We present both numerical and analytical results from a spatial stability analysis of the coupled gas–liquid hydrodynamic equations governing the first wind-induced (FWI) liquid-jet break-up regime. Our study shows that an accurate evaluation of the growth rate of instabilities developing in a liquid jet discharging into a still gaseous atmosphere requires gas viscosity to be included in the stability equations even for low ${\it We}_g$, where ${\it We}_g{=}\rho_gU_l^2R_0/\sigma$, and $\rho_g, U_l, R_0$ and $\sigma$ are the gas density, the liquid injection velocity, the jet radius and the surface tension coefficient, respectively. The numerical results of the complete set of equations, in which the effect of viscosity in the gas perturbations is treated self-consistently for the first time, are in accordance with recently reported experimental growth rates. This permits us to conclude that the simple stability analysis presented here can be used to predict experimental results. Moreover, in order to throw light on the physical role played by the gas viscosity in the liquid-jet break-up process, we have considered the limiting case of very high Reynolds numbers and performed an asymptotic analysis which provides us with a parameter, $\alpha$, that measures the relative importance of viscous effects in the gas perturbations. The criterion $|\alpha|{\ll} 1$, with $\alpha$ computed a priori using only the much simpler inviscid stability results is a guide to assess the accuracy of a stability analysis in which viscous diffusion is neglected. We have also been able to explain the origin of the ad hoc constant 0.175 introduced by Sterling & Sleicher (J. Fluid Mech. vol. 68, 1975, p. 477) to correct the discrepancies between Weber's results (Z. Angew. Math. Mech. vol. 11, 1931, p. 136) and the experimental ones.

This work can be viewed as a reprise of Head & Bandyopadhyay's (J. Fluid Mech. vol. 107, p. 297) original boundary-layer visualization study although in this instance we make use of stereo particle image velocimetry (PIV), techniques to obtain a quantitative view of the turbulent structure. By arranging the laser light-sheet and image plane of a stereo PIV system in inclined spanwise/wall-normal planes (inclined at both $45^{\circ}$ and $135^{\circ}$ to the streamwise axis) a unique quantitative view of the turbulent boundary layer is obtained. Experiments are repeated across a range of Reynolds numbers, $Re_{\tau} \,{\approx}\, 690\hbox{--}2800$. Despite numerous experimental challenges (due to the large out-of-plane velocity components), mean flow and Reynolds stress profiles indicate that the salient features of the turbulent flow have been well resolved. The data are analysed with specific attention to a proposed hairpin eddy model. In-plane two-dimensional swirl is used to identify vortical eddy structures piercing the inclined planes. The vast majority of this activity occurs in the $135^{\circ}$ plane, indicating an inclined eddy structure, and Biot-Savart law calculations are carried out to aid in the discussion. Conditional averaging and linear stochastic estimation results also support the presence of inclined eddies, arranged about low-speed regions. In the $135^{\circ}$ plane, instantaneous swirl patterns exhibit a predisposition for counter-rotating vortex pairs (arranged with an ejection at their confluence). Such arrangements are consistent with the hairpin packet model. Correlation and scaling results show outer-scaling to be the correct way to quantify the characteristic spanwise length scale across the log and wake regions of the boundary layers (for the range of Reynolds numbers tested). A closer investigation of two-point velocity correlation contours indicates the occurrence of a distinct two-regime behaviour, in which contours (and hence streamwise velocity fluctuations) either appear to be ‘attached’ to the buffer region, or ‘detaching’ from it. The demarcation between these two regimes is found to scale well with outer variables. The results are consistent with a coherent structure that becomes increasingly uncoupled (or decorrelated) from the wall as it grows beyond the logarithmic region, providing additional support for a wall–awake description of turbulent boundary layers.

We present two classes of exact solutions of the Navier–Stokes equations, which describe steady vortex structures with two-dimensional symmetry in an infinite fluid. The first is a class of similarity solutions obtained by conformal mapping of the Burgers vortex sheet to produce wavy sheets, stars, flowers and other vorticity patterns. The second is a class of non-similarity solutions obtained by continuation and mapping of the classical solution to steady advection–diffusion around a finite circular absorber in a two-dimensional potential flow, resulting in more complicated vortex structures that we describe as avenues, fishbones, wheels, eyes and butterflies. These solutions exhibit a transition from ‘clouds’ to ‘wakes’ of vorticity in the transverse flow with increasing Reynolds number. Our solutions provide useful test cases for numerical simulations, and some may be observable in experiments, although we expect instabilities at high Reynolds number. For example, vortex avenues may be related to counter-rotating vortex pairs in transverse jets, and they may provide a practical means to extend jets from dilution holes, fuel injectors, and smokestacks into crossflows.

We investigate the aerodynamics of freely falling plates in a quasi-two-dimensional flow at Reynolds number of $10^{3}$, which is typical for a leaf or business card falling in air. We quantify the trajectories experimentally using high-speed digital video at sufficient resolution to determine the instantaneous plate accelerations and thus to deduce the instantaneous fluid forces. We compare the measurements with direct numerical solutions of the two-dimensional Navier–Stokes equation. Using inviscid theory as a guide, we decompose the fluid forces into contributions due to acceleration, translation, and rotation of the plate. For both fluttering and tumbling we find that the fluid circulation is dominated by a rotational term proportional to the angular velocity of the plate, as opposed to the translational velocity for a glider with fixed angle of attack. We find that the torque on a freely falling plate is small, i.e. the torque is one to two orders of magnitude smaller than the torque on a glider with fixed angle of attack. Based on these results we revise the existing ODE models of freely falling plates. We get access to different kinds of dynamics by exploring the phase diagram spanned by the Reynolds number, the dimensionless moment of inertia, and the thickness-to-width ratio. In agreement with previous experiments, we find fluttering, tumbling, and apparently chaotic motion. We further investigate the dependence on initial conditions and find brief transients followed by periodic fluttering described by simple harmonics and tumbling with a pronounced period-two structure. Near the cusp-like turning points, the plates elevate, a feature which would be absent if the lift depended on the translational velocity alone.

We study the dynamics of a rigid card falling in air using direct numerical simulations of the two-dimensional Navier–Stokes equation and a fluid force model based on ordinary differential equations derived from recent experiments and simulations. The system depends on three non-dimensional parameters, i.e. the thickness-to-width ratio, the dimensionless moment of inertia, and the Reynolds number. By increasing the thickness-to-width ratio in the direct numerical simulations and thereby the non-dimensional moment of inertia we observe a transition from periodic fluttering to periodic tumbling with a wide transition region in which the cards flutter periodically but tumble once between consecutive turning points. In the transition region the period of oscillation diverges and the cards fall vertically for distances of up to 50 times the card width. We analyse the transition between fluttering and tumbling in the ODE model and find a heteroclinic bifurcation which leads to a logarithmic divergence of the period of oscillation at the bifurcation point. We further discuss the bifurcation scenario of the ODE model in relation to our direct numerical simulations and the phase diagrams measured by willmarth, Hawk & Harvey (1964) and belmonte, Eisenberg & Moses (1998).

The effect of a plane wall on the orbiting motion of a freely suspended prolate or oblate spheroidal particle in simple shear flow is described by numerical simulation. The particle translational and angular velocities are computed directly by solving an integral equation of the second kind using a spectral boundary-element method for Stokes flow. The presence of the wall modifies the Jeffery orbits executed by the particle director, and causes the particle centre to perform a lateral and transverse periodic motion deviating from the straight longitudinal path. The effect of the wall is moderate for prolate particles and most significant for oblate particles. In the second case, the complete orbiting motion is suppressed when the particle is sufficiently close to the wall, and the particle director precesses around an axis that is tilted with respect to the vorticity and direction of the simple shear flow, revealing that a neutrally stable steady orientation is established when the particle is in contact with the wall.

The effects of an insoluble surfactant on the linear stability of a two-fluid core–annular flow in the thin annulus limit, for axisymmetric disturbances with wavelengths large compared to the annulus thickness, $h_{0}$, are the focus of this investigation. A base shear flow affects the interfacial surfactant distribution, thereby inducing Marangoni forces that, along with capillary forces, affect the fluid–fluid interface stability. The resulting system's stability differs markedly from that of the same system with zero base flow. In the thin-annulus limit (the ratio $\varepsilon$ of the undisturbed annulus thickness to core radius tends to zero), common in applications, a scaling and asymptotic analysis yields a coupled set of equations for the perturbed fluid–fluid interface shape and surfactant concentration. The linear dynamics of the annular film fully determine these equations, i.e. the core dynamics are slaved to the film dynamics. The theory provides a unified view of the mechanism of stability in three different regimes of capillary number ${\it Ca}$ (defined as the product of the core viscosity, $\mu _1$, and the centreline velocity, $W_0 $, divided by the interface tension, $\sigma _0^\ast$, that corresponds to an undisturbed (signified by the subscript 0) uniform surfactant concentration, $\Gamma _0^\ast$). In the absence of a base flow or in the limit of small ${\it Ca}({\ll}\varepsilon ^{2})$, Marangoni forces deriving from non-uniformities in the interface concentration of insoluble surfactants oppose the net capillary forces. These latter forces normally stabilize the longitudinal curvature and destabilize the circumferential curvature of perturbations to the interface. In the limit of large ${\it Ca}({\gg}\varepsilon ^{2})$, Marangoni forces destabilize disturbances with wavelengths that are large compared to the annulus thickness. For moderately small ${\it Ca}({\sim} \varepsilon^2)$, increasing the Marangoni number Ma (defined as the product of $(-\partial\sigma^\ast/\partial \Gamma^\ast )_0$ and $\Gamma_0^\ast $, divided by $\mu _1 W_0 )$ from zero increases the growth rates of all disturbances (with wavelengths ${\gg}h_{0})$ and, consequently, reduces the marginal wavelength below that typical of the capillary instability. However a further increase in Ma eventually reverses these trends. A very large value for Ma stiffens the interface, which opposes any local variation of the tangential velocity along the interface, and this is true whether or not there is a base flow. In the limit of infinite Ma, the growth rate of the instability is 1/4 of that of the clean interface and the marginal wavenumber, non-dimensionalized by the undisturbed core circumference, returns to its clean interface (capillary) value of 1. All trends are explained physically.

In the present studies, the effects of the end conditions of a circular cylinder on its wake at a fairly high Reynolds number of $Re = 1.57\times 10^4$ were studied. The transverse control cylinder technique (TCCT) was previously reported to be able to induce parallel vortex shedding at $Re = O(10^2)$. In the present work, experimental results showed that the TCCT is still effective in inducing parallel vortex shedding at $Re = O(10^4)$. Initially, before the inclusion of the control cylinders, vortices shed by the main cylinder were curved (all shapes referred to are time-averaged shapes) owing to the influence of the cylinder end conditions. Later, two larger control cylinders of diameter D were included and were located normal and upstream of the main cylinder near its ends to change its end conditions. By manipulating the control distance (the gap between the control cylinders and the main cylinder), different vortex-shedding patterns could be induced. With both control cylinders fixed at the optimum control distance of $L_1 = L_2 = L_0 = 1.26D$, the main cylinder was induced to shed parallel vortices. For the cases of curved vortex shedding (without control cylinders) and parallel vortex shedding (with control cylinders at the optimum distance of $L_1 = L_2 = L_0 = 1.26D)$, various aerodynamic parameters of the main cylinder were measured and compared. Results showed that the inclusion of the control cylinders speeded up the flow velocity at the ends of the main cylinder and led to a more uniform pressure distribution over the central span of the main cylinder, which finally resulted in parallel vortex shedding. Aerodynamic parameters such as drag coefficient and Strouhal number associated with parallel vortex shedding were found to be larger than their curved shedding counterparts. However, extra caution should be exercised in interpreting their implications as these data were under the influence of additional wind-tunnel blockage caused by the presence of the control cylinders. Preliminary and approximate calculations had shown that blockage effects were likely to be responsible for a significant part in the change in the aerodynamic parameters such as the drag coefficient and Strouhal number when the control cylinders were installed. When the control cylinders were symmetrically placed, but not at the optimum distance $(L_1 = L_2\ne L_0)$, the vortex-shedding pattern became curved, and was concave or convex downstream at $L_1 = L_2 < L_0$ or $L_1 = L_2 > L_0$, respectively. When the control cylinders were asymmetrically placed $(L_1\ne L_2)$, oblique vortex shedding was induced, with the oblique vortex slanting in the same way as the straight line joining the centres of the control cylinders. The relation between the Strouhal numbers for parallel and oblique vortex shedding was found to still follow the cosine law. The present work confirms earlier finding by other workers that a non-uniform spanwise base pressure distribution was the cause of spanwise base flow, which led to curved or oblique vortex shedding.

In this paper we study the steady uniform flows that develop when granular material is released from a hopper on top of a static pile in a channel. More specifically, we focus on the role of sidewalls by carrying out experiments in set-up of different widths, from narrow channels 20 particle diameters wide to channels 600 particle diameters wide. Results show that steady flows on pile are entirely controlled by sidewall effects. A theoretical model, taking into account the wall friction and based on a simple local constitutive law recently proposed for other granular flow configurations, gives predictions in quantitative agreement with the measurements. This result gives new insights into our understanding of free-surface granular flows and strongly supports the relevance of the constitutive law proposed.

We present time-dependent governing equations and boundary conditions for the mushy-zone free-boundary problem that are valid in an arbitrary frame of reference. The model for time-evolving mushy zones is more complicated than in the steady case because the interface velocity $\bm{w}$ can be distinct from both the velocity of the dendrites $\bm{v}$ and the fluid velocity $\bm{u}$. We consider the limit of negligible solutal diffusivity, where there are four types of boundary condition at the mush–liquid interface, depending on both the direction of flow across the interface and the direction of the interface motion relative to the solid phase. We illustrate these boundary conditions by examining a family of one-dimensional problems in which a binary material is chilled from a fixed cold point in the laboratory frame of reference while fluid is pumped through the resulting mushy layer at a rate $Q$ and the mushy layer itself is translated at a rate $V$. This allows us to exhibit three of the four types of mushy-layer interfaces. We show that the fourth type cannot occur in this scenario.

Although potential flow, including viscous dissipation, explains quite well the flow around individual bubbles of about 1mm radius rising in water, and e.g. predicts their drag quite accurately, this model cannot explain the homogeneous rise of a bubbly suspension. From numerical and analytical work it follows that eventually all bubbles cluster together. On the other hand it has been shown that velocity fluctuations of the bubbles of sufficient intensity, expressed in terms of a critical (pseudo) temperature, prevents clustering.

Bubbles with radius above 0.8 mm rising in water perform zigzag or spiralling motions. Recently experimental and numerical work has made it clear that such bubbles have a wake behind them consisting of twin vortical threads carrying vorticity of opposite sign in the direction of motion. It is the purpose of this contribution to make an estimate of the velocity fluctuations induced by these trailing vortices in neighbouring bubbles. To this end the two-threaded wake is represented as a horseshoe vortex similar to the wake behind an airfoil. A pair of bubbles is considered and first the velocity induced by the horseshoe vortex behind one of the pair at the centre of the other is calculated. After this the force exerted on the latter based on the induced velocity and on the relative velocity of the bubbles, due to hydrodynamic interaction is calculated. Then the motion of one bubble in the pair is analysed under the influence both of this force and the hydrodynamic forces already there in the absence of the horseshoe vortex. Using these results and appropriate averaging, an estimate is made of the intensity of the velocity fluctuations of bubbles, and the corresponding temperature.

This paper presents a combined numerical and analytical investigation into various problems involving two symmetric interacting constant-pressure bubbles evolving in two-dimensional Stokes flow. The bubbles have constant surface tension on their boundaries and are taken to be in an ambient straining flow. First, a novel numerical method based on conformal mappings is presented to compute the free-surface evolution. Then, a special class of time-evolving exact solutions to the problem is derived and used to check the numerical code. These solutions reveal that, for bubbles with shrinking area, a competition between the imposed strain and surface tension can lead to either a slit or a point as the limiting shape. Numerical solutions of fixed-area bubbles are then computed and reveal that when they are forced together by a straining flow, a thin lubrication layer forms. In the absence of surface tension, large-curvature regions develop at the bubble edges and these are smoothed out by capillary effects. Further, motivated by the viscous sintering application, a study of interaction effects on the pure surface-tension-driven shrinkage of circular bubbles is investigated and compared, in an appropriate limit, to a recently derived ‘elliptical-pore model’.

Hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls are studied under creeping-flow conditions. The many-particle friction matrix in this system is evaluated using our novel numerical algorithm based on transformations between Cartesian and spherical representations of Stokes flow. The cartesian representation is used to describe the interaction of the fluid with the walls and the spherical representation is used to describe the interaction with the particles. The transformations between these two representations are given in a closed form, which allows us to evaluate the coefficients in linear equations for the induced-force multipoles on particle surfaces. the friction matrix is obtained from these equations, supplemented with the superposition lubrication corrections. we have used our algorithm to evaluate the friction matrix for a single sphere, a pair of spheres, and for linear chains of spheres. The friction matrix exhibits a crossover from a quasi-two-dimensional behaviour (for systems with small wall separation H) to THE three-dimensional behaviour (when the distance H is much larger than the interparticle distance L). the crossover is especially pronounced for a long chain moving in the direction normal to its orientation and parallel to the walls. in this configuration, a large pressure build-up occurs in front of the chain for small values of the gapwidth H, which results in a large hydrodynamic friction force. a standard wall superposition approximation does not capture this behaviour.

We report experiments on the shape and motion of millimetre-sized drops sliding down a plane in a situation of partial wetting. When the Bond number based on the component of gravity parallel to the plane $\Bo_{\alpha}$ exceeds a threshold, the drops start moving at a steady velocity which increases linearly with $\Bo_{\alpha}$. When this velocity is increased by tilting the plate, the drops change their aspect ratio: they become longer and thinner, but maintain a constant, millimetre-scale height. As their aspect ratio changes, a threshold is reached at which the drops are no longer rounded but develop a ‘corner’ at their rear: the contact line breaks into two straight segments meeting at a singular point or at least in a region of high contact line curvature. This structure then evolves such that the velocity normal to the contact line remains equal to the critical value at which the corner appears, i.e. to a maximal speed of dewetting. At even higher velocities new shape changes occur in which the corner changes into a ‘cusp’, and later a tail breaks into smaller drops (pearling transition). Accurate visualizations show four main results. (i) The corner appears when a critical non-zero value of the receding contact angle is reached. (ii) The interface then has a conical structure in the corner regime, the in-plane and out-of-plane angles obeying a simple relationship dictated by a lubrication analysis. (iii) The corner tip has a finite non-zero radius of curvature at the transition to a corner, and its curvature diverges at a finite capillary number, just before the cusp appears. (iv) The cusp transition occurs when the corner opening in-plane half-angle reaches a critical value of about 45°.

Motivated by applications in rapidly rotating machinery, we have previously extended the lubrication model of the thin-film flow on the inside of a rotating circular cylinder to incorporate the effect of a constant shear applied to the free surface of the film and discovered a system rich in film profiles featuring shock structures. In this paper, we extend our model to include the effects of surface tension at leading order and take into account higher-order effects produced by gravity in order to resolve issues regarding existence, uniqueness and stability of such weak solutions to our lubrication model. We find, by analytical and numerical means, a set of feasible steady two-dimensional solutions that fit within a rational asymptotic framework. Having identified mathematically feasible solutions, we study their stability to infinitesimal two-dimensional disturbances. based on our findings, we conjecture which of the possible weak solutions are physically meaningful.

The problem of short wind waves propagating on surface wind drift is considered here. Under the assumption of small monochromatic surface waves on a steady and horizontally uniform surface shear of an inviscid fluid, the governing equation becomes the well-known Rayleigh instability equation. Perturbation solutions exist for the surface-wave problem; however, the conditions for these approximations are violated in the case of short wind waves on wind drift shear. As an alternative approach, the piecewise-linear approximation (PLA) is explored. A proof is given for the rate of convergence of the piecewise-linear approximation for solving the Rayleigh equation without limitations on boundary conditions. The artificial modes of the piecewise-linear flow system are also discussed. The method is numerically efficient and highly accurate. Applying this method, the linear instability of various boundary-layer flows is examined. Short waves propagating with surface shear-flows are stable, while it is possible for waves that are travelling against a shear current to become unstable when the surface speed of the shear is greater than the wavespeed in stagnant fluid. PLA is also applied to examine the applicability of other perturbation approaches to the problem of propagation of short waves on wind drift shear. It is found that the existing approximations cannot fit the whole range of short wind waves. To bridge the gap, new approximations are derived from an implicit form of the exact dispersion relation based upon the variational principle.

An expression for the rate-of-strain tensor on a rigid surface due to Caswell is generalized to an arbitrarily moving and continuously deforming surface or interface between two immiscible fluids. Corresponding expressions for the velocity gradient and vorticity tensors are derived in an inertial frame of reference. A noteworthy feature of the expression for the rate-of-strain tensor is the presence of a tangent-tangent component, which is absent in the case of a rigid surface. Kinematic applications based on numerical solutions of the Navier–Stokes equation for laminar and turbulent flow demonstrate the significance and implications of the derived expressions.

High-Grashof-number turbulent natural convection in the vicinity of vertical walls with heat transfer is analysed asymptotically. The near-wall boundary layer has a viscosity-influenced inner layer and a fully turbulent outer layer, similar to the structure of forced convection boundary layers. Scaling laws and wall functions are found by asymptotic matching of the temperature gradients in the overlap layer. The temperature wall function then is a simple logarithmic function of wall distance whereas the velocity profile in the overlap layer is a more complex correlation. Constants in these wall functions are deduced from high-quality data for large Grashof numbers. Experimental as well as numerical profiles as a whole are very well reproduced by the combination of wall functions and viscous sublayer profiles. Therefore these new asymptotic profiles can be used in CFD codes to avoid very fine grids close to the wall, when Grashof numbers are high.

We consider the evolution of a thin viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions for rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location.