A regularised slender-body theory of non-uniform filaments

Resolving the detailed hydrodynamics of a slender body immersed in highly-viscous
Newtonian fluid has been the subject of extensive research, applicable to a broad
range of biological and physical scenarios. In this work, we expand upon classical
theories developed over the past fifty years, deriving an algebraically-accurate slenderbody theory that may be applied to a wide variety of body shapes, ranging from
biologically-inspired tapering flagella to highly-oscillatory body geometries with only
weak constraints, most-significantly requiring that cross sections be circular. Inspired by
well-known analytic results for the flow around a prolate ellipsoid, we pose an ansatz for
the velocity field in terms of a regular integral of regularised Stokes-flow singularities with
prescribed, spatially-varying regularisation parameters. A detailed asymptotic analysis is
presented, seeking a uniformly-valid expansion of the ansatz integral, accurate at leading
algebraic order in the geometry aspect ratio, to enforce no slip boundary conditions
and thus analytically justify the slender-body theory developed in this framework. The
regularisation within the ansatz additionally affords significant computational simplicity
for the subsequent slender-body theory, with no specialised quadrature or numerical
techniques required to evaluate the regular integral. Furthermore, in the special case of
slender bodies with a straight centreline in uniform flow, we derive a slender-body theory
that is particularly straightforward via use of the analytic solution for a prolate ellipsoid.
We evidence the validity of our simple theory by explicit numerical example for a wide
variety of slender bodies, and highlight a potential robustness of our methodology beyond
its rigorously-justified scope.