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Fluctuating Hydrodynamics


At the molecular scale, fluids are inherently noisy with thermally induced fluctuations playing a key role in the dynamics. When mechanical instabilities, chemical reactions, and other phenomena at the microscopic scale are sensitive to these fluctuations, fluid behavior at larger scales can be significantly affected. The goal of this project is to develop stochastic numerical methods to simulate these types of multiscale problems arising in fluids. Accurate modeling of these types of multiscale phenomena requires the correct decomposition of the component processes for these fluctuations. The correct treatment of fluctuations is especially important for nonlinear systems, such as those undergoing phase transitions, nucleation, barrier-crossing, Brownian motors, noise-driven instabilities, and combustive ignition to name a few.

Over the last few years, we have developed fluctuating hydrodynamics formulations and numerical methods not only for gas mixtures (based on the compressible Landau-Lifshitz Navier-Stokes equations) but also for liquid mixtures (based on the low Mach number formulation). Our formulations are derived from the principles of nonequilibrium thermodynamics and deal with multi-species systems without the need to assume a solvent species. Our numerical methods are based on the staggered-grid finite-volume approach (preserving the discrete fluctuation-dissipation balance).

Our current focus is on extending the existing low-Mach multi-species framework to electrolyte solutions and incorporating an accurate mesoscopic description of reactions. We aim at developing accurate and robust stochastic numerical methods for the realistic simulation of microfluids under thermal fluctuations.

☞ For more information, contact John B. Bell JBBell@lbl.gov.


Low Mach Number Fluctuating Hydrodynamics.

Using a low Mach number approach that eliminates fast sound waves (pressure fluctuations) from the full compressible equations, we have obtained a quasi-incompressible formulation and developed a simulation code for multicomponent mixtures. The animation on the left shows the development of a mixed-mode instability as a layer of salty water is placed on top of a horizontal layer of less-dense sweet water. The animation on the right shows a different configuration where the salty water has been further diluted in water so that denser sweet water lies underneath. Here we see the development of a diffusive layer convection instability. The color plots show the vertically averaged density (horizontal plane) and planar slices of density (vertical planes). The observed giant fluctuations are caused by long-range correlations between fluctuations. ☞ For more information, contact Andy Nonaka AJNonaka@lbl.gov.





Electrolyte Solutions.

Our low Mach code can be used for studying electrolyte systems. The animation on the left shows a type of electrokinetic instability that occurs when an electric field is applied to an inhomogeneous solution. The top region is more dilute than the bottom region. Because of the different behavior of positively and negatively charged species in the presence of the electric field, charged layers appear and become unstable. The animation on the right shows fluctuation-enhanced electric conductivity in an electrolyte solution. When an electric field is applied to an homogeneous solution, charged regions spontaneously appear due to fluctuations and couple with the electric field. This results in enhanced velocity fluctuations (as demonstrated by the velocities shown in the animation) as well as enhanced electric transport. ☞ For more information, contact Jean-Philippe Péraud JPeraud@lbl.gov.

Reaction-Diffusion Problems.

As a first step towards incorporating stochastic reactions, we have developed stochastic simulation methods for reaction-diffusion systems. Our approach combines the rigor of the master equation approach for reactions and the efficiency of the fluctuating hydrodynamics approach for diffusion. The animation on the left shows pattern formation. Compared with the widely-used stochastic simulation algorithm (SSA), our method produces the same results much faster. The animation on the right shows front propagation. This example involves the equivalent of 1012 molecules, which cannot be simulated by SSA. ☞ For more information, contact Changho Kim ChanghoKim@lbl.gov.



J-P. Péraud, A. Nonaka, J. B. Bell, A. Donev, and A. L. Garcia, "Fluctuation-Enhanced Electric Conductivity in Electrolyte Solutions," submitted for publication. [arxiv]

C. Kim, A. Nonaka, J. B. Bell, A. L. Garcia, and A. Donev, "Stochastic Simulation of Reaction-Diffusion Systems: A Fluctuating-Hydrodynamics Approach," J. Chem. Phys. 146, 124110, 2017. [doi] [arxiv]

J-P. Péraud, A. Nonaka, A. Chaudhri, J. B. Bell, A. Donev, and A. L. Garcia, "Low Mach Number Fluctuating Hydrodynamics for Electrolytes," Phys. Rev. Fluids 1, 074103, 2016. [doi] [arxiv]

A. K. Bhattacharjee, K. Balakrishnan, A. L. Garcia, J. B. Bell, and A. Donev, "Fluctuating Hydrodynamics of Multispecies Reactive Mixtures," J. Chem. Phys. 142, 224107, 2015. [doi] [arxiv]

A. Donev, A. Nonaka, A. K. Bhattacharjee, A. L. Garcia, and J. B. Bell, "Low Mach Number Fluctuating Hydrodynamics of Multispecies Liquid Mixtures," Phys. Fluids 27, 037103, 2015. [doi] [arxiv]

A. Nonaka, Y. Sun, J. B. Bell, and A. Donev, "Low Mach Number Fluctuating Hydrodynamics of Binary Liquid Mixtures," Comm. App. Math. Comp. Sci. 10, 163-204, 2015. [doi] [arxiv]

A. Chaudhri, J. B. Bell, A. L. Garcia, and A. Donev, "Modeling Multi-Phase Flow Using Fluctuating Hydrodynamics", Phys. Rev. E 90, 033014, 2014. [doi] [arxiv]

M. Cai, A. Nonaka, B. E. Griffith, J. B. Bell, and A. Donev, "Efficient Variable-Coefficient Finite-Volume Stokes Solvers," Commun. Comput. Phys. 16, 1263-1297, 2014. [doi] [arxiv]

A. Donev, A. Nonaka, Y. Sun, T. Fai, A. Garcia, and J. B. Bell, "Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids," Comm. App. Math. Comp. Sci. 9, 47-105, 2014. [doi] [arxiv]

K. Balakrishnan, A. Garcia, A. Donev, and J. B. Bell, "Fluctuating Hydrodynamics of Multispecies Nonreactive Mixtures," Phys. Rev. E 89, 013017, 2014. [doi] [arxiv]

F. Balboa Usabiaga, J. B. Bell, R. Delgado-Buscalioni, A. Donev, T. Fai, B. Griffith, and C. Peskin, "Staggered Schemes for Fluctuating Hydrodynamics," Multiscale Model. Simul. 10, 1360-1408, 2012. [doi] [arxiv]

K. Balakrishnan, J. B. Bell, A. Donev, and A. Garcia, "Fluctuating Hydrodynamics and Direct Simulation Monte Carlo," 28th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. 1501, 695-704, 2012. [pdf]

A. Garcia, A. Donev, J. B. Bell, and B. Alder, "Hydrodynamic Fluctuations in a Particle-Continuum Hybrid for Complex Fluids," 27th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. 1333, 551-556, 2011. [pdf]

A. Donev, A. de la Fuente, J. B. Bell, and A. L. Garcia, "Enhancement of Diffusive Transport by Nonequilibrium Thermal Fluctuations," JSTAT 2011, P06014, (2011). [doi] [arxiv]

A. Donev, A. de la Fuente, J. B. Bell, and A. L. Garcia, "Diffusive Transport by Thermal Velocity Fluctuations," Phys. Rev. Lett. 106, 204501, 2011. [doi] [arxiv]

A. Donev, J. B. Bell, A. L. Garcia, and B. J. Alder, "A Hybrid Particle-Continuum Method for Hydrodynamics of Complex Fluids," Multiscale Model. Simul. 8, 871-911, 2010. [doi [arxiv]

A. Donev, E. Vanden-Eijnden, A. Garcia, and J. Bell, "On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics," Comm. App. Math. Comp. Sci. 5, 149-197, 2010. [doi] [arxiv]

J. B. Bell, A. L. Garcia, and S. A. Williams, "Computational Fluctuating Fluid Dynamics," ESAIM: Mathematical Modelling and Numerical Analysis 44, 1085-1105, 2010. [doi] [pdf]

A. Donev, A. L. Garcia, and B. J. Alder, "A Thermodynamically-Consistent Non-Ideal Stochastic Hard Sphere Fluid," J. Stat. Mech., P11008, 2009. [doi] [arxiv]

S. A. Williams, J. B. Bell, and A. L. Garcia, "Algorithm Refinement for Fluctuating Hydrodynamics," Multiscale Model. Simul. 6, 1256-1280, 2008. [doi] [pdf]

J. B. Bell, A. L. Garcia, and S. A. Williams, "Numerical Methods for the Stochastic Landau-Lifshitz Navier-Stokes Equations," Phys. Rev. E 76, 016708, 2007. [doi] [arxiv]

J. B. Bell, J. Foo, and A. L. Garcia, "Algorithm Refinement for the Stochastic Burgers' Equation," J. Comp. Phys. 223, 451-468, 2007. [doi] [pdf]

A. L. Garcia, J. B. Bell, W. Y. Crutchfield, and B. J. Alder, "Adaptive Mesh and Algorithm Refinement Using Direct Simulation Monte Carlo," J. Comp. Phys. 154, 134-155, 1999. [doi] [ps.gz]