Coupling an Incompressible Fluctuating Fluid with
Suspended Structures
Aleksandar Donev
Courant Institute, New York University
&
Rafael Delgado-Buscalioni, UAM
Florencio Balboa Usabiaga, UAM
Boyce Griffith, Courant
SIAM Conference on Mathematical Aspects of Materials Science
Philadelphia, June 2013
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Outline
1
Incompressible Inertial Coupling
2
Numerics
3
Results
4
Outlook
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Levels of Coarse-Graining
Figure: From Pep Español, “Statistical Mechanics of Coarse-Graining”
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Incompressible Inertial Coupling
Fluid-Structure Coupling
We want to construct a bidirectional coupling between a fluctuating
fluid and a small spherical Brownian particle (blob).
Macroscopic coupling between flow and a rigid sphere:
No-slip boundary condition at the surface of the Brownian particle.
Force on the bead is the integral of the (fluctuating) stress tensor over
the surface.
The above two conditions are questionable at nanoscales, but even
worse, they are very hard to implement numerically in an efficient and
stable manner.
We saw already that fluctuations should be taken into account at
the continuum level.
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Incompressible Inertial Coupling
Brownian Particle Model
Consider a Brownian “particle” of size a with position q(t) and
velocity u = q̇, and the velocity field for the fluid is v(r, t).
We do not care about the fine details of the flow around a particle,
which is nothing like a hard sphere with stick boundaries in reality
anyway.
Take an Immersed Boundary Method (IBM) approach and describe
the fluid-blob interaction using a localized smooth kernel δa (∆r) with
compact support of size a (integrates to unity).
Often presented as an interpolation function for point Lagrangian
particles but here a is a physical size of the particle (as in the Force
Coupling Method (FCM) of Maxey et al [1]).
We will call our particles “blobs” since they are not really point
particles.
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Incompressible Inertial Coupling
Local Averaging and Spreading Operators
Postulate a no-slip condition between the particle and local fluid
velocities,
Z
q̇ = u = [J (q)] v = δa (q − r) v (r, t) dr,
where the local averaging linear operator J(q) averages the fluid
velocity inside the particle to estimate a local fluid velocity.
The induced force density in the fluid because of the particle is:
f = −λδa (q − r) = − [S (q)] λ,
where the local spreading linear operator S(q) is the reverse (adjoint)
of J(q).
The physical volume of the particle ∆V is related to the shape and
width of the kernel function via
Z
−1
−1
2
∆V = (JS) =
δa (r) dr
.
(1)
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Incompressible Inertial Coupling
Fluid-Structure Direct Coupling
The equations of motion in our coupling approach are postulated to
be [2]
ρ (∂t v + v · ∇v) = −∇π − ∇ · σ − [S (q)] λ + ’thermal’ drift
me u̇ = F (q) + λ
s.t. u = [J (q)] v and ∇ · v = 0,
where λ is the fluid-particle force, F (q) = −∇U (q) is the
externally applied force, and me is the excess mass of the particle.
The stress tensor σ = η ∇v + ∇T v + Σ includes viscous
(dissipative) and stochastic contributions. The stochastic stress
Σ = (kB T η)1/2 W + W T
drives the Brownian motion. Note momentum is conserved.
In the existing (stochastic) IBM approach [3] inertial effects are
ignored, me = 0 and thus λ = −F.
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Incompressible Inertial Coupling
Effective Inertia
Eliminating λ we get the particle equation of motion
mu̇ = ∆V J (∇π + ∇ · σ) + F + blob correction,
where the effective mass m = me + mf includes the mass of the
“excluded” fluid
mf = ρ∆V = ρ (JS)−1 .
For the fluid we get the effective equation
∂
ρeff ∂t v = − ρ (v · ∇) + me S u ·
J v − ∇π − ∇ · σ + SF
∂q
where the effective mass density matrix (operator) is
ρeff = ρ + me PSJP,
where P is the L2 projection operator onto the linear subspace
∇ · v = 0, with the appropriate BCs.
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Incompressible Inertial Coupling
Fluctuation-Dissipation Balance
One must ensure fluctuation-dissipation balance in the coupled
fluid-particle system.
We can eliminate the particle velocity using the no-slip constraint, so
only v and q are independent DOFs.
This really means that the stationary (equilibrium) distribution must
be the Gibbs distribution
P (v, q) = Z −1 exp [−βH]
where the Hamiltonian (coarse-grained free energy) is
Z
u2
v2
+ ρ dr.
H (v, q) = U (q) + me
2
2
Z T
v ρeff v
= U (q) +
dr
2
No entropic contribution to the coarse-grained free energy because
our formulation is isothermal and the particles do not have internal
structure.
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Incompressible Inertial Coupling
contd.
A key ingredient of fluctuation-dissipation balance is that that the
fluid-particle coupling is non-dissipative, i.e., in the absence of
viscous dissipation the kinetic energy H is conserved.
Crucial for energy conservation is that J(q) and S(q) are adjoint,
S = J? ,
Z
Z
(Jv) · u = v · (Su) dr = δa (q − r) (v · u) dr.
(2)
The dynamics is not incompressible in phase space and “thermal
drift” correction terms need to be included [4], but they turn out to
vanish for incompressible flow (gradient of scalar).
The spatial discretization should preserve these properties: discrete
fluctuation-dissipation balance (DFDB).
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Numerics
Numerical Scheme
Both compressible (explicit) and incompressible schemes have been
implemented by Florencio Balboa (UAM) on GPUs.
Spatial discretization is based on previously-developed staggered
schemes for fluctuating hydro [5] and the IBM kernel functions of
Charles Peskin.
Temporal discretization follows a second-order splitting algorithm
(move particle + update momenta), and is limited in stability only by
advective CFL.
The scheme ensures strict conservation of momentum and (almost
exactly) enforces the no-slip condition at the end of the time step.
Continuing work on temporal integrators that ensure the correct
equilibrium distribution and diffusive (Brownian) dynamics.
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Numerics
Spatial Discretization
IBM kernel functions of Charles Peskin are used to average
( d
)
X Y
Jv ≡
φa [qα − (rk )α ] vk .
k∈grid
α=1
Discrete spreading operator S = (∆Vf )−1 J?
( d
)
Y
(SF)k = (∆x∆y ∆z)−1
φa [qα − (rk )α ] F.
α=1
The discrete kernel function φa gives translational invariance
X
X
φa (q − rk ) = 1 and
(q − rk ) φa (q − rk ) = 0,
k∈grid
X
k∈grid
φ2a (q
− rk ) = ∆V
−1
= const.,
(3)
k∈grid
independent of the position of the (Lagrangian) particle q relative to
the underlying (Eulerian) velocity grid.
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Numerics
Temporal Discretization
Predict particle position at midpoint:
1
∆t n n
qn+ 2 = qn +
J v .
2
Solve the coupled constrained momentum conservation
1
equations for vn+1 and un+1 and the Lagrange multipliers π n+ 2 and
1
λn+ 2 (hard to do efficiently!)
ρ
n+ 12
1
1
1
vn+1 − vn
+ ∇π n+ 2 = −∇ · ρvvT + σ
− Sn+ 2 λn+ 2
∆t
1
1
me un+1 = me un + ∆t Fn+ 2 + ∆t λn+ 2
∇ · vn+1 = 0
1
1
un+1 = Jn+ 2 vn+1 + Jn+ 2 − Jn vn ,
(4)
Correct particle position,
qn+1 = qn +
A. Donev (CIMS)
∆t n+ 1 n+1
J 2 v
+ vn .
2
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Numerics
Temporal Integrator (sketch)
Predict particle position at midpoint:
∆t n n
J v .
2
Solve unperturbed fluid equation using stochastic Crank-Nicolson
for viscous+stochastic:
1
1
ṽn+1 − vn
η 2 n+1
ρ
+ ∇π̃ =
∇ ṽ
+ vn + ∇ · Σn + Sn+ 2 Fn+ 2 + adv.
∆t
2
∇ · ṽn+1 = 0,
1
qn+ 2 = qn +
where we use the Adams-Bashforth method for the advective
(kinetic) fluxes, and the discretization of the stochastic flux is
described in Ref. [5],
i
kB T η 1/2 h n
n
Σ =
(W ) + (Wn )T ,
∆V ∆t
where Wn is a (symmetrized) collection of i.i.d. unit normal variates.
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Numerics
contd.
Solve for inertial velocity perturbation from the particle ∆v (too
technical to present), and update:
vn+1 = ṽn+1 + ∆v.
If neutrally-buyoant me = 0 this is a non-step, ∆v = 0.
Update particle velocity in a momentum conserving manner,
1
un+1 = Jn+ 2 vn+1 + slip correction.
Correct particle position,
qn+1 = qn +
A. Donev (CIMS)
∆t n+ 1 n+1
J 2 v
+ vn .
2
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Numerics
Implementation
With periodic boundary conditions all required linear solvers (Poisson,
Helmholtz) can be done using FFTs only.
Florencio Balboa has implemented the algorithm on GPUs using
CUDA in a public-domain code (combines compressible and
incompressible algorithms):
https://code.google.com/p/fluam
Our implicit algorithm is able to take a rather large time step size, as
measured by the advective and viscous CFL numbers:
V ∆t
ν∆t
, β=
,
(5)
∆x
∆x 2
where V is a typical advection speed.
Note that for compressible flow there is a sonic CFL number
αs = c∆t/∆x α, where c is the speed of sound.
Our scheme should be used with α . 1. The scheme is stable for any
β, but to get the correct thermal dynamics one should use β . 1.
α=
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Results
Equilibrium Radial Correlation Function
RDF g(r)
1.5
1
me=0
me=mf
0.5
0
0
Monte Carlo
1
2
r/σ
3
4
Figure: Equilibrium radial distribution function g2 (r) for a suspension of blobs
interacting with a repulsive LJ (WCA) potential.
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Results
Hydrodynamic Interactions
20
me=mf, α=0.01
me=mf, α=0.1
me=mf, α=0.25
Rotne-Prager (RP)
RP + Lubrication
RPY
F / FStokes
16
12
8
4
0.5
1
2
Distance d/RH
4
8
Figure: Effective hydrodynamic force between two approaching blobs at small
2F0
Reynolds numbers, FFSt = − 6πηR
.
H vr
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Results
Velocity Autocorrelation Function
We investigate the velocity autocorrelation function (VACF) for
the immersed particle
C (t) = hu(t0 ) · u(t0 + t)i
From equipartition theorem C (0) = hu 2 i = d kBmT .
However, for an incompressible fluid the kinetic energy of the particle
that is less than equipartition,
hu i = 1 +
2
mf
(d − 1)m
−1 kB T
d
,
m
as predicted also for a rigid sphere a long time ago, mf /m = ρ0 /ρ.
Hydrodynamic persistence (conservation) gives a long-time
power-law tail C (t) ∼ (kT /m)(t/tvisc )−3/2 not reproduced in
Brownian dynamics.
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Results
Numerical VACF
Rigid sphere
c=16
c=8
c=4
c=2
c=1
Incompress.
C(t) / (kT/m)
1
0.8
0.6
10
0
0.4
-2
0.2
10
10
10
-4
-1
10
10
-3
0
1
10
10
-2
t / tν
10
-1
10
0
10
1
Figure: VACF for a blob with me = mf = ρ∆V .
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Results
Diffusive Dynamics
At long times, the motion of the
particle is diffusive with a diffusion
R∞
coefficient χ = limt→∞ χ(t) = t=0 C (t)dt, where
∆q 2 (t)
1
=
h[q(t) − q(0)]2 i.
2t
2dt
The Stokes-Einstein relation predicts
χ(t) =
χ=
kB T
kB T
(Einstein) and χSE =
(Stokes),
µ
6πηRH
(6)
where for our blob with the 3-point kernel function RH ≈ 0.9∆x.
The dimensionless Schmidt number Sc = ν/χSE controls the
separation of time scales between v (r, t) and q(t).
Self-consistent theory [6] predicts a correction to Stokes-Einstein’s
relation for small Sc ,
χ
kB T
χ ν+
=
.
2
6πρRH
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Results
Stokes-Einstein Corrections
χ / χSE
1
0.9
Self-consistent theory
From VACF integral
From mobility
0.8
1
2
4
8 16 32 64 128 256
Schmidt number Sc
Figure: Corrections to Stokes-Einstein with changing viscosity ν = η/ρ,
me = mf = ρ∆V .
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Results
Stokes-Einstein Corrections (2D)
1
3pt kernel
4pt kernel
χ (ν+χ/2) = const
χ (ν+χ) = const
χ / SE
0.9
0.8
0.7
0.6
1
2
4
8
32
16
Approximate Sc=ν/χSE
64
128
256
Figure: Corrections to Stokes-Einstein with changing viscosity ν = η/ρ,
me = mf = ρ∆V .
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Outlook
Overdamped Limit (me = 0)
[With Eric Vanden-Eijnden] In the overdamped limit, in which
momentum diffuses much faster than the particles, the motion of the
blob at the diffusive time scale can be described by the fluid-free
Stratonovich stochastic differential equation
q̇ = µF + J (q) ◦ v (r, t)
where the random advection velocity is a white-in-time process is the
solution of the steady Stokes equation
p
∇π = ν∇2 v + ∇ ·
2νρ−1 kB T W such that ∇ · v = 0,
and the blob mobility is given by the Stokes solution operator L−1 ,
µ (q) = −J (q) L−1 S (q) .
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Outlook
Brownian Dynamics (BD)
For multi-particle suspensions the mobility matrix M (Q) = µij
depends on the positions of all particles Q = {qi }, and the limiting
equation in the Ito formulation is the usual Brownian Dynamics
equation
p
1
∂
f
Q̇ = MF + 2kB T M 2 W+kB T
·M .
∂Q
It is possible to construct temporal integrators for the overdamped
1
f (work in progress).
equations, without ever constructing M 2 W
The limiting equation when excess inertia is included has not been
derived though it is believed inertia does not enter in the overdamped
equations.
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Outlook
BD without Green’s Functions
The following algorithm can be shown to solve the Brownian Dynamics
SDE:
Solve a steady-state Stokes problem (here δ 1)
Gπ n = η∇2 vn + ∇ · Σn + Sn F (qn )
δ fn
kB T
n
S q + W − S qn −
+
δ
2
Dvn = 0.
δ fn
W
2
fn
W
Predict particle position:
q̃n+1 = qn + ∆tJn vn .
Correct particle position,
qn+1 = qn +
A. Donev (CIMS)
∆t n n+1 n
J +J̃
v .
2
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Outlook
Conclusions
Fluctuating hydrodynamics seems to be a very good coarse-grained
model for fluids, and can be coupled to immersed particles to model
Brownian suspensions.
The minimally-resolved blob approach provides a low-cost but
reasonably-accurate representation of rigid particles in flow.
Particle inertia can be included in the coupling between blob
particles and a fluctuating incompressible fluid.
Stokes-Einstein’s relation only holds for large Schmidt numbers.
Overdamped limit can be handled just by changing the temporal
integrator.
More complex particle shapes can be built out of a collection of
blobs.
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Outlook
References
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Force-coupling method for particulate two-phase flow: Stokes flow.
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F. Balboa Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, and A. Donev.
Inertial Coupling Method for particles in an incompressible fluctuating fluid.
Submitted, code available at https://code.google.com/p/fluam, 2013.
P. J. Atzberger, P. R. Kramer, and C. S. Peskin.
A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales.
J. Comp. Phys., 224:1255–1292, 2007.
P. J. Atzberger.
Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations.
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F. Balboa Usabiaga, J. B. Bell, R. Delgado-Buscalioni, A. Donev, T. G. Fai, B. E. Griffith, and C. S. Peskin.
Staggered Schemes for Incompressible Fluctuating Hydrodynamics.
SIAM J. Multiscale Modeling and Simulation, 10(4):1369–1408, 2012.
A. Donev, A. L. Garcia, Anton de la Fuente, and J. B. Bell.
Enhancement of Diffusive Transport by Nonequilibrium Thermal Fluctuations.
J. of Statistical Mechanics: Theory and Experiment, 2011:P06014, 2011.
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