Simultaneous capillary–gravity solitary waves (simultons or quadratic solitons) are shown to be possible in a rectangular liquid channel of arbitrary finite depth bounded below by a solid plate and above with a free deformable surface with constant surface tension. A second-harmonic resonance between two waveguide modes (fundamental and second-harmonic waves) is studied with the inclusion of dispersion in the system. The nonlinearly coupled amplitude equations for the two slowly varying envelopes of the fundamental and the second-harmonic wave components are derived using the method of multiple scales. Two types of capillary–gravity simulton solutions are explicitly obtained and an experiment for observing such hydrodynamic simultons is suggested.

Direct numerical simulations of the motion of up to 216 three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a deformable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is about 12–30, depending on the void fraction; deformations of the bubbles are small. Although the motion of the individual bubbles is unsteady, the simulations are carried out for a sufficient time that the average behaviour of the system is well defined. Simulations with different numbers of bubbles are used to explore the dependence of the statistical quantities on the size of the system. Examination of the microstructure of the bubbles reveals that the bubbles are dispersed approximately homogeneously through the flow field and that pairs of bubbles tend to align horizontally. The dependence of the statistical properties of the flow on the void fraction is analysed. The dispersion of the bubbles and the fluctuation characteristics, or ‘pseudo-turbulence’, of the liquid phase are examined in Part 2.

Direct numerical simulations of the motion of up to 216 three-dimensional buoyant bubbles in periodic domains are presented. The bubbles are nearly spherical and have a rise Reynolds number of about 20. The void fraction ranges from 2% to 24%. Part 1 analysed the rise velocity and the microstructure of the bubbles. This paper examines the fluctuation velocities and the dispersion of the bubbles and the ‘pseudo-turbulence’ of the liquid phase induced by the motion of the bubbles. It is found that the turbulent kinetic energy increases with void fraction and scales with the void fraction multiplied by the square of the average rise velocity of the bubbles. The vertical Reynolds stress is greater than the horizontal Reynolds stress, but the anisotropy decreases when the void fraction increases. The kinetic energy spectrum follows a power law with a slope of approximately −3.6 at high wavenumbers.

We examine surfactant transport on a highly viscous film overlying a much less viscous layer, aimed at situations within the smaller pulmonary airways, wherein the highly viscous film corresponds to the mucus layer, that rides atop the Periciliary liquid layer (PCL). To this end, we generate a variant of the lubrication approximation by promoting terms which would have otherwise been neglected within standard lubrication theory; this is reminiscent of theories involving free films, viscous jets and threads. We also account for the presence of van der Waals forces, which could rupture the thin bilayer fluid coating the small airways. This is a potential difficulty for surfactant replacement therapy (SRT) since rupture will leave behind pools of stranded surfactant thereby restricting the levels reaching the smallest airways and potentially leading to clinical failure. In the present study, the presence of the mucus for a wide range of system parameters using analytical techniques and numerical simulations.

This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.

A core–annular flow, the concurrent axial flow of two immiscible fluids in a circular tube or pore with one fluid in the core and the other in the wetting annular region, is frequently used to model technologically important flows, e.g. in liquid–liquid displacements in secondary oil recovery. Most of the existing literature assumes that the pores in which such flows occur are uniform circular cylinders, and examine the interfacial stability of such systems as a function of fluid and interfacial properties. Since real rock pores possess a more complex geometry, the companion paper examined the linear stability of core–annular flows in axisymmetric, corrugated pores in the limit of asymptotically weak corrugation. It found that short-wave disturbances that were stable in straight tubes could couple to the wall's periodicity to excite unstable long waves. In this paper, we follow the evolution of the axisymmetric, linearly unstable waves for fluids of equal densities in a corrugated tube into the weakly nonlinear regime. Here, we ask whether this continual generation of new disturbances by the coupling to the wall's periodicity can overcome the nonlinear saturation mechanism that relies on the nonlinear (kinematic-condition-derived) wave steepening of the Kuramoto–Sivashinsky (KS) equation. If it cannot, and the unstable waves still saturate, then do these additional excited waves make the KS solutions more likely to be chaotic, or does the dispersion introduced into the growth rate correction by capillarity serve to regularize otherwise chaotic motions?

We find that in the usual strong surface tension limit, the saturation mechanism of the KS mechanism remains able to saturate all disturbances. Moreover, an additional capillary-derived nonlinear term seems to favour regular travelling waves over chaos, and corrugation adds a temporal periodicity to the waves associated with their periodical traversing of the wall's crests and troughs. For even larger surface tensions, capillarity dominates over convection and a weakly nonlinear version of Hammond's no-flow equation results; this equation, with or without corrugation, suggests further growth. Finally, for a weaker surface tension, the leading-order base flow interface follows the wall's shape. The corrugation-derived excited waves appear able to push an otherwise regular travelling wave solution to KS to become chaotic, whereas its dispersive properties in this limit seem insufficiently strong to regularize chaotic motions.

We investigate hydrodynamic instability of a steady planar detonation wave propagating in a circular tube to three-dimensional linear perturbations, using the normal mode approach. Spinning instability is identified and its relevance to the well-known spin detonation is discussed. The neutral stability curves in the plane of heat release and activation energy exhibit bifurcations from low-frequency to high-frequency spinning modes as the heat release is increased at fixed activation energy. With a simple Arrhenius model for the heat release rate, a remarkable qualitative agreement with experiment is obtained with respect to the effects of dilution, initial pressure and tube diameter on the behaviour of spin detonation. The analysis contributes to the explanation of spin detonation which has essentially been absent since the discovery of the phenomenon over seventy years ago.

Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, ν, and thermal diffusivity, κ, are lowered to zero, with σ ≡ ν/κ fixed, then the energy dissipation per unit mass, κ, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number σ sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not ‘truly turbulent’ because ε→0 in the inviscid limit.

In a recent study, Sotiropoulos et al. (2001) studied numerically the chaotic particle paths in the interior of stationary vortex-breakdown bubbles that form in a closed cylindrical container with a rotating lid. Here we report the first experimental verification of these numerical findings along with new insights into the dynamics of vortex-breakdown bubbles. We visualize the Lagrangian transport within the bubbles using planar laser-induced fluorescence (LIF) and show that even though the flow fields are steady – from the Eulerian standpoint – the spatial distribution of the dye tracer varies continuously, and in a seemingly random manner, over very long observation intervals. This finding is consistent with the arbitrarily long šil'nikov transients of upstream-originating orbits documented numerically by Sotiropoulos et al. (2001). Sequences of instantaneous LIF images also show that the steady bubbles exchange fluid with the outer flow via random bursting events during which blobs of dye exit the bubble through the spiral-in saddle. We construct experimental Poincaré maps by time-averaging a sufficiently long sequence of instantaneous LIF images. Ergodic theory concepts (Mezić & Sotiropoulos 2002) can be used to formally show that the level sets of the resulting time-averaged light intensity field reveal the invariant sets (unmixed islands) of the flow. The experimental Poincaré maps are in good agreement with the numerical computations. We apply this method to visualize the dynamics in the interior of the vortex-breakdown bubble that forms in the wake of the first bubble for governing parameters in the steady, two-bubble regime. In striking contrast with the asymmetric image obtained for the first bubble, the time-averaged light intensity field for the second bubble is remarkably axisymmetric. Numerical computations confirm this finding and further reveal that the apparent axisymmetry of this bubble is due to the fact that orbits in its interior exhibit quasi-periodic dynamics. We argue that this stark contrast in dynamics should be attributed to differences in the swirl-to-axial velocity ratio in the vicinity of each bubble. By studying the bifurcations of a simple dynamical system, with manifold topology resembling that of a vortex-breakdown bubble, we show that sufficiently high swirl intensities can stabilize the chaotic orbits, leading to quasi-periodic dynamics.

The effect of gravity modulation on long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is analysed. The surfactants are assumed to be insoluble so that variations in their concentration along the free surface produce Marangoni flows in the underlying liquid. Lubrication theory is applied to obtain nonlinear partial differential equations that describe the behaviour of the free surface height and surfactant concentration, and the stability of these equations to small-amplitude disturbances is examined by applying Floquet theory. It is found that long-wavelength disturbances are destabilized by gravity modulation when surfactants are present, whereas such disturbances are stable when surfactants are absent. Results from additional calculations indicate that the instability becomes more difficult to excite as the Marangoni forces, body forces, capillary forces and surfactant diffusivity increase, and becomes easier to excite as the van der Waals forces increase.

Laser-induced cavitation bubbles near a curved rigid boundary are observed experimentally using high-speed photography. An image theory is applied to obtain information on global bubble motion while a boundary integral method is employed to gain a more detailed understanding of the behaviour of a liquid jet that threads a collapsing bubble, creating a toroidal bubble. Comparisons between the theory and experiment show that when a comparable sized bubble is located near a rigid boundary the bubble motion is significantly influenced by the surface curvature of the boundary, which is characterized by a parameter ζ, giving convex walls for ζ < 1, concave walls for ζ > 1 and a flat wall when ζ = 1. If a boundary is slightly concave, the most pronounced migration occurs at the first bubble collapse. The velocity of a liquid jet impacting on the far side of the bubble surface tends to increase with decreasing parameter ζ. In the case of a convex boundary, the jet velocity is larger than that generated in the flat boundary case. Although the situation considered here is restricted to axisymmetric motion without mean flow, this result suggests that higher pressures can occur when cavitation bubbles collapse near a non-flat boundary. Bubble separation, including the pinch-off phenomenon, is observed in the final stage of the collapse of a bubble, with the oblate shape at its maximum volume attached to the surface of a convex boundary, followed by bubble splitting which is responsible for further bubble proliferation.

We present the results of a combined theoretical and experimental investigation of laminar vertical jets impinging on a deep fluid reservoir. We consider the parameter regime where, in a pure water system, the jet is characterized by a stationary field of capillary waves at its base. When the reservoir is contaminated by surfactant, the base of the jet is void of capillary waves, cylindrical and quiescent: water enters the reservoir as if through a rigid pipe. A theoretical description of the resulting fluid pipe is deduced by matching extensional plug flow upstream of the pipe onto entry pipe flow within it. Theoretical predictions for the pipe height are found to be in excellent accord with our experimental results. An analogous theoretical description of the planar fluid pipe expected to arise on a falling fluid sheet is presented.

Theoretical arguments suggest that progressive gravity waves incident on a vertical wall can produce periodic standing waves only if the incident wave steepness ak is quite small, certainly less than 0.284. Laboratory experiments are carried out in which an incident wave train of almost uniform amplitude meets a vertical barrier. At wave steepnesses greater than 0.236 the resulting motion near the barrier is non-periodic. A growing instability is observed in which every third wave crest is steeper than its neighbours. The steep waves develop sharp crests, or vertical jets. The two neighbouring crests are rounded, at-topped, or of intermediate form. The instability grows by a factor of about 2.2 for every three wave periods, almost independently of the incident wave steepness.

This paper is a numerical analysis of the flow over a exible sail with the usual two-dimensional model of ideal weightless incompressible fluid. The sail is assumed to be impervious, inelastic and weightless, and may or may not be mounted on a mast. Separated or attached flows are considered at any angle of attack. Our method is validated by numerical and experimental results, i.e. the sail shape and velocity field are determined by particle imaging velocimetry, and lift and drag by aerodynamic balance. Despite the simplicity of the wake model we use (the Helmholtz model), the computed free streamline geometry and especially the sail shape are in good agreement with the experimental and numerical data.

The aim of this work is to investigate the tensorial filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of a magnetic field. The seepage law under a magnetic field is obtained by upscaling the flow at the pore scale. The macroscopic magnetic field and electric flux are also obtained. We use the method of multiple-scale expansions which gives rigorously the macroscopic behaviour without any preconditions on the form of the macroscopic equations. For finite Hartmann number, i.e. ε [Lt ] Ha [Lt ] ε−1, and finite load factor, i.e. ε [Lt ] [Kscr ] [Lt ] ε−1, where ε characterizes the separation of scales, the macroscopic mass flow and electric current are coupled and both depend on the macroscopic gradient of pressure and the electric field. The effective coefficients satisfy the Onsager relations. In particular, the filtration law is shown to resemble Darcy's law but with an additional term proportional to the electric field. The permeability tensor, which strongly depends on the magnetic induction, i.e. Hartmann number, is symmetric, positive and satisfies the filtration analogue of the Hall effect.

This study focuses on the effect of spatial non-uniformity in the ambient flow on the forces acting on a spherical particle at moderate particle Reynolds numbers. A scaling analysis is performed to obtain conditions under which such effects are important. A direct numerical simulation, based on spectral methods, is used to compute the three-dimensional time-dependent flow past a stationary sphere subject to a uniform flow plus a planar straining flow. The particle Reynolds number, Re, in the range 10 to 300 covering different flow regimes, from unseparated flow to unsteady vortex shedding, is considered. A variety of strain magnitudes and orientations are investigated. A systematic comparison with the potential flow results and axisymmetric strain results is given. Under elongational strain, both the planar and axisymmetric cases are found to stabilize the sphere wake and delay the onset of unsteadiness, while compressional strain leads to instability. In terms of separation angles, length of the recirculation eddy and topology of the surface streamlines, planar and axisymmetric strains yield nearly the same results. The drag force appears to have a linear relation with strain magnitude in both cases, as predicted by the potential flow. However, contrary to the potential flow results, the drag in planar strain is higher than that in axisymmetric strain. The generation of higher drag is explained using the surface pressure and vorticity distributions. Planar strain oriented at an angle with the oncoming uniform flow is observed to break the symmetry of the wake and results in a lift or side force. The variation of the drag and lift forces may be quite complex, and unlike the potential flow results they may not be monotonic with strain magnitude. The direction of the lift force may be opposite to that predicted by the inviscid and low Reynolds number (Re [Lt ] 1) theories. This behaviour is dictated by the presence or absence of a recirculation eddy. In the absence of a recirculation region at low Reynolds numbers (Re < 20), or at a very high strain magnitude when the recirculation region is suppressed, the results follow somewhat the pattern observed in potential flow. However, with the presence of a recirculation region, results opposite to those predicted by the potential theory are observed.