The rupture of soap films is traditionally described by a law accounting for a balance between momentum and surface tension forces, derived independently by Taylor and Culick in the 1960s. This law is highly relevant to the dynamics of thin liquid films of jets when viscous effects are negligible. However, the minute amounts of surfactant molecules present in soap films play a major role in interfacial rheology, and may result in complex behaviour. Petit et al. (J. Fluid Mech., vol. 774, 2015, R3) challenge standard thin film dynamics via intriguing experiments conducted with highly elastic surfactants. Unexpected structures reminiscent of faults are observed.

A mathematical model is presented describing the deformation, under the combined effects of surface tension and draw tension, of an array of channels in the drawing of a broad class of slender viscous fibres. The process is relevant to the fabrication of microstructured optical fibres, also known as MOFs or holey fibres, where the pattern of channels in the fibre plays a crucial role in guiding light along it. Our model makes use of two asymptotic approximations, that the fibre is slender and that the cross-section of the fibre is a circular disc with well-separated elliptical channels that are not too close to the outer boundary. The latter assumption allows us to make use of a suitably generalised ‘elliptical pore model (EPM)’ introduced previously by one of the authors (Crowdy, J. Fluid Mech., vol. 501, 2004, pp. 251–277) to quantify the axial variation of the geometry during a steady-state draw. The accuracy of the elliptical pore model as an approximation is tested by comparison with full numerical simulations. Our model provides a fast and accurate reduction of the full free-boundary problem to a coupled system of nonlinear ordinary differential equations. More significantly, it also allows a regularisation of an important ill-posed inverse problem in MOF fabrication: how to find the initial preform geometry and the experimental parameters required to draw MOFs with desired cross-plane geometries.

A similarity reduction of the Navier–Stokes equations for the motion of an infinite rotating disk above an air-bearing table yields a coupled pair of ordinary differential equations governed by a Reynolds number $Re=Wh/{\it\nu}$ and a rotation parameter $S=\sqrt{2}h{\it\Omega}/W$, where $h$ is the float height, $W$ is the air levitation velocity, ${\it\Omega}$ is the disk rotation rate, and ${\it\nu}$ is the kinematic viscosity of air. After deriving the small- and large-Reynolds-number behaviour of solutions, the equations are numerically integrated over a wide range of $Re{-}S$ parameter space. Zero-lift boundaries are computed as well as the boundaries separating pure outward flow from counter-flow in the gap. The theory is used to model the steady float height of a finite-radius air hockey disk under the assumption that the float height is small relative to the diameter of the disk and the flow is everywhere laminar. The steady results are tested against direct numerical simulation (DNS) of the unsteady axisymmetric Navier–Stokes equations for the cases where the disk rotates at constant angular velocity but is either at a fixed height or free to move axially. While a constant shift in the gap pressure conforms closely to that found using steady theory, the interaction of the radial jet emanating from the gap with a vertical transpiration field produces vortex rings which themselves propagate around to interact with the jet. Although these structures diffuse as they propagate up and away from the gap, they induce a departure from the steady-flow assumption of atmospheric pressure at the gap exit, thus inducing small irregular axial oscillations of the floating disk.

We investigate the characteristics of the oscillatory motion and wake of confined bubbles freely rising in a thin-gap cell ($h=3.1~\text{mm}$ width). Once the diameter $d$ of the bubble in the plane of the cell is known, the mean vertical velocity of the bubble $V_{b}$ is proportional to the gravitational velocity $(h/d)^{1/6}\sqrt{gd}$, where $g$ is the gravitational acceleration. This velocity is used to build the Reynolds number $Re=V_{b}d/{\it\nu}$ that characterizes the flow induced by the bubble in the surrounding liquid (of kinematic viscosity ${\it\nu}$), and which determines at leading order the mean deformation of the bubble given by the aspect ratio ${\it\chi}$ of the ellipse equivalent to the bubble contour. We then show that in the reference frame associated with the bubble (having a fixed origin and axes corresponding to the minor and major axes of the equivalent ellipse) the characteristics of its oscillatory motion in the plane of the cell display remarkable properties in the range $1200<Re<3000$ and $h/d<0.4$. In particular, the velocity of the bubble presents along its path an almost constant component along its minor axis (fluctuations in time of approximately 5 %), given by $V_{a}/V_{b}\simeq 0.92$ for all $Re$. The dimensionless amplitude of oscillation of the angular velocity is also constant for all $Re$, $\tilde{r}d/V_{b}\simeq 0.75$, while that of the transverse velocity of the bubble (along its major axis) is given by $\tilde{V}_{t}/V_{b}\simeq 0.32{\it\chi}$, reaching values comparable to those of the axial velocity $V_{a}$ for the most deformed bubbles (${\it\chi}\approx 3$). Furthermore, the frequency $f$ of oscillation scales with the inertial time scale based on the transverse velocity of the bubble $\tilde{V}_{t}$, corresponding to a constant Strouhal number $St^{\ast }=fd/\tilde{V}_{t}\simeq 0.27$. Using high-frequency particle image velocimetry, we investigate in detail the properties of the wake associated with the oscillatory motion of sufficiently confined bubbles. We observe that vortex shedding occurs for a maximal transverse velocity $V_{t}$ of the bubble, corresponding to a maximal drift angle of the bubble. Furthermore, the measured vorticity of the vortex at detachment corresponds to the estimation $V_{b}{\it\chi}^{3/2}/d$ of the vorticity produced at the bubble surface. Three stages then emerge concerning the evolution in time of the wake generated by the bubble. For one to two periods of oscillation $T_{x}$ following the release of a vortex, a rapid decay of the vorticity of the released vortex is observed. Meanwhile, the released vortex located initially at a distance of approximately one diameter from the bubble centre moves outwards from the bubble path and expands. At intermediate times, the vortex street undergoes vortex pairing. When viscous effects become predominant at a time of the order of the viscous time scale ${\it\tau}_{{\it\nu}}=h^{2}/(4{\it\nu})$, the vortex street becomes frozen and decays exponentially in place.

Two of the most widespread and fascinating patterns observed on cave walls and icefalls – karst and ice flutings – are demonstrated to share the same morphogenesis, whose core is a water film-induced locking mechanism. Creeping flow-based parallel and non-parallel stability analyses are developed through a numerical and analytical approach. These instabilities are shown to develop at inverted overhung conditions. A sharp transition between fluting and ripple-like patterns is presented. The non-parallel problem is solved with the use of Papkovich–Neuber solutions in order to obtain a finite wavelength selection close to the critical conditions. The method and results can be extended to similar problems where the temporal evolution of the interface is linearly related to the film depth.

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

An axisymmetric particle sedimenting in an otherwise quiescent Newtonian fluid, in the Stokes regime, retains its initial orientation. For the special case of a spheroidal geometry, we examine analytically the effects of weak inertia and viscoelasticity in driving the particle towards an eventual steady orientation independent of initial conditions. The generalized reciprocal theorem, together with a novel vector spheroidal harmonics formalism, is used to find closed-form analytical expressions for the $O(\mathit{Re})$ inertial torque and the $O(\mathit{De})$ viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here, $\mathit{Re}=UL/{\it\nu}$ is the Reynolds number, with $U$ being the sedimentation velocity, $L$ the semi-major axis and ${\it\nu}$ the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number, $\mathit{De}=({\it\lambda}U)/L$, is a dimensionless measure of the fluid viscoelasticity, with ${\it\lambda}$ being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit $\mathit{Re},\mathit{De}\ll 1$, and the effects of viscoelasticity are therefore modelled using the constitutive equation of a second-order fluid. The inertial torque always acts to turn the spheroid broadside-on, while the final orientation due to the viscoelastic torque depends on the ratio of the magnitude of the first ($N_{1}$) to the second normal stress difference ($N_{2}$), and the sign (tensile or compressive) of $N_{1}$. For the usual case of near-equilibrium complex fluids – a positive and dominant $N_{1}$ ($N_{1}>0$, $N_{2}<0$ and $|N_{1}/N_{2}|>1$) – both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in ${\it\kappa}$ in the slender-fibre limit (${\it\kappa}\gg 1$), where ${\it\kappa}=L/b$ is the aspect ratio, $b$ being the radius of the spheroid (semi-minor axis). The angular dependence of the inertial and viscoelastic torques turn out to be identical, and one may then characterize the long-time orientation of the sedimenting spheroid based solely on a critical value ($\mathit{El}_{c}$) of the elasticity number, $\mathit{El}=\mathit{De}/\mathit{Re}$. For $\mathit{El}<\mathit{El}_{c}~({>}\mathit{El}_{c})$, inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that $\mathit{El}_{c}\sim O[(1/\text{ln}\,{\it\kappa})]$ for ${\it\kappa}\gg 1$, and the viscoelastic torque thus dominates for a slender rigid fibre. For a slender fibre alone, we also briefly analyse the effects of elasticity on fibre orientation outside the second-order fluid regime.

We consider sound source mechanisms involving the acoustic and instability modes of dual-stream isothermal supersonic jets with the inner nozzle buried within an outer shroud-like nozzle. A particular focus is scattering into radiating sound waves at the shroud lip. For such jets, several families of acoustically coupled instability waves exist, beyond the regular vortical Kelvin–Helmholtz mode, with different shapes and propagation characteristics, which can therefore affect the character of the radiated sound. In our model, the coaxial shear layers are vortex sheets while the incident acoustic disturbances are the propagating shroud modes. The Wiener–Hopf method is used to compute their scattering at the sharp shroud edge to obtain the far-field radiation. The resulting far-field directivity quantifies the acoustic efficiency of different mechanisms, which is particularly important in the upstream direction, where the results show that the scattered sound is more intense than that radiated directly by the shear-layer modes.

Stereoscopic three-component particle image velocimetry (3C-PIV) measurements have been made in a turbulent round jet to investigate the spatio-temporal correlations that are the origin of aerodynamic noise. Restricting attention to subsonic, isothermal jets, measurements were taken in a water flow experiment where, for the same Reynolds number and nozzle size, the shortest time scale of the dynamically important turbulent structures is more than an order of magnitude greater that in equivalent airflow experiments, greatly facilitating time-resolved PIV measurements. Results obtained (for a jet nozzle diameter and velocity of 40 mm and $1~\text{m}~\text{s}^{-1}$, giving $\mathit{Re}=4\times 10^{4}$) show that, on the basis of both single-point statistics and two-point quantities (correlation functions, integral length scales) the present incompressible flow data are in excellent agreement with published compressible, subsonic airflow measurements. The 3C-PIV data are first compared to higher-spatial-resolution 2C-PIV data and observed to be in good agreement, although some deterioration in quality for higher-order correlations caused by high-frequency noise in the 3C-PIV data is noted. A filter method to correct for this is proposed, based on proper orthogonal decomposition (POD) of the 3C-PIV data. The corrected data are then used to construct correlation maps at the second- and fourth-order level for all velocity components. The present data are in accordance with existing hot-wire measurements, but provide significantly more detailed information on correlation components than has previously been available. The measured relative magnitudes of various components of the two-point fourth-order turbulence correlation coefficient ($R_{ij,kl}$) – the fundamental building block for free shear flow aerodynamic noise sources – are presented and represent a valuable source of validation data for acoustic source modelling. The relationship between fourth-order and second-order velocity correlations is also examined, based on an assumption of a quasi-Gaussian nearly normal p.d.f. for the velocity fluctuations. The present results indicate that this approximation shows reasonable agreement for the measured relative magnitudes of several correlation components; however, areas of discrepancy are identified, indicating the need for work on alternative models such as the shell turbulence concept of Afsar (Eur. J. Mech. (B/Fluids), vol. 31, 2012, pp. 129–139).

Steady seepage is determined by a head drop upstream/downstream of a water-retaining wall. Due to its erratic variations, hydraulic log-conductivity $Y=\ln K$ is modelled as a stationary random space function (RSF). We deal with a highly anisotropic porous formation, i.e. an axisymmetric medium where the horizontal correlation integral scale of $Y$ is much larger than the vertical one. The goal of computing the resulting flow field within a stochastic framework is complicated by non-uniformity of the mean flow. Simple (closed-form) expressions for the correlation functions of the flow variables as well as the mean head are derived. We use these results to quantify the impact of spatial variability of $Y$ upon the probability that the exit volumetric flow rate downstream of the wall is greater than that obtained by regarding the formation as homogeneous (with constant hydraulic conductivity). In particular, we show that the spatial variability of $Y$ may lead to predictions (and consequently to design choices) which significantly differ from those achieved by regarding the porous formation as homogeneous.

A major obstacle in designing a perfect cloak for objects in shallow-water waves is that the linear transformation media scheme (also known as transformation optics) requires spatial variations of two independent medium properties. In the Maxwell’s equation and for the well-studied problem of electromagnetic cloaking, these two properties are permittivity and permeability. Designing an anisotropic material with both variable permittivity and variable permeability, while challenging, is achievable. On the other hand, for long gravity waves, whose governing equation maps one-to-one to the single polarization Maxwell’s equations, the two required spatially variable properties are the water depth and the gravitational acceleration; in this case changing the gravitational acceleration is simply impossible. Here we present a nonlinear transformation that only requires the change in one of the medium properties, which, in the case of shallow-water waves, is the water depth, while keeping the gravitational acceleration constant. This transformation keeps the governing equation perfectly intact and, if the cloak is large enough, asymptotically satisfies the necessary boundary conditions. We show that with this nonlinear transformation an object can be cloaked from any wave that merely satisfies the long-wave assumption. The presented transformation can be applied as well for the design of non-magnetic optical cloaks for electromagnetic waves.

The dynamic behaviour of vortex cavitation on marine propellers may cause inboard noise and vibration, but is not well understood. The main goal of the present study is to experimentally analyse the dynamics of an isolated tip vortex cavity generated at the tip of a wing of elliptical planform. Detailed high-speed video shadowgraphy was used to determine the cavity deformations in combination with force and sound measurements. The cavity deformations can be divided in different modes, each of which show a distinct dispersion relation between frequency and wavenumber. The dispersion relations show good agreement with an analytical formulation. Finally, experimental support is given to the hypothesis that the resonance frequency of the cavity volume variation is related to a zero group velocity.

The effects of uniform vorticity on a train of ‘gentle’ and ‘steep’ deep-water waves interacting with underlying flows are investigated through a fully nonlinear boundary integral method. It is shown that wave blocking and breaking can be more prominent depending on the magnitude and direction of the shear flow. Reflection continues to occur when sufficiently strong adverse currents are imposed on ‘gentle’ deep-water waves, though now affected by vorticity. For increasingly positive values of vorticity, the induced shear flow reduces the speed of right-going progressive waves, introducing significant changes to the free-surface profile until waves are completely blocked by the underlying current. A plunging breaker is formed at the blocking point when ‘steep’ deep-water waves interact with strong adverse currents. Conversely negative vorticities augment the speed of right-going progressive waves, with wave breaking being detected for strong opposing currents. The time of breaking is sensitive to the vorticity’s sign and magnitude, with wave breaking occurring later for negative values of vorticity. Stopping velocities according to nonlinear wave theory proved to be sufficient to cause wave blocking and breaking.

By combining Biot’s theory of poro-elasticity with standard shallow-layer scalings, a theoretical model is developed to describe axisymmetric gravity-driven flow through a shallow deformable porous medium. Motivated in part by observations of surface uplift around $\text{CO}_{2}$ sequestration sites, the model is used to explore the injection of a dense fluid into a horizontal, deformable porous layer that is initially saturated with another, less dense, fluid. The layer lies between a rigid base and a flexible overburden, both of which are impermeable. As the injected fluid spreads under gravity, the matrix deforms and the overburden lifts up. The coupled model predicts the location of the injected fluid as it spreads and the resulting uplift of the overburden due to deformation of the solid matrix. In general, the uplift spreads diffusively far ahead of the injected fluid. If fluid is injected with a constant flux and the medium is unbounded, both the uplift and the injected fluid spread in a self-similar fashion with the same similarity variable $\propto r/t^{1/2}$. The asymptotic form of this spreading is established. Results from a series of laboratory experiments, using polyacrylamide hydrogel particles to create a soft poro-elastic material, are compared qualitatively with the predictions of the model.

A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet’s bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.

Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.

This paper presents a stability analysis of a mixed convection problem in an inclined parallel-plate channel with uniform heating (or cooling) from the top and bottom. The channel is filled with a saturated homogeneous porous medium and the momentum equation is given by Darcy’s model. A forced through-flow is prescribed across the channel. Linear stability analysis is thus employed to determine the onset of thermoconvective instability. The channel inclination is shown to play an important role in the stability of the problem, where two different regimes can be present: a buoyancy-assisted regime and a buoyancy-opposed regime. The analysis of the problem leads to a differential eigenvalue problem composed of a system of four complex-valued equations that are used to determine the critical values of the Rayleigh number leading to an instability under different problem configurations. This eigenproblem is solved by employing the generalised integral transform technique (GITT), in which simpler real eigenfunction bases are used to expand the complex eigenproblem. The results indicate that the longitudinal rolls are always more unstable than oblique and transverse rolls. For a buoyancy-opposed regime, even with a very small channel inclination angle, the basic through-flow is always unstable. This result has an important implication for experimental research, as it shows that a perfect alignment must be employed for horizontal mixed-convection experiments to avoid instabilities that arise in the buoyancy-opposed regime.

In a recent paper, Deguchi & Hall (J. Fluid Mech., vol. 752, 2014a, pp. 602–625) described a new kind of exact coherent structure which sits at the edge of an asymptotic suction boundary layer at high values of the Reynolds number $Re$. At a distance $\ln Re$ from the wall, the structure is driven by the fully nonlinear interaction of tiny rolls, waves and streaks convected downstream at almost the free-stream speed. The interaction problem satisfies the unit-Reynolds-number three-dimensional Navier–Stokes equations and is localized in a layer of the same depth as the unperturbed boundary layer. Here, we show that the interaction problem is generic to any boundary layer that approaches its free-stream form through an exponentially small correction. It is shown that away from the layer where it is generated the induced roll–streak flow is dominated by non-parallel effects which now play a major role in the streamwise evolution of the structure. The similarity with the parallel boundary layer case is restricted only to the layer where it is generated. It is shown that non-parallel effects cause the structure to persist only over intervals of finite length in any growing boundary layer and lead to a flow structure reminiscent of turbulent boundary layer simulations. The results found shed light on a possible mechanism to couple near-wall streaks with coherent structures located towards the edge of a turbulent boundary layer. Some discussion of how the mechanism adapts to a three-dimensional base flow is given.

The present paper initiates a systematic computational analysis of the rise dynamics of high viscosity droplets in a viscous ambient liquid. This represents a relevant intermediate case between free rigid particles and bubbles since their shape adjusts to outer forces while almost no inner circulation is present. As a prototype system, we study corn oil droplets rising in pure water with diameters ranging from 0.5 to 16 mm. Since we are interested in the droplet dynamics from the viewpoint of a bifurcation scenario with increasingly complex droplet behaviour, we perform fully three-dimensional numerical simulations, employing the in-house volume-of-fluid (VOF)-code FS3D. The smallest droplets (0.5–2 mm) rise in steady vertical paths, where for the smallest droplet (0.5 mm) the flow field, as well as the terminal velocity, can be described by the Taylor and Acrivos approximate solution, despite the Reynolds number being well above one. Larger droplets (3.2 mm) rise in an oblique path and display a bifid wake, and those with diameters in the range (3.7–8 mm) rise in intermittently oblique paths, showing an intermittent bifid wake of alternating vorticity. The droplets’ shapes in this range change from spherical into oblate ellipsoids of increasing eccentricity, followed by bi-ellipsoidal shapes with higher curvature on the downstream side. Even larger droplets (10–16 mm) rise in oscillatory, essentially vertical paths with drastically different wake structures, including deadzones and aperiodic or periodic vortex shedding. The largest considered droplets (diameter of 14 and 16 mm) display significant shape oscillations and vortex shedding is accompanied by a complex evolution of coherent vortex structures. Their rise paths are best described as zigzagging, but the bifurcation scenario seems to be substantially different from that leading to the zigzagging of air bubbles. In contrast to the rise behaviour of bubbles, helical paths are not observed in the present study.

A theoretical and experimental analysis of non-Newtonian gravity currents in porous media with variable properties is presented. A mound of a power-law fluid of flow behaviour index $n$ is released into a semi-infinite saturated porous medium above a horizontal bed, and can drain freely out of the formation at the origin. The porous medium permeability varies along the vertical as $z^{({\it\omega}-1)}$, porosity varies along the vertical as $z^{({\it\gamma}-1)}$, $z$ being the vertical coordinate and ${\it\omega}$ and ${\it\gamma}$ constant numerical coefficients. A self-similar solution describing the space–time evolution of the resulting gravity current is derived for shear-thinning fluids with $n<1$, generalizing earlier results for Newtonian fluids. The solution conserves a generalized dipole moment of the mound. The spreading of the current front is proportional to $t^{{\it\gamma}n/(2+{\it\omega}(n+1))}$. Expressions for the time evolution of the outgoing flux at the origin and of the current volume are derived in closed form. The Hele-Shaw analogue is derived for flow of a power-law fluid in a porous medium with vertically variable properties. Results from laboratory experiments conducted in two Hele-Shaw cells confirm the constancy of the dipole moment, and compare well with the theoretical formulation.