function of simple fluids in finite systems. J. Stat. Phys. 52 (1988) 295–309. CrossRef
Mansour, M.M., Van den Broeck, C., Bena, I. and Baras, F., Spurious diffusion in particle simulations of the Kolmogorov flow. Europhys. Lett. 47 (1999) 8–13. CrossRef
Mareschal, M., Mansour, M.M., Sonnino, G. and Kestemont, E., Dynamic structure factor in a nonequilibrium fluid: a molecular-dynamics approach. Phys. Rev. A 45 (1992) 7180–7183. CrossRef
Meurs, P., Van den Broeck, C. and Garcia, A.L., Rectification of thermal fluctuations in ideal gases. Phys. Rev. E 70 (2004) 051109. CrossRef
Moro, E., Hybrid method for simulating front propagation in reaction-diffusion systems. Phys. Rev. E 69 (2004) 060101. CrossRef
Moseler, M. and Landman, U., Formation, stability, and breakup of nanojets. Science 289 (2000) 1165–1169. CrossRef
Nieuwoudt, J.C. and Law, B.M., Theory of light scattering by a nonequilibrium binary mixture. Phys. Rev. A 42 (1989) 2003–2014. CrossRef
Nowakowski, B. and Lemarchand, A., Stochastic effects in a thermochemical system with newtonian heat exchange. Phys. Rev. E 64 (2001) 061108.
Nowakowski, B. and Lemarchand, A., Sensitivity of explosion to departure from partial equilibrium. Phys. Rev. E 68 (2003) 031105. CrossRef
Oster, G., Darwin's motors. Nature 417 (2002) 25. CrossRef
R.K. Pathria, Statistical Mechanics. Butterworth-Heinemann, Oxford (1996).
Quentin, G. and Rehberg, I., Direct measurement of hydrodynamic fluctuations in a binary mixture. Phys. Rev. Lett. 74 (1995) 1578–1581. CrossRef
L. Rayleigh, Scientific Papers II. Cambridge University Press, Cambridge (1900) 200–207.
Schmitz, R., Fluctuations in nonequilibrium fluids. Phys. Rep. 171 (1988) 1–58. CrossRef
Sengers, J.V. and de Zárate, J.M.O., Thermal fluctuations in non-equilibrium thermodynamics. J. Non-Equilib. Thermodyn. 32 (2007) 319–329. CrossRef
Sharp, D.H., An overview of Rayleigh-Taylor instability. Phys. D 12 (1984) 3–18. CrossRef
Y. Sone, Kinetic Theory and Fluid Dynamics. Springer (2002).
Taylor, G.I., The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. London Ser. A 201 (1950) 192–196. CrossRef
Van den Broeck, C., Kawai, R. and Meurs, P., Exorcising a Maxwell demon. Phys. Rev. Lett. 93 (2004) 090601. CrossRef
Vladimirova, N., Malagoli, A. and Mauri, R., Diffusion-driven phase separation of deeply quenched mixtures. Phys. Rev. E 58 (1998) 7691–7699. CrossRef
Williams, S.A., Bell, J.B. and Garcia, A.L., Algorithm refinement for fluctuating hydrodynamics. Multiscale Model. Simul. 6 (2008) 1256–1280. CrossRef
Wu, M., Ahlers, G. and Cannell, D.S., Thermally induced fluctuations below the onset of Rayleigh-Bénard convection. Phys. Rev. Lett. 75 (1995) 1743–1746. CrossRef