An electrokinetic model is proposed to describe a slight drop deformation which is induced by a weak external electric field. The fluids forming the system are considered Newtonian incompressible dielectric liquids containing free electric charge carriers. According to the model, the charge carriers take part in migration, diffusion and convection transport and there is no solute adsorption at the interface. Thermodynamic quasi-equilibrium at the interface is assumed for the charge carriers in the contacting liquids. The interfacial thermodynamic equilibrium is described using a common distribution coefficient for all the carriers. The problem is simplified by assuming equal diffusion coefficients for the different charge carriers within the same liquid. An analytical expression is obtained for slight drop deformation which is proportional to the second power of the applied field strength magnitude. The expression derived represents the drop deformation as a function of the parameters employed in previous theories (O’Konski & Thacher 1953; Allan & Mason 1962; Taylor 1966) as well as two additional parameters. The additional parameters are the ratios of the drop radius to the Debye lengths of the outer and inner liquids, respectively. The expression obtained for the drop deformation is valid for arbitrary values of these parameters. According to the theory prediction, with an increase in the drop radius, the drop deformation monotonically changes from that obtained by O’Konski & Thacher (1953) and Allan & Mason (1962) for perfect dielectric liquids to that obtained by Taylor (1966) for leaky dielectric liquids. Two simplified versions of the general expression are suggested to describe particular cases of a conducting drop in a perfect dielectric liquid and of a perfect dielectric drop in a conducting liquid.

The stability of fluid flow past a membrane of infinitesimal thickness is analysed in the limit of zero Reynolds number using linear and weakly nonlinear analyses. The system consists of two Newtonian fluids of thickness R* and HR*, separated by an infinitesimally thick membrane, which is flat in the unperturbed state. The dynamics of the membrane is described by its normal displacement from the flat state, as well as a surface displacement field which provides the displacement of material points from their steady-state positions due to the tangential stress exerted by the fluid flow. The surface stress in the membrane (force per unit length) contains an elastic component proportional to the strain along the surface of the membrane, and a viscous component proportional to the strain rate. The linear analysis reveals that the fluctuations become unstable in the long-wave (α→0) limit when the non-dimensional strain rate in the fluid exceeds a critical value Λt, and this critical value increases proportional to α2 in this limit. Here, α is the dimensionless wavenumber of the perturbations scaled by the inverse of the fluid thickness R*−1, and the dimensionless strain rate is given by Λt = (γ˙*R*η*/Γ*), where η* is the fluid viscosity, Γ* is the tension of the membrane and γ˙* is the strain rate in the fluid. The weakly nonlinear stability analysis shows that perturbations are supercritically stable in the α→0 limit.

We present the results of an integrated experimental, numerical and theoretical examination of spin-up from rest of a stratified fluid. A vertical cylindrical container of radius R and height 2H containing fluid of viscosity ν and characterized by a constant buoyancy frequency N is set impulsively to rotate about its symmetry axis with angular speed Ω = f/2. The characteristic Ekman number E = ν/ΩR2 is small and the Schmidt number S = ν/Ds (where Ds is the diffusivity of salt) is large. The investigation is focused on elucidating the initial stage of spin-up, which is characterized by an axisymmetric circulation driven by nonlinear Ekman layers adjoining the horizontal boundaries. Fluid is drawn by the boundary layers from the stationary, stratified interior and transported into corner regions. It is shown that the corner regions are restricted to a height of approximately 0.3Rf/N from the horizontal boundaries, above which the fluid remains unperturbed apart from that spun up by diffusion of momentum from the sidewall boundary. Two distinct regions thus emerge: rotating corner regions, and a quiescent stratified core. After a time 1.3/(E1/2N), the corner regions cover the bulk of the horizontal boundaries and the boundary layer suction is suppressed. Our study provides a framework for understanding the subsequent evolution of the spin-up process, which may be characterized by axisymmetry-breaking instabilities of the stratified core.

The results of an experimental study of stability, receptivity and transition of the flat-plate laminar boundary layer at Mach 3 are discussed. With a relatively low free-stream disturbance level (∼0.1%), spectra, growth rates and amplitude distributions of naturally occurring boundary layer waves were measured using hot wires. Physical (mass-flux) amplitudes in the boundary layer and free stream are reported and provide stability and receptivity results against which predictions can be directly compared. Comparisons are made between measurements of growth rates of unstable high-frequency waves and theoretical predictions based on a non-parallel, mode-averaging stability theory and receptivity assumptions; good agreement is found. In contrast, it was found that linear stability theory does not account for the measured growth of low-frequency disturbances. A detailed investigation of the disturbance fields in the free stream and on the nozzle walls provides the basis for a discussion of the source and the development of the measured boundary layer waves. Attention is drawn to the close matching in streamwise wavelengths for instability waves and the free-stream acoustic disturbances. It was also found that a calibration of the hot wire in the free stream yields a double-peak boundary layer disturbance amplitude distribution, as has been found by previous investigators, which is not consistent with the predictions of linear stability theory. This double peak was found to be an experimental anomaly which resulted from assumptions that are frequently made in the free-stream calibration procedure. A single-peak amplitude distribution across the boundary layer was established only when the hot-wire voltage was calibrated against the mean boundary layer profile. Finally, the late stages of transition, at a higher Reynolds number with a higher free-stream disturbance level, were explored. Calibrated amplitude levels are provided at locations where nonlinearities are first detected and where the mean boundary layer profile is first observed to depart from the laminar similarity solution. A qualitative discussion of the character of ensuing nonlinearities is also included.

A nonlinear equation based on the hydrodynamic equations is solved analytically using perturbation expansions to calculate the flow field of a steady isothermal, compressible and laminar gas flow in either a circular or a planar microchannel. The solution takes into account slip-flow effects explicitly by utilizing the classical velocity-slip boundary condition, assuming the gas properties are known. Consistent expansions provide not only the cross-stream but also the streamwise evolution of the various flow parameters of interest, such as pressure, density and Mach number. The slip-flow effect enters the solution explicitly as a zero-order correction comparable to, though smaller than, the compressible effect. The theoretical calculations are verified in an experimental study of pressure-driven gas flow in a long microchannel of sub-micron height. Standard micromachining techniques were utilized to fabricate the microchannel, with integral pressure microsensors based on the piezoresistivity principle of operation. The integrated microsystem allows accurate measurements of mass flow rates and pressure distributions along the microchannel. Nitrogen, helium and argon were used as the working fluids forced through the microchannel. The experimental results support the theoretical calculations in finding that acceleration and non-parabolic velocity profile effects were found to be negligible. A detailed error analysis is also carried out in an attempt to expose the challenges in conducting accurate measurements in microsystems.

High-speed observations clearly show that though a collapsing cavitation bubble approaches its minimum size as a coherent single volume, it usually reappears in the first rebounding frame as a cloud of much smaller bubbles or as a highly distorted single volume. This paper explores two mechanisms that may be responsible for that bubble fission process, one invoking a Rayleigh–Taylor stability analysis and the other using the so-called microjet mechanism. Both approaches are shown to lead to qualitatively similar values for the number of fission fragments and the paper investigates the flow parameters that effect that number. Finally, the additional damping of the Rayleigh–Plesset single-bubble calculation caused by the fission process is investigated; it is shown that the fission damping dominates other contributions normally considered and is consistent with the number of collapses and rebounds that are observed to occur in experiments.

This paper reports on an experimental investigation to determine the structure and mean flow quantities of round zero-net-mass-flux (ZNMF) jets. These jets are generated by a piston oscillating in a cavity behind a circular orifice. Several different flow patterns were observed with dye flow visualization and a parameter map of these was generated. Cross-correlation digital particle image velocimetry was used to measure instantaneous two-dimensional in-plane velocity fields in a plane containing the orifice axis. These velocity fields are used to investigate the existence of a self-preserving velocity profile in the far field of the ZNMF jet. The mean flow quantities and turbulent statistics of the ZNMF jets were compared with measurements for ‘equivalent’ continuous jets in the same apparatus. Phase-averaged velocity measurements were obtained in the near field of the ZNMF jets and were used to determine the radial entrainment. The out-of-plane vorticity fields were also investigated to gain an understanding of the mechanisms responsible for the difference in spreading rate of ZNMF jets compared to conventional continuous jets. A conceptual model of the ZNMF jet structure in the near field for Strouhal numbers much less than one is proposed that explains the observed behaviour of these ZNMF jets.

The gravity-driven flow of non-neutrally buoyant suspensions is shown to be unstable to spanwise perturbations when the shearing motion generates a density profile that increases with height. The instability is simply due to having heavier material over light – a Rayleigh–Taylor-like instability. The wavelength of the fastest growing disturbance is on the order of the thickness of the suspension layer. The parameters important to the problem are the angle of inclination of the layer relative to gravity, the relative density difference between the particles and the fluid, the ratio of the particle size to the thickness of the layer and the bulk volume fraction of particles. The instability is illustrated for a range of these parameters and shown to be most pronounced at intermediate values thereof. This instability mechanism may play an important role in pattern formation in multiphase flows.

Laboratory experiments are described which investigate the dynamics of a vortex dipole moving in water towards a planar sloping beach inclined at an angle α to the horizontal. Results are compared with those of a vortex ring model first developed by Peregrine (1996). The vortices separate as they travel up the beach, eventually moving in opposite directions and nearly parallel to the shoreline. The ranges of conditions examined are 3°[les ]α[les ]45°, 1×103[les ]Re[les ]6×103 and 0.05[les ]Fr[les ]0.14, where Re and Fr are the on-slope Reynolds number and the Froude number of the vortices, respectively. The minimum distance from the shoreline reached by the vortices and their along-shore speed are in general agreement with the predictions when, respectively, R*i<3 (where R*i is the non-dimensional initial distance of the vortices from the shore-line) and Re[gsim ]1500. The vortex ring model is likely to have useful applications to the study of the dynamics of the near-shore zone.

A transition scenario initiated by streamwise low- and high-speed streaks in a flat-plate boundary layer is studied. In many shear flows, the perturbations that show the highest potential for transient energy amplification consist of streamwise-aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. In a previous investigation (Andersson et al. 2001), the stability of these streaks in a zero-pressure-gradient boundary layer was studied by means of Floquet theory and numerical simulations. The sinuous instability mode was found to be the most dangerous disturbance. We present here the first simulation of the breakdown to turbulence originating from the sinuous instability of streamwise streaks. The main structures observed during the transition process consist of elongated quasi-streamwise vortices located on the flanks of the low-speed streak. Vortices of alternating sign are overlapping in the streamwise direction in a staggered pattern. The present scenario is compared with transition initiated by Tollmien–Schlichting waves and their secondary instability and by-pass transition initiated by a pair of oblique waves. The relevance of this scenario to transition induced by free-stream turbulence is also discussed.

For many years there has been debate regarding why shock wave reflection off a solid surface has allowed regular reflection to persist beyond the incidence angles where it becomes theoretically impossible. Theory predicts that at some limiting angle the reflection point will move away from the wall and Mach reflection will occur. Previous studies have suggested that the paradox could be due to the presence of the growing viscous boundary layer immediately behind the point of reflection, and some numerical studies support this view. This paper takes the approach of establishing an experimental facility in which the theoretical assumptions regarding the surface of reflection are met, i.e. that the reflecting surface is perfectly smooth, perfectly rigid, and adiabatic. This is done by constructing a bifurcated shock tube facility in which a shock wave is split into two plane waves that are then allowed to reflect off each other at the trailing edge of wedge. The plane of symmetry between the waves then acts as the perfect reflection surface.

Through a careful set of visualization experiments, and the use of multivariate analysis to take account of the uncertainty in shock Mach number, triple-point trajectory angle, and slightly different shock wave arrival times at the trailing edge, the current work shows that the transition from one type of reflection to the other does indeed occur at the theoretical value. Conventional tests of reflection off a solid wall show significantly different transition results. Furthermore, it is also shown that the thermal boundary layer plays an important role in this regard. It is thus confirmed that viscous and thermal effects are the reason for the paradox. Reasons are also suggested for the counter-intuitive behaviour of the reflected shock wave angle.

The surfactant concentration distribution on a planar uniform flow with a surface-piercing barrier was measured via the nonlinear optical technique of second-harmonic generation. The measurements were performed for an insoluble surfactant monolayer on the air/water interface. A theoretical model balancing surface elasticity and bulk shear at the interface was developed to predict the concentration profile for any insoluble monolayer. Measured equations of state, relating the surface tension to the surfactant concentration, were used in the model along with velocity data obtained using boundary-fitted digital particle image velocimetry. Theoretical concentration profiles were in agreement with experimental results. Additionally, global predictions from the model for four different insoluble surfactant systems also showed agreement with experimental measurements.

The fluid dynamic behaviour of a reactive chemical in a stream can be greatly influenced by the presence of sorbing suspended particles. In this case, a kinetically controlled mass transfer is established between sorbed and dissolved phases and complex interactions emerge between fluid dynamical transport processes, sorption–desorption kinetics and chemical reactions. These conditions often occur in rivers, where both suspended sediment and reactive substances are frequently present. This paper deals with the important case in which the chemical reactions are nonlinear decay phenomena that often affect chemical or biological substances. A vertical two-dimensional mathematical model is formulated to take into account advection, turbulent diffusion, particle sedimentation, exchange kinetics between sorbed and dissolved phases, and decay. The decay is modelled for the case in which two different nonlinear decay reactions affect the dissolved and sorbed phases. The main result of the work is to obtain analytically a one-dimensional differential model of the vertically averaged concentration of the dissolved phase, this being conceptually similar to the classical advection–dispersion–decay equation. However, in this case we include the effects of (i) the kinetics with the phase sorbed by suspended particles and (ii) the influence of the two different decay processes. For this purpose, the multiple-scale method of homogenization is applied to the two-dimensional model. The resultant one-dimensional differential model shows how suspended load and decay phenomena affect the pollutant transport mechanisms to a great extent in a non-intuitive way and that the links are nonlinear. Some quantitative results show that these influences are, in general, not negligible.

The intrusion of a polydisperse suspension of particles over a horizontal, rigid boundary is investigated theoretically using both an integral (‘box’) model and the shallow-water equations. The flow is driven by the horizontal pressure gradient associated with the density difference between the intrusion and the surrounding fluid, which is progressively diminished as suspended particles sediment from the flow to the underlying boundary. Each class of particles in a polydisperse suspension has a different settling velocity. The effects of both a discrete and continuous distribution of settling velocities on the propagation of the current are analysed and the results are compared in detail with results obtained by treating the suspension as monodisperse with an average settling velocity. For both models we demonstrate that in many regimes it is insufficient to deduce the behaviour of the suspension from this average, but rather one can characterize the flow using the variance of the settling velocity distribution as well. The shallow-water equations are studied analytically using a novel asymptotic technique, which obviates the need for numerical integration of the governing equations. For a bidisperse suspension we explicitly calculate the flow speed, runout length and the distribution of the deposit, to reveal how the flow naturally leads to a vertical and streamwise segregation of particles even from an initially well-mixed suspension. The asymptotic results are confirmed by comparison with numerical integration of the shallow-water equations and the predictions of this study are discussed in the light of recent experimental results and field observations.

The impact of a drop onto a liquid film with a relatively high impact velocity, leading to the formation of a crown-like ejection, is studied theoretically. The motion of a kinematic discontinuity in the liquid film on the wall due to the drop impact, the formation of the upward jet at this kinematic discontinuity and its elevation are analysed. Four main regions of the drop and film are considered: the perturbed liquid film on the wall inside the crown, the unperturbed liquid film on the wall outside the crown, the upward jet forming a crown, and the free rim bounding this jet. The theory of Yarin & Weiss (1995) for the propagation of the kinematic discontinuity is generalized here for the case of arbitrary velocity vectors in the inner and outer liquid films on the wall. Next, the mass, momentum balance and Bernoulli equations at the base of the crown are considered in order to obtain the velocity and the thickness of the jet on the wall. Furthermore, the dynamic equations of motion of the crown are developed in the Lagrangian form. An analytical solution for the crown shape is obtained in the asymptotic case of such high impact velocities that the surface tension and the viscosity effects can be neglected in comparison to inertial effects. The edge of the crown is described by the motion of a rim, formed due to the surface tension.

Three different cases of impact are considered: normal axisymmetric impact of a single drop, oblique impact of a single drop, and impact and interaction of two drops. The theoretical predictions of the height of the crown in the axisymmetric case are compared with experiments. The agreement is quite good in spite of the fact that no adjustable parameters are used.

We investigate magnetic Taylor–Couette flow in the presence of an imposed axial magnetic field. First we calculate nonlinear steady axisymmetric solutions and determine how their strength depends on the applied magnetic field. Then we perturb these solutions to find the critical Reynolds numbers for the appearance of wavy modes, and the related wave speeds, at increasing magnetic field strength. We find that values of imposed magnetic field which alter only slightly the transition from circular-Couette flow to Taylor-vortex flow, can shift the transition from Taylor-vortex flow to wavy modes by a substantial amount. The results are compared to those for onset in the absence of a magnetic field.