We study, using both experiment and theory, the coupling of transport and shape dynamics for elastomeric fibres moving through an inhomogeneous flow. The cellular flow, created electromagnetically in our experiment, comprises many identical cells of counter-rotating vortices, with a global flow geometry characterized by a backbone of stable and unstable manifolds connecting hyperbolic stagnation points. Our mathematical model is based upon slender-body theory for the Stokes equations, with the fibres modelled as inextensible elastica. Above a certain threshold of the control parameter, the elasto-viscous number, transport of fibres is mediated by their episodic buckling by compressive stagnation point flows, lending an effectively chaotic component to their dynamics. We use simulations of the model to construct phase diagrams of the fibre state (buckled or not) near stagnation points in terms of two variables that arise in characterizing the transport dynamics. We show that this reduced statistical description quantitatively captures our experimental observations. By carefully reproducing the experimental protocols and time scales of observation within our numerical simulations, we also quantitatively explain features of the measured buckling probability curve as a function of the effective flow forcing. Finally, we show within both experiment and simulation the existence of short and long time scales in the evolution of fibre conformation.

We investigate the trajectories of rigid fibres as they are transported in a pressure-driven flow, at low Reynolds number, in shallow Hele-Shaw cells. The transverse confinement and the resulting viscous friction on these elongated objects, as well as the lateral confinement (i.e. the presence of lateral walls), lead to complex fibre trajectories that we characterize with a combination of microfluidic experiments and simulations using modified Brinkman equations. We show that the transported fibre behaves as an oscillator for which we obtain and analyse a complete state diagram.

We present a mathematical model and corresponding series of microfluidic experiments examining the flow of a viscous fluid past an elastic fibre in a three-dimensional channel. The fibre’s axis lies perpendicular to the direction of flow and its base is clamped to one wall of the channel; the sidewalls of the channel are close to the fibre, confining the flow. Experiments show that there is a linear relationship between deflection and flow rate for highly confined fibres at low flow rates, which inspires an asymptotic treatment of the problem in this regime. The three-dimensional problem is reduced to a two-dimensional model, consisting of Hele-Shaw flow past a barrier, with boundary conditions at the barrier that allow for the effects of flexibility and three-dimensional leakage. The analysis yields insight into the competing effects of flexion and leakage, and an analytical solution is derived for the leading-order pressure field corresponding to a slit that partially blocks a two-dimensional channel. The predictions of our model show favourable agreement with experimental results, allowing measurement of the fibre’s elasticity and the flow rate in the channel.