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A Hybrid Particle-Continuum Method Coupling aFluctuating Fluid
with Suspended Structures
Aleksandar Donev1
Courant Institute, New York University&
Alejandro L. Garcia, San Jose State UniversityJohn B. Bell,
Lawrence Berkeley National Laboratory
1This work performed in part under the auspices of the U.S.
Department of Energy byLawrence Livermore National Laboratory under
Contract DE-AC52-07NA27344.
AMS von Neumann SymposiumSnowbird, UtahJuly 6th, 2011
A. Donev (CIMS) Hybrid 7/2011 1 / 40
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Outline
1 Introduction
2 Particle Methods
3 Fluctuating Hydrodynamics
4 Hybrid Particle-Continuum Method
5 The Importance of Thermal FluctuationsBrownian BeadAdiabatic
Piston
6 Fluctuation-Enhanced Diffusion
7 Conclusions
A. Donev (CIMS) Hybrid 7/2011 2 / 40
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Introduction
Micro- and nano-hydrodynamics
Flows of fluids (gases and liquids) through micro- (µm)
andnano-scale (nm) structures has become technologically
important,e.g., micro-fluidics, microelectromechanical systems
(MEMS).
Biologically-relevant flows also occur at micro- and nano-
scales.
An important feature of small-scale flows, not discussed here,
issurface/boundary effects (e.g., slip in the contact line
problem).
Essential distinguishing feature from “ordinary” CFD:
thermalfluctuations!
I focus here not on the technical details of hybrid methods,
butrather, on using our method to demonstrate the general
conclusionthat fluctuations should be taken into account at the
continuumlevel.
A. Donev (CIMS) Hybrid 7/2011 4 / 40
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Introduction
Example: DNA Filtering
Fu et al., NatureNanotechnology 2 (2007) H. Craighead, Nature
442 (2006)
How to coarse grain the fluid (solvent) and couple it to
thesuspended microstructure (e.g., polymer chain)?
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Introduction
Levels of Coarse-Graining
Figure: From Pep Español, “Statistical Mechanics of
Coarse-Graining”
A. Donev (CIMS) Hybrid 7/2011 6 / 40
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Introduction
This talk: Particle/Continuum Hybrid
Figure: Hybrid method for a polymer chain.
A. Donev (CIMS) Hybrid 7/2011 7 / 40
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Particle Methods
Particle Methods for Complex Fluids
The most direct and accurate way to simulate the interaction
betweenthe solvent (fluid) and solute (beads, chain) is to use a
particlescheme for both: Molecular Dynamics (MD)
mr̈i =∑
j
f ij (rij )
The stiff repulsion among beads demands small time steps,
andchain-chain crossings are a problem.
Most of the computation is “wasted” on the unimportant
solventparticles!
Over longer times it is hydrodynamics (local momentum and
energyconservation) and fluctuations (Brownian motion) that
matter.
We need to coarse grain the fluid model further:
Replacedeterministic interactions with stochastic collisions.
A. Donev (CIMS) Hybrid 7/2011 9 / 40
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Particle Methods
Direct Simulation Monte Carlo (DSMC)
(MNG)
Tethered polymer chain inshear flow.
Stochastic conservative collisions ofrandomly chosen nearby
solventparticles, as in DSMC (also related toMPCD/SRD and DPD).
Solute particles still interact with bothsolvent and other
solute particles ashard or soft spheres.
No fluid structure: Viscous ideal gas.
One can introduce biased collisionmodels to give the fluids
consistenstructure and a non-ideal equationof state. [1].
A. Donev (CIMS) Hybrid 7/2011 10 / 40
Graphics/TetheredPolymer.DSMC.2D.mng
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Fluctuating Hydrodynamics
Continuum Models of Fluid Dynamics
Formally, we consider the continuum field of conserved
quantities
U(r, t) =
ρje
∼= Ũ(r, t) = ∑i
mimiυimiυ
2i /2
δ [r − ri (t)] ,where the symbol ∼= means that U(r, t)
approximates the trueatomistic configuration Ũ(r, t) over long
length and time scales.
Formal coarse-graining of the microscopic dynamics has
beenperformed to derive an approximate closure for the
macroscopicdynamics [2].
This leads to SPDEs of Langevin type formed by postulating
awhite-noise random flux term in the usual
Navier-Stokes-Fourierequations with magnitude determined from
thefluctuation-dissipation balance condition, following Landau
andLifshitz.
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Fluctuating Hydrodynamics
Compressible Fluctuating Hydrodynamics
Dtρ =− ρ∇ · vρ (Dtv) =−∇P + ∇ ·
(η∇v + Σ
)ρcp (DtT ) =DtP + ∇ · (µ∇T + Ξ) +
(η∇v + Σ
): ∇v,
where the variables are the density ρ, velocity v, and
temperature Tfields,
Dt� = ∂t� + v ·∇ (�)∇v = (∇v + ∇vT )− 2 (∇ · v) I/3
and capital Greek letters denote stochastic fluxes:
Σ =√
2ηkBT W .〈Wij (r, t)W?kl (r′, t ′)〉 = (δikδjl + δilδjk −
2δijδkl/3) δ(t − t ′)δ(r − r′).
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Fluctuating Hydrodynamics
Landau-Lifshitz Navier-Stokes (LLNS) Equations
The non-linear LLNS equations are ill-behaved stochastic
PDEs,and we do not really know how to interpret the nonlinearities
precisely.
Finite-volume discretizations naturally impose a
grid-scaleregularization (smoothing) of the stochastic forcing.
A renormalization of the transport coefficients is also
necessary [3].
We have algorithms and codes to solve the compressible
equations(collocated and staggered grid), and recently also the
incompressibleones (staggered grid) [4, 5].
Solving the LLNS equations numerically requires paying attention
todiscrete fluctuation-dissipation balance, in addition to the
usualdeterministic difficulties [4].
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Fluctuating Hydrodynamics
Finite-Volume Schemes
ct = −v ·∇c + χ∇2c + ∇ ·(√
2χW)
= ∇ ·[−cv + χ∇c +
√2χW
]Generic finite-volume spatial discretization
ct = D[(−Vc + Gc) +
√2χ/ (∆t∆V )W
],
where D : faces→ cells is a conservative discrete divergence,G :
cells→ faces is a discrete gradient.Here W is a collection of
random normal numbers representing the(face-centered) stochastic
fluxes.
The divergence and gradient should be duals, D? = −G.Advection
should be skew-adjoint (non-dissipative) if ∇ · v = 0,
(DV)? = − (DV) if (DV) 1 = 0.
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Fluctuating Hydrodynamics
Weak Accuracy
Figure: Equilibrium discrete spectra (static structure factors)
Sρ,ρ(k) ∼ 〈ρ̂ρ̂?〉(should be unity for all discrete wavenumbers) and
Sρ,v(k) ∼ 〈ρ̂v̂?x 〉 (should bezero) for our RK3 collocated
scheme.
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Hybrid Particle-Continuum Method
Fluid-Structure Coupling using Particles
MNG
Split the domain into a particle and acontinuum (hydro)
subdomains,with timesteps ∆tH = K∆tP .
Hydro solver is a simple explicit(fluctuating) compressible
LLNScode and is not aware of particlepatch.
The method is based on AdaptiveMesh and Algorithm
Refinement(AMAR) methodology for conservationlaws and ensures
strict conservationof mass, momentum, and energy.
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Hybrid Particle-Continuum Method
Continuum-Particle Coupling
Each macro (hydro) cell is either particle or continuum. There
isalso a reservoir region surrounding the particle subdomain.
The coupling is roughly of the state-flux form:
The continuum solver provides state boundary conditions for
theparticle subdomain via reservoir particles.The particle
subdomain provides flux boundary conditions for thecontinuum
subdomain.
The fluctuating hydro solver is oblivious to the particle
region: Anyconservative explicit finite-volume scheme can trivially
be substituted.
The coupling is greatly simplified because the ideal particle
fluid hasno internal structure.
”A hybrid particle-continuum method for hydrodynamics of complex
fluids”, A.Donev and J. B. Bell and A. L. Garcia and B. J. Alder,
SIAM J. MultiscaleModeling and Simulation 8(3):871-911, 2010
A. Donev (CIMS) Hybrid 7/2011 19 / 40
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Hybrid Particle-Continuum Method
Our Hybrid Algorithm
1 The hydro solution uH is computed everywhere, including the
particlepatch, giving an estimated total flux ΦH .
2 Reservoir particles are inserted at the boundary of the
particle patchbased on Chapman-Enskog distribution from kinetic
theory,accounting for both collisional and kinetic viscosities.
3 Reservoir particles are propagated by ∆t and collisions are
processed,giving the total particle flux Φp.
4 The hydro solution is overwritten in the particle patch based
on theparticle state up.
5 The hydro solution is corrected based on the more accurate
flux,uH ← uH −ΦH + Φp.
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Hybrid Particle-Continuum Method
Other Hybrid Algorithms
For molecular dynamics (non-ideal particle fluids) the insertion
ofreservoir particles is greatly complicated by the need to account
forthe internal structure of the fluid and requires an overlap
region.
A hybrid method based on a flux-flux coupling between
moleculardynamics and isothermal compressible fluctuating
hydrodynamics hasbeen developed by Coveney, De Fabritiis,
Delgado-Buscalioni andco-workers [6].
Some comparisons between different forms of coupling
(state-state,state-flux, flux-state, flux-flux) has been performed
by Ren [7].
Reaching relevant time scales ultimately requires a
stochasticimmersed structure approach coupling immersed structures
directlyto a fluctuating solver (work in progresss).
A. Donev (CIMS) Hybrid 7/2011 21 / 40
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The Importance of Thermal Fluctuations Brownian Bead
Brownian Bead
Themal fluctuations push a sphere of size a and density ρ′
suspendedin a stationary fluid with density ρ and viscosity η
(Brownian walker)with initial velocity Vth ≈
√kT/M, M ≈ ρ′a3.
The classical picture of Brownian motion indicates
threewidely-separated timescales:
Sound waves are generated from the sudden compression of the
fluidand they take away a fraction of the kinetic energy during a
sonic timetsonic ≈ a/c, where c is the (adiabatic) sound
speed.Viscous dissipation then takes over and slows the
particlenon-exponentially over a viscous time tvisc ≈ ρa2/η, where
η is theshear viscosity.Thermal fluctuations get similarly
dissipated, but their constantpresence pushes the particle
diffusively over a diffusion timetdiff ≈ a2/D, where D ∼
kT/(aη).
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The Importance of Thermal Fluctuations Brownian Bead
Velocity Autocorrelation Function
We investigate the velocity autocorrelation function (VACF) for
aBrownian bead
C (t) = 2d−1 〈v(t0) · v(t0 + t)〉
From equipartition theorem C (0) = kBT/M.
For a neutrally-boyant particle, ρ′ = ρ, incompressible
hydrodynamictheory gives C (0) = 2kBT/3M because one third of the
kineticenergy decays at the sound time scale.
Hydrodynamic persistence (conservation) gives a
long-timepower-law tail C (t) ∼ (kBT/M)(t/tvisc)−3/2 that can be
quantifiedusing fluctuating hydrodynamics.
The diffusion coefficient is the integral of the VACF and
isstrongly-affected by the tail.
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The Importance of Thermal Fluctuations Brownian Bead
VACF
0.01 0.1 1
t / tvisc
1
0.1
0.01
M C
(t)
/ k
BT
Stoch. hybrid (L=2)
Det. hybrid (L=2)
Stoch. hybrid (L=3)
Det. hybrid (L=3)
Particle (L=2)
Theory
0.01 0.1 1
t cs / R
1
0.75
0.5
0.25
tL=2
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The Importance of Thermal Fluctuations Adiabatic Piston
The adiabatic piston problem
MNG
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The Importance of Thermal Fluctuations Adiabatic Piston
Relaxation Toward Equilibrium
0 2500 5000 7500 10000t
6
6.25
6.5
6.75
7
7.25
7.5
7.75x(
t)ParticleStoch. hybridDet. hybrid
0 250 500 750 10006
6.5
7
7.5
8
Figure: Massive rigid piston (M/m = 4000) not in mechanical
equilibrium: Thedeterministic hybrid gives the wrong answer!
A. Donev (CIMS) Hybrid 7/2011 27 / 40
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The Importance of Thermal Fluctuations Adiabatic Piston
VACF for Piston
0 1 2 3
0
5×10-4
1×10-3ParticleStoch. wP=2
Det. wP=2
Det. wP=4
Det. wP=8
0 50 100 150 200 250t
-5.0×10-4
-2.5×10-4
0.0
2.5×10-4
5.0×10-4
7.5×10-4
1.0×10-3C(t)
ParticleStoch. hybridDet. (wP=4)
Det. x10
Figure: The VACF for a rigid piston of mas M/m = 1000 at thermal
equilibrium:Increasing the width of the particle region does not
help: One mustinclude the thermal fluctuations in the continuum
solver!
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Fluctuation-Enhanced Diffusion
Nonequilibrium Fluctuations
When macroscopic gradients are present, steady-state
thermalfluctuations become long-range correlated.
Consider a binary mixture of fluids and consider
concentrationfluctuations around a steady state c0(r):
c(r, t) = c0(r) + δc(r, t)
The concentration fluctuations are advected by the
randomvelocities v(r, t) = δv(r, t), approximately:
∂t (δc) + (δv) ·∇c0 = χ∇2 (δc) +√
2χkBT (∇ ·Wc)
The velocity fluctuations drive and amplify the
concentrationfluctuations leading to so-called giant fluctuations
[8].
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Fluctuation-Enhanced Diffusion
Fractal Fronts in Diffusive Mixing
Figure: Snapshots of concentration in a miscible mixture showing
the developmentof a rough diffusive interface between two miscible
fluids in zero gravity [3, 8, 5].
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Fluctuation-Enhanced Diffusion
Giant Fluctuations in Experiments
Figure: Experimental results by A. Vailati et al. from a
microgravity environment[8] showing the enhancement of
concentration fluctuations in space (box scale ismacroscopic: 5mm
on the side, 1mm thick).
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Fluctuation-Enhanced Diffusion
Fluctuation-Enhanced Diffusion Coefficient
The nonlinear concentration equation includes a contribution to
themass flux due to advection by the fluctuating velocities,
∂t (δc) + (δv) ·∇c0 = ∇ · [− (δc) (δv) + χ∇ (δc)] + . . .
Simple (quasi-linear) perturbative theory suggests that
concentrationand velocity fluctuations become correlated and
−〈(δc) (δv)〉 ≈ (∆χ)∇c0.
The fluctuation-renormalized diffusion coefficient is χ+
∆χ(think of eddy diffusivity in turbulent transport).
Because fluctuations are affected by boundaries, ∆χ is
system-sizedependent.
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Fluctuation-Enhanced Diffusion
Fluctuation-Enhanced Diffusion Coefficient
Consider the effective diffusion coefficient in a system of
dimensionsLx × Ly × Lz with a concentration gradient imposed along
the y axis.In two dimensions, Lz � Lx � Ly , linearized
fluctuatinghydrodynamics predicts a logarithmic divergence
χ(2D)eff ≈ χ+
kBT
4πρ(χ+ ν)Lzln
LxL0
In three dimensions, Lx = Lz = L� Ly , χeff converges as L→∞to
the macroscopic diffusion coefficient,
χ(3D)eff ≈ χ+
α kBT
ρ(χ+ ν)
(1
L0− 1
L
)We have verified these predictions using particle (DSMC)
simulationsat hydrodynamic scales [3].
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Fluctuation-Enhanced Diffusion
Particle Simulations
4 8 16 32 64 128 256 512 1024
Lx / λ
3.65
3.675
3.7
3.725
3.75
χKinetic theory
χeff
(System A)
χ0 (System A)
χeff
(System B)
χ0 (System B)
χeff
(SPDE, A)
Theory χ0 (A)
Theory χ0 (B)
Theory χeff
(a)
Figure: Divergence of diffusion coefficient in two dimensions.A.
Donev (CIMS) Hybrid 7/2011 35 / 40
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Fluctuation-Enhanced Diffusion
Microscopic, Mesoscopic and Macroscopic Fluid Dynamics
Instead of an ill-defined “molecular” or “bare” diffusivity, one
shoulddefine a locally renormalized diffusion coefficient χ0 that
dependson the length-scale of observation.
This coefficient accounts for the arbitrary division between
continuumand particle levels inherent to fluctuating
hydrodynamics.
A deterministic continuum limit does not exist in two
dimensions, andis not applicable to small-scale finite systems in
three dimensions.
Fluctuating hydrodynamics is applicable at a broad range of
scalesif the transport coefficient are renormalized based on the
cutoff scalefor the random forcing terms.
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Conclusions
Conclusions
Coarse-grained particle methods can be used to
acceleratehydrodynamic calculations at small scales.
Hybrid particle continuum methods closely reproduce
purelyparticle simulations at a fraction of the cost.
It is necessary to include fluctuations in the continuum solver
inhybrid methods.
Thermal fluctuations affect the macroscopic transport in
fluids.
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Conclusions
Future Directions
Improve and implement stochastic particle methods (parallelize,
addchemistry, analyze theoretically).
Direct fluid-structure coupling between fluctuating
hydrodynamicsand microstructure.
Develop numerical schemes for Low-Mach Number
fluctuatinghydrodynamics.
Ultimately we require an Adaptive Mesh and AlgorithmRefinement
(AMAR) framework that couples a particle model(micro), with
compressible fluctuating Navier-Stokes (meso), andincompressible or
low Mach solver (macro).
A. Donev (CIMS) Hybrid 7/2011 39 / 40
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Conclusions
References
A. Donev, A. L. Garcia, and B. J. Alder.
Stochastic Hard-Sphere Dynamics for Hydrodynamics of Non-Ideal
Fluids.Phys. Rev. Lett, 101:075902, 2008.
P. Español.
Stochastic differential equations for non-linear
hydrodynamics.Physica A, 248(1-2):77–96, 1998.
A. Donev, A. L. Garcia, Anton de la Fuente, and J. B. Bell.
Diffusive Transport Enhanced by Thermal Velocity
Fluctuations.Phys. Rev. Lett., 106(20):204501, 2011.
A. Donev, E. Vanden-Eijnden, A. L. Garcia, and J. B. Bell.
On the Accuracy of Explicit Finite-Volume Schemes for
Fluctuating Hydrodynamics.CAMCOS, 5(2):149–197, 2010.
F. Balboa, J. Bell, R. Delgado-Buscallioni, A. Donev, T. Fai, A.
Garcia, B. Griffith, and C. Peskin.
Staggered Schemes for Incompressible Fluctuating
Hydrodynamics.Submitted, 2011.
G. De Fabritiis, M. Serrano, R. Delgado-Buscalioni, and P. V.
Coveney.
Fluctuating hydrodynamic modeling of fluids at the
nanoscale.Phys. Rev. E, 75(2):026307, 2007.
W. Ren.
Analytical and numerical study of coupled atomistic-continuum
methods for fluids.J. Comp. Phys., 227(2):1353–1371, 2007.
A. Vailati, R. Cerbino, S. Mazzoni, C. J. Takacs, D. S. Cannell,
and M. Giglio.
Fractal fronts of diffusion in microgravity.Nature
Communications, 2:290, 2011.
A. Donev (CIMS) Hybrid 7/2011 40 / 40
IntroductionParticle MethodsFluctuating HydrodynamicsHybrid
Particle-Continuum MethodThe Importance of Thermal
FluctuationsBrownian BeadAdiabatic Piston
Fluctuation-Enhanced DiffusionConclusions