Limitation on the use of slender-body theory in Stokes flow: Falling needle viscometer errata

An attempt to model the “brush” flow, studied experimentally by Tachie et al. [Phys. Fluids 16, 445 (2004)], by means of slender-body theory revealed that a successful truncation procedure requires that a few orthogonal functions be sufficient to describe the singularity distributions. In previous applications, this was easily achieved. However, when walls and/or other slender bodies are in close proximity, a large number of basis functions are likely to be needed and a loss of diagonal dominance, indeed the eventual appearance of negative diagonal coefficients, precludes a successful truncation of the infinite system of linear equations. It was also noticed, but after the proofs had been checked, that a sign error, eradicating the above difficulty, had been made in the falling needle viscometer presentation [A. M. J. Davis and H. Brenner, Phys. Fluids 13, 3086 (2001)]. Fortunately, in that flow the bounding cylinder wall is sufficiently distant from the thin needle for a convergent truncation to be quickly achieved, and the modified numerical results, listed below, exhibit fairly limited changes.