Vortex-current filaments have been used to study phenomena such as coronal loops and solar flares as well as tokamaks, and recent experimental work has demonstrated dynamics akin to vortex-current filaments on a table-top plasma focus device. While MHD vortex dynamics and related applications to turbulence have attracted consideration in the literature due to a wide variety of applications, not much analytical progress has been made in this area, and the analysis of such vortex-current filament solutions under various geometries may motivate further experimental efforts. To this end, we consider the motion of open, isolated vortex-current filaments in the presence of magnetohydrodynamic (MHD) as well as the standard hydrodynamic effects. We begin with the vortex-current model of Yatsuyanagi, Hatori & Kato (J. Phys. Soc. Japan, vol. 65, 1996, pp. 745–759) giving the self-induced motion of a vortex-current filament. We give the ‘cutoff’ formulation of the Biot–Savart integrals used in this model, to avoid the singularity at the vortex core. We then study the motion of a variety of vortex-current filaments, including helical, planar and self-similar filament structures. In the case where MHD effects are weak relative to hydrodynamic effects, the filaments behave as expected from the pure hydrodynamic theory. However, when MHD effects are strong enough to dominate, then we observe structural changes to the filaments in all cases considered. The most common finding is reversal of vortex-current filament orientation for strong enough MHD effects. Kelvin waves along a vortex filament (as seen for helical and self-similar structures) will reverse their translational and rotational motion under strong MHD effects. Our findings support the view that vortex-current filaments can be studied in a manner similar to classical hydrodynamic vortex filaments, with the primary role of MHD effects being to change the filament motion, while preserving the overall geometric structure of such filaments.

A Lagrangian fluctuation–dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh–Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time $\unicode[STIX]{x1D70F}_{mix}$ for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an ‘ultimate regime’ with the Nusselt number scaling predicted by Spiegel (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374–1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time $\unicode[STIX]{x1D70F}_{free}$. Precisely, we show that $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=(RaPr)^{1/2}/Nu,$ with $Ra$ the Rayleigh number, $Pr$ the Prandtl number, and $Nu$ the Nusselt number. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal ‘mixing zone’, which is much wider than the standard thermal boundary layer. Kraichnan–Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time $\unicode[STIX]{x1D70F}_{mix}$, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.

Spectral content and spatial organization of momentum transport events are investigated in a turbulent boundary layer at the Reynolds number $(Re_{\unicode[STIX]{x1D70F}})=2700$, with time-resolved planar particle image velocimetry. The spectral content of the Reynolds-shear-stress fluctuations reveals that the largest range of time and length scales can be observed in proximity to the wall, while this range becomes progressively more narrow when the wall distance increases. Farther from the wall, longer time and larger length scales exhibit an increasing spectral content. Wave velocities of transport events are estimated from wavenumber–frequency power spectra at different wall-normal locations. Wave velocities associated with ejection events (Q2) are lower than the local average streamwise velocity, while sweep events (Q4) are characterized by wave velocities larger than the local average velocity. These velocity deficits are almost insensitive to the wall distance, which is also confirmed from time tracking the intense transport events. The vertical advection velocities of the intense ejection and sweep events are on average a small fraction of the friction velocity $U_{\unicode[STIX]{x1D70F}}$, different from previous observations in a channel flow. In the range of wall-normal locations $60<y^{+}<600$, sweeps are considerably larger than ejections, which could be because the ejections are preferentially located in between the legs of hairpin packets. Finally, it is observed that negative quadrant events of the same type tend to appear in groups over a large spatial streamwise extent.

This paper describes the hysteresis in the torque for Taylor–Couette flow in the turbulent flow regime for different shear Reynolds numbers, aspect ratios and boundary conditions. The hysteresis increases with decreasing shear Reynolds number and becomes more pronounced as the aspect ratio is increased from 22 to 88. Measurements conducted in two different Taylor–Couette set-ups depict the effect of the flow conditions at the ends of the cylinders on the flow hysteresis by showing reversed hysteresis behaviour. In addition, the flow structure in the different branches of the hysteresis loop was investigated by means of stereoscopic particle image velocimetry. The results show that the dominant flow structures differ in shape and magnitude depending on the branch of the hysteresis loop. Hence, it can be concluded that the geometry could have an effect on the hysteresis behaviour of turbulent Taylor–Couette flow, but its occurrence is related to a genuine change in the flow dynamics.

The translation of a bubble under the action of an acoustic forcing finds applications in fields ranging from drug delivery to sonoluminescence. This phenomenon has been widely studied for cases where the amplitude of the forcing remains constant over time. However, in many practical applications, the duration of the forcing is not long enough for the bubble to attain a constant translational velocity, mainly due to the effect of the history force. Here, we develop a formulation, valid in the limit of very viscous flow and small-amplitude acoustic forcing, that allows us to describe the transient dynamics of bubbles driven by acoustic pulses consisting of finite numbers of cycles. We also present an asymptotic solution to this theory for the case of a finite-duration sinusoidal pressure pulse. This solution takes into account both the history integral term and the transient period that the bubble needs to achieve steady radial oscillations, the former being dominant during most of the acceleration process. Moreover, by introducing some additional assumptions, we derive a simplified formula that describes the time evolution of the bubble velocity fairly well. Using this solution, we show that the convergence to the steady translational velocity, given by the so-called Bjerknes force, occurs rather slowly, namely as $\unicode[STIX]{x1D70F}^{-1/2}$, where $\unicode[STIX]{x1D70F}$ is the time made dimensionless with the viscous time scale of the bubble, which explains the slow convergence of the bubble velocity and stresses the importance of taking the history force into account.

We study non-equilibrium pair interactions between microscopic particles moving through a model shear-thinning fluid. Prior efforts to model pair interactions in non-Newtonian fluids have largely focused on constitutive models derived from polymer-chain kinetic theories focusing on conformational degrees of freedom, but neglecting the details of microstructural evolution beyond a single polymer length scale. To elucidate the role of strong structural distortion in mediating pair interactions in Brownian suspensions, we formulate and solve a Smoluchowski equation describing the detailed evolution of the particle configuration between and around a pair of microscopic probes driven at fixed velocity by an external force through a colloidal dispersion. To facilitate analysis, we choose a model system of Brownian hard spheres that do not interact hydrodynamically; while simple, this ‘freely draining’ model permits insight into connections between microstructure and rheology. The flow induces a non-equilibrium particle density gradient that gives rise to both viscous drag and an interactive force between the probes. The drag force acts to slow the centre-of-mass velocity of the pair, while the interactive force arising from osmotic pressure gradients can lead to attraction or repulsion, as well as deterministic reorientation of the probes relative to the external force. The degree to which the microstructure is distorted, and the shape of that distortion, depend on the arrangement of the probes relative to one another and their orientation to the driving force. It also depends on the magnitude of probe velocity relative to the Brownian velocity of the suspension. When only thermal fluctuations set probe velocity, the equilibrium depletion attraction is recovered. For weak forcing, long-ranged interactions mediated via the bath-particle flux give rise to entropic forces on the probes. The linear response is a viscous drag that slows forward motion; only the weakly nonlinear response can produce relative motion–attraction, repulsion or reorientation of the probes. We derive entropic coupling tensors, similar in ethos to pair hydrodynamic tensors, to describe this behaviour. The structural symmetry that permits this analogy is lost when forcing becomes strong, revealing instabilities in system behaviour. Far from equilibrium, the interactive force depends explicitly on the initial probe separation, orientation and strength of forcing; widely spaced probes interact through the distorted microstructure, whereas the behaviour of closely spaced probes is largely set by excluded-volume effects. In this regime, a pair of closely spaced probes sedimenting side-by-side tend to attract and reorient to permit alignment of their line-of-centres with the flow, while widely spaced probes fall without reorienting. Our results show qualitative agreement with experimental observations and provide a potential connection to the observed column instability in shear-thinning fluids.

In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$.

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

Hydraulic fracturing for production of oil and gas from shale formations releases fluid waste, by-products that must be managed carefully to avoid significant harm to human health and the environment. These fluids are presumed to result from a variety of fracture relaxation processes, and are commonly referred to as ‘flowback’ and ‘produced water’, depending primarily on the time scale of their appearance. Here, a model is presented for investigating the dynamics of backflows caused by the elastic relaxation of a pre-strained medium, namely a single fracture and two model fracture network systems: a single bifurcated channel and its generalization for $n$ bifurcated fracture generations. Early- and late-time asymptotic solutions are obtained for the model problems and agree well with numerical solutions. In the late-time period, the fracture apertures and backflow rates exhibit a time dependence of $t^{-1/3}$ and $t^{-4/3}$, respectively. In addition, the pressure distributions collapse to universal curves when scaled by the maximum pressure in the system, which we calculate as a function of $n$. The pressure gradient along the network is steepest near the outlet while the bulk of the network serves as a ‘reservoir’. Fracture networks with larger $n$ are less efficient at evicting fluids, manifested through a longer time required for a given fractional reduction of the initial volume. The developed framework may be useful for informing engineering design and environmental regulations.

In Part 3 of this series of papers (Needham et al., Q. J. Mech. Appl. Maths, vol. 61, 2008, pp. 581–614), we studied the free surface flow generated in a horizontal layer of inviscid fluid when a flat, rigid plate, inclined at an external angle $\unicode[STIX]{x1D6FC}$ to the horizontal, is driven into the fluid with a constant, horizontal acceleration. We found that the most interesting behaviour occurs when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x03C0}/2$ (the plate leaning into the fluid). When $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D6FC}_{c}$, with $\unicode[STIX]{x1D6FC}_{c}\approx 102.6^{\circ }$, we were able to find the small-time asymptotic solution structure, and solve the leading-order problem numerically. When $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{c}$, we found numerical evidence that a $120^{\circ }$ corner exists on the free surface, at leading order as $t\rightarrow 0$. For $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$, we could find no numerical solution of the leading-order problem as $t\rightarrow 0$, and hypothesised that the solution does not exist for any $t>0$ for these values of $\unicode[STIX]{x1D6FC}$. At the present time, there is no rigorous proof of this hypothesis. In this paper, we demonstrate that the likely non-existence of a solution for any $t>0$ when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$ can be reconciled with the physics of the system by including the effect of surface tension in the model. Specifically, we find that for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$, the solution exists for $0\leqslant t<t_{c}$, with $t_{c}\sim Bo^{-1/(3\unicode[STIX]{x1D6FE}-1)}\unicode[STIX]{x1D70F}_{c}$ and $\unicode[STIX]{x1D70F}_{c}=O(1)$ as $Bo^{-1}\rightarrow 0$, where $Bo$ is the Bond number ($Bo^{-1}$ is the square of the ratio of the capillary length to the fluid depth) and $\unicode[STIX]{x1D6FE}\equiv 1/(1-\unicode[STIX]{x03C0}/4\unicode[STIX]{x1D6FC})$. The solution does not exist for $t\geqslant t_{c}$ due to a topological transition driven by a nonlinear capillary wave, i.e. the free surface pinches off (self-intersects) when $t=t_{c}$. We are also able to compare this asymptotic solution with experimental results which show that, in an experimental case where the contact angle remains approximately constant (the modelling assumption that we make in this paper), the asymptotic solution is in good agreement. In general, the inclusion of surface tension leads to the generation of capillary waves ahead of the wavecrest, which decay as $t$ increases for $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{c}$ but dominate the flow and lead to pinch-off for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$. These capillary waves are an unsteady analogue of the parasitic capillary waves that can be generated ahead of steadily propagating, periodic capillary–gravity waves, e.g. Lin & Rockwell (J. Fluid Mech., vol. 302, 1995, pp. 29–44).

The evolution of vortex rings in isodensity and isoviscosity fluid has been studied analytically using a novel mathematical model. The model predicts the spatiotemporal variation in peak vorticity, circulation, vortex size and spacing based on instantaneous vortex parameters. This proposed model is quantitatively verified using experimental measurements. Experiments are conducted using high-speed particle image velocimetry (PIV) and laser induced fluorescence (LIF) techniques. Non-buoyant vortex rings are generated from a nozzle using a constant hydrostatic pressure tank. The vortex Reynolds number based on circulation $(\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708})$ is varied in the range 100–1500 to account for a large range of operating conditions. Experimental results show good agreement with theoretical predictions. However, it is observed that neither Saffman’s thin-core model nor the thick-core equations could correctly explain vortex evolution for all initial conditions. Therefore, a transitional theory is framed using force balance equations which seamlessly integrate short- and long-time asymptotic theories. It is found that the parameter $A=(a/\unicode[STIX]{x1D70E})^{2}$, where $a$ is the vortex half-spacing and $\unicode[STIX]{x1D70E}$ denotes the standard deviation of the Gaussian vorticity profile, governs the regime of vortex evolution. For higher values of $A$, evolution follows short-time behaviour, while for $A=O(1)$, long-time behaviour is prominent. Using this theory, many reported anomalous observations have been explained.

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

The effect of turbulence on the mass transfer between a fluid and embedded small heavy inertial particles that experience surface reactions is studied. For simplicity, the surface reaction, which takes place when a gas phase reactant is converted to a gas phase product at the external surface of the particles, is unimolar and isothermal. Two effects are identified. The first effect is due to the relative velocity between the fluid and the particles, and a model for the relative velocity is presented. The second effect is due to the clustering of particles, where the mass transfer rate is inhibited due to the rapid depletion of the consumed species inside the dense particle clusters. This last effect is relevant for large Damköhler numbers, where the Damköhler number is defined as the ratio of the turbulent and chemical time scales, and it may totally control the mass transfer rate for Damköhler numbers larger than unity. A model that describes how this effect should be incorporated into existing simulation tools that utilize the Reynolds averaged Navier–Stokes approach is presented.

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date. In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.

In this work an experimental investigation of the near-field pressure of a compressible jet is presented. The proper orthogonal decomposition (POD) of the pressure fluctuations measured by a linear array of microphones is performed in order to provide the streamwise evolution of the jet structure. The wavenumber–frequency spectrum of the space–time pressure fields re-constructed using each POD mode is computed in order to provide the physical interpretation of the mode in terms of hydrodynamic/acoustic nature. Specifically, non-radiating hydrodynamic, radiating acoustic and ‘hybrid’ hydro-acoustic modes are found based on the phase velocity associated with the spectral energy bumps in the wavenumber–frequency domain. Furthermore, the propagation direction in the far field of the radiating POD modes is detected through the cross-correlation with the measured far-field noise. Modes associated with noise emissions from large/fine scale turbulent structures radiating in the downstream/sideline direction in the far field are thus identified.

Geophysical flows occur over a large range of scales, with Reynolds numbers and Richardson numbers varying over several orders of magnitude. For this study, jets of different densities were ejected vertically into a large ambient region, considering conditions relevant to some geophysical phenomena. Using particle image velocimetry, the velocity fields were measured for three different gases exhausting into air – specifically helium, air and argon. Measurements focused on both the jet core and the entrained ambient. Experiments considered relatively low Reynolds numbers from approximately 1500 to 10 000 with Richardson numbers near 0.001 in magnitude. These included a variety of flow responses, notably a nearly laminar jet, turbulent jets and a transitioning jet in between. Several features were studied, including the jet development, the local entrainment ratio, the turbulent Reynolds stresses and the eddy strength. Compared to a fully turbulent jet, the transitioning jet showed up to 50 % higher local entrainment and more significant turbulent fluctuations. For this condition, the eddies were non-axisymmetric and larger than the exit radius. For turbulent jets, the eddies were initially smaller and axisymmetric while growing with the shear layer. At lower turbulent Reynolds number, the turbulent stresses were more than 50 % higher than at higher turbulent Reynolds number. In either case, the low-density jet developed faster than a comparable non-buoyant jet. Quadrant analysis and proper orthogonal decomposition were also utilized for insight into the entrainment of the jet, as well as to assess the energy distribution with respect to the number of eigenmodes. Reynolds shear stresses were dominant in Q1 and Q3 and exhibited negligible contributions from the remaining two quadrants. Both analysis techniques showed that the development of stresses downstream was dependent on the Reynolds number while the spanwise location of the stresses depended on the Richardson number.

Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients – which are steep near interfaces – drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work. Such activity can be modelled with judicious additional terms in the equations of motion for simple or liquid-crystalline binary fluids. Throughout, the emphasis of the article is on presenting the theoretical tools needed to address a wide range of physical phenomena. Examples include the kinetics of fluid–fluid demixing from an initially uniform state; the result of imposing a steady macroscopic shear flow on this demixing process; and the diffusive coarsening, Brownian motion and coalescence of emulsion droplets. We discuss strategies to create long-lived emulsions by adding trapped species, solid particles, or surfactants; to address the latter, we outline the theory of bending energy for interfacial films. In emulsions where one of the components is liquid-crystalline, ‘anchoring’ terms can create preferential orientation tangential or normal to the fluid–fluid interface. These allow droplets of an isotropic fluid in a liquid crystal (or vice versa) to support a variety of topological defects, which we describe, altering their interactions and stability. Addition of active terms to the equations of motion for binary simple fluids creates a model of ‘motility-induced’ phase separation, where demixing stems from self-propulsion of particles rather than their interaction forces, altering the relation between interfacial structure and fluid stress. Coupling activity to binary liquid crystal dynamics creates models of active liquid-crystalline emulsion droplets. Such droplets show various modes of locomotion, some of which strikingly resemble the swimming or crawling motions of biological cells.

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

Rough surfaces have been widely used as an efficient way to enhance the heat-transfer efficiency in turbulent thermal convection. In this paper, however, we show that roughness does not always mean a heat-transfer enhancement, but in some cases it can also reduce the overall heat transport through the system. To reveal this, we carry out numerical investigations of turbulent Rayleigh–Bénard convection over rough conducting plates. Our study includes two-dimensional (2D) simulations over the Rayleigh number range $10^{7}\leqslant Ra\leqslant 10^{11}$ and three-dimensional (3D) simulations at $Ra=10^{8}$. The Prandtl number is fixed to $Pr=0.7$ for both the 2D and the 3D cases. At a fixed Rayleigh number $Ra$, reduction of the Nusselt number $Nu$ is observed for small roughness height $h$, whereas heat-transport enhancement occurs for large $h$. The crossover between the two regimes yields a critical roughness height $h_{c}$, which is found to decrease with increasing $Ra$ as $h_{c}\sim Ra^{-0.6}$. Through dimensional analysis, we provide a physical explanation for this dependence. The physical reason for the $Nu$ reduction is that the hot/cold fluid is trapped and accumulated inside the cavity regions between the rough elements, leading to a much thicker thermal boundary layer and thus impeding the overall heat flux through the system.