The objective of this work is to develop techniques to predict the results of experiments involving chaotic dispersion of passive tracers in two-dimensional low Reynolds number flows. We present the design of a flow apparatus which allows the unobstructed observation of the entire flow region. Whenever possible we compare the experimental results with those of computations. Conventional tracking of the boundaries of the tracer is inefficient and works well only for low stretches (order 102 at most). However, most mixing experiments involve extremely large perturbations from steady flow since this is where the best mixing occurs. The best prediction of widespread mixing and large stretching (order 104–105) is provided by lineal stretching plots; surprisingly the technique also works for relatively low numbers of periods (as low as 2 or 3). The second best prediction is provided by a combination of low-order unstable manifolds – which indicate where the tracer goes, especially for short times – and the eigendirections of low-order hyperbolic periodic points – which indicate the alignment of striations in the flow. On the other hand, Poincaré sections provide only a gross picture of the spreading: they can be used primarily to detect what regions are inaccessible to the dye. Comparison of computations and experiments invariably reveals that bifurcations within islands have little impact in the mixing process.

When buoyant fluid is released into the base of a crack in an elastic medjura the crack will propagate upwards, driven by the buoyancy of the fluid. Viscous fluid flow in such a fissure is described by the equations of lubrication theory with the pressure given by the sum of the hydrostatic pressure of the fluid and the elastic pressures exerted by the walls of the crack. The elastic pressure and the width of the crack are further coupled by an integro-differential equation derived from the theory of infinitesimal dislocations in an elastic medium. The steady buoyancy-driven propagation of a two-dimensional fluid-filled crack through an elastic medium is analysed and the governing equations for the pressure distribution and the shape of the crack are solved numerically using a collocation technique. The fluid pressure in the tip of an opening crack is shown to be very low. Accordingly, a region of relatively inviscid vapour or exsolved volatiles in the crack tip is predicted and allowed for in the formulation of the problem. The solutions show that the asymptotic width of the crack, its rate of ascent and the general features of the flow are determined primarily by the fluid mechanics; the strength of the medium and the vapour pressure in the crack tip affect only the local structure near the advancing tip of the crack. When applied to the transport of molten rock through the Earth's lithosphere by magma-fracture, this conclusion is of fundamental importance and challenges the geophysicist's usual emphasis on the controlling influence of fracture mechanics rather than that of fluid mechanics.

A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker–Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Σ can be computed; Vc and D are required in the cell conservation equation, and Σ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ε; λ is the ratio of the rotary diffusion time D-1r to the torque relaxation time B (balancing gravitational and viscous torques), while ε is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker–Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s−1). The continuum model for small ε is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Σ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions.

Simultaneous planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) yield measurements of two-dimensional jet fluid concentration and velocity fields in turbulent crossflowing jets. The jet-to-crossflow velocity ratio is $r = 5.7$ and the jet exit Reynolds number is approximately 5000. The measurements are focused on the developing region of the flow. Two flow configurations are studied, one in which the jet nozzle is flush with the tunnel wall and the other where the nozzle protrudes into the uniform region of the tunnel flow. The jet nozzle in both cases is a simple pipe. The averaged scalar and velocity fields show a strong similarity in growth rates and centreline decay rates between the two nozzle configurations when using the centreline downstream coordinate $s$. This suggests that the flow geometry is less important than, for example, jet exit conditions in determining the initial flow development. The velocity fields show indications of jet-like scaling in the near field, and a possible approach to wake-like scaling in the far field. These canonical scalings are less evident in the scalar fields. Results are presented for the averaged scalar variance, $\langle C^{'2}\rangle$, the scalar flux components, $\langle u_{i}'C'\rangle$, the turbulent normal stresses, $\langle u^{\prime 2}\rangle$ and $\langle v^{\prime 2}\rangle$, and the turbulent shear stress, $\langle u'v'\rangle$. We also discuss the resolution issues bearing on the determination of small-scale fluctuations.

An investigation was undertaken to improve our understanding of low-Reynolds-number turbulent boundary layers flowing over a smooth flat surface in nominally zero pressure gradients. In practice, such flows generally occur in close proximity to a tripping device and, though it was known that the flows are affected by the actual low value of the Reynolds number, it was realized that they may also be affected by the type of tripping device used and variations in free-stream velocity for a given device. Consequently, the experimental programme was devised to investigate systematically the effects of each of these three factors independently. Three different types of device were chosen: a wire, distributed grit and cylindrical pins. Mean-flow, broadband-turbulence and spectral measurements were taken, mostly for values of Rθ varying between about 715 and about 2810. It was found that the mean-flow and broadband-turbulence data showed variations with Rθ, as expected. Spectra were plotted using scaling given by Perry, Henbest & Chong (1986) and were compared with their models which were developed for high-Reynolds-number flows. For the turbulent wall region, spectra showed reasonably good agreement with their model. For the fully turbulent region, spectra did show some appreciable deviations from their model, owing to low-Reynolds-number effects. Mean-flow profiles, broadband-turbulence profiles and spectra were found to be affected very little by the type of device used for Rθ ≈ 1020 and above, indicating an absence of dependence on flow history for this Rθ range. These types of measurements were also compared at both Rθ ≈ 1020 and Rθ ≈ 2175 to see if they were dependent on how Rθ was formed (i.e. the combination of velocity and momentum thickness used to determine Rθ). There were noticeable differences for Rθ ≈ 1020, but these differences were only convincing for the pins, and there was a general overall improvement in agreement for Rθ ≈ 2175.

We present an experimental investigation of the flow structure and vorticity field in the wake of a NACA-0012 airfoil pitching sinusoidally at small amplitude and high reduced frequencies. Molecular tagging velocimetry is used to quantify the characteristics of the vortex array (circulation, peak vorticity, core size, spatial arrangement) and its downstream evolution over the first chord length as a function of reduced frequency. The measured mean and fluctuating velocity fields are used to estimate the mean force on the airfoil and explore the connection between flow structure and thrust generation.

Results show that strong concentrated vortices form very rapidly within the first wavelength of oscillation and exhibit interesting dynamics that depend on oscillation frequency. With increasing reduced frequency the transverse alignment of the vortex array changes from an orientation corresponding to velocity deficit (wake profile) to one with velocity excess (reverse Kármán street with jet profile). It is found, however, that the switch in the vortex array orientation does not coincide with the condition for crossover from drag to thrust. The mean force is estimated from a more complete control volume analysis, which takes into account the streamwise velocity fluctuations and the pressure term. Results clearly show that neglecting these terms can lead to a large overestimation of the mean force in strongly fluctuating velocity fields that are characteristic of airfoils executing highly unsteady motions. Our measurements show a decrease in the peak vorticity, as the vortices convect downstream, by an amount that is more than can be attributed to viscous diffusion. It is found that the presence of small levels of axial velocity gradients within the vortex cores, levels that can be difficult to measure experimentally, can lead to a measurable decrease in the peak vorticity even at the centre of the flow facility in a flow that is expected to be primarily two-dimensional.

We develop a phase-field multiphase lattice Boltzmann model to systematically investigate the dynamic behaviour of a droplet passing through a microfluidic T-junction, especially focusing on the non-breakup of the droplet. Detailed information on the breakup and non-breakup is presented, together with the quantitative evolutions of driving and resistance forces as well as the droplet deformation characteristics involved. Through comparisons between cases of non-breakup and breakup, we find that the appearance of tunnels (the lubricating film between droplet and channel walls) provides a precondition for the final non-breakup of droplets, which slows down the droplet deformation rate and even induces non-breakup. The vortex flow formed inside droplets plays an important role in determining whether they break up or not. In particular, when the strength of vortex flow exceeds a critical value, a droplet can no longer break up. Additionally, more effort has been devoted to investigating the effects of viscosity ratio between disperse and continuous phases and width ratio between branch and main channels on droplet dynamic behaviours. It is found that a large droplet viscosity results in a small velocity gradient in a droplet, which restricts vortex generation and thus produces lower deformation resistance. Consequently, it is easier to break up a droplet with larger viscosity. Our work also reveals that a droplet in small branch channels tends to obstruct the channels and have small vortex flows, which induces easier breakup too. Eventually, several phase diagrams for droplet flow patterns are provided, and the corresponding power-law correlations ($l_{0}/w=\unicode[STIX]{x1D6FD}Ca^{b}$, where $l_{0}/w$ is dimensionless initial droplet length and $Ca$ is capillary number) are fitted to describe the boundaries between different flow patterns.

In Part 1 a study is made of the internal solitary wave on the pycnocline of a continuously stratified fluid. A Korteweg–de Vries (KdV) equation for the ‘interfacial’ displacement is developed following Benney's method for long nonlinear waves. Experiments were conducted in a long wave tank with the pycnocline at several different depths below the free surface, while keeping the total depth approximately constant. A step-like pool of light water, trapped behind a sliding gate, served as the initial disturbance condition. The number of solitons generated was verified to satisfy the prediction of inverse-scattering theory. The fully developed soliton was found to satisfy the KdV theory for all ratios of upper-layer thickness to total depth.

In Part 2 of this study we investigate experimentally the evolution and breaking of an internal solitary wave as it shoals on a sloping bottom connecting the deeper region where the waves were generated to a shallower shelf region. It is found through quantitative measurements that the onset of wave-breaking was governed by shear instability, which was initiated when the local gradient Richardson number became less than ¼. The internal solitary wave of depression was found to steepen at the back of the wave before breaking, in contrast with waves of elevation. Two slopes were used, with ratios 1:16 and 1:9, and the fluid was a Boussinesq fluid with weak stratification using brine solutions.

In this paper we present an experimental study of the long-surface-wave instability that can develop when a granular material flows down a rough inclined plane. The threshold and the dispersion relation of the instability are precisely measured by imposing a controlled perturbation at the entrance of the flow and measuring its evolution down the slope. The results are compared with the prediction of a linear stability analysis conducted in the framework of the depth-averaged or Saint-Venant equations. We show that when the friction law proposed in Pouliquen (1999a) is introduced in the Saint-Venant equations, the theory is able to predict quantitatively the stability threshold and the phase velocity of the waves but fails in predicting the observed cutoff frequency. The instability is shown to be of the same nature as the long-wave instability observed in classical fluids but with characteristics that can dramatically differ due to the specificity of the granular rheology.

In an experimental investigation of the transient processes that occur when a single droplet of butane at the superheat limit vaporizes explosively, short-exposure photographs and fast-response pressure measurements have been used to construct a description of the complete explosion process. It is observed that only a single bubble forms within the drop during each explosion, and that the growth proceeds on a microsecond time scale. An interfacial instability driven by rapid evaporation has been observed on the surface of the bubbles. It is suggested that the Landau mechanism of instability, originally described in connection with the instability of laminar flames, also applies to rapid evaporation at the superheat limit.

The photographic evidence and the pressure data are used to estimate the evapora- tive mass flux across the liquid-vapour interface after the onset of instability. The rate of evaporation is shown to be two orders of magnitude greater than would be predicted by conventional bubble-growth theories that do not account for the effects of instability. An estimate of the mean density within the bubbles during the evaporative stage indicates that it is more than one half of the critical density of butane.

Additional interesting dynamical effects that are observed include a series of toroidal waves that form on the interface between the butane vapour and the external host liquid in the bubble column apparatus after the bubble has grown large enough to contact the outer edge of the drop, and violent oscillations of the bubble that occur on a millisecond time scale, after evaporation of the liquid butane is complete, that cause the disintegration of the bubble into a cloud of tiny bubbles by Rayleigh–Taylor instability.

Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a woven mesh surface at Reynolds numbers approximately equal to those for the smooth wall. Fully rough conditions were achieved. The present work focuses on turbulence structure, as documented through spectra of the fluctuating velocity components, swirl strength, and two-point auto- and cross-correlations of the fluctuating velocity and swirl. The present results are in good agreement, both qualitatively and quantitatively, with the turbulence structure for smooth-wall boundary layers documented in the literature. The boundary layer is characterized by packets of hairpin vortices which induce low-speed regions with regular spanwise spacing. The same types of structure are observed for the rough- and smooth-wall flows. When the measured quantities are normalized using outer variables, some differences are observed, but quantitative similarity, in large part, holds. The present results support and help to explain the previously documented outer-region similarity in turbulence statistics between smooth- and rough-wall boundary layers.

We have applied the methods of classical statistical mechanics to derive the inviscid equilibrium states for one- and two-layer nonlinear quasi-geostrophic flows, with and without bottom topography and variable rotation rate. In the one-layer case without topography we recover the equilibrium energy spectrum given by Kraichnan (1967). In the two-layer case, we find that the internal radius of deformation constitutes an important dividing scale: at scales of motion larger than the radius of deformation the equilibrium flow is nearly barotropic, while at smaller scales the stream functions in the two layers are statistically uncorrelated. The equilibrium lower-layer flow is positively correlated with bottom topography (anticyclonic flow over seamounts) and the correlation extends to the upper layer at scales larger than the radius of deformation. We suggest that some of the statistical trends observed in non-equilibrium flows may be looked on as manifestations of the tendency for turbulent interactions to maximize the entropy of the system.

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

When two liquid drops come into contact, they coalesce rapidly, owing to the large curvature and unbalanced surface-tension forces in the neck region. We use an ultra-high-speed video camera to study the coalescence of a pendent and a sessile drop, over a range of drop sizes and liquid viscosities. For low viscosity, the outward motion of the liquid contact region is successfully described by a dynamic capillary-inertial model based on the local vertical spacing between the two drop surfaces. This model applies even when the drops are of different sizes. Increasing viscosity slows down the coalescence when the Reynolds number $\hbox{\it Re}_v \,{=}\,\rho R_{\hbox{\scriptsize\it ave}}\sigma/\mu^2\,{<}\,5000$, where $R_{\hbox{\scriptsize\it ave}}$ is the average of the tip radii of the two similar size drops, $\rho$ is the liquid density, $\sigma$ is the surface tension and $\mu$ the dynamic viscosity. At $\hbox{\it Re}_v\,{\simeq}\,50$, the growth-rate of the neck radius has reduced by a half, which for water corresponds to a drop diameter of only 2\,$\umu$m. For the largest viscosities, the neck region initially grows in size at a constant velocity. The neck curvature also becomes progressively sharper with increasing viscosity. The results are compared to previously predicted power laws, finding slight, but significant deviations from the predicted exponents. These deviations are most probably caused by the finite initial contact radius.

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to a uniform electric field. Both liquids are taken to be leaky dielectrics. The jump in electrical properties creates an electric stress balanced by hydrodynamic and capillary stresses. Assuming creeping flow conditions and axisymmetry of the problem, the electric and flow fields are solved numerically withboundary integral techniques. The system is characterized by the physical property ratios R (resistivities), Q (permitivities) and λ (dynamic viscosities). Depending on these parameters, the drop deforms into a prolate or an oblate spheroid. The relative importance of the electric stress and of the drop/medium interfacial tension is measured by the dimensionless electric capillary number, Cae. For λ = 1, we present a survey of the various behaviours obtained for a wide range of R and Q. We delineate regions in the (R,Q)-plane in which the drop either attains a steady shape under any field strength or reaches a fold-point instability past a critical Cae. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Various break-up modes are identified, as well as more complex behaviours such as bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast can stabilize the drop or advance break-up in the different situations encountered for λ = 1.

Direct numerical simulations (DNS) are presented of three-dimensional, Rayleigh–Taylor instability (RTI) between two incompressible, miscible fluids, with a 3:1 density ratio. Periodic boundary conditions are imposed in the horizontal directions of a rectangular domain, with no-slip top and bottom walls. Solutions are obtained for the Navier–Stokes equations, augmented by a species transport-diffusion equation, with various initial perturbations. The DNS achieved outer-scale Reynolds numbers, based on mixing-zone height and its rate of growth, in excess of 3000. Initial growth is diffusive and independent of the initial perturbations. The onset of nonlinear growth is not predicted by available linear-stability theory. Following the diffusive-growth stage, growth rates are found to depend on the initial perturbations, up to the end of the simulations. Mixing is found to be even more sensitive to initial conditions than growth rates. Taylor microscales and Reynolds numbers are anisotropic throughout the simulations. Improved collapse of many statistics is achieved if the height of the mixing zone, rather than time, is used as the scaling or progress variable. Mixing has dynamical consequences for this flow, since it is driven by the action of the imposed acceleration field on local density differences.

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value of R does not depend on S (for some range of S) or vice versa. This situation pertains when the critical horizontal wave-number is zero.

A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.

Thrust performance and wake structure were investigated for a rigid rectangular panel pitching about its leading edge in a free stream. For ReC = O(104), thrust coefficient was found to depend primarily on Strouhal number St and the aspect ratio of the panel AR. Propulsive efficiency was sensitive to aspect ratio only for AR less than 0.83; however, the magnitude of the peak efficiency of a given panel with variation in Strouhal number varied inversely with the amplitude to span ratio A/S, while the Strouhal number of optimum efficiency increased with increasing A/S. Peak efficiencies between 9% and 21% were measured. Wake structures corresponding to a subset of the thrust measurements were investigated using dye visualization and digital particle image velocimetry. In general, the wakes divided into two oblique jets; however, when operating at or near peak efficiency, the near wake in many cases represented a Kármán vortex street with the signs of the vortices reversed. The three-dimensional structure of the wakes was investigated in detail for AR = 0.54, A/S = 0.31 and ReC = 640. Three distinct wake structures were observed with variation in Strouhal number. For approximately 0.20 < St < 0.25, the main constituent of the wake was a horseshoe vortex shed by the tips and trailing edge of the panel. Streamwise variation in the circulation of the streamwise horseshoe legs was consistent with a spanwise shear layer bridging them. For St > 0.25, a reorganization of some of the spanwise vorticity yielded a bifurcating wake formed by trains of vortex rings connected to the tips of the horseshoes. For St > 0.5, an additional structure formed from a perturbation of the streamwise leg which caused a spanwise expansion. The wake model paradigm established here is robust with variation in Reynolds number and is consistent with structures observed for a wide variety of unsteady flows. Movies are available with the online version of the paper.

An approximate numerical method for calculating flow profiles in arteries is developed. The theory takes into account the nonlinear terms of the Navier-Stokes equations as well as the nonlinear behaviour and large deformations of the arterial wall. Through the locally measured values of the pressure, pressure gradient and pressure–radius function the velocity distribution and wall shear at a given location along the artery can be determined. The computed results agree well with the corresponding experimental data.