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Theoretical Justification and Error Analysis for Slender Body Theory
Author(s) -
Mori Yoichiro,
Ohm Laurel,
Spirn Daniel
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21872
Subject(s) - mathematics , slender body theory , fiber , mathematical analysis , body force , neumann boundary condition , stokes flow , constant (computer programming) , bounded function , surface (topology) , boundary value problem , flow (mathematics) , geometry , classical mechanics , physics , chemistry , organic chemistry , computer science , programming language
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 1D curve does not result in a well‐posed boundary value problem for the Stokes equations in 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, allows 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. © 2019 Wiley Periodicals, Inc.