Deformation and breakup of a viscous drop in a potential vortex are numerically simulated. Capillary number, Reynolds number, and viscosity and density ratios are varied to investigate their effects on the drop dynamics. The vortex locally gives rise to an extensional flow near the drop with the axis of extension rotating at a constant rate, as the drop revolves around the vortex centre. The rotation of the axis plays a critical role in the competing dynamics between the flow-induced stretching and the interfacial tension. The relation between the rotating extensional flow and a shear flow is explored. For low capillary numbers, a periodic state is reached, where the drop deforms into an ellipsoidal shape and undergoes steady rotation with a distinct phase lag behind the imposed flow. For density-and-viscosity-matched drops, increased interfacial tension results in decreased deformation and reduced phase lag. Increased inertia promotes deformation. In the presence of inertia, decreasing capillary number leads to a negative phase lag. The rotation of the extension axis inhibits deformation at low values of the Reynolds number. But at high Reynolds numbers, rotation-induced centrifugal forces promote deformation. At low and high viscosity ratios, an increase in viscosity ratio leads to enhancement and reduction in deformation, respectively. At density ratios larger than unity, the drop deformation displays resonance in that it varies non-monotonically with a distinct peak with variation of interfacial tension and density ratio. The peak corresponds to the natural frequency of the drop deformation matching with the frequency of rotation due to the vortex. A simple physical model is used to explain various observations including asymptotic scalings. We also explore different mechanisms for drop breakup at different Reynolds number, and provide critical capillary numbers as functions of other parameters. In particular, vortex-induced resonance offers an alternative mechanism for size-selective drop breakup. Details of flow fields and transients are also presented and discussed.

The stability of fluid through a channel subject to a system rotation of constant rate about the spanwise axis is considered. In contrast to previous studies, a strongly nonlinear bifurcation approach is used to solve for a family of two-dimensional, steady, streamwise-orientated vortex flows. A stability analysis of these flows is also performed. All of the two-dimensional flows considered lose stability to an Eckhaus (streamwise-independent) secondary disturbance in a steady bifurcation to another member of the solution set. This property, given also that lower-order primary and secondary disturbance modes can become unstable, leads to a rich structure of bifurcation relationships between the secondary flows. With increasing Reynolds number, the secondary flow arising from the linear critical point first loses stability to the Eckhaus instability, and then loses stability to a fundamental spanwise mode with small streamwise wavenumber. With a further increase in Reynolds number, the secondary flow then also becomes unstable to a disturbance of subharmonic spanwise and $O(1)$ streamwise wavenumber, and finally (on the upper solution branch) a disturbance of fundamental spanwise and $O(1)$ streamwise wavenumber. Other types of bifurcation for possible tertiary flows are also identified. By superimposing the secondary disturbance onto the secondary flow, visualizations of the possible structure of the bifurcating tertiary flows are obtained. The visualizations show low-speed streaks in the streamwise velocity component lying between a set of staggered vortices for superharmonic bifurcations, and between aligned vortices for subharmonic bifurcations. Excellent qualitative and quantitative agreement is found with previous experimental results and direct-numerical-simulation-based stability studies, and good overall agreement with previous DNS studies was also found.

We derive an exact theory of three-dimensional steady separation and reattachment using nonlinear dynamical systems methods. Specifically, we obtain criteria for separation points and separation lines on fixed no-slip boundaries in compressible flows. These criteria imply that there are only four basic separation patterns with well-defined separation surfaces. We also derive a first-order prediction for the separation surface using wall-based quantities; we verify this prediction using flow models obtained from local expansions of the Navier–Stokes equations.

We consider the spreading of a droplet of soluble surfactant, at concentrations beyond its critical micelle concentration (CMC), on a pre-existing thin liquid layer. Lubrication theory is used to derive a coupled system of four two-dimensional nonlinear evolution equations for the film thickness and surfactant concentration at the interface and in the bulk as both monomers and micelles. These equations are parameterized by a number of dimensionless groups that reflect the relative importance of Marangoni stresses, surface and bulk diffusion, capillarity, surfactant solubility, sorption kinetics and the nonlinearity of the equation of state. Our results for the base state indicate that two parameters in particular exert a significant influence on the flow profiles: the dimensionless mass of surfactant deposited, $M$, and a parameter reflecting the preference of the surfactant to form micelles, $R$. For a fixed value of $R$, increasing $M$ leads to the development of a protuberance that appears at the edge of the drop; upon increasing $M$ further, this protuberance separates from the drop to form a distinct secondary front that lies behind a leading front that usually accompanies the spreading process. Our examination of the linear and nonlinear stability of the system through a transient growth analysis and transient numerical simulations, respectively, indicates that these features are vulnerable to transverse perturbations, leading to the formation of fingers. The results obtained in the present work are in qualitative agreement with recently available experimental data.

Ventilation of adjacent, connected chambers, forced in one chamber by an isolated point source of buoyancy is investigated. There are floor- and ceiling-level external openings in the forced and unforced chambers, respectively, while the partition between the chambers has both a floor- and ceiling-level opening. The flow evolves on the time scale over which the volume flux associated with the plume at the ceiling would fill both chambers. The steady state in the forced chamber is analogous to the single chamber flow described by Linden, Lane-Serff & Smeed (J. Fluid Mech., vol. 212, 1990, p. 309), with a well-mixed buoyant upper layer which is deeper than in the single chamber flow due to the extra pressure drop at the upper interior opening. The steady state in the unforced chamber inevitably exhibits vertical stratification, and depends on the transient flow, all the opening areas, and the relative plan area of the two chambers, as is verified by laboratory experiments. When the upper interior opening is relatively large, the buoyant layer in the unforced chamber is deeper than the buoyant layer in the forced chamber, which contradicts model predictions based on the assumption that the layers are always well-mixed.

A theoretical model describing the response of a single-phase near-critical fluid to small-amplitude, high-frequency translational vibrations is developed on the basis of the multiple-scale and averaging methods. Additionally to the usual terms of thermal convection, the equations and boundary conditions contain new terms responsible for the generation of pulsating and average flows due to vibrational forcing and fluid compressibility. The effect of compressibility is taken into account both in the boundary layers and in the bulk. A number of classical problems of convection and thermoacoustics are considered on the basis of the new model.

The governing equations and effective boundary conditions to describe thermal vibrational convection in a near-critical fluid are derived with the help of the multiple-scale method and averaging procedure. In contrast to Part 1, this paper focuses on the effects of density non-homogeneities caused not by external heating but by vibrational and gravity stratifications due to the divergent mechanical compressibility of near-critical media. It is shown that vibrations generate non-homogeneities in the average temperature, which result in the onset of thermal convection even under isothermal boundary conditions. An agreement with the results of previous numerical and asymptotical analyses and with experiments is found.

Direct numerical simulations are used to study mixing and dispersion in decaying stably stratified turbulence from a Lagrangian perspective. The change in density of fluid particles owing to small-scale mixing is extracted from the simulations to provide insight into the mixing process. These changes are driven by temporally and spatially intermittent events that are strongly suppressed as the stratification increases and overturning motions disappear. This occurs for times $Nt \,{>}\, 2\upi$, i.e. after one buoyancy period, where $N$ is the buoyancy frequency. The role of small-scale mixing processes in the density (or buoyancy) flux is analysed. After an initial transient, we find that diapycnal displacements due to mixing dominate the dispersion of fluid particles, even in weak stratification. The relationship between the diapycnal diffusivity and vertical dispersion coefficients is found to be strongly dependent on stratification. Models for the mixing following fluid particles are investigated. The time scale for the density changes due to small-scale mixing is shown to be approximately independent of $N$ and instead remains linked to the energy decay time scale which is relatively insensitive to stratification. There are large changes in the structure of these flows as they evolve under the influence of buoyancy forces. We investigate these changes and their relationship to mixing. We find that strong mixing events are closely linked to the presence of overturning regions in the flow, and that they occur close to (but not within) these regions. The results reported here have implications for the development of improved models of diffusion in stably stratified turbulence.

A three-dimensional boundary-integral algorithm is developed to study the squeezing of a deformable drop through a tight constriction formed by several solid particles rigidly held in space. The drop is freely suspended and driven by a flow that is uniform away from the solid obstacles. Particular emphasis is on the trapping mechanism and flow conditions close to critical, when the drop squeezes with high resistance. The problem is a close prototype of drop–solid interactions for emulsion flow through a granular material; such interactions are much more lubrication-sensitive than drop–drop interactions and require advanced numerical tools to succeed. The algorithm is based on the Hebeker representation for the solid–particle contribution, leading to a well-behaved system of second-kind integral equations, combined with novel regularization techniques for singular and near-singular boundary integrals; high-order near-singularity subtraction for the solid-to-drop double-layer contribution is the most crucial element. Simulations are performed for drop squeezing between (i) two close spheres, (ii) two parallel spheroidal disks, and (iii) three close spheres forming an equilateral triangle (including the case of close solid–solid contact). The drop non-deformed diameter is from two to several times larger than the inner constriction diameter and, in some simulations, the drop decelerates $10^3$–$10^4$ times in the throat before being able to pass through. The effects of the constriction type, capillary number, and viscosity ratio on the drop velocity in the throat, exit time, and drop–solid spacing (of the order of 1% of the particle size) are explored in detail; critical capillary numbers (below which trapping occurs) are accurately determined. Even for a substantially supercritical capillary number, the drop has to nearly coat solid particles to be able to pass through a tight constriction. The ability of the algorithm to simulate both supercritical and subcritical conditions (when the drop is trapped, with a small but non-zero drop–solid spacing) is vital for future applications to large-scale simulations of emulsion flow through granular media.

Mean flow measurements are presented for fully developed turbulent pipe flow over a Reynolds number range of $57\,{\times}\,10^3$ to $21\,{\times}\,10^6$ where the flow exhibits hydraulically smooth, transitionally rough, and fully rough behaviours. The surface of the pipe was prepared with a honing tool, typical of many engineering applications, achieving a ratio of characteristic roughness height to pipe diameter of 1 : 17000. Results for the friction factor show that in the transitionally rough regime this surface follows a Nikuradse (1933)-type inflectional relationship rather than the monotonic Colebrook (1939) relationship used in the Moody diagram. This result supports previous suggestions that the Moody diagram in the transitional regime must be used with caution. Outer scaling of the mean velocity data shows excellent collapse and strong evidence for Townsend's outer layer similarity hypothesis for rough-walled flows. Finally, the pipe exhibited smooth behaviour for scaled roughness height $k_s^+ \,{\le}\, 3.5$, which supports the suggestion by Zagarola & Smits (1998) that their pipe was hydraulically smooth for $Re_D\,\,{\leq}\, 24\,{\times}\,10^6$.

The problem of Poiseuille flow in a fluid overlying a porous medium saturated with the same fluid is studied. A careful linear instability analysis is carried out. It is shown that there are three modes of instability, two belong to one eigenvalue and persist in small ranges of parameters, while beyond these parameter ranges a third corresponding to another eigenvalue prevails. These three modes are of different stability characteristics, but are triggered by the shear stress of the Poiseuille flow in the fluid layer.

Uniform approximations for the combined permanent and transient start-up waves produced by the impulsive motion of a body in a density-stratified fluid are calculated. These new approximations to the wave field remove the unphysically diverging fluid velocities and phase shifts near the boundary of the causality envelope associated with previous non-uniform approximations. The calculated wave field is compared to experimental measurements of the wave field generated by towing a spherical body through a linearly stratified fluid. Synthetic schlieren, a technique previously used only to analyse two-dimensional or axisymmetric flow, is developed further to allow comparison with this fully three-dimensional wave field. In particular, breadth averaging across the tank is employed. Good agreement is found between the theoretical and experimental results in the far field.

In immiscible two-phase flows, the contact line denotes the intersection of the fluid–fluid interface with the solid wall. When one fluid displaces the other, the contact line moves along the wall. A classical problem in continuum hydrodynamics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a non-integrable singularity. The recently discovered generalized Navier boundary condition (GNBC) offers an alternative to the no-slip boundary condition which can resolve the moving contact line conundrum. We present a variational derivation of the GNBC through the principle of minimum energy dissipation (entropy production), as formulated by Onsager for small perturbations away from equilibrium. Through numerical implementation of a continuum hydrodynamic model, it is demonstrated that the GNBC can quantitatively reproduce the moving contact line slip velocity profiles obtained from molecular dynamics simulations. In particular, the transition from complete slip at the moving contact line to near-zero slip far away is shown to be governed by a power-law partial-slip regime, extending to mesoscopic length scales. The sharp (fluid–fluid) interface limit of the hydrodynamic model, together with some general implications of slip versus no slip, are discussed.

The interaction of a dilute dispersion of small heavy particles with homogeneous and isotropic air turbulence has been investigated. Stationary turbulence (at Taylor micro-scale Reynolds number of 230) with small mean flow was created in a nearly spherical sealed chamber by means of eight synthetic jet actuators. Two-dimensional particle image velocimetry was used to measure global turbulence statistics in the presence of spherical glass particles that had a diameter of 165 $\umu$m, which was similar to the Kolmogorov length scale of the flow. Experiments were conducted at two different turbulence levels and particle mass loadings up to 0.3. The particles attenuated the fluid turbulence kinetic energy and viscous dissipation rate with increasing mass loadings. Attenuation levels reached 35–40% for the kinetic energy (which was significantly greater than previous numerical studies) and 40–50% for the dissipation rate at the highest mass loadings. The main source of fluid turbulence kinetic energy production in the chamber was the speakers, but the loss of potential energy of the settling particles also resulted in a significant amount of production of extra energy. The sink of fluid energy in the chamber was due to the ordinary viscous dissipation and extra dissipation caused by particles. The extra dissipation was greatly underestimated by conventional models.

We theoretically and experimentally investigate the settling velocity and deformation of a leaky dielectric liquid drop in a second leaky dielectric liquid subject to a uniform electric field, $E$. Both shape distortion and charge convection, when coupled with the asymmetric velocity profiles, will produce a net drag and a shift in the settling speed. Perturbation methods for small shape distortion and small charge convection are used to solve the problem. Corrections to the settling velocity from both contributions are combined linearly at the lowest order, and show a dependence on the drop size. The shape distortion due to charge convection is known to be asymmetric. Experiments are performed to measure the settling velocity and deformation of phenylmethylsiloxane-dimethylsiloxane (PMM) drops in castor oil. The experimental results are in qualitative agreement with the theory: the symmetric and asymmetric deformations and the change in settling velocity are all proportional to $E^{2}$, as predicted, and the settling speed shows the correct trends with drop size. Quantitative agreement is lacking, presumably due to the imprecision of the fluid properties, but the theory can fit all the data with reasonable choices for these properties.

Electrothermal motion in an aqueous solution arises from the action of an electric field on inhomogeneities in the liquid induced by temperature gradients. The temperature field can be produced by the applied electric field through Joule heating, or caused by external sources, such as strong illumination. Electrothermal flows in microsystems are usually observed at applied signal frequencies around 1 MHz and voltages around 10 V. In this work, we present self-similar solutions for the motion of an aqueous solution in a constant temperature gradient placed on top of: (a) two coplanar electrodes subjected to an a.c. potential difference, and (b) four coplanar electrodes subjected to a four-phase a.c. signal, generating a rotating field. The first case produces two-dimensional rolls whereas the second case produces a liquid whirl. Finally, we present experimental results of electrothermal liquid flows generated by alternating and rotating electric fields under strong illumination, and these experiments are compared to the analytical solutions. The induced rotating flow could be used in the mixing of analytes and of liquids in microsystems.

Motivated by work on tilted convection (Sheremet, J. Fluid Mech., vol. 506, 2004, p. 217), a set of experiments is presented here using the same set-up of a tilted tank attached to a rotating centrifuge with a 2.5 m arm. Within the tank small, almost neutrally buoyant, spheres are released, and their trajectories are recorded. Thus the forces acting on a sphere can be analysed in the case of misalignment between the buoyancy force and the axis of rotation.

The angles of descent characterizing the trajectory are compared with inviscid linear theory developed by Stewartson (Q. J. Math. Appl. Mech., vol. 6, 1953, p. 141), and the agreement is found to be good. The angles should be independent of the density anomaly of the spheres compared to their environment. Using the descent velocity from non-rotating experiments, the density of the spheres is estimated and used to determine the drag acting on them in the rotating experiments. It is found that the drag is up to 50% larger than expected from Stewartson's theory. The agreement is best, not for infinitesimal, but for small Rossby numbers. The results are consistent with observations recorded by Maxworthy (J. Fluid Mech., vol. 40, 1970, p. 453).

We investigate the decay of freely evolving isotropic turbulence. There are two canonical cases: $E(k \,{\to}\, 0)\,{\sim}\, Lk^2$ and $E(k \,{\to}\, 0)\,{\sim}\, Ik^4,$$L$ and $I$ being the Saffman and Loitsyansky integrals respectively. We focus on the second of these. Numerical simulations are performed in a periodic domain whose dimensions, $l_{box}$, are much larger than the integral scale of the turbulence, $l$. We find that, provided that $l_{box} \,{\gg}\, l$ and $\hbox{\textit{Re}}\,{\gg}\, 1$, the turbulence evolves to a state in which $I$ is approximately constant and Kolmogorov's classical decay law, $u^2\,{\sim}\, t^ {{- 10} / {7}}$, holds true. The approximate conservation of $I$ in fully developed turbulence implies that the long-range interactions between remote eddies, as measured by the triple correlations, are very weak. This finding seems to be at odds with the non-local nature of the Biot-Savart law.