Stokes流中不可伸长细长光纤的积分谱方法
An integral-based spectral method for inextensible slender fibers in Stokes flow
摘要：
每个动物细胞都充满了细胞骨架，这是一种由不可伸展的纤维（如微管、肌动蛋白纤维和中间纤维）组成的动态凝胶，所有这些纤维都悬浮在粘性流体中。这种凝胶的数值模拟是具有挑战性的，因为纤维纵横比可以大到$10^4$。本文提出了一种快速计算不可伸长细长纤维在剪切Stokes流中动力学的新方法。的动态的细丝是由一个非局部的细长体理论，我们部分重新制定的Rotne-Prager-Yamakawa流体动力张量。为了加强不可扩展性，我们参数化的空间不可扩展的纤维运动和严格限制的动力学流形的不可扩展的配置。为此，我们引入了一组拉格朗日乘子的张力密度的细丝，并施加约束，没有虚功在$L^2$弱意义。我们增加了这种方法的局部和非局部细长体理论运营商的未知数是线性的，并提供了改进的空间精度的方法的基础上解决线张力方程的谱离散。对于动力学，我们开发了一个二阶半隐式时间积分器，它最多需要对非局部流体动力学进行一些评估和每个时间步进行一些块对角线性求解。在通过数值例子证明了我们的方法的精度和鲁棒性的提高后，我们将我们的配方应用于背景振荡剪切流中的永久交联肌动蛋白网格。我们观察到一个特征频率，在该频率下，网络从准静态，主要是弹性行为转变为动态，主要是粘性行为。我们发现，非局部流体力学增加的粘性模量高达25%，即使是半稀释的纤维悬浮液。
英文摘要：
Every animal cell is filled with a cytoskeleton, a dynamic gel made of inextensible fibers, such as microtubules, actin fibers, and intermediate filaments, all suspended in a viscous fluid. Numerical simulation of this gel is challenging because the fiber aspect ratios can be as large as $10^4$. We describe a new method for rapidly computing the dynamics of inextensible slender filaments in periodically-sheared Stokes flow. The dynamics of the filaments are governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To enforce inextensibility, we parameterize the space of inextensible fiber motions and strictly confine the dynamics to the manifold of inextensible configurations. To do this, we introduce a set of Lagrange multipliers for the tensile force densities on the filaments and impose the constraint of no virtual work in an $L^2$ weak sense. We augment this approach with a spectral discretization of the local and nonlocal slender body theory operators which is linear in the number of unknowns and gives improved spatial accuracy over approaches based on solving a line tension equation. For dynamics, we develop a second-order semi-implicit temporal integrator which requires at most a few evaluations of nonlocal hydrodynamics and a few block diagonal linear solves per time step. After demonstrating the improved accuracy and robustness of our approach through numerical examples, we apply our formulation to a permanently cross-linked actin mesh in a background oscillatory shear flow. We observe a characteristic frequency at which the network transitions from quasi-static, primarily elastic behavior to dynamic, primarily viscous behavior. We find that nonlocal hydrodynamics increases the viscous modulus by as much as 25%, even for semi-dilute fiber suspensions.
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