Simulations of chemical transport and reaction in a suspension of cells I: an augmented forcing point method for the stationary case

Abstract

SUMMARY A novel augmented forcing point method is presented to solve the problem of chemical transport in the fluid outside of a collection of suspended cells coupled with chemical reactions on the surfaces of the cells. In this method, the chemical concentrations and the forcing function values are determined simultaneously from an augmented system of equations. The method is more stable and accurate than predictor‐corrector‐type forcing point methods, yields the same solution as a corresponding ghost cell method with much less computational cost, and provides solutions that are pointwise second‐order accurate in space and time, even for closely‐spaced cells. Copyright © 2011 John Wiley & Sons, Ltd.

A study of different modeling choices for simulating platelets within the immersed boundary method

“…The computation of the force requires the approximation to the normal vectors and an approximation to the mean curvature (32). For the values of E, F , and G in the mean curvature computation (see (33)), we use the approximations (38) and (39). For the values of e, f , and g, we use the approximations…”

An immersed boundary method for two-fluid mixtures

“…An alternative approach to our penalty method, which might allow a larger time step, is a constraint-based approach in which constraint forces are determined to ensure that the two fluids have the same velocity on each of the immersed boundaries [35, 36, 37], an approach which we have found effective in another context [38]. While exploration of these approaches for the current problem is warranted, their application here is not straightforward.…”

A radial basis function (RBF) finite difference method for the simulation of reaction–diffusion equations on stationary platelets within the augmented forcing method

“…In the AFM as implemented in [15], for each forcing point, the boundary condition at the corresponding boundary point (see Section 3) and the concentrations at five nearby fluid points are used to construct a bivariate quadratic interpolant tht satisfies the boundary condition at the boundary Since the fluid concentrations are still to be determined, this gives an implicit relationship between the forcing point concentration and those at the five fluid points. This relationship is used to populate one row of the matrix E. With this approach, if two platelets are close to one another or the shape of the platelet is concave, there may not be a sufficient number of points necessary to perform this interpolation.…”

“…The advantage of model 2 over model 1, of course, is that one has greater flexibility in model 2, in terms of selecting initial conditions for C u and C b , and different coefficients of diffusion as well. The Augmented Forcing Method (AFM) was developed in [15] for the simulation of chemical transport in a stationary fluid in the presence of irregular boundaries (platelets). In that work, an ODE model for chemistry on platelet surfaces was also presented, with the ODEs contributing boundary conditions to the fluid-phase chemical diffusion equation and the fluid-phase chemical diffusion equation contributing to the ODEs.…”