A minimally-resolved immersed boundary model for reaction-diffusion problems

Abstract

We develop an immersed-boundary approach to modeling reaction-diffusion processes in dispersions of reactive spherical particles, from the diffusion-limited to the reaction-limited setting. We represent each reactive particle with a minimally-resolved "blob" using many fewer degrees of freedom per particle than standard discretization approaches. More complicated or more highly resolved particle shapes can be built out of a collection of reactive blobs. We demonstrate numerically that the blob model can provide an accurate representation at low to moderate packing densities of the reactive particles, at a cost not much larger than solving a Poisson equation in the same domain. Unlike multipole expansion methods, our method does not require analytically-computed Green's functions, but rather, computes regularized discrete Green's functions on the fly by using a standard grid-based discretization of the Poisson equation. This allows for great flexibility in implementing different boundary conditions, coupling to fluid flow or thermal transport, and the inclusion of other effects such as temporal evolution and even nonlinearities. We develop multigrid-based preconditioners for solving the linear systems that arise when using implicit temporal discretizations or studying steady states. In the diffusion-limited case the resulting linear system is a saddlepoint problem, the efficient solution of which remains a challenge for suspensions of many particles. We validate our method by comparing to published results on reaction-diffusion in ordered and disordered suspensions of reactive spheres.

An immersed boundary method for rigid bodies

“…The problem of solving a linear systems similar in structure to (21) appears in many other methods for hydrodynamics of suspensions, including Brownian [74,75] and Stokesian [15] dynamics, the method of regularized Stokeslets [21,22], computations based on bead models of rigid bodies [17][18][19][20], and first-kind boundary integral formulations of Stokes flow [76]. Similar matrices appear in static Poisson problems such as electrostatics or reaction-diffusion models [50], and there is a substantial ongoing work that can be applied to our problem. Notably, the approximate mobility matrix M is dense but has a well-understood low-rank structure that can be exploited.…”

The Stokes-Einstein relation at moderate Schmidt number

“…After accounting for finite-size effects due to the finite length of the periodic box, in three dimensions we numerically estimate 17,30 the effective hydrodynamic radius to be R 3pt H = (0.91 ± 0.01) h the three-point kernel, 53 where h is the grid spacing, and R 4pt H = (1.255 ± 0.005) h for the 4pt kernel. 46 In two dimensions, the effective (rigid disk) hydrodynamic radii are estimated to be R 3pt H = (0.72 ± 0.01) h and R 4pt H = (1.04 ± 0.005) h. Note that the spatial discretization we use is not perfectly translationally invariant and there is a small variation of R H (quoted above as an error bar) as the particle moves relative to the underlying fixed fluid grid. 17,46 We use a relatively small grid of 32 2 cells in two dimensions and 32 3 cells in three dimensions in order to be able to perform sufficiently long runs even with the larger Schmidt numbers.…”

“…46 In two dimensions, the effective (rigid disk) hydrodynamic radii are estimated to be R 3pt H = (0.72 ± 0.01) h and R 4pt H = (1.04 ± 0.005) h. Note that the spatial discretization we use is not perfectly translationally invariant and there is a small variation of R H (quoted above as an error bar) as the particle moves relative to the underlying fixed fluid grid. 17,46 We use a relatively small grid of 32 2 cells in two dimensions and 32 3 cells in three dimensions in order to be able to perform sufficiently long runs even with the larger Schmidt numbers. In two dimensions we use a neutrally buoyant particle (m e = 0), while in three dimensions we use a particle twice denser than the surrounding fluid (m e = ρ V ), in order to confirm that the excess mass (density) does not (significantly) affect the conclusions of our investigations.…”

Inertial coupling method for particles in an incompressible fluctuating fluid

“…It is possible to include additional transport processes in the minimally-resolved approach we employ here, however, the specifics of how to express the surface boundary conditions to a volumetric blob condition are very problem specific and need to be carefully constructed on a case by case basis. In this respect, recently, reaction-diffusion processes have been included in the type of method studied here [49], and compressible blobs have been considered and found to adequately describe coupling between ultrasound waves and small particles [50].…”