Free drops of uncharged and charged inviscid, conducting fluids subjected to small-amplitude perturbations undergo linear oscillations (Rayleigh, Proc. R. Soc. London, vol. 29, no. 196–199, 1879, pp. 71–97; Rayleigh, Philos. Mag., vol. 14, no. 87, 1882, pp. 184–186). There exist a countably infinite number of oscillation modes, $n=2, 3, \ldots$, each of which has a characteristic frequency and mode shape. Presence of charge ($Q$) lowers modal frequencies and leads to instability when $Q>Q_R$ (Rayleigh limit). The $n=0$ and $n=1$ modes are disallowed because they violate volume conservation and cause centre of mass (COM) motion. Thus, the first mode to become unstable is the $n=2$ prolate–oblate mode. For free drops, there is a one-to-one correspondence between mode number and shape (Legendre polynomial $P_n$). Recent research has shifted to studying oscillations of spherical drops constrained by solid rings. Pinning the drop introduces a new low-frequency mode of oscillation ($n=1$), one associated primarily with COM translation of the constrained drop. We analyse theoretically the effect of charge on oscillations of constrained drops. Using normal modes and solving a linear operator eigenvalue problem, we determine the frequency of each oscillation mode. Results demonstrate that for ring-constrained charged drops (RCCDs), the association between mode number and shape is lost. For certain pinning locations, oscillations exhibit eigenvalue veering as $Q$ increases. While slightly charged RCCDs pinned at zeros of $P_2$ have a first mode that involves COM motion and a second mode that entails prolate–oblate oscillations, the modes flip as $Q$ increases. Thereafter, prolate–oblate oscillations of RCCDs adopt the role of being the first mode because they exhibit the lowest vibration frequency. At the Rayleigh limit, the first eigenmode – prolate–oblate oscillations – loses stability while the second – involving COM motion – remains stable.

We present a detailed comparison of the rheological behaviour of sheared sediment beds in a pressure-driven, straight channel configuration based on data that were generated by means of fully coupled, grain-resolved direct numerical simulations and experimental measurements previously published by Aussillous et al. (J. Fluid Mech., vol. 736, 2013, pp. 594–615). The highly resolved simulation data allow us to compute the stress balance of the suspension in the streamwise and vertical directions and the stress exchange between the fluid and particle phases, which is information needed to infer the rheology, but has so far been unreachable in experiments. Applying this knowledge to the experimental and numerical data, we obtain the statistically stationary, depth-resolved profiles of the relevant rheological quantities. The scaling behaviour of rheological quantities such as the shear and normal viscosities and the effective friction coefficient are examined and compared to data coming from rheometry experiments and from widely used rheological correlations. We show that rheological properties that have previously been inferred for annular Couette-type shear flows with neutrally buoyant particles still hold for our set-up of sediment transport in a Poiseuille flow and in the dense regime we found good agreement with empirical relationships derived therefrom. Subdividing the total stress into parts from particle contact and hydrodynamics suggests a critical particle volume fraction of 0.3 to separate the dense from the dilute regime. In the dilute regime, i.e. the sediment transport layer, long-range hydrodynamic interactions are screened by the porous medium and the effective viscosity obeys the Einstein relation.

While studies on unsteady airfoils in a uniform free stream are abundant, the quest for efficient man-made propulsion and energy harvesting calls for an improved understanding and predictive capability of unsteady airfoils encountering external flow disturbances. In this paper, we conduct experimental and theoretical investigations of the interactions between an airfoil engaged in unsteady motion and external flow disturbances generated by an upstream source. The flow field interactions are experimentally studied using particle-image velocimetry and finite-time Lyapunov exponent techniques. An interesting outcome of the interactions is an interruption of leading-edge vortex (LEV) shedding from the airfoil and a consequent modulation of the lift history, which are dependent on the phase of the disturbances relative to the airfoil kinematics. A low-order model for an unsteady airfoil encountering the disturbances is built upon the leading-edge suction parameter (LESP)-modulated discrete-vortex method (LDVM) developed by Ramesh et al. (J. Fluid Mech., vol. 751, 2014, pp. 500–538). The LDVM distils the determination of the LEV shedding characteristics of unsteady airfoils to a single parameter, the LESP. We show that the LDVM, modified for the current work, is able to predict the effect of the disturbances on the LEV shedding characteristics of the airfoil and the associated lift history in good agreement with experimental observations. In addition to being a predictive tool, the LDVM also augments the experimental study by providing a theoretical framework and various graphical approaches to analyse the flow phenomena from a fundamental perspective and elucidate the role of different factors governing the flow field evolution.

Ray-finned fish are able to adapt the curvature of their fins through musculature at the base of the fin. In this work we numerically investigate the effects of such leading-edge curvature actuation on the hydrodynamic performance of a heaving and pitching fin. We present a geometric and numerical framework for constructing the shape of ray-membrane-type fins with imposed leading-edge curvatures, under the constraint of membrane inextensibility. This algorithm is coupled with a three-dimensional Navier–Stokes solver, enabling us to assess the hydrodynamic performance of such fins. To determine the space of possible shapes, we present a simple model for leading-edge curvature actuation through two coefficients that determine chordwise and spanwise curvature, respectively. We systematically vary these two parameters through regimes that mimic both passive elastic deformations and active actuation against the hydrodynamic loading, and compute thrust and power coefficients, as well as hydrodynamic efficiency. Our results demonstrate that both thrust and efficiency are predominantly affected by chordwise curvature, with some small additional benefits of spanwise curvature on efficiency. The main improvements in performance are explained by the altered trailing-edge kinematics arising from leading-edge curvature actuation, which can largely be reproduced by a rigid fin whose trailing-edge kinematics follow that of the curving fin. Changes in fin camber, for fixed trailing-edge kinematics, mostly benefit efficiency. Based on our results, we discuss the use of leading-edge curvature actuation as a robust and versatile way to improve flapping fin performance.

Flow of a shear-thinning fluid over curved and moving geometries demands special attention owing to its industrial and biological relevance. The present study reports a numerical simulation of the motion of a Carreau fluid over a transversely oscillating cylinder for a range of oscillation amplitudes and frequencies at Reynolds number $Re = 100$, and compares the flow behaviour with that of a Newtonian fluid. The numerical solver uses the sharp-interface immersed boundary method to reconstruct the flow variables at the intercepted cells in the vicinity of the moving cylinder geometry. Synchronized vortex shedding is observed during the locked-in oscillation mode, whereas paired counter-rotating vortices are shed during the non-locked-in mode. The generation of vorticity and enstrophy variations in the Carreau and Newtonian fluids are compared. The vortex patterns in the Carreau fluid show a strong difference from those of the Newtonian fluid at higher amplitude of oscillation ($A^{*} = 1.2$). This difference in the flow structures is explained through the behaviour of the diffusion terms in the vorticity transport equation. Further, the effect of the Carreau fluid properties on the flow structures and vorticity dynamics are also discussed.

Motivated by observations of turbulence in the strongly stratified ocean thermocline, we use direct numerical simulations to investigate the interaction of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite strong nonlinearities in the flow and a lack of scale separation, we find that linear ray-tracing theory is qualitatively useful in describing the early development of the flow as the wave is refracted by the shear. Consistent with the linear theory, the energy of the wave accumulates in regions of negative mean shear where we observe evidence of convective and shear instabilities. Streamwise-aligned convective rolls emerge the fastest, but their contribution to irreversible mixing is dwarfed by shear-driven billow structures that develop later. Although the wave strongly distorts the buoyancy field on which these billows develop, the mixing efficiency of the subsequent turbulence is similar to that arising from Kelvin–Helmholtz instability in a stratified shear layer. We run simulations at Reynolds numbers Re of 5000 and 8000, and vary the initial amplitude of the internal gravity wave. For high values of initial wave amplitude, the results are qualitatively independent of $Re$. Smaller initial wave amplitudes delay the onset of the instabilities, and allow for significant laminar diffusion of the internal wave, leading to reduced turbulent activity. We discuss the complex interaction between the mean flow, internal gravity wave and turbulence, and its implications for internal wave-driven mixing in the ocean.

Microorganisms may exhibit rich swimming behaviours in anisotropic fluids, such as liquid crystals, which have direction-dependent physical and rheological properties. Here we construct a two-dimensional computation model to study the undulatory swimming mechanisms of microswimmers in a solution of rigid, rodlike liquid crystal polymers. We describe the fluid phase using Doi's $Q$-tensor model, and treat the swimmer as a finite-length flexible fibre with imposed propagating travelling waves on the body curvature. The fluid–structure interactions are resolved via an immersed boundary method. Compared with the swimming dynamics in Newtonian fluids, we observe non-Newtonian behaviours that feature both enhanced and retarded swimming motions in lyotropic liquid crystal polymers. We reveal the propulsion mechanism by analysing the near-body flow fields and polymeric force distributions, together with asymptotic analysis for an idealized model of Taylor's swimming sheet.