Numerical simulations are performed of a compressible oxidizer gas laden with fuel droplets. The carrier phase is considered in the Eulerian context and is simulated via direct numerical simulation (DNS). The fuel droplets are tracked in the Lagrangian frame and interactions between the two phases are taken into account in a realistic two-way coupled formulation. It is assumed that combustion takes place in the vapour phase, resulting in a ‘homogeneous’ reaction described by fuel + oxidizer → products + energy. Several simulations are performed within the configuration of low-Mach-number homogeneous shear turbulence to investigate the effects of the mass loading ratio, the droplet time constant, the Damköhler number, and the heat release coefficient. Initial mass loading ratios up to 0.8 and initial Stokes numbers (based on the Kolmogorov time scale) of 1.23 and 2.46 are considered. The results of these simulations along with those from non-reacting cases are utilized to analyse the droplet size distribution, the fuel vapour, the oxidizer, and the reaction rate and zone. An analysis of the statistics of the two-phase flow indicates that various fields are accurately resolved and the assumptions invoked in the formulation of the problem are satisfied. The mean evaporation rate (normalized with the initial mass of the droplets) decreases with the increase of either the mass loading ratio or the droplet time constant. It is shown that the droplet size distribution can be reasonably approximated by a Gaussian probability density function (p.d.f.) for all of the cases. The joint p.d.f. of the fuel vapour and the oxidizer mass fractions exhibits the features of a premixed reaction. The values of the Taylor microscale of the fuel vapour and the oxidizer are closer in the presence of the chemical reaction than in the evaporating but non-reacting case. The reaction rate exhibits higher values in the regions of the flow containing the droplets while experiencing moderate increase in the high-strain-rate regions. The evaporation rate (per mass of the droplet) is smaller for larger droplets but an opposite trend is observed for the reaction rate. The reaction zone tends to align with the streamwise direction due to the effects of the mean flow on the droplets. The alignment is enhanced with either the increase of the mass loading ratio or the decrease of the droplet time constant, or the decrease of the Damköhler number. The alignment of the fuel vapour and the oxidizer with the mean flow direction decreases and increases, respectively, as a result of the chemical reaction.

The electrohydrodynamic stability of a liquid bridge was studied in steady and oscillatory axial electric fields with a novel apparatus aboard a space shuttle. To avoid interphase transport, which complicates matters in terrestrial, matched-density systems, the experiments focused on a liquid column surrounded by a dielectric gas. The micro-gravity acceleration level aboard the spacecraft kept the Bond number small; so interface deformation by buoyancy was negligible. To provide microgravity results for comparison with terrestrial data, the behaviour of a castor oil bridge in a silicone oil matrix liquid was studied first. The results from these experiments are in excellent agreement with earlier work with isopycnic systems as regards transitions from a perfect cylinder to the amphora shape and the separation of an amphora into drops. In addition, the location of the amphora bulge was found to be correlated with the field direction, contrary to the leaky dielectric model but consistent with earlier results from terrestrial experiments. Next, the behaviour of a bridge surrounded by a dielectric gas, sulphur hexa fluoride (SF6), was investigated. In liquid–gas experiments, electrohydrodynamic ejection of liquids from ‘Taylor cones’ was used to deploy fluid and form bridges by remote control. Experiments with castor oil bridges in SF6 identified the conditions for two transitions: cylinder–amphora, and pinch-off. In addition, new behaviour was uncovered with liquid–gas interfaces. Contrary to expectations based on perfect dielectric behaviour, castor oil bridges in SF6 could not be stabilized in AC fields. On the other hand, a low-conductivity silicone oil bridge, which could not be stabilized by a DC field, was stable in an AC field.

An oscillatory instability mechanism is identified for a horizontal liquid layer with undeformable open surface heated from the air side. Although buoyancy and surface tension gradients are expected to play a stabilizing role in this situation, we show that, acting together, they may lead to the instability of the motionless state of the system. The instability is a consequence of the coupling between internal and surface waves, whose resonant interaction and resulting mode mixing are discussed. Predictions amenable to experimental test are given together with a thorough analytical and numerical study of the problem.

Marangoni convection in a cavity with differentially heated sidewalls has been investigated. The analysis includes the complete effects of interface deformation. The results determined for large Biot and zero Marangoni (zero Prandtl) numbers show that steady convection may exist for Reynolds numbers Re larger than, and for capillary numbers Ca and cavity lengths L smaller than, certain critical values. The main factor limiting the existence of steady convection involves the interface becoming tangential to the hot wall at the contact point (tangency condition). Unsteady analysis shows that the tangency condition defines the limit point for the system; its violation is most likely to lead to the formation of a dry spot at the hot wall. The critical values of Re, Ca, and L are mutually dependent and change with the heating rate (they reach a minimum for instantaneous heating). For a certain range of parameters, multiple (i.e. steady and oscillatory) states are possible. The oscillatory state has a form consisting of the steady mode with a simple harmonic sloshing motion superposed on it. A reduction in the heating rate permits heating of the liquid without triggering the oscillatory state. Transition between the steady and the oscillatory states involves a nonlinear instability process.

Marangoni convection in a cavity subject to point (concentrated) heating has been investigated. The analysis includes the complete effects of the interface deformation. The results determined for large Biot and zero Marangoni (zero Prandtl) numbers show that steady convection may exist only for a limited range of Reynolds numbers Re (bounded from above and from below), and for capillary numbers Ca and cavity lengths L smaller than certain critical values. The main factor limiting the existence of steady convection involves the interface approaching the bottom of the cavity. Unsteady analysis shows that when the conditions guaranteeing the existence of steady convection are not met, an interface rupture process sets in leading, eventually, to the formation of a dryout at the bottom of the cavity. The initial stages of the rupture process are characterized by a rapidly accelerating growth of the interface deformation. The critical values of Re, Ca and L, which guarantee the existence of steady convection, are mutually dependent and change with the heating rate; they reach a minimum for instantaneous heating. Too rapid heating produces an oscillatory transient which always decays in the range of parameters studied.

In this article, we considered experimentally the deformation and breakup of Newtonian and non-Newtonian conducting drops in surrounding fluid subjected to a uniform electric field. First, we examined three distinctive cases of Newtonian-fluid pairs with different relative conductivities, namely highly conducting drops, conducting drops and slightly conducting drops. The results on the Newtonian fluids demonstrated that when the conductivity of the drop is very large relative to that of the surrounding fluid, the deformation response of such highly conducting drops is described well by the electrohydrostatic theory, especially with regard to the prediction of the critical point. Specifically, when the ratio of drop to continuous-phase resistivity, R, was less than 10−5, the electrohydrostatic theory was quite satisfactory. Then, the non-Newtonian effect on the drop deformation and breakup was studied for highly conducting drops which satisfied the condition R < O(10−5). The highly conducting drop became stable in a weak or moderate field strength when either the drop or the continuous phase was non-Newtonian. On the other hand, when both the phases were non-Newtonian, more complicated responses were observed depending on the ratio of zero-shear-rate viscosities. Although the effects of the rheological properties are minimal on all features away from the critical conditions for breakup or prior to the instability, the non-Newtonian properties have a significant influence during drop burst, which is accompanied by large velocities and velocity gradients. In particular, when the ratio of the zero-shear-rate viscosity of the drop to that of the ambient fluid was much larger than unity, non-Newtonian properties of the drop phase enhanced the drop stability. Conversely, the elasticity of the continuous phase deteriorated the drop stability. Meanwhile if the zero-shear-rate viscosity ratio was much smaller than unity, the elasticity of the continuous phase produced a stabilizing effect. The effects of resistivity and viscosity ratios on the breakup modes were also investigated. When at least one of the two contiguous phases possessed considerable non-Newtonian properties, tip streaming appeared.

The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than the true global value by an O(1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex s-plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the global criterion on a related two-parameter family of models, which includes the spherical convection problem for particular O(1) values of his parameters. Since he used one of them as the basis of a small-parameter expansion, his results are necessarily approximate for our problem.

Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Yano's relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the leading-order value of the critical Rayleigh number together with its first-order correction for various values of the Prandtl number.

Numerical solutions of the relevant PDEs have also been found, for values of the Ekman number down to 10−6; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.

A Boussinesq-type model is derived which is accurate to O(kh)4 and which retains the full representation of the fluid kinematics in nonlinear surface boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity potential. In order to achieve a (4,4) Padé representation of the dispersion relationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of internal kinematics is greatly improved over existing O(kh)2 approximations, especially in the intermediate to deep water range. The model equations are first examined for their ability to represent weakly nonlinear wave evolution in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a random sea as well as nonlinear dispersion and stability effects in the nonlinear Schrödinger equation for a narrow-banded sea state. We then examine numerical properties of solitary wave solutions in shallow water, and compare model performance to the full solution of Tanaka (1986) as well as the level 1, 2 and 3 solutions of Shields & Webster (1988).

The instability of an annular layer coated on the interior side of an outer circular tube and surrounding another annular layer coated on the exterior side of an inner circular tube, is studied in the absence of an imposed flow due to a pressure gradient or boundary motion. As the radius of the inner cylinder tends to vanish and the radius of the outer cylinder tends to infinity, the inner layer reduces to a liquid thread suspended in a quiescent infinite ambient fluid. The fluids are separated by a membrane that exhibits constant surface tension and develops elastic tensions due to deformation from the unstressed cylindrical shape. The surface tension is responsible for the Rayleigh capillary instability, but the elastic tensions resist the deformation and slow down or even prevent the growth of small perturbations. In the first part of this paper, we formulate the linear stability problem for axisymmetric perturbations, and derive a nonlinear eigenvalue system whose solution produces the complex phase velocity of the normal modes. When inertial effects are negligible, there are two normal modes; one is stable under any conditions, and the second may be unstable when the interfacial elasticity is sufficiently small compared to surface tension, and the wavelength of the perturbation is sufficiently long. Stability graphs are presented to illustrate the properties of the normal modes and their dependence on the ratio of the viscosity of the outer to inner fluid, the interfacial elasticity, and the ratios of the cylinders' radii to the interface radius. The results show that as the interfacial elasticity tends to vanish, the unconditionally stable mode becomes physically irrelevant by requiring extremely large ratios of axial to lateral displacement of material points along the trace of the membrane in an azimuthal plane. In the second part of this paper, we investigate the nonlinear instability of an infinite thread in the limit of vanishing Reynolds numbers by dynamical simulation based on a boundary-integral method. In the problem formulation, the elastic tensions derive from a constitutive equation for a thin sheet of an incompressible isotropic elastic solid described by Mooney's constitutive law. The numerical results suggest that the interfacial elasticity ultimately restrains the growth of disturbances and leads to slowly evolving periodic shapes, in agreement with laboratory observations.

Alternating shear flow over a self-similar, rhythmically expanding hemispherical depression is investigated. It provides a fluid-mechanical model for an alveolated respiratory unit, by means of which the effect of lung rhythmical expansion on gas mixing as well as aerosol dispersion and deposition can be studied. The flow is assumed to be very slow and governed by the quasi-steady linear Stokes equations. Consequently, superposition of the following two cases provides an easy route toward characterizing the aforementioned flow field. The first case treats the flow field that is generated by a rhythmically expanding spherical cap (the alveolus). The cap is attached at its rim to a circular opening in an expanding unbounded plane bounding a semi-infinite fluid region. The rate of expansion of the cap and the plane are chosen such as to maintain the system's configurational self-similarity. The second case addresses the flow disturbance that is generated by an alternating shear flow encountering a rigid hemispherical cavity in a plane bounding a semi-infinite fluid domain.

For the first case, a stream-function representation employing toroidal coordinates furnishes an analytical solution, whereas the second case was solved numerically by Pozrikidis (1994). Linear superposition of the two flow cases results in particularly rich streamline maps. In the symmetry plane (bisecting the cap and parallel to the mean shear flow), for a certain range of shear to expansion-rate ratios, the streamline maps are self-similar and display closed orbits and two internal stagnation points. One of the stagnation points is a ‘centre’ surrounded by closed streamlines whereas the other constitutes a ‘saddle point’. For other planes, no stagnation points exist in the field, but the streamlines associated with the saddle point display complex looping patterns. These unique flow structures, when subjected to a small perturbation (e.g. a small asynchrony between ductal and alveolar entering flows) cause highly complex stochastic particle trajectories even in the quasi-static Stokes alveolar flow. The observed irreversible flow phenomena in a rhythmically expanding alveolus may be partially responsible for the ‘stretch-and-fold’ flow mixing patterns found in our recent flow visualization studies performed in excised animal lung acini.

Various observations of layering and intrusions in the ocean strongly suggest that such structures and motions are produced and driven by horizontal and vertical gradients of temperature and salinity, i.e. by double-diffusive processes. Much of the laboratory work in this field has concentrated on one-dimensional problems, with the neglect of two-dimensional phenomena. The latter are addressed explicitly in the present paper, using the salt–sugar analogue system in a simple geometry, but with the aim of establishing some more widely applicable general principles. Two sources of salt or sugar solution were fed in at opposite ends of a 750 mm long tank, with an overflow tube drawing fluid from a point at the centre of the tank. With two salt sources of different concentrations and densities, a stratification built up through the ‘filling box’ process, and the total density range lay within that of the input solutions. For one salt and one sugar source, a much larger density gradient could be set up, with the range lying outside that of the inputs. The flows were monitored using various experimental techniques: photographs of dye streaks with still and video cameras; a polarimeter to monitor sugar concentration; and the withdrawal of samples for the measurement of density and refractive index, from which the separate contributions of salt and sugar to the density could be calculated.

Three related experiments with simple input conditions were particularly instructive, and these will be described first. Both inputs and the withdrawal tube were located at mid-depth, and the tank fluid and the salt and sugar supplies had the same density. The only difference between runs was the initial composition of the solution in the tank: pure salt, pure sugar, and a 50[ratio ]50 mixture of the two. Following an initial transient response which was different in the three experiments, they all tended to the same asymptotic distributions of salt, sugar and density after about 100 h, with a sharp central interface and weakly stratified upper and lower layers. This state corresponded approximately to the one-dimensional ‘rundown’ of a layer of salt solution above sugar solution, with a slightly higher, unstable concentration of salt in the top layer compared to the bottom and a very stable sugar distribution, with a much larger concentration in the bottom layer than in the top one. This distribution cannot be produced by ‘finger’ rundown, and it corresponds to the maximum release of potential energy. It was, however, achieved through the action of many intrusions, which remained active in the dynamic final state, and maintained a strong communication between the two ends of the tank. A comparable experiment was carried out using a tank 1820 mm long. With this larger aspect ratio there was a predominantly local influence of the sources at each end of the tank. Other runs have explored a variety of geometries of the sources and sinks, and the final state has been shown to be sensitive to these boundary conditions.

Transitions of flow past a row of square bars placed across a uniform flow are investigated by numerical simulations and the bifurcation analysis of the numerical results. The flow is assumed two-dimensional and incompressible. It is already known that jets coming through gaps between square bars are independent of each other when the pitch-to-side-length ratio of the row is large, whereas the confluence of two or three jets occurs due to a first pitchfork bifurcation from the flow with independent jets when the pitch-to-side-length ratio is small. It is found that confluence of four jets occurs in consequence of the second pitchfork bifurcation from the flow with pairs of jets joined to each other. Bifurcation diagrams of the flow are obtained, which include confluences of double, triple and quadruple jets. Lengths of the twin vortices are evaluated for each flow pattern. The confluences of two, three and four jets are qualitatively confirmed experimentally by flow visualizations.

The nonlinear development of stationary crossflow vortices over a 45° swept NLF(2)-0415 airfoil is studied. Previous investigations indicate that the linear stability theory (LST) is unable to accurately describe the unstable flow over crossflow-dominated configurations. In recent years the development of nonlinear parabolized stability equations (NPSE) has opened new pathways toward understanding unstable boundary-layer flows. This is because the elegant inclusion of nonlinear and non-parallel effects in the NPSE allows accurate stability analyses to be performed without the difficulties and overhead associated with direct numerical simulations (DNS). NPSE results are presented here and compared with experimental results obtained at the Arizona State University Unsteady Wind Tunnel. The comparison shows that the saturation of crossflow disturbances is responsible for the discrepancy between LST and experimental results for cases with strong favourable pressure gradient. However, for cases with a weak favourable pressure gradient the stationary crossflow disturbances are damped and nonlinearity is unimportant. The results presented here clearly show that for the latter case curvature and non-parallel effects are responsible for the previously observed discrepancies between LST and experiment. The comparison of NPSE and experimental results shows excellent agreement for both configurations.

Through this work, a detailed quantitative comparison and validation of NPSE with a careful experiment has now been provided for three-dimensional boundary layers. Moreover, the results validate the experiments of Reibert et al. (1996), and Radeztsky et al. (1993, 1994) suggesting that their databases can be used by future researchers to verify theories and numerical schemes. The results show the inadequacy of linear theories for modelling these flows for significant crossflow amplitude and demonstrate the effects of weak curvature to be more significant than slight changes in basic state, especially near neutral-stability locations.