Second kind integral equation formulation of Stokes' flows past a particle of arbitary shape
SIAM Journal on Applied Mathematics
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
An efficient numerical method for studying interfacial motion in two-dimensional creeping flows
Journal of Computational Physics
Interfacial dynamics for Stokes flow
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Simulating the dynamics and interactions of flexible fibers in Stokes flows
Journal of Computational Physics
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains
Journal of Computational Physics
Petascale Direct Numerical Simulation of Blood Flow on 200K Cores and Heterogeneous Architectures
Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
A fast algorithm for simulating vesicle flows in three dimensions
Journal of Computational Physics
Accurate computation of Stokes flow driven by an open immersed interface
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.47 |
We consider numerical algorithms for the simulation of hydrodynamics of two-dimensional vesicles suspended in a viscous Stokesian fluid. The motion of vesicles is governed by the interplay between hydrodynamic and elastic forces. Continuum models of vesicles use a two-phase fluid system with interfacial forces that include tension (to maintain local ''surface'' inextensibility) and bending. Vesicle flows are challenging to simulate. On the one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives in the bending term. On the other hand, implicit time-stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step. Our method is an extension of the work of Veerapaneni et al. [S.K. Veerapaneni, D. Gueyffier, D. Zorin, G. Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334-2353], in which a semi-implicit time-marching scheme based on a boundary integral formulation of the Stokes problem for vesicles in an unbounded medium was proposed. In this paper, we consider two important generalizations: (i) confined flows within arbitrary-shaped stationary/moving geometries; and (ii) flows in which the interior (to the vesicle) and exterior fluids have different viscosity. In the rest of the paper, we will refer to this as the ''viscosity contrast''. These two problems require solving additional integral equations and cause nontrivial modifications to the previous numerical scheme. Our method does not have severe time-step stability constraints and its computational cost-per-time-step is comparable to that of an explicit scheme. The discretization is pseudo-spectral in space, and multistep BDF in time. We conduct numerical experiments to investigate the stability, accuracy and the computational cost of the algorithm. Overall, our method achieves several orders of magnitude speed-up compared to standard explicit schemes. As a preliminary validation of our scheme, we study the dependence of the inclination angle of a single vesicle in shear flow on the viscosity contrast and the reduced area of the vesicle, the lateral migration of vesicles in shear flow, the dispersion of two vesicles, and the effective viscosity of a dilute suspension of vesicles.