Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a one-dimensional curve does not result in a well-posed boundary value problem for the Stokes equations in $\mathbb{R}^3$. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allow us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.

中文翻译：

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细长体理论的理论论证与误差分析

细长体理论通过仅使用纤维中心线曲线的几何形状和沿其的线力密度来近似每根纤维，从而促进了浸入粘性流体中的细纤维的计算模拟。然而，目前尚不清楚细长体理论实际上如何近似关于细而真正的三维纤维的斯托克斯流，部分原因是简单地沿一维曲线指定数据并不会导致适定边界$\mathbb{R}^3$ 中斯托克斯方程的值问题。在这里，我们引入了一个 PDE 问题，细长体理论 (SBT) 提供了一个近似值，从而将 SBT 置于稳固的理论基础上。细长体偏微分方程是斯托克斯流的一种新型边值问题，其中沿纤维表面处处指定部分 Dirichlet 和部分 Neumann 条件。仅给定沿闭合纤维的 1D 力密度，我们表明细纤维外部的流场是通过施加纤维完整性条件唯一确定的：纤维上的表面速度场沿着与纤维中心线正交的横截面必须是恒定的. 此外，对残差的仔细估计，连同 PDE 适定框架提供的稳定性估计，使我们能够在细长体近似和上述问题的精确解之间建立误差估计。