TY - JOUR

T1 - Shape of a tethered filament in various low-Reynolds-number flows

AU - Kurzthaler, Christina

AU - Brandão, Rodolfo

AU - Schnitzer, Ory

AU - Stone, Howard A.

N1 - Publisher Copyright:
© 2023 authors. Published by the American Physical Society.

PY - 2023/1

Y1 - 2023/1

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevFluids.8.014101

DO - 10.1103/PhysRevFluids.8.014101

M3 - Article

AN - SCOPUS:85146366696

SN - 2469-990X

VL - 8

JO - Physical Review Fluids

JF - Physical Review Fluids

IS - 1

M1 - 014101

ER -