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.
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Note that the mobility is symmetric so there is no notational ambiguity in
$\partial _{{\bm q}}\cdot \bm{\mathcal {M}}$
.97.
Note that this is an approximation and in practice the mobility is not pairwise additive if higher-order multipoles such as stresslets are accounted for, see Appendix B.
98.
Perhaps a more relevant diffusive length scale to use is the typical inter-particle gap, which can be substantially smaller than a for dense suspensions.
99.
In Green's function's based approaches there is no difficulty in dealing with unbounded three dimensional systems (at rest at infinity) since the Oseen tensor is the required response function and is easy to compute. Note however that when simulating periodic domains (e.g., colloidal suspensions) one requires the Green's functons for Stokes flow in a periodic domain, and these are not simple to compute,28,85 unlike in our approach, which handles boundary conditions naturally.
100.
Note, however, that Metropolization of even the simple midpoint scheme is a rather nontrivial task because the mobility matrix is never formed or factorized in FIB.
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