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Slender body theory facilitates computational simulations of thin fibersimmersed in a viscous fluid by approximating each fiber using only the geometryof the fiber centerline curve and the line force density along it. However, ithas been unclear how well slender body theory actually approximates Stokes flowabout a thin but truly three-dimensional fiber, in part due to the fact thatsimply prescribing data along a one-dimensional curve does not result in awell-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 anapproximation, thereby placing SBT on firm theoretical footing. The slenderbody PDE is a new type of boundary value problem for Stokes flow where partialDirichlet and partial Neumann conditions are specified everywhere along thefiber surface. Given only a 1D force density along a closed fiber, we show thatthe flow field exterior to the thin fiber is uniquely determined by imposing afiber integrity condition: the surface velocity field on the fiber must beconstant along cross sections orthogonal to the fiber centerline. Furthermore,a careful estimation of the residual, together with stability estimatesprovided by the PDE well-posedness framework, allow us to establish errorestimates between the slender body approximation and the exact solution to theabove problem. The error is bounded by an expression proportional to the fiberradius (up to logarithmic corrections) under mild regularity assumptions on the1D force density and fiber centerline geometry.

Theoretical justification and error analysis for slender body theory