Deformation of a flexible fiber in a viscous flow past an obstacle

TL;DR

This study investigates how elastic fibers deform and move in viscous flows with curved streamlines, revealing a key dimensionless parameter, the Sperm number, that governs the balance between elastic and viscous forces.

Contribution

The paper introduces the Sperm number as a dimensionless parameter to describe fiber deformation in viscous flows, linking flow parameters and fiber properties.

Findings

Fiber curvature closely follows local flow velocity and orientation.

The curvature ratio depends on the Sperm number, showing a linear increase for low values.

Fibers reach streamline curvature at high Sperm numbers.

Abstract

We study the deformation and transport of elastic fibers in a viscous Hele-Shaw flow with curved streamlines. The variations of the global velocity and orientation of the fiber follow closely those of the local flow velocity. The ratios of the curvatures of the fibers by the corresponding curvatures of the streamlines reflect a balance between elastic and viscous forces: this ratio is shown experimentally to be determined by a dimensionless {\it Sperm number} $Sp$ combining the characteristic parameters of the flow (transverse velocity gradient, viscosity, fiber diameter/cell gap ratio) and those of the fiber (diameter, effective length, Young's modulus). For short fibers, the effective length is that of the fiber; for long ones, it is equal to the transverse characteristic length of the flow. For $S\_p \lesssim 250$, the ratio of the curvatures increases linearly with $Sp$; For $S\_p \gtrsim 250$, the fiber reaches the same curvature as the streamlines.