Stokes流中扭曲、弯曲、不可伸长纤维的流体力学
The hydrodynamics of a twisting, bending, inextensible fiber in Stokes flow
摘要：
在游动的微生物和细胞骨架中，不可伸展的纤维抵抗弯曲和扭曲，并与周围的流体相互作用以引起或抵抗大规模的流体运动。在本文中，我们开发了一种新的数值方法，通过扩展我们以前对不可伸展弯曲纤维的工作[Maxian等人，Phys. Rev. Fluids 6（1），014102]中的纤维与具有扭转弹性的纤维的连接。在我们的“欧拉”模型中，扭转是一个标量函数，它测量纤维截面相对于无扭转框架的偏差，纤维仅对流体施加平行于中心线的扭矩，并且旋转流体速度的垂直分量被丢弃以有利于平移速度。在本文的第一部分中，我们证明这个模型通过比较它的另一个常用的“基尔霍夫”公式，其中的纤维施加垂直和平行扭矩的流体，垂直的角流体速度需要与平移流体速度一致。然后，我们开发了一种谱数值方法的欧拉模型的流体动力学。我们定义流体动力学流动算子使用积分的Rotne-Prager-Yamakawa张量，并评估这些积分通过一个新的细长体求积，这需要在10个点的顺序沿纤维获得几位数的精度。我们证明，这种选择的流动性消除了非物理的负特征值的摩擦平移流动性与渐近细长体理论，并确保纤维速度的强收敛性和纤维约束力的弱收敛性。我们配对的空间离散与半隐式时间积分确认扭曲弹性弯曲纤维的松弛动力学的贡献可以忽略不计，并研究了旋转纤维的不稳定性。
英文摘要：
In swimming microorganisms and the cell cytoskeleton, inextensible fibers resist bending and twisting, and interact with the surrounding fluid to cause or resist large-scale fluid motion. In this paper, we develop a novel numerical method for the simulation of cylindrical fibers by extending our previous work on inextensible bending fibers [Maxian et al., Phys. Rev. Fluids 6 (1), 014102] to fibers with twist elasticity. In our "Euler" model, twist is a scalar function that measures the deviation of the fiber cross section relative to a twist-free frame, the fiber exerts only torque parallel to the centerline on the fluid, and the perpendicular components of the rotational fluid velocity are discarded in favor of the translational velocity. In the first part of this paper, we justify this model by comparing it to another commonly-used "Kirchhoff" formulation where the fiber exerts both perpendicular and parallel torque on the fluid, and the perpendicular angular fluid velocity is required to be consistent with the translational fluid velocity. We then develop a spectral numerical method for the hydrodynamics of the Euler model. We define hydrodynamic mobility operators using integrals of the Rotne-Prager-Yamakawa tensor, and evaluate these integrals through a novel slender-body quadrature, which requires on the order of 10 points along the fiber to obtain several digits of accuracy. We demonstrate that this choice of mobility removes the unphysical negative eigenvalues in the translation-translation mobility associated with asymptotic slender body theories, and ensures strong convergence of the fiber velocity and weak convergence of the fiber constraint forces. We pair the spatial discretization with a semi-implicit temporal integrator to confirm the negligible contribution of twist elasticity to the relaxation dynamics of a bent fiber and study the instability of a twirling fiber.
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