Shape of a tethered filament in various low-Reynolds-number flows

Abstract

We consider the steady-state deformation of an elastic filament in various unidirectional, low-Reynolds-number flows, with the filament either clamped at one end, perpendicular to the flow, or tethered at its center and deforming symmetrically about a plane parallel to the flow. We employ a slender-body model [Pozrikidis, J. Fluids Struct. 26, 393 (2010)0889-974610.1016/j.jfluidstructs.2010.01.008] to describe the filament shape as a function of the background flow and a nondimensional compliance η characterizing the ratio of viscous to elastic forces. For η1, we describe the small deformation of the filament by means of a regular perturbation expansion. For η≫1, the filament strongly bends such that it is nearly parallel to the flow except close to the tether point; we analyze this singular limit using boundary-layer theory, finding that the radius of curvature near the tether point, as well as the distance of the parallel segment from the tether point, scale like η-1/2 for flow profiles that do not vanish at the tether point, and like η-1/3 for flow profiles that vanish linearly away from the tether point. We also use a Wentzel-Kramers-Brillouin approach to derive a leading-order approximation for the exponentially small slope of the filament away from the tether point. We compare numerical solutions of the model over a wide range of η values with closed-form predictions obtained in both asymptotic limits, focusing on particular uniform, shear and parabolic flow profiles relevant to experiments.