A hydrodynamic description of dilute binary gas mixtures comprising smooth inelastic spheres interacting by binary collisions with a random coefficient of restitution is presented. Constitutive relations are derived using the Chapman–Enskog perturbative method, associated with a computer-aided method to allow high-order Sonine polynomial expansions. The transport coefficients obtained are checked against DSMC simulations. The resulting equations are applied to the analysis of a vertically vibrated system. It is shown that differences in the shape of the distributions of the coefficient of restitution are sufficient to produce partial segregation.

Transition to turbulence in shear flows is often subcritical, thus the dynamics of the flow strongly depends on the shape and amplitude of the perturbation of the laminar state. In the state space, initial perturbations which directly relaminarize are separated from those that go through a chaotic trajectory by a hypersurface having a very small number of unstable dimensions, known as the edge of chaos. Even for the simple case of plane Couette flow in a small domain, the edge of chaos is characterized by a fractal, folded structure. Thus, the problem of determining the threshold energy to trigger subcritical transition consists in finding the states on this complex hypersurface with minimal distance (in the energy norm) from the laminar state. In this work we have investigated the minimal-energy regions of the edge of chaos, by developing a minimization method looking for the minimal-energy perturbations capable of approaching the edge state (within a prescribed tolerance) in a finite target time $T$. For sufficiently small target times, the value of the minimal energy has been found to vary with $T$ following a power law, whose best fit is given by $E_{min}\propto T^{-1.75}$. For large values of $T$, the minimal energy achieves a constant value which corresponds to the energy of the minimal seed, namely the perturbation of minimal energy asymptotically approaching the edge state (Rabin et al., J. Fluid Mech., vol. 738, 2012, R1). For $T\geqslant 40$, all of the symmetries of the edge state are broken and the minimal perturbation appears to be localized in space with a basic structure composed of scattered patches of streamwise velocity with inclined streamwise vortices on their flanks. Finally, we have found that minimal perturbations originate in a small low-energy zone of the state space and follow very fast similar trajectories towards the edge state. Such trajectories are very different from those of linear optimal disturbances, which need much higher initial amplitudes to approach the edge state. The time evolution of these minimal perturbations represents the most efficient path to subcritical transition for Couette flow.

Two mechanisms for the generation of standing edge waves over a sloping beach are described using classical linear water-wave theory. The first is an extension of the result of Yih (1984) to a class of localized bottom protrusions on a sloping beach in the presence of a longshore current. The second is a class of longshore surface-pressure distributions over a beach. In both cases it is shown that Ursell-type standing edge-wave modes can be generated in an appropriate frame of reference. Typical curves of the mode shapes are presented and it is shown how in certain circumstances the dominant mode is not the lowest.

This analysis demonstrates theoretically that a lateral bending wave propagating along the walls of a two-dimensional channel filled with a viscous incompressible fluid will induce a mean flow. In addition to this ‘pure’ bending wave, another possible condition is investigated: that of the superposition of an area contraction wave propagating with the same speed as the lateral bending wave. This second condition, called complex wave motion, takes into account the slight occlusion which occurs naturally at the amplitude peaks when a finite amplitude bending wave propagates along the walls of a container. A perturbation solution is found which satisfies Navier—Stokes equations for the case in which wave amplitude/wavelength is ‘small’. However the wave amplitude is finite, in the sense that it is of the same order as the channel width. Under these conditions, the occlusion at the amplitude peaks is allowed to be of the same order as the channel width. For the case of a pure bending wave the motion induced by the peristaltis is found to be of second order in the perturbation parameter, whereas in the more realistic case a first-order pumping effect is obtained.

The eddy-damped quasi-normal Markovian (EDQNM) turbulence theory has been applied to the covariance spectrum of two passive isotropic scalars with different diffusivities in stationary isotropic turbulence. A rigorous application of EDQNM, which introduces no new modelling assumptions or constants, is shown to yield a covariance spectrum that violates the Cauchy–Schwartz inequality over some of the wavenumbers. One approach to this problem is to derive a model based on a stochastic differential equation, as its presence guarantees realizability. For an isotropic scalar, it is possible to construct a Langevin equation for the Fourier transform of the scalar concentrations that is consistent with each EDQNM scalar autocorrelation spectrum. The Langevin equations can then be used to construct a model for the covariance spectrum that is realizable. However, the resulting covariance transfer term does not properly conserve the scalar covariance, and so the model is still not satisfactory. The problem can be traced to the Markovianization step, which leads to the presence of the scalar diffusivities in the transfer functions in an unphysical fashion. A simple fix is described which reconciles the two approaches and gives conservative, realizable results for all time.

Next, we apply the EDQNM theory to a more general system involving the mixing of anisotropic scalars. Anisotropy in this case results from a uniform mean gradient of the two scalar concentrations in one direction. As with the isotropic scalars, direct application of the EDQNM closure results in a covariance spectrum that violates the Cauchy–Schwartz inequality; however, in this case it is not as simple to construct a Langevin model that reproduces all of the spectral interactions that result from the EDQNM procedure. Nevertheless, we show that the same modification of the inverse time scale as is done for the isotropic scalar results in an anisotropic scalar covariance spectrum that is realizable for all times.

We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.

The thermal boundary layer near a heated vertical plate in a poorly conducting liquid is subject to a horizontal d.c. electric field. If the electric field is strong enough, the boundary layer becomes unstable. In this paper a theory is developed to predict the onset of this instability. Experiments measuring the threshold voltage for instability are compared with the theoretical predictions. Other experiments are reported which determine the effect of this instability on the heat transferred from the heated plate.

Large-scale particle-driven gravity currents occur in the atmosphere, often in the form of pyroclastic flows that result from explosive volcanic eruptions. The behaviour of these gravity currents is analysed here and it is shown that compressibility can be important in flow of such particle-laden gases because the presence of particles greatly reduces the density scale height, so that variations in density due to compressibility are significant over the thickness of the flow. A shallow-water model of the flow is developed, which incorporates the contribution of particles to the density and thermodynamics of the flow. Analytical similarity solutions and numerical solutions of the model equations are derived. The gas–particle mixture decompresses upon gravitational collapse and such flows have faster propagation speeds than incompressible currents of the same dimensions. Once a compressible current has spread sufficiently that its thickness is less than the density scale height it can be treated as incompressible. A simple ‘box-model’ approximation is developed to determine the effects of particle settling. The major effect is that a small amount of particle settling increases the density scale height of the particle-laden mixture and leads to a more rapid decompression of the current.

Inspired by recent calculations by Thoraval et al. (Phys. Rev. Lett., vol. 108, 2012, p. 264506) relating to droplet impact, this paper presents an analysis of the perturbations to the free surface caused by small surface tension and viscosity in steady Helmholtz flows. In particular, we identify the regimes in which appreciable vorticity can be shed from the boundary layer to the bulk flow.

An experimental and numerical study was conducted to examine the effects of internal surface curvature and leading-edge angle on the shock waves and steady flow fields produced by axisymmetric ring wedges. Test models with leading-edge-radius-normalised internal radii of curvature of $R_{c}=\{1,1.5,2\}$ and leading-edge angles of $\unicode[STIX]{x1D6FC}=\{0^{\circ },4^{\circ },8^{\circ }\}$ were manufactured and tested. Experimental shadowgraph and schlieren results were obtained for Mach numbers ranging from 2.8 to 3.6 using a blowdown supersonic wind tunnel with accompanying numerical results for additional insight. The higher the internal surface curvature and leading-edge angle, the greater the flow fields were impacted. As a result, steeper compression waves were formed, thus curving the shock wave more noticeably. The internal surface curvature and leading-edge angle were both found to have an effect on the trailing-edge expansion fans. This altered the shape of downstream shock wave structures. The highest curvature models produced steady double reflection patterns due to the imposed internal surface curvature. The effects of conical and curved internal surfaces were explored for the presence of flow-normal curvature and the curving of the attached shock waves.

The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically, the particular system of three stratified layers with two internal fluid–fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared with the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all of the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilization of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearized advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All of these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic–elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto–Sivashinsky-type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularize the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. Such phenomena may be of importance in mixing, mass and heat-transfer applications.

Oceanic internal waves can be decomposed into an infinite set of modes, and the dominant internal mode 1 waves have been extensively investigated. Although mode 2 waves have been observed, they have not received comparable attention, especially the generation mechanisms. In this work, we examine the generation of mode 2 internal waves by the interaction of mode 1 waves with topography. We use a coupled linear long-wave theory with mode coupling through topography, combined with evolution using a Korteweg–de Vries model, to predict the mode 2 wave amplitude, in an ideal three-layer fluid model, in a smooth density stratification and in two realistic oceanic settings. We find that the mode 2 wave amplitude is usually much smaller than the incident mode 1 wave amplitude and is quite sensitive to the pycnocline thickness, topographic slope and background stratification.

We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier–Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier–Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time alone, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: $\D$, the amplitude of the oscillation, and $R$, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small $R$ the flow is synchronous with the forcing and as $R$ increases a Hopf bifurcation takes place. Subsequent dynamics, as $R$ increases, depend on the value of $\Delta$. For small $\Delta$ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring – this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate $\Delta$, the Hopf bifurcation is either quasi-periodic (for the smaller $\Delta$) or subharmonic (for larger $\Delta$), and the solutions tend to a chaotic attractor at sufficiently large $R$; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of $\Delta$, we find that the solution remains time periodic as $R$ increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large $\Delta$.

Numerical methods are discussed for computing the pressure inside a two- or three-dimensional inviscid bubble with negligible density suspended in Stokes flow, subject to a specified rate of expansion. In the case of flow past a solitary two- or three-dimensional bubble, the bubble pressure is found by solving an integral equation of the first kind for the normal derivative of the pressure on the side of the liquid over the free surface, while requiring that the pressure field decays at a rate that is faster than the potential due to a point source. In another approach, an explicit expression for the bubble pressure is derived by applying the reciprocal theorem for the flow around the bubble and the flow due to a point source situated inside the bubble. In the case of flow past, or due to the expansion or shrinkage of, a periodic lattice of bubbles, the bubble pressure is found by solving an integral equation of the second kind for the density of an interfacial distribution of point-source dipoles, while ensuring existence and uniqueness of solution by spectrum deflation. The new methods considerably simplify the computation of the bubble pressure by circumventing the evaluation of the finite part of hypersingular integrals. Results of numerical simulations illustrate the pressure developing inside a solitary two- and three-dimensional incompressible bubble suspended in simple shear flow, and the pressure developing inside a doubly periodic array of gaseous inclusions representing shrinking pores trapped in a sintered medium.

An approximate binary collision procedure is devised for the loaded-sphere molecular model and is used with Bird's direct simulation technique for computer experiments on diatomic gases. The collision procedure is tested by simulating the approach to equilibrium of a constant-speed gas of loaded spheres and good agreement with Jeans's (1904) theoretical prediction is obtained. The procedure is then used to study the structure of normal shock waves in the diatomic gas, giving mean velocity, density and temperature profiles. It is found that the present procedure consumes considerably more computer time than comparable simulations of monatomic gases.

Steady, spatial, algebraically growing eigenfunctions are now known to occur in several important classes of boundary-layer flow, including two-dimensional hypersonic boundary layers and more recently in Blasius boundary layers subject to three-dimensional linearized disturbances, and in more general three-dimensional boundary layers. These spatial eigensolutions are particularly important and intriguing, given that they exist within the broad limits of the classical steady boundary-layer approximation, and as such are independent of Reynolds number.

In this paper we make the natural extension to these previous (stability) analyses by incorporating the effects of unsteadiness into the model for treating disturbances to a quite general class of similarity-type boundary-layer flows. The flow disturbances are inherently non-parallel, but this effect is properly incorporated into the analysis.

A further motivation for this paper is that Duck et al. (1999, 2000) have shown that by permitting a spanwise component of flow within a boundary layer of the appropriate form (in particular, growing linearly with the spanwise coordinate), it is found that new families of solutions exist – even the Blasius boundary layer has a three-dimensional ‘cousin’. Therefore a further aim of this paper is to assess the stability of the different solution branches, using the ideas introduced in this paper, to give some clues as to which of the solutions may be encountered experimentally.

Several numerical methods are presented for tackling various aspects of the problem. It is shown that when algebraically growing, steady eigensolutions exist, their effect remains important in the unsteady context. We show how even infinitesimal, unsteady flow perturbations can provoke extremely large-amplitude flow responses, including in some cases truly unstable flow disturbances which grow algebraically downstream without bound in the linear context. There are some interesting parallels suggested therefore regarding mechanisms perhaps linked to bypass transition in an important class of boundary-layer flows.

The asymptotic behaviour for large Rayleigh number and Prandtl number of O(1) of two-dimensional convection cells in a fluid between horizontal plates heated from below has been discussed by Pillow (1952) and more recently by Robinson (1967). The flow models derived by Pillow and Robinson differ from each other. The purpose of the present paper is to point out the likelihood of an inconsistency in Robinson's flow model, and to discuss qualitatively a few hitherto unnoticed features of the flow.