A yield stress is added to Taylor's (1952, Proc. Royal Soc. A, 211, 225-239) model of a microscopic organism with a wavy cylindrical tail swimming through a viscous fluid. Viscoplastic slender-body theory is employed for the task, generalising existing results for Bingham fluid to the Herschel–Bulkley constitutive model. Numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength, and a Bingham number which describes the strength of the yield stress. Numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. If the wave amplitude is sufficiently small, the yield stress of the material inevitably dominates the flow; the resulting ‘plastic locomotion’ results in swimming speeds that depend strongly on the swimming gait, and can, in some cases, even be negative. Conversely, when the yield stress is large, swimming becomes possible at the wave speed, with the swimmer sliding or burrowing along its centreline with a relatively high efficiency.

This study discusses turbulent suspension flows of non-Brownian, non-colloidal, neutrally buoyant and rigid spherical particles in a Newtonian fluid over porous media with particles too large to penetrate and move through the porous layer. We consider suspension flows with the solid volume fraction ${{\varPhi _b}}$ ranging from 0 to 0.2, and different wall permeabilities, while porosity is constant at 0.6. Direct numerical simulations with an immersed boundary method are employed to resolve the particles and flow phase, with the volume-averaged Navier–Stokes equations modelling the flow within the porous layer. The results show that in the presence of particles in the free-flow region, the mean velocity and the concentration profiles are altered with increasing porous layer permeability because of the variations in the slip velocity and wall-normal fluctuations at the suspension-porous interface. Furthermore, we show that variations in the stress condition at the interface significantly affect the particle near-wall dynamics and migration toward the channel core, thereby inducing large modulations of the overall flow drag. At the highest volume fraction investigated here, ${{\varPhi _b}}= 0.2$, the velocity fluctuations and the Reynolds shear stress are found to decrease, and the overall drag increases due to the increase in the particle-induced stresses.

Astrophysical flows are often subject to both rotation and large-scale background magnetic fields. Individually, each is known to two-dimensionalize the flow in the perpendicular plane. In realistic settings, both of these effects are simultaneously present and, importantly, need not be aligned. In this work, we numerically investigate three-dimensional forced magnetohydrodynamic turbulence subject to the competing effects of global rotation and a perpendicular background magnetic field. We focus on the case of a strong background field and find that increasing the rotation rate from zero produces significant changes in the structure of the turbulent flow. Starting with a two-dimensional inverse energy cascade at zero rotation, the flow first transitions to a forward cascade of kinetic energy, then to a shear-layer dominated regime and finally to a second shear-layer regime where the kinetic energy flux is strongly suppressed and the energy transfer is mediated by the induced magnetic field. We show that the first two transitions occur at distinct values of the Rossby number, and the third occurs at a distinct value of the Lehnert number. The three-dimensional results are confirmed using an asymptotic two-dimensional, three-component model, which allows us to extend our results to the planetary-relevant case of an arbitrary angle between the rotation vector and guide field. More generally, our results demonstrate that, when considering the simultaneous limits of strong rotation and a strong guide field, the order in which those limits are taken matters in the misaligned case.

We study experimentally the dynamics of a water droplet on a tilted and vertically oscillating rigid fibre. As we vary the frequency and amplitude of the oscillations the droplet transitions between different modes: harmonic pumping, subharmonic pumping, a combination of rocking and pumping modes, and a combination of pumping and swinging modes. We characterize these responses and report how they affect the sliding speed of the droplet along the fibre. The swinging mode is explained by a minimal model making an analogy between the droplet and a forced elastic pendulum.

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$. We assume a Debye length $\kappa ^{-1}\ll h_0$, so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$, this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$. If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$, in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).

In this paper, we study inertial and gravity wave transmissions near radiative–convective boundaries on the f-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or the other way around. It has been found that waves prefer to survive at low latitudes when the stable layer is strongly stratified ($N^2/(2\varOmega )^2>1$). When the stable layer is weakly stratified ($N^2/(2\varOmega )^2 < 1$), however, waves can survive at any latitude if the meridional wavenumber is large. Then we have discussed transmission ratios for two buoyancy frequency structures: the uniform stratification and the continuously varying stratification. For the uniform stratification, we have found that the transmission is efficient when the rotation is rapid, or when the wave is near the critical colatitude. For the continuously varying stratification, we have discussed the transmission ratio when the square of buoyancy frequency is an algebraic function $N^2\propto z^{\nu } (\nu >0)$. We have found that the transmission can be efficient when the rotation is rapid, or when the wave is near the critical colatitude, or when the thickness of the stratification layer is far greater than the horizontal wavelength. The transmission ratio does not depend on the configurations (radiative layer sits above convective layer, or vice versa; wave propagates outward or inward), but only on characteristics of the wave (frequency and wavenumber) and the fluid (degree of stratification).

With an immersed-boundary lattice-Boltzmann method, we consider the transit of a three-dimensional initially spherical capsule with a viscoelastic membrane through a cross-slot microchannel. The capsule is released with a small initial off-centre distance in the feeding channel, to mimic experiments where capsules or cells are not perfectly aligned with the centreline. Our main objective is to establish the phase diagram of the capsule's deformation modes as a function of the flow inertia and capsule membrane viscosity. We mainly find three deformation modes in the channel cross-slot. For a capsule with low membrane viscosity, a quasi-steady mode occurs at low Reynolds numbers ($Re$), in which the capsule can reach and maintain a steady ellipsoidal shape near the stagnation point, for a considerable time period. With $Re$ increasing to 20, an overshoot–retract mode is observed. The capsule deformation oscillates on an inertial–elastic time scale, suggesting that the dynamics is mainly driven by the balance of the inertial and membrane elastic forces. The membrane viscosity slows down the capsule deformation and suppresses the overshoot–retract mode. A capsule with high membrane viscosity undergoes a continuous-elongation mode, in which its deformation keeps increasing during most of its journey in the channel cross-slot. We summarise the results in phase diagrams, and propose a scaling model which can predict the deformation modes of a viscoelastic capsule in the inertial flow regime. We also discuss implications of the present findings for practical experiments for mechanical characterisation of capsules or cells.

In two-fluid simulations of gas–solid fluidised beds, the gaseous phase and the particulate phase are modelled as continuous media. The stress exerted by the particulate medium on the container walls should be modelled to predict accurately the bed dynamics. This paper addresses the modelling of sliding particle–wall contacts in two-fluid simulations, based on reference simulations coupling computational fluid dynamics with the discrete element method (CFD-DEM), in which the individual movement of the particles is tracked. The analysis of the CFD-DEM highlights the complex near-wall behaviour of the particles, which is not reproduced by two-fluid models. Nevertheless, the particle–wall shear stress can be expressed based on the total granular pressure within the first cell off the wall. The model is validated for the two-fluid simulation of a bubbling gas–solid fluidised bed of olefin particles in the dense-fluidisation regime.

We study the compressible flow dynamics of two-dimensional, steady detonation wave propagation in a high explosive (HE) confined by aluminium (Al) or stainless steel (SS), outside of which is an air layer. We examine how the thickness of the confinement affects the subsonic detonation driving zone structure (DDZ) and the detonation speed $(D_{0})$, demonstrating a strong dependence on whether the oblique shock-driven flow in the confiner is supersonic, as for SS, or subsonic, as for Al. A characteristic path analysis is used to examine the information flow from the material boundaries through the supersonic flow regions in both the HE and confiner that can impact the sonic surfaces bounding the subsonic flow regions. It is shown that the nature of gas-dynamic wave reflection off the SS–air or Al–air boundary can significantly influence the DDZ and $D_{0}$.

A microswimmer placed inside of a passive lamellar vesicle can hydrodynamically induce directed motion of the vesicle so long as fluid is permitted to pass through the vesicle's surface. With an interest in understanding the underlying theoretical mechanism responsible for this directed motion, we study the low Reynolds number fluid mechanics of a reduced system in which a spherical squirming particle is encapsulated inside of a rigid porous spherical container (membrane). We create a theoretical model for this system and obtain two exact analytical solutions to the Stokes equations which describe the motion of the squirmer and container under porous and non-porous container descriptions. Fluid flow through the container's surface is described using a model similar to Darcy's law where proportionality constants, $R_{\parallel }$ and $R_{\perp }$, parameterize the container's resistance to permeable flow parallel and normal to the container's surface. We numerically simulate trajectories of the squirmer–container system by reformulating the fluid mechanics problem as a coupled set of second kind boundary integral equations (BIEs). This system of BIEs is solved numerically using a Galerkin boundary element discretization on graphics processing units enabled with NVIDIA's Compute Unified Device Architecture. We obtain excellent agreement between the analytical and numerical solutions for the concentric geometry. Trajectories of pusher squirmers show earlier radial spread towards the container's surface, whereas puller squirmers tend to move radially inwards, towards the container's centre. Both the squirmer type (pusher, puller, neutral) and container resistance parameters heavily influence net container motion and early squirmer dynamics.

We investigate the interaction of an aqueous polymeric droplet with a tuneable continuous laser in an acoustically levitated environment. The effect of laser irradiation intensity and polymeric concentration on various spatio-temporal parameters is unearthed using high-speed shadowgraphy and theoretical scaling analysis. We observe four temporal phases: droplet evaporation, vapour bubble growth followed by membrane inflation, bubble/membrane rupture through hole nucleation and droplet breakup. During the initial droplet evaporation phase, concentration build-up at the droplet surface beyond a critical limit leads to the formation of a skin layer. It is revealed that, at a given location inside the droplet, hot spots occur, and the maximum temperature at the hot spots scales linearly with irradiation intensity until a bubble nucleates. The low-intensity laser interaction leads to symmetric membrane inflation that eventually forms holes at droplet poles and cracks on the shell surface. On the contrary, high intensity causes early bubble nucleation followed by asymmetric membrane inflation that eventually ruptures through multiple hole formation. Furthermore, the growth and rupture of the membrane is followed by a catastrophic breakup of the droplet. Two dominant atomization modes are reported at significantly high irradiation intensities: stable sheet collapse and unstable sheet breakup. The evolution of droplets into a stable/unstable sheet follows universally observed ligament and hole dynamics. A regime map is shown to describe the influence of polymer concentration and irradiation intensity on the strength and mode of droplet atomization.

The dissolution of a horizontal salt body is theoretically and numerically investigated by considering the coupled dynamics of the dissolution reaction and the diffusive and convective mass transfer at the salt surface. To take into account the nature of the recessing salt surface and the dissolution-driven convection, a dynamic moving boundary condition coupled with a dissolution reaction are employed By employing a newly derived moving interface condition and parameters, a theoretical analysis predicts the onset of gravitational instability, suggesting scaling relationships between parameters to describe the onset of instability (in transport-dominant (Damkholer number $Da \to \infty $ and Rayleigh number $Ra > {10^5}$), onset time ${\tau _d}\sim R{a^{ - 2/3}}$ and ${\tau _d}\sim (RaDa)^{1/4}$ for reaction-dominant regimes $(Da \to 0)$). The scaling relationships on dissolution (the average height change, $\Delta {h_{avg}}\sim \sqrt \tau $ for the diffusion-dominant regime and $\Delta {h_{avg}}\sim R{a^{0.79/3}}{\tau ^{0.86}}$ for the convection-dominant regime) are also suggested to explain the pattern formation. Under a convective mass transfer condition, the concentration gradient of the solute developed during the recession of the salt surface results in an adverse density gradient, which causes gravitational instability motion and finally determines the formation of a dissolution pattern. In addition, two-dimensional and three-dimensional numerical simulations visualize dissolution-driven convective flow fields, the recession of the liquid-solid interface and pattern formation on the dissolving interface. Theoretical and numerical analyses of the dissolution process suggest that the control of gravitational instability is important to prevent or enhance dissolution pattern formation. This systematic parameter study provides a deep understanding of the effect of gravitational instability on convection-driven dissolution in a horizontal geometry.

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$, where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$, wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$, and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$. The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$, $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$, $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$, while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$.

We examine the response of an inertialess viscoelastic channel flow to localized perturbations. We thus performed an experiment in which we perturbed the flow using a localized velocity pulse and probed the perturbed fluid packet downstream from the perturbation location. While for low Weissenberg numbers the perturbed fluid reaches the measurement location as a single velocity pulse, for sufficiently high Weissenberg numbers and perturbation strengths, a random number of pulses arrive at the measurement location. The average number of pulses observed increases with the Weissenberg number. This observation constitutes a transition to a novel elastic pulse-splitting regime. Our results suggest a possible new direction for studying the elastic instability of viscoelastic channel flows at high elasticity through the growth of localized perturbations.

A large-Reynolds-number asymptotic reduction of the Navier–Stokes equations capable of describing a locally periodic vortex-wave array and the associated large-scale variation of the mean-shear velocity field first suggested in Hall (J. Fluid Mech., vol. 850, 2018, pp. 46–82) is extended. The sustaining process of the locally periodic coherent structures is based on the vortex-wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, 641–666), wherein two-dimensional roll–streak fields are supported by localised nonlinear self-interactions of three-dimensional waves that are largest in size within critical layers of the streak field. The variation of the mean velocity is made possible by incorporating a slow change to the mean profile using a Wentzel–Kramers–Brillouin-type approach. As the first extension, we demonstrate that the local structure corresponds to the asymptotic limit of computations in a shearing box. A variety of solutions with different symmetry properties are found via the hybrid numerical asymptotic approach of Blackburn, Hall & Sherwin (J. Fluid Mech. vol. 721, 2013, 58–85). Moreover, some solutions show generic flow features such as uniform momentum zones and spatial intermittency known to occur in near-wall turbulent boundary layers. We extend the vortex-wave interaction array theory to show that, in addition to a Reynolds-averaged-Navier–Stokes-type relationship between the large-scale vertical variation of the mean flow and local roll–streak scale, a higher-order analysis gives a second constraint on the slow-scale dynamics. Those constraints are used for the first time to derive the logarithmic law of the wall through a closed asymptotic analysis of self-similar local coherent structures, consistent with the attached eddy hypothesis.

Lagrangian averaging has been shown to be more effective than the Eulerian mean in separating waves from slow dynamics in two time scale flows. It also appears in many reduced models that capture the wave feedback on the slow flow. Its calculation, however, requires tracking particles in time, which imposes several difficulties in grid-based numerical simulations or estimation from fixed-point measurements. To circumvent these difficulties, we propose a grid-based iterative method to calculate the Lagrangian mean without tracking particles in time, which also reduces computation, memory footprint and communication between processors in parallelised numerical models. To assess the accuracy of this method several examples are examined and discussed. We also explore an application of this method in the context of shallow-water equations by quantifying the validity of wave-averaged geostrophic balance – a modified form of geostrophic balance accounting for the effect of strong waves on slow dynamics.

Atmospheric entry capsules shaped as spherically blunted, large apex-angle cones are widely used in space missions. In Part 1 of this study (Hornung, Martinez Schramm & Hannemann, J. Fluid Mech., vol. 871, 2019, pp. 1097–1116) we explored flows over the two elements of the capsule shape, the sphere and the sharp cone with detached shock, theoretically and computationally. Using a large number of inviscid, perfect-gas computations, analytical functions of two independent variables, the normal-shock density ratio $\varepsilon$ and a cone-angle parameter $\eta$ (which is a function of $\varepsilon$ and the cone half-angle $\theta$) were found for the dimensionless shock wave stand-off distance and the drag coefficient of a sharp cone. An analytical description was found for the shock stand-off distance in the transition from the 90$^\circ$ cone (flat-faced cylinder) to the sphere. In Part 1, it was speculated that the perfect-gas results have relevance to non-equilibrium situations if the normal-shock density ratio is replaced by the density ratio based on the average density along the stagnation streamline. In Part 2, the investigation is extended to blunted-cone capsule shapes. High-precision force measurements and schlieren image analysis are performed in the High-Enthalpy Shock Tunnel Göttingen (HEG) of the German Aerospace Centre using air as the test gas, at conditions where vibrational non-equilibrium effects are significant. Accordingly, results are compared with viscous numerical predictions using different physico-chemical models. A theoretical model is constructed for the density profile along the stagnation streamline that is determined by the free stream conditions and gives the average density. Comparisons of the experimental and numerical results for the dimensionless shock stand-off distance and the drag coefficient, with the extension of the analytical functions of Part 1 to vibrationally relaxing flow, exhibit very good agreement in all of a range of geometries.