Coupling an Incompressible Fluctuating Fluid with Suspended Structures

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 A. Donev (CIMS) IICM 6/9/2013 1 / 32

Outline 1 Incompressible Inertial Coupling 2 Numerics 3 Results 4 Outlook A. Donev (CIMS) IICM 6/9/2013 2 / 32

Levels of Coarse-Graining Figure: From Pep Español, Statistical Mechanics of Coarse-Graining A. Donev (CIMS) IICM 6/9/2013 3 / 32

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. A. Donev (CIMS) IICM 6/9/2013 5 / 32

Incompressible Inertial Coupling Brownian Particle Model Consider a Brownian particle of size a with position q(t) and velocity u = q, 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 δ a ( 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. A. Donev (CIMS) IICM 6/9/2013 6 / 32

Incompressible Inertial Coupling Local Averaging and Spreading Operators Postulate a no-slip condition between the particle and local fluid velocities, q = u = [J (q)] v = δ a (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 = λδ a (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 [ 1 V = (JS) 1 = δa 2 (r) dr]. (1) A. Donev (CIMS) IICM 6/9/2013 7 / 32

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 m e u = 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 m e 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 = (k B 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, m e = 0 and thus λ = F. A. Donev (CIMS) IICM 6/9/2013 8 / 32

Incompressible Inertial Coupling Effective Inertia Eliminating λ we get the particle equation of motion m u = V J ( π + σ) + F + blob correction, where the effective mass m = m e + m f includes the mass of the excluded fluid m f = ρ V = ρ (JS) 1. For the fluid we get the effective equation [ ( )] ρ eff t v = ρ (v ) + m e S u q J v π σ + SF where the effective mass density matrix (operator) is ρ eff = ρ + m e PSJP, where P is the L 2 projection operator onto the linear subspace v = 0, with the appropriate BCs. A. Donev (CIMS) IICM 6/9/2013 9 / 32

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 u 2 H (v, q) = U (q) + m e 2 + ρ v 2 2 dr. v T ρ = U (q) + eff v dr 2 No entropic contribution to the coarse-grained free energy because our formulation is isothermal and the particles do not have internal structure. A. Donev (CIMS) IICM 6/9/2013 10 / 32

contd. Incompressible Inertial Coupling 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, (Jv) u = v (Su) dr = δ a (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). A. Donev (CIMS) IICM 6/9/2013 11 / 32

Numerical Scheme Numerics 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. A. Donev (CIMS) IICM 6/9/2013 13 / 32

Spatial Discretization Numerics IBM kernel functions of Charles Peskin are used to average Jv { d } φ a [q α (r k ) α ] v k. k grid α=1 Discrete spreading operator S = ( V f ) 1 J (SF) k = ( x y z) 1 { d α=1 φ a [q α (r k ) α ] The discrete kernel function φ a gives translational invariance φ a (q r k ) = 1 and (q r k ) φ a (q r k ) = 0, k grid k grid k grid φ 2 a (q r k ) = V 1 = const., (3) independent of the position of the (Lagrangian) particle q relative to the underlying (Eulerian) velocity grid. A. Donev (CIMS) IICM 6/9/2013 14 / 32 } F.

Numerics Temporal Discretization Predict particle position at midpoint: q n+ 1 2 = q n + t 2 Jn v n. Solve the coupled constrained momentum conservation equations for v n+1 and u n+1 and the Lagrange multipliers π n+ 1 2 (hard to do efficiently!) λ n+ 1 2 ρ vn+1 v n t + π n+ 1 2 = (ρvv T + σ ) n+ 1 2 S n+ 1 2 λ n+ 1 2 m e u n+1 = m e u n + t F n+ 1 2 + t λ n+ 1 2 v n+1 = 0 Correct particle position, u n+1 = J n+ 1 2 v n+1 + q n+1 = q n + t 2 Jn+ 1 2 and ( J n+ 1 2 J n ) v n, (4) ( v n+1 + v n). A. Donev (CIMS) IICM 6/9/2013 15 / 32

Numerics Temporal Integrator (sketch) Predict particle position at midpoint: q n+ 1 2 = q n + t 2 Jn v n. Solve unperturbed fluid equation using stochastic Crank-Nicolson for viscous+stochastic: ρṽn+1 v n + π t = η ( 2 2 ṽ n+1 + v n) + Σ n + S n+ 1 2 F n+ 1 2 + adv. ṽ n+1 = 0, where we use the Adams-Bashforth method for the advective (kinetic) fluxes, and the discretization of the stochastic flux is described in Ref. [5], ( ) Σ n kb T η 1/2 [ ] = (W n ) + (W n ) T, V t where W n is a (symmetrized) collection of i.i.d. unit normal variates. A. Donev (CIMS) IICM 6/9/2013 16 / 32

contd. Numerics Solve for inertial velocity perturbation from the particle v (too technical to present), and update: v n+1 = ṽ n+1 + v. If neutrally-buyoant m e = 0 this is a non-step, v = 0. Update particle velocity in a momentum conserving manner, Correct particle position, u n+1 = J n+ 1 2 v n+1 + slip correction. q n+1 = q n + t ( 2 Jn+ 1 2 v n+1 + v n). A. Donev (CIMS) IICM 6/9/2013 17 / 32

Implementation Numerics 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: α = V t x, β = ν t x 2, (5) where V is a typical advection speed. Note that for compressible flow there is a sonic CFL number α s = 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. A. Donev (CIMS) IICM 6/9/2013 18 / 32

Results Equilibrium Radial Correlation Function Figure: Equilibrium radial distribution function g 2 (r) for a suspension of blobs interacting with a repulsive LJ (WCA) potential. A. Donev (CIMS) IICM 6/9/2013 20 / 32 1.5 RDF g(r) 1 0.5 m e =0 m e =m f Monte Carlo 0 0 1 2 3 4 r / σ

Results Hydrodynamic Interactions 20 F / F Stokes 16 12 8 m e =m f, α=0.01 m e =m f, α=0.1 m e =m f, α=0.25 Rotne-Prager (RP) RP + Lubrication RPY 4 0.5 1 2 4 8 Distance d/r H Figure: Effective hydrodynamic force between two approaching blobs at small F Reynolds numbers, F St = 2F0 6πηR H v r. A. Donev (CIMS) IICM 6/9/2013 21 / 32

Results Velocity Autocorrelation Function We investigate the velocity autocorrelation function (VACF) for the immersed particle C(t) = u(t 0 ) u(t 0 + t) From equipartition theorem C(0) = u 2 = d k BT m. However, for an incompressible fluid the kinetic energy of the particle that is less than equipartition, u 2 = [ ] m 1 ( f 1 + d k ) BT, (d 1)m m as predicted also for a rigid sphere a long time ago, m f /m = ρ /ρ. Hydrodynamic persistence (conservation) gives a long-time power-law tail C(t) (kt /m)(t/t visc ) 3/2 not reproduced in Brownian dynamics. A. Donev (CIMS) IICM 6/9/2013 22 / 32

Numerical VACF Results C(t) / (kt/m) 1 0.8 0.6 0.4 10 0 Rigid sphere c=16 c=8 c=4 c=2 c=1 Incompress. 0.2 10-2 10-1 10 0 10 1 10-4 10-3 10-2 10-1 10 0 10 1 t / t ν Figure: VACF for a blob with m e = m f = ρ V. A. Donev (CIMS) IICM 6/9/2013 23 / 32

Diffusive Dynamics Results At long times, the motion of the particle is diffusive with a diffusion coefficient χ = lim t χ(t) = t=0 C(t)dt, where χ(t) = q2 (t) 2t The Stokes-Einstein relation predicts = 1 2dt [q(t) q(0)]2. χ = k BT µ (Einstein) and χ SE = k BT 6πηR H (Stokes), (6) where for our blob with the 3-point kernel function R H 0.9 x. The dimensionless Schmidt number S c = ν/χ SE 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 S c, ( χ ν + χ ) = k BT. 2 6πρR H A. Donev (CIMS) IICM 6/9/2013 24 / 32

Results Stokes-Einstein Corrections 1 χ / χ SE 0.9 0.8 Self-consistent theory From VACF integral From mobility 1 2 4 8 16 32 64 128 256 Schmidt number S c Figure: Corrections to Stokes-Einstein with changing viscosity ν = η/ρ, m e = m f = ρ V. A. Donev (CIMS) IICM 6/9/2013 25 / 32

Results Stokes-Einstein Corrections (2D) 1 0.9 3pt kernel 4pt kernel χ (ν+χ/2) = const χ (ν+χ) = const χ / SE 0.8 0.7 0.6 1 2 4 8 16 32 64 128 256 Approximate Sc=ν/χ SE Figure: Corrections to Stokes-Einstein with changing viscosity ν = η/ρ, m e = m f = ρ V. A. Donev (CIMS) IICM 6/9/2013 26 / 32

Outlook Overdamped Limit (m e = 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 q = 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 ( ) π = ν 2 v + 2νρ 1 k B 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). A. Donev (CIMS) IICM 6/9/2013 28 / 32

Outlook Brownian Dynamics (BD) For multi-particle suspensions the mobility matrix M (Q) = { } µ ij depends on the positions of all particles Q = {q i }, and the limiting equation in the Ito formulation is the usual Brownian Dynamics equation Q = MF + ( ) 2k B T M 1 2 W+kB T Q M. It is possible to construct temporal integrators for the overdamped equations, without ever constructing M 1 2 W (work in progress). The limiting equation when excess inertia is included has not been derived though it is believed inertia does not enter in the overdamped equations. A. Donev (CIMS) IICM 6/9/2013 29 / 32

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 v n + Σ n + S n F (q n ) + k [ ( BT S q n + δ δ 2 W n) ( S q n δ 2 W n)] Wn Dv n = 0. Predict particle position: Correct particle position, q n+1 = q n + tj n v n. q n+1 = q n + t 2 ( J n + J n+1) v n. A. Donev (CIMS) IICM 6/9/2013 30 / 32

Conclusions Outlook 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. A. Donev (CIMS) IICM 6/9/2013 31 / 32

References Outlook S. Lomholt and M.R. Maxey. Force-coupling method for particulate two-phase flow: Stokes flow. J. Comp. Phys., 184(2):381 405, 2003. 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. J. Comp. Phys., 230:2821 2837, 2011. 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. A. Donev (CIMS) IICM 6/9/2013 32 / 32