Stokes流中若干细长刚性细丝的运动
Motion of several slender rigid filaments in a Stokes flow
AI总结：
本文研究Stokes流中多个细长刚体在厚度趋于零时的极限动力学，证明极限方程退化为仅依赖极限曲线和背景流的解耦一阶ODE，并给出收敛速率和初始指数松弛时间。
摘要：
我们研究在三维稳态Stokes系统驱动的流动中，若干细长刚体在光滑背景流存在时的动力学。更准确地说，我们考虑这些细长刚体的厚度以共同速率$\varepsilon$趋于零的极限，同时保持其体积质量密度固定，从而使这些物体收缩为彼此分离的无质量曲线。虽然对每个正的$\varepsilon$，物体的动力学由Newton方程给出，并对应于关于物体位置的某些耦合二阶常微分方程组，但我们证明极限方程是解耦的一阶常微分方程组，其系数仅依赖于极限曲线和背景流。这些系数通过与每条极限曲线相关联的适当重整化Stokes阻力张量，以及与极限曲线和背景流相关联的重整化Faxén型力和力矩出现。我们建立了曲线的收敛速率为$O(|\log \varepsilon|^{-1/2})$阶。本着浸入边界法的精神，我们还确定了极限曲线对流体的极限效应。对于细丝速度和流体速度的收敛，我们都识别出在$O(\varepsilon^2 |\log \varepsilon|)$时间内的初始指数松弛。
英文摘要：
We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate $\varepsilon$, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive $\varepsilon$, the bodies' dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. These coefficients appear through appropriate renormalized Stokes resistance tensors associated with each limit curve, and through renormalized Faxén-type force and torque associated with the limit curves and the background flow. We establish a rate of convergence of the curves of order $O( \vert \log \varepsilon \vert^{-1/2})$. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method. Both for the convergence of the filament velocities and the fluid velocities we identify an initial exponential relaxation within a $O(\varepsilon^2 \vert \log \varepsilon \vert)$ time.
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