The constitutive relation for the granular flow of smooth, nearly elastic particles ($(1 - e) \ll 1$) is derived in the adiabatic limit, where the length scale for conduction, $\lambda / (1 - e)^{1/2}$, is small compared to the macroscopic scale. Here, $\lambda$ is the mean free path and $e$ is the coefficient of restitution. In this case, energy is convected by the mean flow, and the rate of change of the energy in a Lagrangian reference frame is determined by a balance between the rate of production (due to the shear) and the rate of dissipation (due to inelastic collisions). A Gaussian approximation is used for the velocity distribution function, and the velocity variance is determined from a balance equation for the second moment of the velocity distribution using an asymptotic expansion in the small parameter $\epsilon = (1 - e)^{1/2}$. The stress tensor is then determined from the velocity variance. It is found that the leading-order, $O(\epsilon)$ and $O(\epsilon^{2})$ contributions to the stress tensor are identical in form to those in the Euler, Navier–Stokes and Burnett approximations, and the numerical values of these coefficients are close to those calculated using the Enskog procedure. The stress tensor is used to obtain analytical expressions for the growth rates of the hydrodynamic modes in a linear shear flow in the limit where the wavelength is long compared to the length scale of conduction. In the plane of flow, transverse momentum perturbations are found to be stable, while perturbations in the density and longitudinal momenta grow exponentially at short times and decay in the long-time limit. It is found that the Navier–Stokes approximation captures the leading behaviour of the growth rate in the small-wavenumber limit for perturbations in the plane of flow. In the vorticity direction, it is found that the Navier–Stokes approximation is not adequate to capture the leading behaviour in the small-wavenumber limit, and dominant terms in the growth rates of the transverse modes depend on the Burnett coefficients. Perturbations in this direction are unstable at low density, but become stable as the density is increased. When the wave length is small compared to the conduction length, the rate of conduction of energy is large compared to the rate of dissipation, and the hydrodynamic modes are identical to those for a gas of elastic particles at equilibrium. The hydrodynamic modes are all stable in this case. The transition between these two regimes is examined for a dilute granular flow, and a transition from unstable to stable modes is predicted as the wavenumber is increased. The analysis indicates that the minimal model which accurately captures the dynamics in both limits is one in which the constitutive relation for the stress incorporates the strain-rate-dependent Burnett terms in the stress equation (neglecting the temperature-dependent Burnett terms), and the constitutive relation for the heat flux is the Fourier law for heat conduction (neglecting all the Burnett terms).

Non-axisymmetric simulations verify and extend the results from a previous linear Floquet stability analysis of the wakes of rings. The wakes corresponding to the saturated state of each predicted non-axisymmetric instability mode over the entire aspect ratio range are successfully computed, and isosurface plots are presented elucidating the vortical wake structures. The existence of three non-axisymmetric flow regimes (Modes I, II and III) for the flow past rings with aspect ratios $\textit{Ar\/} \lesssim 3.9$ is verified, as is the existence of non-axisymmetric instabilities of vortex streets in the flow past rings with $\textit{Ar\/} \gtrsim 3.9$. Wakes are computed which correspond to the Mode A and B instabilities found in the flow past a circular cylinder, and a wake is computed which develops from a subharmonic Mode C instability. This wake features an azimuthal wavelength of approximately 1.7 ring cross-section diameters, which is between the azimuthal wavelengths of the Mode A and B instabilities. This mode cannot occur, at least in a pure state, in the flow past a circular cylinder.

Nonlinear transition characteristics are predicted by evaluating coefficients of the truncated Landau equation, and transition hysteresis is verified by studying the mode amplitude variation with Reynolds number in the vicinity of the transitions. The regular Mode I and Mode III transitions are found to occur through supercritical and subcritical bifurcations, respectively, and the secondary Hopf bifurcations to these transitions, as well as the Mode II Hopf transition, are found to be supercritical. We verify that the Mode A and Mode B transitions are subcritical and supercritical, respectively, and we determine that the nature of the Mode C transition is dependent on aspect ratio. Landau constants are evaluated for the Hopf transitions throughout the aspect ratio range studied.

In two earlier papers, we studied the statistical and mechanistic structure of the turbulent boundary layer under a stress-free (clean) free surface. Findings there, such as the presence of inner and outer surface layers, are very much the direct result of the absence of shear stresses at the surface. The latter condition is easily lost when the surface is contaminated and surface elasticity varies with space and time. In this paper we consider the effect of surfactant on features of the free-surface turbulent flow. We perform direct numerical simulations of the Navier–Stokes equations subject to surfactant-laden free-surface boundary conditions for varying Reynolds and Marangoni numbers and low Froude numbers. As expected, the Marangoni effect decreases the horizontal turbulence intensity and normal vorticity at the surface. The direct effect on the turbulent kinetic energy is an increase in the dissipation and viscous diffusion and a decrease in the production near the surface relative to the clean case. The most prominent effect of the presence (of even a small amount) of surfactant is the drastic reduction in the surface divergence and the associated sharp decrease of up- and downwelling at the surface which has direct implications to near-surface turbulent transport. The observed surfactant effects on turbulent kinetic energy budget can be attributed to the generation of Marangoni vorticity at the free surface by approaching hairpin vortices. The Marangoni effect has also a direct effect on the boundary-layer structure, causing an increase of the thickness of the boundary layer and in the maxima of the mean shear near the surface. For moderate values of the Marangoni number, up-/downwelling effectively vanishes and the flow approaches a state independent of the Marangoni number. Guided by these results and to obtain theoretical insight, we develop a similarity solution for the mean flow. The analytic solution agrees well with the numerical data and provides precise measures for the multi-layer structure of the boundary layer. Based on the theoretical model, we derive scaling laws for the thickness of the inner and the outer boundary layers, which are also confirmed by numerical simulations.

A common problem in multi-layer shear flows, especially from the perspective of process engineering, is the occurrence of interfacial instabilities. Here we show how multi-layer duct flows can in fact be made nonlinearly stable, by using a suitable lubricating fluid. First we show how interfacial instabilities may be eliminated through the introduction of a yield stress fluid as the lubricant and by preserving an unyielded layer adjacent to the interface. Second we show how to treat the nonlinear stability of a two-layer flow, allowing finite motion of the domains. We focus on the simplest practically interesting case of visco-plastically lubricated viscous shear flow: a core–annular pipe flow consisting of a central core of Newtonian fluid surrounded by a Bingham fluid. We demonstrate that this flow can be nonlinearly stable at significant Reynolds numbers and produce stability bounds. Our analysis can be straightforwardly generalized to other flows in this class.

Flapping flight is ubiquitous in Nature, yet cilia and flagella, not wings, prevail in the world of micro-organisms. This paper addresses this dichotomy. We investigate experimentally the dynamics of a wing, flapped up and down and free to move horizontally. The wing begins to move forward spontaneously as a critical frequency is exceeded, indicating that ‘flapping flight’ occurs as a symmetry-breaking bifurcation from a pure flapping state with no horizontal motion. A dimensionless parameter, the Reynolds number based on the flapping frequency, characterizes the point of bifurcation. Above this bifurcation, we observe that the forward speed increases linearly with the flapping frequency. Visualization of the flow field around the heaving and plunging foil shows a symmetric pattern below transition. Above threshold, an inverted von Kármán vortex street is observed in the wake of the wing. The results of our model experiment, namely the critical Reynolds number and the behaviour above threshold, are consistent with observations of the flapping-based locomotion of swimming and flying animals.

A thin liquid film experiences additional intermolecular forces when the film thickness $h$ is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by the disjoining pressure $\Pi $. Since $\Pi $ dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, $\Pi = \Pi (h)$ because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope $h_x$ leads to $\Pi = \Pi (h, h_x )$, which allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates $\Pi $ to an excess interaction energy $E = E(h, h_x )$. By considering a fluid wedge on a solid substrate, $E(h,h_x )$ is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit $$\Pi = \frac{B}{h^3}\big(\alpha ^4 - h_x^4 + 2hh_x^2 h_{xx}\big),$$ where $\alpha $ is the contact angle and $B$ is a material constant. The term containing the curvature $h_{xx} $ is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that $h_{xx} = 0$ at the contact line is sufficient to specify the contact angle.

We examine the interaction of shock waves by studying solutions of the two-dimensional Euler equations about a point. The problem is reduced to linear form by considering local solutions that are constant along each ray and thereby exhibit no length scale at the intersection point. Closed-form solutions are obtained in a unified manner for standard gasdynamics problems including oblique shock waves, Prandtl–Meyer flow and Mach reflection. These canonical gas dynamical problems are shown to reduce to a series of geometrical transformations involving anisotropic coordinate stretching and rotation operations. An entropy condition and a requirement for geometric regularity of the intersection of the incident waves are used to eliminate spurious solutions. Consideration of the downstream boundary conditions leads to a formal determination of the allowable downstream matching criteria. By retaining the time-dependent terms, an approach is suggested for future investigation of the open problem of the stability of shock wave interactions.

An exact Craik–Criminale (CC) solution to the incompressible Navier–Stokes (NS) equations describes the instability of an elliptical columnar flow interacting with a single Kelvin wave. These CC solutions are extended to allow multi-harmonic Kelvin waves to interact with any exact ‘base’ solution of the NS equations. The interaction is evaluated along an arbitrarily chosen flowline of the base solution, so exact nonlinear instability in this context is locally convective, rather than absolute. Furthermore, an iterative method called ‘WKB-bootstrapping’ is introduced which successively adds Kelvin waves with incommensurate phases to the extended CC solutions. In illustrating WKB bootstrapping, we construct a succession of extended nonlinear CC solutions consisting of a circular columnar flow interacting progressively with one (Kelvin 1887), two (Fabijonas & Lifschitz 1996), or three waves with incommensurate phases. The phase of each wave packet is frozen into the previous flow and we examine the exact nonlinear convective instability induced at each stage (primary, secondary, tertiary). At each stage, the flow becomes progressively more unstable.

The effect of both vertical and horizontal components of the Earth's rotation on plumes during deep convection in the ocean is studied. In the laboratory, the misalignment, characterized by the angle $\alpha$, between the buoyancy force (‘effective’ free-fall acceleration ${\bm g}_e$) and the rotation axis ${\bm \Omega}$ is produced by using the centrifugal force: an experimental tank was placed at a large distance from the centre of the turntable. The mathematical analogy between the laboratory model and the oceanic environment is presented. For $\alpha\,{=}\,30^\circ$, a number of laboratory experiments spanning a wide range of the buoyancy flux parameter, and correspondingly Reynolds number, is used to illustrate the development of the convective plume from a point source in regimes ranging from weakly to highly turbulent. New features of the flow, as compared to $\alpha\,{=}\,0$, are documented and explained.

The incoming heavier dyed fluid jet disintegrates into fast-sinking coherent blobs (in a low-Reynolds-number regime) or turbulent billows (in a high-Reynolds-number regime) and a more diffuse cloud of highly diluted dyed water. An analysis of the forces acting on an ellipsoid moving in a rotating fluid with the main balance including the buoyancy, Coriolis forces, and the hydrodynamic reaction due to generation of inertial waves correctly predicts the trajectory of a descending blob. It also explains the tendency of the plume to develop in the direction intermediate between ${\bm g}_e$ and ${\bm \Omega}$ and to shift ‘eastward’ (lagging the rotation of the centrifuge) if the plume is envisaged as an ensemble of blobs.

The stretching of the highly diluted dyed water along the absolute vorticity tubes with simultaneous shearing by horizontal quasi-two-dimensional flow produces conspicuous tilted structures or tilted Taylor ‘ink walls’. The misalignment between ${\bm g}_e$ and ${\bm \Omega}$ enhances the turbulent mixing and development of tilted structures by breaking the symmetry and producing motions directed away from the rotation axis.

We argue that the conditions at the sites of ocean deep convection are favourable for the development of tilted structures because of the smallness of the Rossby number and an extreme homogenization of the mixed layer. We hypothesize that the homogenized sublayers observed within actively convecting regions in the ocean may not be horizontal, but in fact analogous to the tilted ‘ink walls’ observed in the laboratory experiments and that they represent the internal structure of a plume on horizontal scales smaller than its depth.

A general method is presented to investigate the hydrodynamic stability of ideal incompressible or barotropic flows described in a Lagrangian representation. Based on the theory of short-wavelength instabilities, the problem is reduced to a transport equation which involves only the distortion matrix of the equilibrium flow. The theory is applied to Gerstner's rotational free-surface gravity waves. It is shown that they are three-dimensionally unstable when their steepness exceeds 1/3.

An open system or open resonator is a domain of wave activity separated from the exterior by a partly open or partly transparent surface. Such open resonators lose energy to infinity through radiation. The numerical computation of the corresponding resonances is complicated by spurious reflections of the outgoing waves at the necessarily finite grid boundaries. These reflections can be reduced to extremely low levels by applying perfectly matched layer (PML) absorbing boundary conditions, which separate the discrete resonances from the continuous spectrum. Using a simple one-dimensional model problem, the influence of the various PML parameters is determined by a numerical error analysis. In addition to one-dimensional open resonators, two-dimensional open resonators as well as various resonating structures in waveguides are considered, and the resonant spectra and selected modes are evaluated. For the first time, leaky modes are computed for several resonating structures in a waveguide in addition to the trapped modes published in the literature. In applications, leaky mode resonances are often more important than trapped mode resonances. Gap tones, observed in a model problem of high-lift configurations, are identified as transversal resonant modes with the lowest radiation losses.

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed.

We examine the decompression and associated flow of water through a high-pressure permeable rock towards a well of low pressure. Using a series of new analytical similarity solutions, and some asymptotic simplifications, we explore the controls on the vaporization as a function of (i) the pressure jump from the far field to the well; (ii) the initial superheat of the reservoir and (iii) the permeability of the system. We find that for a sufficiently large pressure decrease, the liquid becomes superheated and boils, leading to venting of vapour into the well. For a low-permeability system, the flux of liquid from the far field is rate limiting and a sharp boiling interface develops ahead of the well. However, in higher-permeability rock, the flux of water from the far field increases and there is insufficient heat conducted from the far field in order to boil this water. The boiling front is then located at the well, while the fluid ahead of the boiling front may become superheated leading to formation of a two-phase zone which spreads outwards from the well.

A phase-plane approach is used to determine similarity solutions of the axisymmetric shallow-water equations which represent inwardly propagating, inviscid gravity currents. A Froude number condition characterizes the movement of the front. The unique similarity exponent is found numerically as a function of the frontal Froude number and the height and velocity profiles are presented for three different Froude numbers. The fluid speed and height are seen to increase monotonically towards the front except very close to the front where the height decreases. The maxima in both height and speed increase as the Froude number increases, reflecting the change in ambient resistance.

For the Froude number that has been obtained experimentally for lock-exchange Boussinesq flows ($Fr\,{=}\,1.19$) for which the similarity exponent is 0.859094, the similarity solution is compared to the numerical solution of the initial value problem, obtained recently by Hallworth, Huppert & Ungarish (2003). Our similarity solution compares reasonably well with their integration of the shallow-water equations in the neighbourhood of the front and at times close to collapse (when the front reaches the origin); however, near this point their numerics begin to fail. The solution at collapse and the similarity solution after collapse are also found for $Fr\,{=}\,1.19$. This similarity solution describes the formation of a shock, as well as its initial propagation.

Experimental studies were carried out on fully developed and steady electro-osmotic flow in a rectangular channel where the channel height $h$ is comparable to its width and the thickness of the electric double layer characterized by the Debye length is much less than $h$. The nano-particle image velocimetry technique was used to measure the two components of the velocity field parallel to and within about 100 nm of the channel wall for $h\,{\leq}\,25\,\umu$m. The mobility of the particle tracers was calculated from averaged velocity data for various electric field strengths. The experimentally determined mobility values are compared with analytical predictions for dilute aqueous solutions of sodium tetraborate.

Scalar composition and mixing rates within a non-reactive plane shear layer between two uniform water streams were manipulated by surface actuators on the high-speed side of the flow partition. The scalar concentrations were assessed using a thermal analogue by maintaining a time-invariant temperature difference between the uniform streams upstream of the flow partition. The ratio of the smallest velocity and temperature scales was governed by the Prandtl number. Because in water $\hbox{\it Pr}\,{ \approx}\,7$ while the Schmidt number $\hbox{\it Sc}$ for dyes and reactants is O(1000), temperature concentrations in water are a better representation of scalar mixing in air in which $\hbox{\it Sc}\,{=}\,O(1)$. The effects of spanwise-uniform and -non-uniform actuation programs were investigated using arrays of discrete, individually controlled thin-film resistive heating elements that were surface-mounted on the flow partition. Spatial and temporal temperature distributions were measured phase-locked to the actuation waveform using a cross-stream array of closely spaced cold-wire sensors. These data were used to infer both the mixedness and the composition through the onset of mixing transition and quantify local and integral cross-stream mixing performance measures. Actuation programs that hastened mixing and significantly altered the composition of mixed fluid were identified.