We develop a Fluctuating Immersed Boundary (FIB) method for performing Brownian dynamics simulations of confined particle suspensions. Unlike traditional methods which employ analytical Green's functions for Stokes flow in the confined geometry, the FIB method uses a fluctuating finite-volume Stokes solver to generate the action of the response functions “on the fly.” Importantly, we demonstrate that both the deterministic terms necessary to capture the hydrodynamic interactions among the suspended particles, as well as the stochastic terms necessary to generate the hydrodynamically correlated Brownian motion, can be generated by solving the steady Stokes equations numerically only once per time step. This is accomplished by including a stochastic contribution to the stress tensor in the fluid equations consistent with fluctuating hydrodynamics. We develop novel temporal integrators that account for the multiplicative nature of the noise in the equations of Brownian dynamics and the strong dependence of the mobility on the configuration for confined systems. Notably, we propose a random finite difference approach to approximating the stochastic drift proportional to the divergence of the configuration-dependent mobility matrix. Through comparisons with analytical and existing computational results, we numerically demonstrate the ability of the FIB method to accurately capture both the static (equilibrium) and dynamic properties of interacting particles in flow.
We thank Eric Vanden-Eijnden and Anthony Ladd for informative discussions, and Ranojoy Adhikari and Eric Keaveny for helpful comments on the manuscript. S. Delong was supported by the DOE office of Advanced Scientific Computing Research under Grant No. DE-FG02-88ER25053. A. Donev was supported in part by the Air Force Office of Scientific Research under Grant No. FA9550-12-1-0356. B. Griffith acknowledges research support from the National Science Foundation under Award Nos. OCI 1047734 and DMS 1016554. R. Delgado-Buscalioni and F. Balboa acknowledge funding from the Spanish government FIS2010-22047-C05 and from the Comunidad de Madrid MODELICO-CM (S2009/ESP-1691). Collaboration between A. Donev and R. Delgado-Buscalioni was fostered at the Kavli Institute for Theoretical Physics in Santa Barbara, California, and supported in part by the National Science Foundation (NSF) under Grant No. NSF PHY05-51164.
I. INTRODUCTION A. Brownian dynamics B. Mobility matrix 1. Unconfined systems 2. Confined systems II. FLUCTUATING IMMERSED BOUNDARY METHOD A. Fluid-particle interaction B. Overdamped limit C. Relation to Brownian dynamics D. Thermal drift 1. Fixman's method 2. Random finite difference III. SPATIAL DISCRETIZATION A. Discrete local averaging and spreading B. Stokes solver C. Discrete fluctuation dissipation balance IV. TEMPORAL DISCRETIZATION A. Simple midpoint scheme B. Improved midpoint scheme V. RESULTS A. Mobility in a slit channel B. Diffusion coefficient C. Thermodynamic equilibrium 1. Free diffusion 2. Diffusion in a slit channel 3. Colloidal suspension D. Particles in she