Author ' s personal copy Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain , a boundary integral method
Author ' s personal copy Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain , a boundary integral method
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作者个人副本悬浮在任意二维域中的局部不可伸展囊泡的动态模拟，一种边界积分方法
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发表时间：
2010
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通讯作者：
G. Biros
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作者：
Abtin Rahimian;S. Veerapaneni;G. Biros
我们考虑用于模拟悬浮在粘性斯托克斯流体中的二维囊泡的流体动力学的数值算法。囊泡的运动由水动力和弹力之间的相互作用控制。囊泡的连续体模型使用具有界面力的两相流体系统，其中包括张力（以保持局部“表面”不可扩展性）和弯曲。囊泡流的模拟具有挑战性。一方面，由于与弯曲项中的高阶空间导数相关的刚度，显式时间步进方案受到严重的稳定性约束。另一方面，隐式时间步进方案可能很昂贵，因为它们需要在每个时间步求解一组非线性方程。我们的方法是 Veerapaneni 等人的工作的延伸 [S.K. Veerapaneni, D. Gueyffier, D. Zorin, G. Biros, 用于模拟二维粘性流体中悬浮的不可扩展囊泡动力学的边界积分方法，计算物理学杂志 228(7) (2009) 2334–2353]，其中半隐式在本文中，我们提出了基于无界介质中囊泡问题的边界积分公式的时间推进方案：（i）任意形状的静止/移动几何形状内的受限流动；以及（ii）内部（到囊泡）和外部流体具有不同粘度的流动。这两个问题需要求解额外的积分方程，并导致对先前的数值方案进行重大修改。我们的方法没有严格的时间步稳定性约束，并且其每时间步的计算成本与显式方案的计算成本相当。空间上的离散化是伪谱的，时间上的离散化是多步BDF。我们进行数值实验来研究算法的稳定性、准确性和计算成本。总体而言，与标准显式方案相比，我们的方法实现了几个数量级的加速。作为我们方案的初步验证，我们研究了剪切流中单个囊泡的倾斜角对粘度对比和囊泡面积减小的依赖性、剪切流中囊泡的侧向迁移、两个囊泡的分散以及囊泡稀释悬浮液的有效粘度。 2010 爱思唯尔公司。保留所有权利。 0021-9991/$ 请参阅前面的内容 2010 Elsevier Inc. 保留所有权利。 doi:10.1016/j.jcp.2010.05.006 * 通讯作者。电子邮件地址：rahimian@gatech.edu (A. Rahimian)、shravan@ims.nyu.edu (S.K. Veerapaneni)、gbiros@acm.org、biros@seas.upenn.edu (G. Biros)。计算物理学杂志229（2010）6466–6484
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. 2010 Elsevier Inc. All rights reserved. 0021-9991/$ see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2010.05.006 * Corresponding author. E-mail addresses: rahimian@gatech.edu (A. Rahimian), shravan@ims.nyu.edu (S.K. Veerapaneni), gbiros@acm.org, biros@seas.upenn.edu (G. Biros). Journal of Computational Physics 229 (2010) 6466–6484