Capillarity is an important feature in controlling the spreading of liquid drops and in the coating of substrates by liquid films. For thin films and small contact angles, lubrication theory enables the analysis of the motion to be reduced to single evolution equations for the heights of the drops or films, provided the inertia of the liquid can be neglected. In general, the presence of inertia destroys the major simplification provided by lubrication theory, but two special cases that can be treated are identified here. In the first example, the approach of a drop to its equilibrium position is studied. For sufficiently low Reynolds numbers, the rate of approach to the terminal state and the contact angle are slightly reduced by inertia, but, above a critical Reynolds number, the approach becomes oscillatory. In the latter case there is no simple relation connecting the dynamic contact angle and contact-line speed. In the second example, the spreading drop is supported by a plate that is forced to oscillate in its own plane. For the parameter range considered, the mean spreading is unaffected by inertia, but the oscillatory motion of the contact line is reduced in magnitude as inertia increases, and the drop lags behind the plate motion. The oscillatory contact angle increases with inertia, but is not in phase with the plate oscillation.

Turbulent convection induced by heating the bottom boundary of a horizontally homogeneous, linearly (temperature) stratified, rotating fluid layer is studied using a series of laboratory experiments. It is shown that the growth of the convective mixed layer is dynamically affected by background rotation (or Coriolis forces) when the parameter R = (h2Ω3/q0)2/3 exceeds a critical value of Rc ≈ 275. Here h is the depth of the convective layer, Ω is the rate of rotation, and q0 is the buoyancy flux at the bottom boundary. At larger R, the buoyancy gradient in the mixed layer appears to scale as (db/dz)ml = CΩ2, where C ≈ 0.02. Conversely, when R < Rc, the buoyancy gradient is independent of Ω and approaches that of the non-rotating case. The entrainment velocity, ue, for R > Rc was found to be dependent on Ω according to E = [Ri(1 + CΩ2/N2)]−1, where E is the entrainment coefficient based on the convective velocity w∗ = (q0h)1/3, E = ue/w∗, Ri is the Richardson number Ri = N2h2/w2∗, and N is the buoyancy frequency of the overlying stratified layer. The results indicate that entrainment in this case is dominated by non-penetrative convection, although the convective plumes can penetrate the interface in the form of lenticular protrusions.

Particle image velocimetry has been applied to the study of a canonical turbulent boundary layer and to a turbulent boundary layer forced by transversal wall oscillations. This work is part of the research programme at the Politecnico di Torino aerodynamic laboratory with the objective of investigating the response of near-wall turbulence to external perturbations. Results are presented for the optimum oscillation period of 100 viscous time units and for an oscillation amplitude of 320 viscous units. As expected, turbulent velocity fluctuations are considerably reduced by the wall oscillations. Particle image velocimetry has allowed comparisons between the canonical and forced flows in an attempt to find the physical mechanisms by which the wall oscillation influences the near-wall organized motions.

We consider the weakly nonlinear evolution of the Faraday waves produced in a vertically vibrated two-dimensional liquid layer, at small viscosity. It is seen that the surface wave evolves to a drifting standing wave, namely a wave that is standing in a moving reference frame. This wave is determined up to a spatial phase, whose calculation requires consideration of the associated mean flow. This is just the streaming flow generated in the boundary layer attached to the lower plate supporting the liquid. A system of equations is derived for the coupled slow evolution of the spatial phase and the streaming flow. These equations are numerically integrated to show that the simplest reflection symmetric steady state (the usual array of counter-rotating eddies below the surface wave) becomes unstable for realistic values of the parameters. The new states include limit cycles (the array of eddies oscillating laterally), drifted standing waves (patterns that are standing in a uniformly propagating reference frame) and some more complex attractors.

We develop an analysis of the two-dimensional cascade of a tracer (passive or active Lagrangian-conserved scalar), locally in space and time, and establish connections with the modelling of turbulent mixing. We define a local scale-to-scale flux of tracer variance based on the dynamics of tracer increments. This flux reduces at small scales to the production or destruction rate of tracer gradients by stirring as a function of their local orientation with respect to the compressional axis of the strain-rate tensor. The local detailed budget of tracer variance on which this approach is based is compared to the global statistical budget expressed by Yaglom's equation. The spatial pattern of the local transfers produced by a numerical simulation as well as their statistical distribution are discussed.

We then address the problem of the parameterization of turbulent mixing. We consider an anisotropic tensor diffusivity proportional to the velocity gradients. In this model the tracer dissipation involves the axes of the strain-rate tensor and we shall refer to it as strain diffusivity. We show analytically that it locally matches the scale-to-scale flux through the cutoff scale. This matching is studied numerically in decaying two-dimensional turbulence. A comparison is made with eddy diffusivity and hyperdiffusivity. The presence of a numerical instability and ways to suppress it are discussed from numerical and fundamental points of view.

We consider the special case of vorticity, an active scalar in two dimensions. When applied to vorticity, models affect the energy budget. The two-dimensional inverse energy cascade requires that parameterizations conserve energy and we show that strain diffusivity conserves energy. We finally study the sensitivity of the large scales of the flow to the operator used on vorticity in forced stationary simulations. Strain diffusivity is found to produce more realistic spectral features than hyperviscosity.

Surface coating is generally achieved by an active operation: painting with a brush; withdrawing from a bath. Porous imbibition constitutes a more passive way: a porous material just put in contact with a reservoir containing a wetting fluid is spontaneously invaded. In this case, the material can eventually be filled by the fluid. If the aim is to coat only the surface of the pores, the liquid excess must be actively removed. We present an experiment in which a liquid train spontaneously moves in a capillary tube because of the trail it leaves behind. From a practical point of view, this system achieves a coating of the tube. The condition required for this motion and a model for its dynamics are presented. We also show how these trains can drive extremely viscous liquid slugs (or even solid bodies) in narrow tubes. We discuss the thickness of the deposited films. Extensions and limits of this system in more complex geometries are finally described, together with special cases such as the deposition of solid films.

The optimal distribution of steady suction needed to control the growth of single or multiple disturbances in quasi-three-dimensional incompressible boundary layers on a flat plate is investigated. The evolution of disturbances is analysed in the framework of the parabolized stability equations (PSE). A gradient-based optimization procedure is used and the gradients are evaluated using the adjoint of the parabolized stability equations (APSE) and the adjoint of the boundary layer equations (ABLE). The accuracy of the gradient is increased by introducing a stabilization procedure for the PSE. Results show that a suction peak appears in the upstream part of the suction region for optimal control of Tollmien–Schlichting (T–S) waves, steady streamwise streaks in a two-dimensional boundary layer and oblique waves in a quasi-three-dimensional boundary layer subject to an adverse pressure gradient. The mean flow modifications due to suction are shown to have a stabilizing effect similar to that of a favourable pressure gradient. It is also shown that the optimal suction distribution for the disturbance of interest reduces the growth rate of other perturbations. Results for control of a steady cross-flow mode in a three-dimensional boundary layer subject to a favourable pressure gradient show that not even large amounts of suction can completely stabilize the disturbance.

We consider the behaviour of a passive tracer in multiscale velocity field, when there is no separation of scales; the energy spectrum of the velocity field extends into the region of long waves and even can be singular there. We suppose that the velocity field is a superposition of random waves. The turbulence of various ocean or atmospheric waves provides examples. We find anomalous diffusion (sub- and super-diffusion), anomalous drift (super-drift), and anomalous spreading of a passive tracer cloud. For the latter we find the existence of two regimes: (i) ‘close’ passive tracer particles diverge sub- or supper-exponentially in time, and (ii) a ‘large’ passive tracer cloud spreads as a power-law in time. The exponents, as well as the corresponding pre-factors, are found. The theory is confirmed by numerical simulations.

The Lagrangian averaged Navier–Stokes–alpha (LANS-α) model of turbulence is found to possess a Kármán–Howarth (KH) theorem for the dynamics of its second-order autocorrelation functions in homogeneous isotropic turbulence. This KH result implies that alpha-filtering in the LANS-α model of turbulence does not affect the exact Navier–Stokes relation between second and third moments at separation distances large compared to the model's length scale α. Moreover, at separations r that are smaller than α, the KH scaling between energy dissipation rate and longitudinal third-order autocorrelation changes to match the scaling found in two-dimensional incompressible flow. This is consistent with the corresponding change in scaling of the kinetic energy spectrum from k−5/3 for larger scales with kα < 1, which switches to k−3 for smaller scales with kα > 1, as discovered in Foias, Holm & Titi (2001).

Small perturbations of a choked flow through a thin annular nozzle are investigated. Two cases are considered, corresponding to a ‘choked outlet’ and a ‘choked inlet’ respectively. For the first case, either an acoustic or entropy or vorticity wave is assumed to be travelling downstream towards the nozzle contraction. An asymptotic analysis for low frequency is used to find the reflected acoustic wave that is created. The boundary condition found by Marble & Candel (1977) for a compact choked nozzle is shown to apply to first order, even for circumferentially varying waves. The next-order correction can be expressed as an ‘effective length’ dependent on the mean flow (and hence the particular geometry of the nozzle) in a quantifiable way.

For the second case, an acoustic wave propagates upstream and is reflected from a convergent–divergent nozzle. A normal shock is assumed to be present. By considering the interaction of the shock's position and flow perturbations, the reflected propagating waves are found for a compact nozzle. It is shown that a significant entropy disturbance is produced even when the shock is weak, and that for circumferential modes a vorticity wave is also present. Numerical calculations are conducted using a sample geometry and good agreement with the analysis is found at low frequency in both cases, and the range of validity of the asymptotic theory is determined.

We consider the dynamics of convection in a strong vertical magnetic field, and in the presence of rapid rotation. In both these cases, in circumstances which can be realized in the laboratory, the onset of convection is in the form of tall thin cells. Because of this, the dynamics near onset is characterized by an interaction between the cellular modes and the horizontally averaged temperature profile. The effects on the dynamics are slight in the case of a Boussinesq fluid. However in both cases, when the layer is stratified (non-Boussinesq), the convection can lose stability to oscillations close to onset. Properties of the oscillations and their stability to long-wavelength modulation are extensively investigated.

A nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented. The one-dimensional weakly nonlinear equation for shallow water wave propagation in a uniform channel with a floating ice cover is derived. The ice cover is assumed to be a thin uniform elastic plate. The weakly nonlinear equation is a fifth-order KdV equation. Analytical solutions of the nonlinear periodic wave equation are obtained. These solutions show that the shape, wavelength and celerity of the nonlinear waves depend on the wave amplitude. The wave celerity is slightly smaller than the open water wave celerity. The wavelength decreases as the wave amplitude increases. Based on these solutions the fracture of the ice cover is analysed. The spacing between transverse cracks varies form 50 m to a few hundred metres with the corresponding wave amplitude varying from 0.2 to 0.8 m, depending on the thickness and strength of the cover. These results agree well with limited field observations.

A simple model is developed, based on an approximation of the Boussinesq equation, that considers the weakly nonlinear evolution of an initial interface disturbance in a closed basin. The solution consists of the sum of the solutions of two independent Korteweg–de Vries (KdV) equations (one along each characteristic) and a second-order wave–wave interaction term. It is demonstrated that the solutions of the two independent KdV equations over the basin length [0, L] can be obtained by the integration of a single KdV equation over the extended reflected domain [0, 2L]. The main effect of the second-order correction is to introduce a phase shift to the sum of the KdV solutions where they overlap. The results of model simulations are shown to compare qualitatively well with laboratory experiments. It is shown that, provided the damping timescale is slower than the steepening timescale, any initial displacement of the interface in a closed basin will generate three types of internal waves: a packet of solitary waves, a dispersive long wave and a train of dispersive oscillatory waves.

Bounds on the bulk rate of energy dissipation in body-force-driven steady-state turbulence are derived directly from the incompressible Navier–Stokes equations. We consider flows in three spatial dimensions in the absence of boundaries and derive rigorous a priori estimates for the time-averaged energy dissipation rate per unit mass, ε, without making any further assumptions on the flows or turbulent fluctuations. We prove

ε [les ] c1vU2/l2 + c2U3/l,

where v is the kinematic viscosity, U is the root-mean-square (space and time averaged) velocity, and l is the longest length scale in the applied forcing function. The prefactors c1 and c2 depend only on the functional shape of the body force and not on its magnitude or any other length scales in the force, the domain or the flow. We also derive a new lower bound on ε in terms of the magnitude of the driving force F. For large Grashof number Gr = Fl3/v2, we find

c3vFl/λ2 [les ] ε

where λ = √vU2/ε is the Taylor microscale in the flow and the coefficient c3 depends only on the shape of the body force. This estimate is seen to be sharp for particular forcing functions producing steady flows with λ/l ∼ O(1) as Gr → 1. We interpret both the upper and lower bounds on ε in terms of the conventional scaling theory of turbulence – where they are seen to be saturated – and discuss them in the context of experiments and direct numerical simulations.

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the values of the parameters. We classify all distinguished limits for varying values of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with (a) the numerical solution of the original exact problem, (b) some ad hoc approximations in the literature, and (c) some experimental results.

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

In a recent article (Forterre & Pouliquen 2001), we have reported a new instability observed in rapid granular flows down inclined planes that leads to the spontaneous formation of longitudinal vortices. From the experimental observations, we have proposed an instability mechanism based on the coupling between the flow and the granular temperature in rapid granular flows. In order to investigate the relevance of the proposed mechanism, we perform in the present paper a three-dimensional linear stability analysis of steady uniform flows down inclined planes using the kinetic theory of granular flows. We show that in a wide range of parameters, steady uniform flows are unstable under transverse perturbations. The structure of the unstable modes is in qualitative agreement with the experimental observations. This theoretical analysis shows that the kinetic theory is able to capture the formation of longitudinal vortices and validates the instability mechanism.

The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes before merging. Two parameters are varied: a steepness parameter n which measures the steepness of the initial vorticity profiles in a given family of profiles, and the Reynolds number Re (between 500 and 16 000). Two distinct relaxation processes are identified. The first one is non-viscous and corresponds to a rapid adaptation of each vortex to the external (strain) field generated by the other vortex. This adaptation process, which is profile dependent, is described and explained using the damped Kelvin modes of each vortex. The second relaxation process is a slow diffusion phenomenon. It is similar to the relaxation of any non-Gaussian axisymmetrical vortex towards the Gaussian. The quasi-stationary solution evolves on a viscous-time scale toward a single attractive solution which corresponds to the evolution from two initially Gaussian vortices. The attractive solution is analysed in detail up to the merging threshold a/b ≈ 0.22 where a and b are the vortex radius and the separation distance respectively. The vortex core deformations are quantified and compared to those induced by a single vortex in a rotating strain field. A good agreement with the asymptotic predictions is demonstrated for the eccentricity of vortex core streamlines. A weak anomalous Reynolds number dependence of the solution is also identified. This dependence is attributed to the advection–diffusion of vorticity towards the hyperbolic points of the system and across the separatrix connecting these points. A Re1/3 scaling for the vorticity at the central hyperbolic point is obtained. These findings are discussed in the context of a vortex merging criterion.