A higher-order slender-body theory for axisymmetric flow past a particle at moderate Reynolds number

Abstract

Slender-body theory is utilized to derive an asymptotic approximation to the hydrodynamic drag on an axisymmetric particle that is held fixed in an otherwise uniform stream of an incompressible Newtonian fluid at moderate Reynolds number. The Reynolds number, $Re$ , is based on the length of the particle. The axis of rotational symmetry of the particle is collinear with the uniform stream. The drag is expressed as a series in powers of $1/\text{ln}(1/\unicode[STIX]{x1D716})$ , where $\unicode[STIX]{x1D716}$ is the small ratio of the characteristic width to length of the particle; the series is asymptotic for $Re\ll O(1/\unicode[STIX]{x1D716})$ . The drag is calculated through terms of $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ , thereby extending the work of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462) who determined the drag through $O[1/\text{ln}^{2}(1/\unicode[STIX]{x1D716})]$ . The calculation of the $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ term is accomplished via the generalized reciprocal theorem (Lovalenti & Brady, J. Fluid Mech., vol. 256, 1993, pp. 561–605). The first dependence of the inertial contribution to the drag on the cross-sectional profile of the particle is at $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ . Notably, the drag is insensitive to the direction of travel at this order. The asymptotic results are compared to a numerical solution of the Navier–Stokes equations for the case of a prolate spheroid. Good agreement between the two is observed at moderately small values of $\unicode[STIX]{x1D716}$ , which is surprising given the logarithmic error associated with the asymptotic expansion.