The maximal steady frictional exchange flow through a rectangular channel of constant width is analysed within the context of internal hydraulics. A one-dimensional analytical solution to the fully nonlinear problem of two-layer frictional exchange is developed and shown to compare well with experimental and field data. The analytical solution gives the maximal exchange flow rate and the variation in the height of the density interface along the channel for the case of zero barotropic forcing. In contrast to the assumed constant interface slope of previous theoretical formulations of frictional exchange flows, the resulting density interface is found to be nonlinear and asymmetric. Both interfacial and bottom friction play important roles in determining the exchange flow rate. It is shown that the frictional effects are important even in relatively short channels.

Linear waves in bounded inviscid fluids do not generally form normal modes with regular eigenfunctions. Examples are provided by inertial waves in a rotating fluid contained in a spherical annulus, and internal gravity waves in a stratified fluid contained in a tank with a non-rectangular cross-section. For wave frequencies in the ranges of interest, the inviscid linearized equations are spatially hyperbolic and their characteristic rays are typically focused onto wave attractors. When these systems experience periodic forcing, for example of tidal origin, the response of the fluid can become localized in the neighbourhood of a wave attractor. In this paper, I define a prototypical problem of this form and construct analytically the long-term response to a periodic body force in the asymptotic limit of small viscosity. The vorticity of the fluid is localized in a detached shear layer close to the wave attractor in such a way that the total rate of dissipation of energy is asymptotically independent of the viscosity. I further demonstrate that the same asymptotic dissipation rate is obtained if a non-viscous damping force is substituted for the Navier–Stokes viscosity. I discuss the application of these results to the problem of tidal forcing in giant planets and stars, where the excitation and dissipation of inertial waves may make a dominant, or at least important, contribution to the orbital and spin evolution.

The interaction of vortex shedding from a circular cylinder with an inflow which has low-amplitude periodic velocity oscillations (perturbations) superimposed upon it, was investigated experimentally by means of particle image velocimetry. The experiments were made at three perturbation frequencies across the lock-on range in which the vortex shedding frequency is synchronized with the subharmonic of the imposed frequency. The basic wake pattern in this range is antisymmetric vortex shedding, i.e. the familiar 2S mode. The timing of vortex shedding is defined with respect to the cross-flow oscillation of the wake which is found to play a critical role. Quantitative analysis of the phase-referenced patterns of vorticity distribution in the wake shows that a vortex is actually shed from the cylinder when the cross-flow oscillation of the wake is strongest, marked by a sudden drop in the computed vortex strength. At the middle of the lock-on range, shedding occurs near the minimum inflow velocity in the cycle or, equivalently, during the forward stroke of a cylinder oscillating in-line with the flow. It is argued that the imposed timing of vortex shedding relative to the cylinder motion induces a negative excitation from the fluid, which might explain why the in-line response of a freely vibrating cylinder exhibits two positive excitation regions separated by the lock-on region found in forced oscillations.

Field-amplified sample stacking (FASS) uses conductivity gradients and resulting non-uniform electromigration fluxes to effect concentration increases of analyte ions. For cases where the initial sample concentration is much smaller than the background electrolyte (BGE) concentration, the ideal maximum concentration enhancement is equal to $\gamma $, the ratio of conductivity of the sample solution to that of the BGE. However, in practice both molecular diffusion and convective dispersion limit concentration enhancement. We present a theoretical and experimental study of concentration enhancement using FASS. We model the FASS process as electromigration, diffusion, and advection of two background electrolyte ions and multiple sample species across a known initial concentration gradient. Regular perturbation methods and a generalized Taylor dispersion analysis are used to derive area-averaged species conservation and electric field equations. The model predicts the spatial and temporal development of background electrolyte concentration field, electric field, and sample-ion distribution of the FASS process. We have validated this model using on-chip FASS experiments. We use an acidified poly(ethylene oxide) (PEO) coating to minimize dispersion due to electro-osmotic flow (EOF), and thereby evaluate the low- (but finite) dispersion regime of most interest. We have used CCD-based quantitative epifliuorescence imaging to quantify unsteady concentration fields and validate the model. This experimentally validated model is useful in developing optimal designs of sample stacking assay devices.

We have performed direct numerical simulations for a microbubble-laden spatially developing turbulent boundary layer (SDTBL) and compared the amount of skin friction reduction due to the presence of the bubbles for two Reynolds numbers: $Re_{\theta}{=}1430$ and $Re_{\theta}{=}2900$. The results show that increasing the Reynolds number decreases the percentage of drag reduction. Increasing $Re_{\theta}$ ‘squeezes’ the quasi-streamwise vortical structures toward the wall, whereas the microbubbles ‘push them away’ from the wall. The net result of these two opposing effects determines the amount of skin friction reduction by the microbubbles. The displacement of the vortical structures by the microbubbles is a result of the local positive velocity divergence, $\bm\nabla {\bm\cdot} {\bm U}$, created by the bubble concentration gradients. Thus, the volume fraction of bubbles that is responsible for the reduction of skin friction in a SDTBL at a given Reynolds number is not sufficient to produce the same amount of reduction in skin friction at higher Reynolds number.

Laboratory experiments are carried out to determine the nature of internal wave breaking and the limiting wave steepness for progressive, periodic, lowest-mode internal waves in a two-layer, miscible density stratification. Shoaling effects are not considered. The waves investigated here are long relative to the thickness of the density interface separating the two fluid layers. Planar laser-induced fluoresence (PLIF) flow visualization shows that wave breaking most closely resembles a Kelvin–Helmholtz shear instability originating in the high-shear wave crest and trough regions. However, this instability is strongly temporally and spatially modified by the oscillations of the driving wave shear. Unlike a steady stratified shear layer, the wave instability discussed here is not governed by the canonical $\it Ri{=}1/4$ stability limit. Instead, the wave time scale (the time scale of the destabilizing shear) imposes an additional constraint on instability, lowering the critical Richardson number below 1/4. Experiments were carried out to quantify this instability threshold, and show that, for the range of wavenumbers considered in this study, the critical wave steepness at which the wave breaking occurs is wavenumber-dependent (unlike surface waves). The corresponding critical wave Richardson numbers at incipient wave breaking are well below 1/4, in consonance with a modified instability analysis based on results from stratified shear flow instability theory.

We first report a quantitative experimental study of the collision of a spinning disk with water, from a single to many skips. We then focus on the high spin limit and propose a simple model which enables us to discuss both the physical origin of the bounces and the source of the dissipation which fixes the number of skips.

The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.

The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. The evolution of the film is governed by a lubrication equation coupled with the Laplace problem associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to diffusion-limited evaporation for which the gas phase is saturated in vapour in the vicinity of the interface. The stability depends then only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilizing whereas capillarity and evaporation are stabilizing processes. The results of the linear stability analysis are compared with the experiments of Poulard et al. (2003) performed in a different geometry. In order to study the resulting patterns, an amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem affects the amplitude equation unusually.

Flow in a closed-loop thermosyphon heated from below exhibits a sequence of bifurcations with increasing Grashof number. Using the Navier–Stokes equations in the Boussinesq approximation we have derived a model where, in the case of a slender circular loop, the first Fourier modes exactly decouple from all other Fourier modes, leaving a system of three coupled nonlinear partial differential equations that completely describes the flow in the thermosyphon. We have characterized the flow through two bifurcations, identifying stable periodic solutions for flows of Prandtl number greater than 18.5, a much lower value than predicted previously. Because of the quadratic nonlinearity in this system of equations, it is possible to find the global stability limit, and we have proved that it is identical to the first bifurcation point.

The numerical study of the model equations is based on a highly accurate Fourier–Chebyshev spectral method, combined with asymptotic analysis at the various bifurcation points. Three-dimensional computations with a finite element method computational fluid dynamics code (MPSalsa), are also pursued. All three approaches are in close agreement.

We develop asymptotic solutions for passive suspended sediment transport under a flood surge on a uniform slope. Our solutions provide predictions of the net scour under a surge, and simple estimates of the conditions under which it may ‘bulk up’ into a mud or debris flow, as well as illustrating their sensitivity to sediment entrainment rates.

A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some ‘from deep to shore’ computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in Chamberlain & Porter (1995). When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1.

It is also shown that the dominant asymptotic analysis, responsible for the new formula, is valid for all slope angles $\alpha{<}\pi/2$ and not just the special angles considered by Friedrichs.

In this paper we discuss numerical simulations of the generation of large-amplitude solitary waves in a continuously stratified fluid by flow over isolated topography. We employ the fully nonlinear theory for internal solitary waves to classify the numerical results for mode-1 waves and compare with two classes of approximate theories, weakly nonlinear theory leading to the Korteweg–deVries and Gardner equations and conjugate flow theory which makes no approximation with respect to nonlinearity, but neglects dispersion entirely. We find that both weakly nonlinear theories have a limited range of applicability. In contrast, the conjugate flow theory predicts the nature of the limiting upstream propagating response (a dissipationless bore), successfully describes the bore's vertical structure, and gives a value of the inflow speed, $c_j$, above which no upstream propagating response is possible. The numerical experiments demonstrate the existence of a class of large-amplitude response structures that are generated and trapped over the topography when the inflow speed exceeds $c_j$. While similar in structure to fully nonlinear solitary waves, these trapped disturbances can induce isopycnal displacements more than 100% larger than those induced by the limiting solitary wave while remaining laminar. We develop a theory to describe the vertical structure at the crest of these trapped disturbances and describe its range of validity. Finally, we turn to the generation of mode-2 solitary-like waves. Mode-2 waves cannot be truly solitary owing to the existence of a small mode-1 tail that radiates energy downstream from the wave. We demonstrate that, for stratifications dominated by a single pycnocline, mode-2 wave dissipation is dominated by wave breaking as opposed to mode-1 wave radiation. We propose a phenomenological criterion based on weakly nonlinear theory to test whether mode-2 wave generation is to be expected for a given stratification.

The release from a point source of relatively heavy fluid into a saturated porous medium above an impermeable boundary is considered. A theoretical relationship is compared with experimental data for the rate of propagation of the front of the resulting gravity current and its shape. A motivation of the study, the problem of carbon dioxide sequestration, is briefly discussed.

Oblique impact of single drops on deep fluids at various impact angles $\alpha $ is studied for a wide Weber number range (15$\,{\le}\,$We$\,{\le}\,$249), where $\alpha $ is varied between 5.4$^\circ$ and 64.4$^\circ$ with respect to the target surface, and drop and target consist of the same fluid (water/glycerol). In the range of studied parameters, coalescence always takes place. Below a critical level of the Weber number We$_{N}$ formed with the normal velocity component, We$_{N}\,{<}\, 10$, capillary waves are produced on the drop and target surface in the first stage after impact. For We$_{N}\,{>}\, 10$, a lamella is ejected from the target surface and drop volume for which both experimental and numerical evidence is given. For We$\,{<}\, 140$ and $\alpha \,{<}\, 23^\circ$, the drop spreads on the target surface with no visible immersion of drop fluid into the target fluid. Outside this range, a partial immersion of drop fluid can be seen: for We$_{N}\,{>}\,$10, it occurs in the front part of the spreading drop; for $\alpha \,{>}\,$23$^\circ$, in the rear part of the impact crater. The spreading patterns of the drop fluid are visualized with dyed drop fluid once it has come to rest after impact. In many cases, patterns of high complexity and strong sensitivity to the impact conditions can be seen which are very reproducible for similar impact conditions. The mechanisms underlying capillary wave or lamella formation and the partial immersions of drop fluid are discussed and their relevance for pattern formation is taken into account.

A linearized analysis is made of the canonical problem of sound production during the convection of a line vortex of strength $\Gamma$ in steady flow of water over a ‘fence’ of height $h$ on a flat wall in the presence of a vacuous cavity in the wake of the fence. The cavity is assumed to extend sufficiently far downstream for sound waves to be regarded as launched above a non-compact, pressure-release surface. Additional vorticity is released from the tip of the fence in accordance with the Kutta condition, and is convected at the mean stream speed $U$ along the free streamline boundary of the cavity. Sound pressures of opposite phases are generated by the incident and the shed vorticity. The predicted radiation consists of a pressure pulse of amplitude proportional to $\rho_oU\Gamma/h$ ($\rho_o$ being the mean water density) and width approximately equal to fence height/mean flow Mach number, produced as the vortex passes the tip of the fence. The acoustic amplitude decreases rapidly at later times because of destructive interference between the sound generated by the impinging vortex and the shed vorticity.