Matthias Heil

Solid particles immersed in a low-Reynolds-number shear flow generally tend to rotate. It is, however, possible to construct rigid particles that adopt a constant orientation relative to the flow while drifting across the streamlines. A recent study by Roggeveen & Stone (2022 J. Fluid Mech . 939, A23), analysed the existence and stability of such orientations for ‘boomerang’-shaped rigid fibres formed by two straight arms of different lengths. Given that slender fibres are easily deformed by the fluid traction, we extend their study to the case of elastic fibres, allowing for arbitrary stress-free initial shapes. We establish the conditions under which such fibres adopt a constant shape and orientation while drifting steadily across the streamlines. Numerical simulations show that, in other parameter regimes, the fibres perform Jeffery-like rotations while undergoing large-amplitude deformations. In such regimes, the fibres are advected by the flow but have a zero net drift across the streamlines, implying that a shear flow might be used to separate dilute suspensions of such fibres based on their shapes and/or elasticity. We explore the transition between steady (non-rotating) and Jeffery-like (rotating) states and show that the transition between these regimes arises through a degenerate sniper bifurcation. The behaviour is found to be generic for a wide range of fibre shapes.

The accurate characterization of the three-dimensional deformations of slender fibers and thin sheets in flow is a key experimental challenge in the study of particle-laden flows. We propose a high-resolution, single-camera method to visualize nonintrusively the shape of a transparent crumpled sheet, as it translates, rotates, and deforms. We perform periodic scans of the crumpled shape by illuminating it with a sequence of stacked light sheets at a rate much faster than its deformation and image the scattered light signal in a plane near orthogonal to the plane of lighting. Processing of the data using a pinhole camera model yields a noisy spatiotemporal dataset of the strongly deformed time-evolving surface of the sheet, which we reconstruct in three dimensions using a neural autoencoder. We validate the robustness of the shape reconstruction algorithm to noise using synthetic data sets, and demonstrate the accurate reconstruction of laboratory sedimentation experiments with elastic disks. We find that the inclusion of isometricity-enforcing penalties into the cost function of the autoencoder enables us to robustly reconstruct highly folded shapes, where different regions of the sheet overlap.

There has been much recent interest in the chiral motion of achiral particles that sediment in a viscous fluid in a regime where inertial effects can be neglected. This occurs in a broad range of applications such as those involving biological objects like algae, ultrathin graphene flakes, or colloidal suspensions. It is known that particles with two planes of symmetry can be categorized as “settlers,” “drifters,” or “flutterers,” where the latter sediment along chiral trajectories despite their achiral shapes. Previous work investigated the sedimentation of circular disks bent into a U-shape and identified them as “flutterers.” In this work we analyze the change in behavior of such particles when we break one of their symmetries by pinching the disks isometrically along their axis, a shape change that can arise during the sedimentation of thin elastic sheets. The “fluttering” behavior is found to be robust to such shape changes, with the trajectories now evolving toward helical paths. However, the behavior changes when the degree of pinching becomes too strong, at which point the particles become “drifters” which sediment steadily without rotation. We establish criteria for the transition between the two types of behavior and confirm our predictions in experiments. Finally, we discuss the implications of our observations for the dispersion of dilute suspensions made of such particles.

The critical points of vorticity in a two-dimensional viscous flow are essential for identifying coherent structures in the vorticity field. Their bifurcations as time progresses can be associated with the creation, destruction or merging of vortices, and we analyse these processes using the equation of motion for these points. The equation decomposes the velocity of a critical point into advection with the fluid and a drift proportional to viscosity. Conditions for the drift to be small or vanish are derived, and the analysis is extended to cover bifurcations. We analyse the dynamics of vorticity extrema in numerical simulations of merging of two identical vortices at Reynolds numbers ranging from 5 to 1500 in the light of the theory. We show that different phases of the merging process can be identified on the basis of the balance between advection and drift of the critical points, and identify two types of merging, one for low and one for high values of the Reynolds number. In addition to local maxima of positive vorticity and minima of negative vorticity, which can be considered centres of vortices, minima of positive vorticity and maxima of negative vorticity can also exist. We find that such anti-vortices occur in the merging process at high Reynolds numbers, and discuss their dynamics.

We study the sedimentation of U-shaped circular disks in the Stokes limit of vanishing inertia. We simulate the flow past such disks using a finite-element-based solution of the three-dimensional Stokes equations, accounting for the integrable singularities that develop along their edges. We show that the purely vertical sedimentation of such disks in their upright (upside-down) U orientation is unstable to perturbations about their pitching (rolling) axes. The instability is found to depend only weakly on the size of the container in which the disks sediment, allowing us to analyse their behaviour based on the resistance matrix which governs the evolution of the disk's six rigid-body degrees of freedom in an unbounded fluid. We show that the governing equations can be reduced to two ordinary differential equations which describe the disk's inclination against the direction of gravity. A phase-plane analysis, the results of which are in good agreement with experiments, reveals that the two instabilities generally cause the disk to sediment along complex spiral trajectories while it alternates between pitching- and rolling-dominated motions. The chirality of the trajectories is set by the initial conditions rather than the (non-chiral) shape of the disk. For certain initial orientations, the disk retains its inclination and sediments along a perfectly helical path. The observed behaviour is fundamentally different from that displayed by flat circular disks which sediment without any reorientation. We therefore study the effect of variations in the disk's curvature to show how in the limit of vanishing curvature the behaviour of a flat disk is recovered.

We examine the mechanisms responsible for the onset of the three-dimensional mode B instability in the wake behind a circular cylinder. We show that it is possible to explicitly account for the stabilising effect of spanwise viscous diffusion and then demonstrate that the remaining mechanisms involved in this short-wavelength instability are preserved in the limit of zero wavelength. Using the resulting simplified equations, we show that perturbations in different fluid particles interact only through the in-plane viscous diffusion which turns out to have a destabilising effect. We also show that in the presence of viscous diffusion, the closed trajectories which had been conjectured to play a crucial role in the onset of the mode B instability are not actually a prerequisite for the growth of mode B type perturbations. We combine these observations to identify the three essential ingredients for the development of the mode B instability: (i) the amplification of perturbations in the braid regions due to the stretching mechanism; and the spreading of perturbations through (ii) viscous diffusion, and (iii) cross-flow advection which transports fluid between the two braid regions on either side of the cylinder. Finally, we develop a simple criterion that allows the prediction of the regions where three-dimensional short-wavelength perturbations are amplified by the stretching mechanism. The approach used in our study is general and has the potential to give insights into the onset of three-dimensionality via short-wavelength instabilities in other flows.

When particles sediment in a viscous fluid, the character of their trajectories depends sensitively on the particles’ shape. Here we study the sedimentation of U-shaped rigid disks in a regime where inertia can be neglected. We show that, unlike the case of planar disks which settle in a fixed orientation relative to the direction of gravity, U-shaped disks tend to perform a periodic sequence of pitching and rolling motions which cause their centre of mass to sediment along complex trajectories that range from quasi-periodic spirals to helices. Thus, we demonstrate that particles of achiral shape can sediment along chiral paths whose handedness is determined by their initial orientation rather than their geometry. Our analysis provides a framework in which to interpret the motion of sedimenting particles of arbitrary shape.

We study the effect of inflation on the swelling-induced wrinkling of thin elastic membranes in a set-up that is commonly used to create microchannels in lab-on-chip applications.

We study the onset of the three-dimensional mode A instability in the near wake behind a circular cylinder. We show that long-wavelength perturbations organise in a time-shifting pattern such that the in-plane velocity in each streamwise slice corresponds to the base flow solution at shifted times. This observation introduces an additional unifying characteristic for certain mode A type instabilities. We then analyse the mechanisms which control the growth or decay of these perturbations and highlight the crucial role played by the tilting mechanism which operates via non-local interactions in a manner similar to Biot–Savart induction. We characterise its domain of influence using a Green's function-based approach which allows us to rationalise the non-trivial dependence of the growth rate on the spanwise wavenumber. We connect this behaviour to the subtle balance between the local growth of the perturbations as they are swept along by the flow and the feedback on the perturbations that are generated during the next period of the time-periodic base flow. Finally, we discuss generalisations of our findings to other types of flows.