A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

Abstract

We extend "A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D", Veerapaneni et al. Journal of Computational Physics, 228(7), 2009 to the case of three dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows. Although the main components of the algorithm are similar in spirit to the 2D case-spectral approximation in space, semi-implicit time-stepping scheme-the main differences are that the bending and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived, a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction, and a novel numerical scheme for the evaluation of the 3D Stokes single-layer potential on an axisymmetric surface is necessary to speed up the calculations. By introducing these novel components, we obtain a time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme.To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and three vesicles with a background Poiseuille flow.

A fast algorithm for simulating vesicle flows in three dimensions

“…Another option is to use a fully implicit scheme. In our previous work on 3D axisymmetric vesicle flows [61], we observed experimentally that a line-search, single-level, Newton scheme in which the Jacobian is approximated using the semi-implicit scheme linearization scheme does not result in computational savings.…”

“…The main features of the method of [62] are an integral equation formulation, spectral discretization in space, and a semi-implicit time-stepping scheme. In [61], we described an extension of [62] to the axisymmetric case. There are several challenges specific to numerical simulation of nonaxisymmetric vesicles flows in 3D: (i) What should the spatial discretization scheme be to maximize accuracy, computational efficiency, and numerical stability?…”

“…(ii) Unlike 2D, using a Lagrangian frame of reference for spatial discretization of vesicle boundary results in extreme distortions, leading to numerical instability; how can this instability be controlled? (iii) Bending and tension forces in the 3D case have a more complex nonlinear form; can we design a time-stepping scheme that addresses the severe stiffness while having similar per-time-step complexity to the explicit scheme (similar to our schemes for 2D and axisymmetric flows [61,62])? We summarize our contributions below.…”

“…A few closely related papers are our work on 2D and 3D axisymmetric flows ( [62] , [61]), Graham and Sloan's singular quadratures (for the scalar Helmholtz operator, but are directly applicable to our case) [30], Zinchenko and Davis' surface reparametrization schemes for drops and deformable particles [72], and the very recent work of Zhao et al [68] on boundary integral equation based simulations of red blood cells in shear flow, presenting a spherical harmonic discretization of membranes with bending and shear resistance and and explicit time discretization for particulate flows of this type. We discuss these papers in more detail below.…”

“…In this paper, we present an algorithm for the simulation of general three-dimensional vesicle flows, extending our recent work on 2D [62] and 3D axisymmetric vesicles [61]. (To demonstrate the capabilities of our code, we depict a few time-snapshots from a twenty-vesicle simulation in Figure 1.…”