Fractional dynamics of tethered particles in oscillatory Stokes flows

A mechanistic model for the low-Reynolds-, high-Strouhal-number behaviour of a system consisting of a spherical particle attached to an inelastic tether under uniform sinusoidal cross-flow is presented. Unsteady history drag and virtual mass effects are considered for both the sphere and the tether. The mechanics of the problem is such that the resulting coupled fractional differential equations are linear and solvable analytically. The stationary solutions obtained in this work show that there are limiting dimensions for the length and thickness of the tether when compared to the radius of the particle that allow for the motion of the particle–tether system to simulate the motion of a free particle. These conditions exist for the range of small oscillation amplitudes that are required for keeping the particle Reynolds number smaller than unity while oscillating the particle–tether system at high frequencies (Strouhal numbers larger than unity). The fractional order model for the particle–tether system is compared against detailed experimental results for tethered particles for a wide range of experimental frequencies, including the low-frequency range where tether effects are measurable.