The addition of even minute amounts of a polymer to a liquid can modify some of its properties drastically, like the ways in which jets of the liquid break into droplets or the dynamics of dripping droplets at an orifice. The elasticity conferred by the polymers can delay the separation of the drops considerably, which hesitate between falling or running back to the orifice, ‘gobbling’ along their way the thread to which they are attached. Other free-surface flows are also of a hesitant nature.

This paper discusses the breakup of capillary jets of dilute polymer solutions and the dynamics associated with the transition from dripping to jetting. High-speed digital video imaging reveals a new scenario of transition and breakup via periodic growth and detachment of large terminal drops. The underlying mechanism is discussed and a basic theory for the mechanism of breakup is also presented. The dynamics of the terminal drop growth and trajectory prove to be governed primarily by mass and momentum balances involving capillary, gravity and inertial forces, whilst the drop detachment event is controlled by the kinetics of the thinning process in the viscoelastic ligaments that connect the drops. This thinning process of the ligaments that are subjected to a constant axial force is driven by surface tension and resisted by the viscoelasticity of the dissolved polymeric molecules. Analysis of this transition provides a new experimental method to probe the rheological properties of solutions when minute concentrations of macromolecules have been added.

The paper considers the moderately turbulent natural convection flow of a stably stratified fluid along an infinite vertical plate (wall). Attention is restricted to statistically stationary flow driven by constant surface forcing (heating), with Prandtl number of unity. The flow is controlled by the surface energy production rate Fs, molecular viscosity/diffusivity ν and ambient stratification in terms of the Brunt–Väisälä (buoyancy) frequency N. Following the transition from a laminar to a turbulent regime, the simulated flow enters a quasi-stationary oscillatory phase. In this phase, turbulent fluctuations gradually fade out with distance from the wall, while periodic laminar oscillations persist over much larger distances before they fade out. The scaled mean velocity, scaled mean buoyancy and scaled second-order turbulence statistics display a universal behaviour as functions of distance from the wall for given value of dimensionless combination Fs/(νN2) that may be interpreted as an integral Reynolds number. In the conducted numerical experiments, this number varied in the range from 2000 to 5000.

Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

We analyse the low-mode structure of internal tides generated in laboratory experiments and numerical simulations by a two-dimensional ridge in a channel of finite depth. The height of the ridge is approximately half of the channel depth and the regimes considered span sub- to supercritical topography. For small tidal excursions, of the order of 1% of the topographic width, our results agree well with linear theory. For larger tidal excursions, up to 15% of the topographic width, we find that the scaled mode 1 conversion rate decreases by less than 15%, in spite of nonlinear phenomena that break down the familiar wave-beam structure and generate harmonics and inter-harmonics. Modes two and three, however, are more strongly affected. For this topographic configuration, most of the linear baroclinic energy flux is associated with the mode 1 tide, so our experiments reveal that nonlinear behaviour does not significantly affect the barotropic to baroclinic energy conversion in this regime, which is relevant to large-scale ocean ridges. This may not be the case, however, for smaller scale ridges that generate a response dominated by higher modes.

We analyse the linear stability and nonlinear evolutions of circular hetons under the quasi-geostrophic approximation. We compare results obtained with a three-layer model and with a model based on a continuous density stratification. Though the models also differ by the vertical boundary conditions, they show a remarkable similarity in the stability properties of the hetons (threshold values of vortex radius for baroclinic instability, dominant modes, growth rates, etc.), and in their nonlinear evolutions (spatial reorganization of potential vorticity by nonlinear processes, end-states of the simulations). The hetons prone to baroclinic instability often break into two hetons drifting in opposite directions, and in more hetons, for wider initial structures. In both models, instability is quite sensitive to the vertical gap between the opposite-signed vortices: as it increases, the instability decreases and shifts to lower azimuthal modes. Finally, though modes l ≥ 2 (i.e. elliptical and shorter wave deformations) prevail in most of the parameter space, the mode l = 1 perturbation (a vertical tilt of the vortex column) exists for hetons with small vertical gaps. Such perturbations are concentrated vertically near the gap, and can only be evidenced in the continuously stratified model.

Laboratory experiments have been conducted that test the predictions of Holmboe (Geofys. Publ., vol. 24, 1962, pp. 67–112). Symmetric Holmboe instabilities are observed during steady, maximal two-layer exchange flow in a long laboratory channel of rectangular cross-section. Internal hydraulic controls at each end of the channel isolate the subcritical region within the channel from disturbances in the reservoirs. Inside the channel, the instabilities form cusp-like waves that propagate in both directions. The phase speed of the instabilities is consistent with Holmboe's theory and increases along the length of the channel as a result of the gradual acceleration of each layer. This acceleration causes the wavelength of any given instability to increase in the flow direction until it is approximately twice the most amplified wavelength. At this point new waves develop with the result that the average wavelength is almost constant along the length of the channel.

We study the drawing of a Newtonian viscous sheet under the influence of cooling with temperature dependence of the viscosity. Classically this problem has an instability called draw resonance, when the draw ratio Dr, which is the ratio of the outlet velocity relative to the inlet velocity, is beyond a critical value Drc. The heat transfer from the surface compared to the bulk energy advection is conveniently measured by the Stanton number St. Usual descriptions of the problem are one-dimensional and rigorously apply for St ≤ 1. The model presented here accounts for variations of the temperature across the sheet and has therefore no restriction on St. Stability analysis of the model shows two different cooling regimes: the ‘advection-dominated’ cooling for St ≪ 1 and the ‘transfer-dominated’ cooling for St ≫ 1. The transition between those two regimes occurs at St = O(1) where the stabilizing effect due to cooling is most efficient, and for which we propose a mechanism for stabilization, based on phase shifts between the tension and axial-averaged flow quantities. Away from this transition, the sheet is always shown to be unstable at smaller draw ratios. Additionally, in the limit of St → ∞, the heat exchange is such that the temperature of the fluid obtains the far-field temperature, which hence corresponds to a ‘prescribed temperature’ regime. This dynamical situation is comparable to the isothermal regime in the sense that the temperature perturbation has no effect on the stability properties. Nevertheless, in this regime, the critical draw ratio for draw resonance can be below the classical value of Drc = 20.218 obtained in isothermal conditions.

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics is compatible with the stress and insulating boundary conditions at the free surface but argue that unless the flow is driven at fixed Galilei and capillary numbers, respectively parameterizing gravity and surface tension, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field and the associated Hartmann steady state affect the soft and hard instability modes of free-surface flow, working in the low-magnetic-Prandtl-number regime of conducting laboratory fluids (Pm ≤ 10−4). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode exhibits similar behaviour as the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral-stability curve and the dependence of its critical Reynolds number Rec on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but non-zero, Pm and their counterparts in the inductionless limit, Pm ↘ 0. As derived from large-wavelength approximations and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is non-zero the Lorentz force originating from the steady-state current leads to a modification of Rec(Ha) to either a sub-linearly increasing or a decreasing function of Ha, respectively for problems with insulating or perfectly conducting walls. In insulating-wall problems we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transferred from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.

We study the Navier–Stokes steady states for a low density monodisperse hard sphere granular gas (i.e a hard sphere ideal monatomic gas with inelastic inter-particle collisions). We present a classification of the uniform steady states that can arise from shear and temperature (or energy input) applied at the boundaries (parallel walls). We consider both symmetric and asymmetric boundary conditions and find steady states not previously reported, including sheared states with linear temperature profiles. We provide explicit expressions for the hydrodynamic profiles for all these steady states. Our results are validated by the numerical solution of the Boltzmann kinetic equation for the granular gas obtained by the direct simulation Monte Carlo method, and by molecular dynamics simulations. We discuss the physical origin of the new steady states and derive conditions for the validity of Navier–Stokes hydrodynamics.

We propose a two-phase model having a Newtonian rheology for the fluid phase and friction for the particle phase to describe bed-load transport in the laminar viscous regime. We have applied this continuum model to sediment transport by viscous shearing flows. The equations are shown to reduce to the momentum equation for the mixture and the Brinkman equation for the fluid velocity. This modelling is able to provide a description of the flow of the mobile granular layer. At some distance from threshold of particle motion, where the continuum approach is more realistic as the mobile layer is larger than one particle diameter, there is very little slip between the two phases and the velocities inside the mobile bed have approximately a parabolic profile. When the Poiseuille (or Couette) flow is not significantly perturbed, simple analytical results of the particle flux varying cubically with the Shields number and of the bed-load thickness varying linearly with it can then be obtained. These predictions compare favourably with experimental observations of bed-load transport in pipe flows.

We present a phase diagram of the different dune patterns observed when a bed composed of spherical particles is subjected to a pipe flow. While the threshold for incipient motion is determined by the Shields number, that for dune formation seems to be controlled by the Reynolds number. A simple linear stability analysis based on a particle flux derived by Ouriemi, Aussillous & Guazzelli (J. Fluid Mech., 2009) accounts reasonably well for the experimental observations.

The flow in a plane asymmetric diffuser with an opening angle of 8.5° has been studied experimentally using time-resolving stereoscopic particle image velocimetry. The inlet condition is fully developed turbulent channel flow at a Reynolds number based on the inlet channel height and bulk velocity of Re = 38000. All mean velocity and Reynolds stress components have been measured. A separated region is found on the inclined wall with a mean separation point at 7.4 and a mean reattachment point at 30.5 inlet channel heights downstream the diffuser inlet (the inclined wall ends 24.8 channel heights downstream the inlet). Instantaneous flow reversal never occurs upstream of five inlet channel heights but may occur far downstream the point of reattachment. A strong shear layer in which high rates of turbulence production are found is located in a region outside the separation. The static wall pressure through the diffuser is presented and used in an analysis of the balance between pressure forces and momentum change. It is demonstrated that production of turbulence causes a major part of the losses of mean flow kinetic energy. The character of the large turbulence structures is investigated by means of time-resolved sequences of velocity fields and spatial auto-correlation functions. Pronounced inclined structures are observed in the spanwise velocity and it is suggested that these are due to the legs of hairpin-like vortices.

In plants, osmotically driven flows are believed to be responsible for translocation of sugar in the pipe-like phloem cell network, spanning the entire length of the plant – the so-called Münch mechanism. In this paper, we present an experimental and theoretical study of transient osmotically driven flows through pipes with semi-permeable walls. Our aim is to understand the dynamics and structure of a ‘sugar front’, i.e. the transport and decay of a sudden loading of sugar in a water-filled pipe which is closed in both ends. In the limit of low axial resistance (valid in our experiments as well as in many cases in plants) we show that the equations of motion for the sugar concentration and the water velocity can be solved exactly by the method of characteristics, yielding the entire flow and concentration profile along the tube. The concentration front decays exponentially in agreement with the results of Eschrich, Evert & Young (Planta (Berl.), vol. 107, 1972, p. 279). In the opposite case of very narrow channels, we obtain an asymptotic solution for intermediate times showing a decay of the front velocity as M−1/3t−2/3 with time t and dimensionless number M ~ ηκL2r−3 for tubes of length L, radius r, permeability κ and fluid viscosity η. The experiments (which are in the small M regime) are in good quantitative agreement with the theory. The applicability of our results to plants is discussed and it is shown that it is probable that the Münch mechanism can account only for the short distance transport of sugar in plants.

The low-frequency dynamics of the shock-induced separation region in a Mach 2 compression ramp interaction is investigated by performing high-speed particle image velocimetry (HSPIV) measurements, at a rate of 6kHz, in a streamwise–spanwise plane. The HSPIV measurements made in the upstream turbulent boundary layer indicate the presence of spanwise strips of elongated regions of uniform streamwise velocity that extend to lengths greater than 30δ, validating previous results based on planar laser scattering measurements obtained by Ganapathisubramani, Clemens & Dolling (J. Fluid Mech., vol. 585, 2007, p. 369). At a wall normal-location of y/δ=0.2, a surrogate for separation based on a velocity threshold is found to fluctuate over a streamwise range of ±1.2δ, consistent with previous studies. The amplitude of unsteadiness has contributions from at least two sources that are related to the incoming boundary layer. First, the velocity threshold based surrogate separation line exhibits large-scale undulations along the spanwise direction that conform to the passage of elongated low- and high-speed regions in the upstream boundary layer. This motion is classified as the local influence of the upstream boundary layer. Second, the spanwise-averaged surrogate separation is found to respond to the overall change in streamwise velocity in the incoming boundary layer and is classified as the global influence of the upstream boundary layer. However, this global influence includes the contributions from the elongated low- and high-speed regions. Preliminary findings based on statistical analysis suggest that the local influence contributes nearly 50% more than the global influence. Regardless, the low-frequency unsteadiness of the separation-region can be attributed to the local and global influences of the incoming boundary layer.

This paper proposes statistical tools adapted to study highly unsteady and inhomogeneous flows, such as vortex bursting. For this, we use the wavelet representation in which each coefficient keeps track of both location and scale, in contrast to Fourier representation which requires keeping the phase of all coefficients to preserve the spatial structure of the flow. Based on the continuous wavelet transform, we propose several diagnostics, such as the local spectrum and the local intermittency measure. We also use the orthogonal wavelet transform to split each flow realization into coherent and incoherent contributions, which are then analysed independently and from which we define the coherency measure. We apply these wavelet tools to analyse the bursting of a three-dimensional stretched vortex immersed in a steady laminar channel flow. The time evolution of the velocity field is measured by particle image velocimetry during several successive bursts.

We examine the linear stability of a capillary rivulet under the assumption that it is shallow enough to be described by the lubrication approximation. It is shown that rivulets on a sloping plate are stable regardless of their parameters, whereas rivulets on the underside of a plate can be either stable or unstable, depending on their widths and the plate's slope. For the case of a horizontal plate, sufficiently narrow rivulets are shown to be stable and sufficiently wide ones unstable, with the threshold width being π/2(σ/gρ)1/2(ρ and σ are the liquid's density and surface tension, g is the acceleration due to gravity).

It is also shown that, even though the plate's slope induces in a rivulet a sheared flow (which would normally be viewed as a source of instability) – in the present problem, it is a stabilizing factor. The corresponding stability criterion involving the rivulet's width and the plate's slope is computed, and it is demonstrated that, if the latter is sufficiently strong, all rivulets are stable regardless of their widths.

We study ‘surface switching’ quantitatively in flows driven by the constant rotation of the endwall of an open cylindrical vessel reported by Suzuki, Iima & Hayase (Phys. Fluids, vol. 18, 2006, p. 101701): the deformed free surface switches between axisymmetric and non-axisymmetric shapes accompanied by irregular vertical oscillation. Detailed simultaneous measurements showed that the magnitude of the velocity fluctuations (turbulent intensity) temporally varies greatly and are strongly correlated with the surface height, suggesting that dynamic switching between laminar and turbulent states is accompanied by vessel-scale surface shape changes. The study also identified clear hysteresis in the turbulent intensity arising from changes in the Reynolds number; the bifurcation diagram consists of two overlapping branches representing a high-intensity (turbulent) state and a low-intensity (laminar) state. Based on the results, a switching mechanism is suggested.

An idealized model is proposed to explain the appearance of the long bed waves that have been recently observed in shallow tidal seas. The model assumes that these bedforms grow due to tide–topography interaction. The water motion is described by means of the depth-averaged shallow water equations and the bottom evolution is governed by conservation of sediment mass. The sediment transport formulation includes a critical bottom stress below which no sediment moves. Also, anisotropic sediment transport, due to local bottom slopes in the longitudinal and transverse directions, is taken into account. A linear stability analysis of the flat bottom configuration reveals that different bottom patterns can emerge. In accordance with previous analyses, for strong tidal currents, the fastest growing modes are sand banks. However, if the tidal currents are elliptical and the maximum bottom stress is just above its threshold value for the initiation of sediment motion, the model shows the presence of further growing modes which resemble the long bed waves observed in the field.