Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
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DOI：
10.1002/cpa.21872
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发表时间：
2018-06
影响因子：
2.5
通讯作者：
Yoichiro Mori;Laurel Ohm;Daniel Spirn
中科院分区：
文献类型：
--
作者：
Yoichiro Mori;Laurel Ohm;Daniel Spirn
细长体理论是粘性流体中细纤维计算模型中常用的近似，特别是在模拟游动微生物中纤毛或鞭毛的运动时。在Mori等人中。(Commun Pure App Math，2018)Arxiv：1807.00178)，我们开发了一个偏微分方程组框架，用于分析等半径光纤的细长体近似引入的误差。ε\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setLength{\oddsidemargin}{-69pt}\Begin{文档}$\varepsilon$\end{Document}，并证明了我们的闭环偏微分方程解和细长身体近似之间的差异由与ε|LOGε|\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{amsbsy}\usepackage{upgreek}\setLong{\oddsidemargin}{-69pt}\Begin{Document}$\varepsilon|\log\varepsilon|$\END{Document}的表达式限定。在这里，我们将细长身体PDE框架扩展到自由端点设置，从建模的角度来看，这更具物理意义，但比闭合环路分析更具技术要求。在自由终点设置中出现的主要新困难是定义终点几何形状、确定一维细长体力密度的范围以及确定适定性常数如何取决于非常数纤维半径。假定细长纤维在纤维端点处满足一定的几何约束，且一维力密度满足一个端点衰减条件，我们证明了在自由端点条件下，细长体偏微分方程组的解与细长体近似解之间的差值的一个界。界限是出现在闭合循环设置中的相同ε|ε|\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\Usepackage{amsFonts}\Usepackage{amssymb}\Usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setLong{\oddsidemargin}{-69pt}\Begin{Document}$\varepsilon|\Log\varepsilon|$$\end{Document}术语和与ε\DocumentClass[12pt]{}\usepackage{amsath}成比例的终结点术语的总和Usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\varepsilon$\end{Document}，其中，ε\DocumentClass[12pt]{Minimal}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\varepsilon$$\end{Document}现在是最大光纤半径。
Slender body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178), we developed a PDE framework for analyzing the error introduced by the slender body approximation for closed-loop fibers with constant radius ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to ε|logε|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon |\log \varepsilon |$$\end{document}. Here we extend the slender body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D slender body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the slender body PDE and the slender body approximation in the free endpoint setting. The bound is a sum of the same ε|logε|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon |\log \varepsilon |$$\end{document} term appearing in the closed loop setting and an endpoint term proportional to ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is now the maximum fiber radius.