Explicit formulas are given for the hydrodynamic slip lengths associated with longitudinal shear flow over a superhydrophobic grating where the menisci have partially invaded the cavities and are only weakly curved. For flat menisci that have depinned from the top of the grating and have displaced downwards into the cavities, the axial velocity is determined analytically and the slip length extracted from it. This solution is then combined with an integral identity to determine the first-order correction to the slip length when the displaced menisci bow weakly into the cavity. It is argued that the new formulas provide useful upper bounds for quantifying slip in microchannel flows involving partially filled cavities. The new solutions are natural extensions of prior results due to Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372) for shear flow over mixed no-slip/no-shear surfaces and due to Bechert & Bartenwerfer (J. Fluid Mech., vol. 206, 1989, pp. 105–129) for shear flow over blade-shaped riblets.

In jet engines and gas turbines, the annular shape of the combustion chamber allows the appearance of self-oscillating azimuthal thermoacoustic modes. We report experimental evidence of a new type of modal dynamics characterised by periodic switching of the spinning direction and develop a theoretical model that fully reproduces this phenomenon and explains the underlying mechanisms. It is shown that tiny asymmetries of the geometry, the mean temperature field, the thermoacoustic response of the flames or the acoustic impedance of the walls, present in any real systems, can induce these heteroclinic orbits. The model also explains experimental observations showing a statistically dominant spinning direction despite the absence of swirling flow, or pairs of preferred nodal line directions.

This paper reports a new phenomenon in the near-free-surface region of open channel flows (OCFs): namely that higher relative population densities of spanwise vortices are observed than are seen at equivalent positions in turbulent boundary layers. The presence of additional retrograde spanwise vortices in OCFs throughout the region $y/h>0.2$ (where $y$ is the distance to the wall and $h$ is the water depth) prompts us to re-examine the wall-normal extent that the free-surface effect can reach, which is classically expected to be limited within the surface and blockage layers. Finally, possible mechanisms for the phenomenon of additional spanwise vortices in OCFs are provided.

This work examines consequences of modelling approximation errors made within the context of the Navier–Stokes–Boussinesq system. Starting from a canonical Boussinesq model, where density fluctuations are allowed to interact with all accelerative terms of the incompressible Navier–Stokes equations in arbitrarily accelerating reference frames, a unified treatment is developed that provides a straightforward way to identify buoyancy forcing associated with gravitational effects, centrifugal forcing associated with frame rotation, as well as centrifugal-type forcing due to variations in flow kinetic energy. The results of the cases studied in inertial, rotating and mixed reference frames demonstrate that in general it may be important to apply buoyancy effects to all non-local accelerative terms, including non-gradient terms such as Coriolis acceleration. Additionally, it is shown that the common practice of ignoring terms representing interaction between density fluctuation and local fluid acceleration can lead to non-negligible error in Boussinesq modelling of highly unsteady flows. These findings have special significance for accurate simulation of flows with density variations in which there may be both background rotation and localised regions of strong swirl, but are also relevant for studies conducted in the inertial frame of reference.

We experimentally demonstrate quadrupolar electro-osmotic flows around charged dielectric microspheres immersed in an electrolyte when subjected to an alternating current electric field. We present an electrokinetic model that predicts the flow characteristics based on the phenomena of surface conductance and polarization of the electrolyte concentration around the particles. We refer to these flows as concentration polarization electro-osmosis. We anticipate that these flows may play a major role in the electric-field-induced assembly of colloids and on the electrokinetic manipulation of dielectric micro- and nanoparticles.

Viscoelastic flow through an abrupt planar contraction geometry above a certain Weissenberg number ($Wi$) is well known to become unstable upstream of the contraction plane via a central jet separating from the walls and forming vortices in the salient corners. Here, for the first time, we consider three-dimensional (3-D) viscoelastic contraction flows in a microfabricated glass square–square contraction geometry. We employ state-of-the-art microtomographic particle image velocimetry to produce time-resolved and volumetric quantification of the 3-D viscoelastic instabilities arising in a dilute polymer solution driven through the geometry over a wide range of $Wi$ but at negligible Reynolds number. Based on our observations, we describe new insights into the growth, propagation and transient dynamics of an elastic vortex formed upstream of the 3-D microcontraction due to flow jetting towards the contraction. At low $Wi$ we observe vortex growth for increasing $Wi$, followed by a previously unreported vortex growth plateau region. In the plateau region, the vortex circulates around the jet with a period that decreases with $Wi$ but an amplitude that is independent of $Wi$. In addition, we report new out-of-plane asymmetric jetting behaviour with a phase-wise dependence on $Wi$. Finally, we resolve the rate-of-strain tensor $\boldsymbol{\mathsf{D}}$ and ascribe local gradients in $\boldsymbol{\mathsf{D}}$ as the underlying driver of circulation via strain hardening of the fluid in the wake of the jet.

Non-Newtonian fluids are characterized by complex rheological behaviour that affects the hydrodynamic features, such as the flow rate–pressure drop relation. While flow rate–pressure drop measurements of such fluids are common in the literature, a comparison of experimental data with theory is rare, even for shear-thinning fluids at low Reynolds number, presumably due to the lack of analytical expressions for the flow rate–pressure drop relation covering the entire range of pressures and flow rates. Such a comparison, however, is of fundamental importance as it may provide insight into the adequacy of the constitutive model that was used and the values of the rheological parameters. In this work, we present a theoretical approach to calculating the flow rate–pressure drop relation of shear-thinning fluids in long, narrow channels that can be used for comparison with experimental measurements. We utilize the Carreau constitutive model and provide a semi-analytical expression for the flow rate–pressure drop relation. In particular, we derive three asymptotic solutions for small, intermediate and large values of the dimensionless pressures or flow rates, which agree with distinct limits previously known and allow us to approximate analytically the entire flow rate–pressure drop curve. We compare our semi-analytical and asymptotic results with the experimental measurements of Pipe et al. (Rheol. Acta, vol. 47, 2008, pp. 621–642) and find excellent agreement. Our results rationalize the change in the slope of the flow rate–pressure drop data, when reported in log–log coordinates, at high flow rates, which cannot be explained using a simple power-law model.

We investigate with a two-fluid model the convections developing in a gas–liquid two-phase flow, which consists of a horizontal liquid layer subject to the injection of monodisperse gas bubbles at the bottom. The convections develop in either whole- or multi-layered modes, once the gas injection flux exceeds a critical value (Nakamura et al., Phys. Rev. E, vol. 102, 2020, 053102). We determine the nonlinear evolution of these modes of flows with varying injection flux and find that the whole- and multi-layered modes develop through subcritical and supercritical bifurcations, respectively. The formation of gas plumes is observed in both cases when the nonlinearity is significant. Examining energy transfer from base to perturbation flows, we show that the lift forces on bubbles play a key role in the bifurcations. While they impede the convections in both subcritical and supercritical bifurcations at weak nonlinearity, the lift forces turn to driving the convections in the subcritical bifurcation as nonlinearity increases.

We experimentally study the nonlinear fluid damping of a rigid but elastically mounted pitching wing in the absence of a free-stream flow. The dynamics of the elastic mount are simulated using a cyber-physical system. We perturb the wing and measure the fluid damping coefficient from damped oscillations over a large range of pitching frequencies, pitching amplitudes, pivot locations and sweep angles. A universal fluid damping scaling is proposed to incorporate all these parameters. Flow fields obtained using particle image velocimetry are analysed to explain the nonlinear behaviours of the fluid damping.

Families of exact coherent states in elliptical pipe flow obtained from the travelling-wave solutions in circular pipe flow by a continuation approach are found. Results are given in a regime of the aspect ratio $A$ in which the laminar flow is linearly stable. The results suggest the possibility of two distinct classes of solutions of elliptical travelling waves at higher values of $A$: (i) rotationally symmetric centre-mode states that collapse towards the pipe centre and (ii) rotationally asymmetric vortex–wave interaction states with additional mirror symmetry exhibiting organization of the waves around a critical layer. These are the first calculations of three-dimensional travelling waves in elliptical pipes. Investigation of these states has the potential to provide fresh insight into the relationship between exact coherent structures in Poiseuille flow in pipes and channels.

We study the effect of random perturbations of body positions in large uniformly spaced arrays of axisymmetric wave energy converters (WECs). We perform systematic computational simulations of ensembles of randomized array configurations that are obtained by introducing zero-mean position perturbations (characterized by randomness parameter $\varepsilon$) to line arrays of uniform spacing $d$. Of special interest are the conditions under which these randomized arrays can extract more energy than the underlying uniform arrays. We find that random WEC arrays achieve substantial energy extraction gains over the same number of isolated bodies in monochromatic and irregular incident seas. Introducing $\varepsilon >0$ acts to smooth out the uneven performance of uniformly spaced arrays over varying incident wavenumber. In the low-scattering regime, the standard deviation of array gain grows with the square of wavenumber, for which we provide a theoretical explanation. We show that the uniform line array with spacing optimized for a given incident spectrum generally outperforms randomized arrays of any mean $d$ and $\varepsilon$ in that spectrum, and we offer a heuristic explanation. This holds for a wide range of incident spectra.

Rotations of spheroidal particles immersed in turbulent flows reflect the combined effects of fluid strain and vorticity, as well as the time history of these quantities along the particle's trajectory. Conversely, particle rotation statistics in turbulence provide a way to characterise the Lagrangian properties of velocity gradients. Particle rotations are also important for a range of environmental and industrial processes where particles of various shapes and sizes are immersed in a turbulent flow. In this study, we investigate the rotations of inertialess spheroidal particles that follow Lagrangian fluid trajectories. We perform direct numerical simulations (DNS) of homogeneous isotropic turbulence and investigate the dynamics of different particle shapes at different scales in turbulence using a filtering approach. We find that the mean-square particle angular velocity is nearly independent of particle shape across all scales from the Kolmogorov scale to the integral scale. The particle shape does determine the relative split between different modes of rotation (spinning vs tumbling), but this split is also almost independent of the filter scale suggesting a Lagrangian scale-invariance in velocity gradients. We show how the split between spinning and tumbling can be quantitatively related to the particle's alignment with respect to the fluid vorticity.

We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a cylindrical Taylor–Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using direct numerical simulation and find that its emerging butterfly-type structure – a spatio-temporal variation in the form of axially upward and downward travelling waves – is in a very good agreement with the linear analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern.

The ability of microswimmers to deploy optimal propulsion strategies is of paramount importance for their locomotory performance and survival at low Reynolds numbers. Although for perfectly spherical swimmers minimum dissipation requires a neutral-type swimming, any departure from the spherical shape may lead the swimmer to adopt a new propulsion strategy, namely those of puller- or pusher-type swimming. In this study, by using the minimum dissipation theorem for microswimmers, we determine the flow field of an optimal nearly spherical swimmer, and show that indeed depending on the shape profile, the optimal swimmer can be a puller, pusher or neutral. Using an asymptotic approach, we find that amongst all the modes of the shape function, only the third mode determines, to leading order, the swimming type of the optimal swimmer.

Shedding of vortices can be observed in the wake of a fixed cylinder at Reynolds numbers larger than $Re=47$. This might give the impression that a vortex-induced vibration (VIV), which occurs when the frequency of vortex shedding in the wake of a flexibly mounted cylinder synchronizes with the natural frequency of the structure, could be observed only at Reynolds numbers larger than $Re=47$. Recent numerical simulations and theoretical work, however, have shown that it is possible to observe VIV at subcritical Reynolds numbers, i.e. Reynolds numbers smaller than $Re=47$. In these studies, a VIV has been observed numerically at Reynolds numbers as low as $Re=22$. In the present work, the first experimental evidence of VIV at subcritical Reynolds number is presented. We have designed and built an experimental set-up that makes it possible to conduct VIV experiments at subcritical Reynolds numbers, and at a constant Reynolds number over the entire lock-in range (i.e. the range for which oscillations are observed). Using this experimental set-up, we have confirmed experimentally that VIV can indeed be observed at subcritical Reynolds numbers, by observing VIV at Reynolds numbers as low as $Re=19$. We have observed subcritical VIV both when the Reynolds number stays constant over the entire lock-in range, and when the Reynolds number increases with increasing reduced velocity, while staying within the subcritical range.

We study experimentally the primary instability of a visco-plastic film flow down an inclined plane. The experimental set-up is a channel with a varying slope angle, in which a permanent flow of a Herschel–Bulkley fluid (carbopol or kaolin) is established. Controlled perturbations are imposed at the entrance of the channel to generate surface waves, and their downstream evolution is observed with a laser sensor system measuring the local fluid thickness. Growth rates and cutoff frequencies are obtained after processing the thickness signal, and experimental critical Reynolds and Bingham numbers are deduced. We find that the experimental stability map obtained is well described by the pseudo-plug model of Balmforth & Liu (J. Fluid Mech., vol, 519, 2004, pp. 33–54), a model obtained after neglecting the film thickness compared with its length. This is not the case for dispersion effects (growth rates, cutoff frequencies, phase speeds), for which a more accurate model is needed.

We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac 12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \frac {1}{2}(\frac {1}{2}+ \frac {1}{\sqrt {3}} ) - 1.6552\, {\textit {R}}^{-(1/3)}$ uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $ {\textit {R}}$) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on $ {\textit {R}}$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac 12$ at high $ {\textit {R}}$, which corresponds to the top boundary being hotter than the bottom one on average.

This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals.

Oscillatory flows have the potential to overcome long-standing limitations encountered when using steady flows for inertial focusing due to low particle Reynolds numbers. The periodic displacement generated by oscillatory flow increases the total path length travelled by a suspended particle with no net displacement within a channel or the need for increased channel length. The effects of unsteady inertial forces on inertial focusing, however, have not been thoroughly examined. Here, we present a combined theoretical and experimental study on the effect of Womersley number on inertial focusing in planar pulsatile flows. The migration velocity for a small and weakly inertial particle was evaluated for different oscillation frequencies using the method of matched asymptotics. Using experiments in a custom-built microchannel, we show that oscillatory flows are remarkably efficient for inertial focusing, even at low particle Reynolds numbers. For appropriately selected oscillation amplitude and frequency, inertial focusing is achieved in only a fraction of the channel length (1 % to 10 %) compared to what would be required in a steady flow. We show that the Womersley number influences the equilibrium focusing position and the overall focusing efficiency. In fact, above a critical Womersley number, inertial focusing does not occur despite increasing particle Reynolds number. Lastly, the application of oscillatory inertial focusing for the direct measurement of particle migration velocity is demonstrated.

We identify an exceptional point in the numerical stability map of a combustion experiment operated with laminar premixed flames. A low-order model of the experiment allows one to investigate the interplay between the system modes and an exceptional point. The latter is located in the unstable region of the complex eigenvalue plane. Under gradual variation of the operating parameters, the branch-switching characteristic of the exceptional point can facilitate abrupt changes between strong instability and stable operation. The results also indicate that intrinsic thermoacoustic feedback plays an important role in the stability characteristics of this experiment.