论文标题
细长的身体理论的精度，薄纤维在粘性流体上施加的近似力
Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid
论文作者
论文摘要
我们考虑了纤维纤维在规定纤维速度时近似薄纤维在周围粘性流体上施加的力的准确性。我们认为这是细长的身体反问题，众所周知，当规定力时细长的身体理论会收敛到良好的PDE溶液，并且纤维速度尚不清楚。从PDE的角度来看，细长的身体逆问题仅仅是stokes方程的问题，但是从近似角度来看，非局部细长的身体理论表现出很高的波数不稳定性，使分析变得复杂。在这里，我们考虑了将细长身体近似正规化的两种方法：光谱截断和Tornberg and Shelley（2004）的$δ$调查。对于具有恒定半径$ε> 0 $的直纤维，我们明确计算了操作员映射纤维速度的频谱，以强制PDE和近似值。我们表明，原始细长体近似的频谱与低波数在高频下的PDE溶液非常吻合，但在高频上有所不同，从而使我们能够使用波数截止值$ \ sim1/ε$来定义截断的近似值。对于截短和$δ$调数的近似值，我们获得了与PDE解决方案的类似收敛结果，例如$ε\ to0 $：具有$ h^1 $规律性的纤维速度可提供$ o（ε）$收敛，而纤维速度至少具有$ h^2 $ h^2 $ juromential narrifity to $ o o（ε^2 $ o（ε^2）$ clengence。此外，我们确定了$δ$调查误差估计对正则化参数$δ$的依赖性。
We consider the accuracy of slender body theory in approximating the force exerted by a thin fiber on the surrounding viscous fluid when the fiber velocity is prescribed. We term this the slender body inverse problem, as it is known that slender body theory converges to a well-posed PDE solution when the force is prescribed and the fiber velocity is unknown. From a PDE perspective, the slender body inverse problem is simply the Dirichlet problem for the Stokes equations, but from an approximation perspective, nonlocal slender body theory exhibits high wavenumber instabilities which complicate analysis. Here we consider two methods for regularizing the slender body approximation: spectral truncation and the $δ$-regularization of Tornberg and Shelley (2004). For a straight, periodic fiber with constant radius $ε>0$, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We show that the spectrum of the original slender body approximation agrees closely with the PDE solution at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff $\sim1/ε$. For both the truncated and $δ$-regularized approximations, we obtain similar convergence results to the PDE solution as $ε\to0$: a fiber velocity with $H^1$ regularity gives $O(ε)$ convergence, while a fiber velocity with at least $H^2$ regularity yields $O(ε^2)$ convergence. Moreover, we determine the dependence of the $δ$-regularized error estimate on the regularization parameter $δ$.
