A liquid drop placed on a vibrating diaphragm will burst into a fine spray of smaller secondary droplets if it is driven at the proper frequency and amplitude. The process begins when capillary waves appear on the free surface of the drop and then grow in amplitude and complexity as the acceleration amplitude of the diaphragm is slowly increased from zero. When the acceleration of the diaphragm rises above a well-defined critical value, small secondary droplets begin to be ejected from the free-surface wave crests. Then, quite suddenly, the entire volume of the drop is ejected from the vibrating diaphragm in the form of a spray. This event is the result of an interaction between the fluid dynamical process of droplet ejection and the vibrational dynamics of the diaphragm. During droplet ejection, the effective mass of the drop–diaphragm system decreases and the resonance frequency of the system increases. If the initial forcing frequency is above the resonance frequency of the system, droplet ejection causes the system to move closer to resonance, which in turn causes more vigorous vibration and faster droplet ejection. This ultimately leads to drop bursting. In this paper, the basic phenomenon of vibration-induced drop atomization and drop bursting will be introduced, demonstrated, and characterized. Experimental results and a simple mathematical model of the process will be presented and used to explain the basic physics of the system.

Vibration-induced droplet ejection is a novel way to create a spray. In this method, a liquid drop is placed on a vertically vibrating solid surface. The vibration leads to the formation of waves on the free surface. Secondary droplets break off from the wave crests when the forcing amplitude is above a critical value. When the forcing frequency is small, only low-order axisymmetric wave modes are excited, and a single secondary droplet is ejected from the tip of the primary drop. When the forcing frequency is high, many high-order non-axisymmetric modes are excited, the motion is chaotic, and numerous small secondary droplets are ejected simultaneously from across the surface of the primary drop. In both frequency regimes a crater may form that collapses to create a liquid spike from which droplet ejection occurs. An axisymmetric, incompressible, Navier–Stokes solver was developed to simulate the low-frequency ejection process. A volume-of-fluid method was used to track the free surface, with surface tension incorporated using the continuum-surface-force method. A time sequence of the simulated interface shape compared favourably with an experimental sequence. The dynamics of the droplet ejection process was investigated, and the conditions under which ejection occurs and the effect of the system parameters on the process were determined.

This article shows how Euclidean invariance can be preserved in the so-called algebraic Reynolds stress model (ARSM) approximation. This approximation is used to reduce the transport equation for the Reynolds stresses to an explicit algebraic relation. A number of known models, which make use of this approximation, are not form-invariant under transformations to rotating coordinate systems. A simple extension is presented to show how this artifact can be removed.

The bulk rheology of bidisperse mixtures of granular materials is examined under homogeneous shear flow conditions using the event-driven simulation method. The granular material is modelled as a system of smooth inelastic disks, interacting via the hard-core potential. In order to understand the effect of size and mass disparities, two cases were examined separately, namely, a mixture of different sized particles with particles having either the same mass or the same material density. The relevant macroscopic quantities are the pressure, the shear viscosity, the granular energy (fluctuating kinetic energy) and the first normal stress difference.

Numerical results for pressure, viscosity and granular energy are compared with a kinetic-theory constitutive model with excellent agreement in the low dissipation limit even at large size disparities. Systematic quantitative deviations occur for stronger dissipations. Mixtures with equal-mass particles show a stronger shear resistance than an equivalent monodisperse system; in contrast, however, mixtures with equal-density particles show a reduced shear resistance. The granular energies of the two species are unequal, implying that the equi-partition principle assumed in most of the constitutive models does not hold. Inelasticity is responsible for the onset of energy non-equipartition, but mass disparity significantly enhances its magnitude. This lack of energy equipartition can lead to interesting non-monotonic variations of the pressure, viscosity and granular energy with the mass ratio if the size ratio is held fixed, while the model predictions (with the equipartition assumption) suggest a monotonic behaviour in the same limit. In general, the granular fluid is non-Newtonian with a measurable first normal stress difference (which is positive if the stress is defined in the compressive sense), and the effect of bidispersity is to increase the normal stress difference, thus enhancing the non-Newtonian character of the fluid.

Turbulent shear flows, such as those occurring in the wall region of turbulent boundary layers, show a substantial increase of intermittency in comparison with isotropic conditions. This suggests a close link between anisotropy and intermittency. However, a rigorous statistical description of anisotropic flows is, in most cases, hampered by the inhomogeneity of the field. This difficulty is absent for homogeneous shear flow. For this flow the scale-by-scale budget is discussed here by using the appropriate form of the Kámán–Howarth equation, to determine the range of scales where the shear is dominant. The resulting generalization of the four-fifths law is then used to extend to shear-dominated flows the Kolmogorov–Oboukhov theory of intermittency. The procedure leads naturally to the formulation of generalized structure functions, and the description of intermittency thus obtained reduces to the K62 theory for vanishing shear. The intermittency corrections to the scaling exponents are related to the moments of the coarse-grained energy dissipation field. Numerical experiments give indications that the dissipation field is statistically unaffected by the shear, supporting the conjecture that the intermittency corrections are universal. This observation together with the present reformulation of the theory gives a reason for the increased intermittency observed in the classical longitudinal velocity increments.

The problem of a drop of arbitrary density and viscosity moving close to a vertical wall under the effect of buoyancy is analysed theoretically. The case where the suspending fluid is at rest far from the drop and that of a linear shear flow are both considered. Effects of inertia and deformation are assumed to be small but of comparable magnitude, so that both of them contribute to the lateral migration of the drop. Expressions for the drag, deformation and migration valid down to separation distances from the wall of a few drop radii are established and discussed. Inertial and deformation-induced corrections to the drag force and slip velocity of a buoyant drop moving in a linear shear flow near a horizontal wall are also derived.

Motivated by the recent paper of Hills & Moffatt (2000), we investigate the Stokes flow in a trihedral corner formed by three mutually orthogonal planes, induced by a non-zero velocity distribution over one of the walls of the corner. It is shown that the local behaviour of the velocity field near the edges of the corner, where a discontinuity of the boundary velocity is assumed, coincides with the Goodier–Taylor solution for a two-dimensional wedge. Analysis of the streamline patterns confirms the existence of eddies near the stationary edge in the flow, induced either by uniform translation of one of the walls of the corner in the direction perpendicular to its bisectrix or by uniform rotation of a side about the vertex of the corner. These flows are shown to be quasi-two-dimensional. If the wall rotates about a centre displaced from the vertex, the induced flow is essentially three-dimensional. In the antisymmetric velocity field, a stagnation line appears composed of stagnation points of different types. Otherwise the three-dimensionality manifests itself in a non-closed spiral shape of the streamlines.

The problem of direct initiation of detonation, where a powerful ignition source drives a blast wave into a gaseous combustible mixture to generate a Chapman–Jouguet (CJ) detonation, is investigated numerically by using a three-step chain-branching chemical kinetic model. The reaction scheme consists sequentially of a chain-initiation and a chain-branching step, followed by a temperature-independent chain termination. The three regimes of direct initiation i.e. subcritical, critical and supercritical, are numerically simulated for planar, cylindrical and spherical geometries using the present three-step chemical kinetic model. It is shown that the use of a more detailed reaction mechanism allows a well-defined value for the critical initiation energy to be determined. The numerical results demonstrate that detonation instability plays an important role in the initiation process. The effect of curvature for cylindrical and spherical geometries has been found to enhance the instability of the detonation wave and thus influence the initiation process. The results of these simulations are also used to provide further verification of some existing theories of direct initiation of detonation. It appears that these theories are satisfactory only for stable detonation waves and start to break down for highly unstable detonations because they are based on simple blast wave theory and do not include a parameter to model the detonation instability. This study suggests that a stability parameter, such as the ratio between the induction and reaction length, should be considered and a more complex chemistry should be included in future development of a more rigorous theory for direct initiation of detonation.

The advection of a passive scalar blob in the deformation field of an axisymmetric vortex is a simple mixing protocol for which the advection–diffusion problem is amenable to a near-exact description. The blob rolls up in a spiral which ultimately fades away in the diluting medium. The complete transient concentration field in the spiral is accessible from the Fourier equations in a properly chosen frame. The concentration histogram of the scalar wrapped in the spiral presents unexpected singular transient features and its long time properties are discussed in connection with real mixtures.

In this paper we investigate the three-dimensional laminar incompressible steady flow along a corner formed by joining two similar quarter-infinite unswept wedges along a side-edge. We show that a four-region construction of the potential flow arises naturally for this flow problem, the formulation being generally valid for a corner of an arbitrary angle (π−2α), including the limiting cases of semi- and quarter-infinite flat-plate configurations. This construction leads to five distinct three-dimensional boundary-layer regions, whereby both the spanwise length and velocity scales of the blending intermediate layers are O(δ), with Re−1/2 [Lt ] δ [Lt ] 1, Re being a reference Reynolds number supposed to be large. This reveals crucial differences between concave and convex corner flows. For the latter flow regime, the corner-layer motion is shown to be mainly controlled by the secondary flow which effectively reduces to that past sharp wedges with solutions being unique and existing only for favourable streamwise pressure gradients. In this regime, the corner-layer thickness is shown to be O(Re−0.5+α/π/δ2α/π), −½π [les ] α [les ] 0, which is much smaller than O(Re−1/2) for concave corner flows.

Crucially, our numerical results show conclusively that, for α ≠ 0, closed streamwise symmetrically disposed vortices are generated inside the intermediate layers, confirming thus the prediction made by Moore (1956) for a rectangular corner, which has so far remained unconfirmed in the literature.

For almost planar corners, three-dimensional corner boundary-layer features are shown, as in (Smith 1975), to arise when α ∼ O(1/ln Re). On the other hand, we show that the flow past a quarter-infinite flat plate would be attained when both values of the streamwise pressure gradient and external corner angle (π+2α) become O(1/ln Re) or smaller.

Numerical results for all these flow regimes are presented and discussed.

The evolution from a linear temperature gradient to a detonation is investigated for combustible materials whose chemistry is governed by chain-branching kinetics, using a combination of high-activation-temperature asymptotics and numerical simulations. A two-step chemical model is used, which captures the main properties of detonations in chain-branching fuels. The first step is a thermally neutral induction time, representing chain initiation and branching, which has a temperature-sensitive Arrhenius form of the reaction rate. At the end of the induction time is a transition point where the fuel is instantaneously converted into chain-radicals. The second step is the main exothermic reaction, representing chain termination, assumed to be temperature insensitive. Emphasis is on comparing and contrasting the results with previous studies that used simple one-step kinetics. It is shown that the largest temperature gradient for which a detonation can be successfully ignited depends on the heat release rate of the main reaction. The slower the heat release compared to the initial induction time, the shallower the gradient has to be for successful ignition. For example, when the rate of heat release is moderate or slow on the initial induction time scale, it was found that the path of the transition point marking the end of the induction stage should move supersonically, in which case its speed is determined only by the initial temperature gradient. For steeper gradients such that the transition point propagates subsonically from the outset, the rate of heat release must be very high for a detonation to be ignited. Detonation ignition for the two-step case apparently does not involve the formation of secondary shocks, unlike some cases when one-step kinetics is used.

The Orr–Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

The effect of boundary conditions on the ‘critical dynamics’ at the onset of Taylor vortices is investigated in a combined numerical and experimental study. Numerical calculations of Navier–Stokes equations with ‘stress-free’ boundary conditions show that the Landau amplitude equation provides a good model of the transient dynamics. However, this rapidly breaks down when the ‘no-slip’ condition is approached. Apparent ‘critical’ behaviour observed in experiments is shown to have a surprising dependence on the length of the system.

The problem of a two-dimensional inviscid compressible bubble evolving in Stokes flow is considered. By generalizing the work of Tanveer & Vasconcelos (1995) it is shown that for certain classes of initial condition the quasi-steady free boundary problem for the bubble shape evolution is reducible to a finite set of coupled nonlinear ordinary differential equations, the form of which depends on the equation of state governing the relationship between the bubble pressure and its area. Recent numerical calculations by Pozrikidis (2001) using boundary integral methods are retrieved and extended. If the ambient pressures are small enough, it is shown that bubbles can expand significantly. It is also shown that a bubble evolving adiabatically is less likely to expand than an isothermal bubble.

In this article, the multipolar vortex instability of the flow in a finite cylinder is addressed. The experimental study uses a rotating elastic deformable tube filled with water which is elliptically or triangularly deformed by two or three rollers. The experimental control parameters are the cylinder aspect ratio and the Reynolds number based on the angular frequency.

For Reynolds numbers close to threshold, different instability modes are visualized using anisotropic particles, according to the value of the aspect ratio. These modes are compared with those predicted by an asymptotic stability theory in the limit of small deformations and large Reynolds numbers. A very good agreement is obtained which confirms the instability mechanism; for both elliptic and triangular configurations, the instability is due to the resonance of two normal modes (Kelvin modes) of the underlying rotating flow with the deformation field. At least four different elliptic instability modes, including combinations of Kelvin modes with azimuthal wavenumbers m = 0 and m = 2 and Kelvin modes m = 1 and m = 3 are visualized. Two different triangular instability modes which are a combination of Kelvin modes m = −1 and m = 2 and a combination of Kelvin modes m = 0 and m = 3 are also evidenced.

The nonlinear dynamics of a particular elliptic instability mode, which corresponds to the combination of two stationary Kelvin modes m = −1 and m = 1, is examined in more detail using particle image velocimetry (PIV). The dynamics of the phase and amplitude of the instability mode is shown to be predicted well by the weakly nonlinear analysis for moderate Reynolds numbers. For larger Reynolds number, a secondary instability is observed. Below a Reynolds number threshold, the amplitude of this instability mode saturates and its frequency is shown to agree with the predictions of Kerswell (1999). Above this threshold, a more complex dynamic develops which is only sustained during a finite time. Eventually, the two-dimensional stationary elliptic flow is reestablished and the destabilization process starts again.

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.