Unsteady drag force on an immersed sphere oscillating near a wall

TL;DR

This study investigates the unsteady hydrodynamic drag on an oscillating sphere near a wall through experiments, theory, and simulations, revealing how the force depends on the Womersley number and proximity to the wall.

Contribution

It combines experimental AFM measurements with theoretical and numerical analysis to characterize unsteady drag forces near a wall, highlighting the Womersley number's role.

Findings

Resonance frequency shifts depend on gap distance and Womersley number.

A crossover from positive to negative frequency shift occurs with increasing Womersley number.

Theoretical and numerical models agree well with experimental results.

Abstract

The unsteady hydrodynamic drag exerted on an oscillating sphere near a planar wall is addressed experimentally, theoretically, and numerically. The experiments are performed by using colloidal-probe Atomic Force Microscopy (AFM) in thermal noise mode. The natural resonance frequencies and quality factors are extracted from the measurement of the power spectrum density of the probe oscillation for a broad range of gap distances and Womersley numbers. The shift in the natural resonance frequency of the colloidal probe as the probe goes close to a solid wall infers the wall-induced variations of the effective mass of the probe. Interestingly, a crossover from a positive to a negative shift is observed as the Womersley number increases. In order to rationalize the results, the confined unsteady Stokes equation is solved numerically using a finite-element method, as well as asymptotic calculations.The in-phase and out-of-phase terms of the hydrodynamic drag acting on the sphere are obtained and agree well to the experimental results. All together, the experimental, theoretical, and numerical results show that the hydrodynamic force felt by an immersed sphere oscillating near a wall is highly dependent on the Womersley number.