Equilibrium shapes and stability of annular fluid menisci held together by surface tension are analysed by applying asymptotic and computer-aided techniques from bifurcation theory. The shapes and locations of the menisci are governed by the Young–Laplace equation. These shapes are grouped together into families of like symmetry that branch from the basic family of annular shapes at specific values of the aspect ratio, α. Multiple equilibrium shapes exist over certain values of α. The inner, outer or both the inner and outer interfaces may possess either a cylindrical or sinusoidal equilibrium shape. Changes in applied pressure, fluid volume, or gravitational Bond number break families of the same symmetry which now develop limit points. Numerical calculations rely on a finite-element representation of the interfaces and the results compare very well with asymptotic analysis which is valid for small deformations. The results are important for the blow moulding process and are invaluable in understanding its dynamics. These dynamics are expected to be considerably different from the dynamics of a liquid jet first analysed by Rayleigh.

A numerical and experimental study has been made of the three-dimensional flow and heat transfer by natural convection in a closed, rectangular thermosyphon. At low Rayleigh numbers, the flows in the two halves of the cavity remain separate, with heat transfer across the mid-height plane occurring only by conduction. At increasing Rayleigh numbers, an exchange process of increasing complexity occurs. The numerical solutions were used to explore this process and to predict flow patterns which were found to resemble closely those observed during previous investigations of cylindrical thermosyphons. The results were verified by a flow visualization study. Heat-transfer rates are presented and augment previous data for higher values of the governing parameters.

The parabolic approximation which has recently been found to be useful in other physical contexts, is extended to head-sea diffraction of short waves by a slender raft on deep water. In particular, it is a much more direct way of getting the inner approximation of the outer solution in a scheme of matched asymptotics than the original method of Faltinsen (1971). The present results are compared with a more involved integral equation method and are found to be remarkably accurate even when the raft length is comparable to the wavelength. Finally, the asymptotic method is modified for a compliant raft which absorbs wave power by suitably controlled impedance. Optimum efficiency and other performance characteristics are predicted.

The development of secondary instability on streamwise vortex structures generated in a hypersonic shock layer on a flat plate is experimentally studied for the flow with Mach number M∞ = 21 and unit Reynolds number Re1 = 6 × 105 m−1. The study is performed using the electron-beam method. The generation of weak unsteady vortices and steady streamwise vortex structures with finite-amplitude perturbations imposed onto them is studied in detail. Complex data on the characteristics of density fluctuations developed on quasi-steady and unsteady streamwise vortex structures are obtained. It is shown that the characteristics of the natural fluctuations of density developing in the shock layer on a flat plate are qualitatively similar to density fluctuations induced by weak unsteady vortex perturbations introduced into the shock layer. The possibility of existence of parametric resonance between the fundamental frequency and its harmonic and between harmonics for steady streamwise vortex structure is shown.

The linear stability of finite-amplitude surface solitary waves with respect to long-wavelength transverse perturbations is examined by asymptotic analysis for small wavenumbers of perturbations. The sufficient condition for the transverse instability is explicitly derived, and it is found that there exist transversely unstable surface solitary waves whose amplitude-to-depth ratio is greater than 0.713. This critical ratio is well below that for the one-dimensional instability (${=}\,0.781$) obtained by Tanaka (Phys. Fluids, 1986, vol. 29, p. 650).

The formation and transient dynamics of Stewartson layers are examined herein. The results are applied to a transient rotating fluid flow for which (i) Stewartson layers are presumed to be important, and (ii) the Ekman layers are found to be non-divergent, and hence unable to induce the crucial secondary circulation occurring in the spin-up process. The steady state is nevertheless reached after a spin-up time.

Tropical cyclogenesis is studied in the context of idealized three-dimensional Boussinesq dynamics with perhaps the simplest possible model for bulk cloud physics. With low-altitude input of water vapour on realistic length and time scales, numerical simulations capture the formation of vortical hot towers. From measurements of water vapour, vertical velocity, vertical vorticity and rain, it is demonstrated that the structure, strength and lifetime of the hot towers are similar to results from models including more detailed cloud microphysics. The effects of low-altitude vertical shear are investigated by varying the initial zonal velocity profile. In the presence of weak low-level vertical shear, the hot towers retain the low-altitude monopole cyclonic structure characteristic of the zero-shear case (starting from zero velocity). Some initial velocity profiles with small vertical shear can have the effect of increasing cyclonic predominance of individual hot towers in a statistical sense, as measured by the skewness of vertical vorticity. Convergence of horizontal winds in the atmospheric boundary layer is mimicked by increasing the frequency of the moisture forcing in a horizontal subdomain. When the moisture forcing is turned off, and again for zero shear or weak low-level shear, merger of cyclonic activity results in the formation of a larger-scale cyclonic vortex. An effect of the shear is to limit the vertical extent of the resulting emergent moist vortex. For stronger low-altitude vertical shear, the individual hot towers have a low-altitude vorticity dipole rather than a cyclonic monopole. The dipoles are not conducive to the formation of larger-scale vortices, and thus sufficiently strong low-level shear prevents the vortical-hot-tower route to cyclogenesis. The results indicate that the simplest condensation and evaporation schemes are useful for exploratory numerical simulations aimed at better understanding of competing effects such as low-level moisture and vertical shear.

Oscillatory flow around a circular cylinder close to a plane boundary is numerically investigated at low-to-intermediate Keulegan–Carpenter ($KC$) and Stokes numbers ($\unicode[STIX]{x1D6FD}$) for different gap-to-diameter ratios ($e/D$). A set of unique flow regimes is observed and classified based on the established nomenclature in the ($KC,\unicode[STIX]{x1D6FD}$)-space. It is found that the flow is not only influenced by $e/D$ but also by the ratio of the thickness of the Stokes boundary layer ($\unicode[STIX]{x1D6FF}$) to the gap size (e). At relatively large $\unicode[STIX]{x1D6FF}/e$ values, vortex shedding through the gap is suppressed and vortices are only shed from the top of the cylinder. At intermediate values of $\unicode[STIX]{x1D6FF}/e$, flow through the gap is enhanced, resulting in horizontal gap vortex shedding. As $\unicode[STIX]{x1D6FF}/e$ is further reduced below a critical value, the influence of $\unicode[STIX]{x1D6FF}/e$ becomes negligible and the flow is largely dependent on $e/D$. A hysteresis phenomenon is observed for the transitions in the flow regime. The physical mechanisms responsible for the hysteresis and the variation of marginal stability curves with $e/D$ are explored at $KC=6$ through specifically designed numerical simulations. The Stokes boundary layer over the plane boundary is found to be responsible for the relatively large hysteresis range over $0.25<e/D<1.0$. Three mechanisms have been identified to the change of the marginal stability curve over $e/D$, which are the blockage effect due to the geometry setting, the favourable pressure gradient over the gap and the location of the leading eigenmode relative to the cylinder.

We study the conditions under which discontinuous mechanical properties of a collapsible tube can induce transcritical flows, i.e. the transition through the critical state where the speed index (analogous to the Mach or the Froude numbers for compressible and free surface flows, respectively) is one. Such a critical transition may strongly modify the flow properties, cause a significant reduction in the cross-sectional area of the tube, and limit the flow. General relationships are obtained for a short segment using a one-dimensional model under steady flow conditions. Marginal curves delimiting the transcritical regions are identified in terms of the speed index and the cross-sectional area ratio. Since there are many examples of such flows in physiology and medicine, we also analyse the specific case of prosthesis (graft or stent) implantation in blood vessels. We then compute transcritical conditions for the case of stiffness and reference area variations, considering a collapsible tube characterized by physiological parameters representative of both arteries and veins. The results suggest that variations in mechanical properties may induce transcritical flow in veins but is unrealistic in arteries.

We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate, and (ii) the full Stokes equations. In either case, the plate is stabilized by a combination of viscous resistance, surface tension and buoyancy relative to an underlying denser fluid. In the limit of vanishing thickness, plates buckle at a shear rate independent of buoyancy, where is the plate thickness, is the average surface tension between the upper and lower surfaces, and is the fluid viscosity. For thicker plates stabilized by an equal surface tension at the upper and lower surfaces, at and above onset, the most unstable mode has moderate wavelength, is stationary in the frame of the centreline, spans the width of the plate with crests and troughs aligned at approximately to the walls, and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) of approximately 0.6 and are stabilized only by internal viscous resistance. We show that the viscous plate model can only accurately describe the onset of buckling for vanishingly thin plates but provides an excellent description of the most unstable mode above onset. Finally, we show that, by modifying the plate model to incorporate advection and make the model material-frame-invariant, it is possible to extend its predictive power to describe relatively short, travelling waves.

Ray and Wentzel–Kramers–Brillouin (WKB) approximations have long been important tools in understanding and modelling propagation of atmospheric waves. However, contradictory claims regarding the applicability and uniqueness of the WKB approximation persist in the literature. Here, we consider linear acoustic–gravity waves (AGWs) in a layered atmosphere with horizontal winds. A self-consistent version of the WKB approximation is systematically derived from first principles and compared to ad hoc approximations proposed earlier. The parameters of the problem are identified that need to be small to ensure the validity of the WKB approximation. Properties of low-order WKB approximations are discussed in some detail. Contrary to the better-studied cases of acoustic waves and internal gravity waves in the Boussinesq approximation, the WKB solution contains the geometric, or Berry, phase. The Berry phase is generally non-negligible for AGWs in a moving atmosphere. In other words, knowledge of the AGW dispersion relation is not sufficient for calculation of the wave phase.

This paper considers the nonlinear optimal control of transition in a boundary layer flow subjected to a pair of free stream vortical perturbations using a receding horizon approach. The optimal control problem is solved using the Lagrange variational technique that results in a set of linearized adjoint equations, which are used to obtain the optimal wall actuation (blowing and suction from a control slot located in the transition region). The receding horizon approach enables the application of control action over a longer time period, and this allows the extraction of time-averaged statistics as well as investigation of the control effect downstream of the control slot. The results show that the controlled flow energy is initially reduced in the streamwise direction and then increased because transition still occurs. The distribution of the optimal control velocity responds to the flow activity above and upstream of the control slot. The control effect propagates downstream of the slot and the flow energy is reduced up to the exit of the computational domain. The mean drag reduction is $55\,\%$ and $10\,\%$ in the control region and downstream of the slot, respectively. The control mechanism is investigated by examining the second-order statistics and the two-point correlations. It is found that in the upstream (left) side of the slot, the controller counteracts the near-wall high-speed streaks and reduces the turbulent shear stress; this is akin to opposition control in channel flow, and because the time-average control velocity is positive, it is more similar to blowing-only opposition control. In the downstream (right) side of the slot, the controller reacts to the impingement of turbulent spots that have been produced upstream and inside the boundary layer (top–bottom mechanism). The control velocity is positive and increases in the streamwise direction, and the flow behaviour is similar to that of uniform blowing.

Viscoplastic fluids do not flow unless they are sufficiently stressed. This property can be exploited in order to produce novel flow features. One example of such flows is viscoplastically lubricated (VPL) flow, in which a viscoplastic fluid is used to stabilize the interface in a multi-layer flow, far beyond what might be expected for a typical viscous–viscous interface. Here we extend this idea by considering the encapsulation of droplets within a viscoplastic fluid, for the purpose of transportation, e.g. in pipelines. The main advantage of this method, compared to others that involve capillary forces is that significantly larger droplets may be stably encapsulated, governed by the length scale of the flow and yield stress of the encapsulating fluid. We explore this set-up both analytically and computationally. We show that sufficiently small droplets are held in the unyielded plug of a Poiseuille flow (pipe or plane channel). As the length or radius of the droplets increases, the carrier fluid eventually yields, potentially breaking the encapsulation. We study this process of breaking and give estimates for the limiting size of droplets that can be encapsulated.

Two-dimensional direct numerical simulation and Floquet stability analysis have been performed at moderate Keulegan–Carpenter number ($KC$) and low Reynolds number ($Re$) for a square cross-section cylinder with its face normal to the oscillatory flow. Based on the numerical simulations a map of flow regimes is formed and compared to the map of flow around an oscillating circular cylinder by Tatsuno & Bearman (J. Fluid Mech., vol. 211, 1990, pp. 157–182). Two new flow regimes have been observed, namely A$^{\prime }$ and F$^{\prime }$. The regime A$^{\prime }$ found at low $KC$ is characterised by the transverse convection of fluid particles perpendicular to the motion; and the regime F$^{\prime }$ found at high $KC$ shows a quasi-periodic feature with a well-defined secondary period, which is larger than the oscillation period. The Floquet analysis demonstrates that when the two-dimensional flow breaks the reflection symmetry about the axis of oscillation, the quasi-periodic instability and the synchronous instability with the imposed oscillation occur alternately for the square cylinder along the curve of marginal stability. This alternate pattern in instabilities leads to four distinct flow regimes. When compared to the vortex shedding in otherwise unidirectional flow, the two quasi-periodic flow regimes are observed when the oscillation frequency is close to the Strouhal frequency (or to half of it). Both the flow regimes and marginal stability curve shift in the $(Re,KC)$-space compared to the oscillatory flow around a circular cylinder and this shift appears to be consistent with the change in vortex formation time associated with the lower Strouhal frequency of the square cylinder.

The tumbling motion of freely falling plates with aspect ratio ranging from 2 to 10 is studied using both experimental measurement and simplified lifting line theory. The trajectories of plates are recorded by high-speed video camera and analysed to obtain instantaneous and average kinematic and aerodynamic parameters, including the descent angle, rotating speed, descent velocities, lift and drag coefficients, and their fluctuation spectrums. A double period rotation with period doubling is observed here for some range of aspect ratio and the double frequency is determined from a Fourier analysis. By adding a correction from the inducing effect of trailing vortices and wing-tip vortices to the corresponding two-dimensional force and torque expressions, a simplified kinematic model is obtained which successfully predicts the qualitative influence of aspect ratio on the averaged kinematics, that the descent angle and the vertical descent velocity component are decreasing functions of the aspect ratio, while the rotation speed and horizontal velocity component are increasing functions of the aspect ratio. For decreasing aspect ratio, the induced drag due to the trailing and tip vortices reduces the lift to drag ratio and thus increases the descent angle, while the dissipative torque due to translational induced drag decreases the rotation speed.

A two-dimensional, unsteady, transonic, irrotational, inviscid flow of a perfect gas with constant specific heats is considered. The analysis involves perturbations from a uniform sonic isentropic flow. The governing perturbation potential equations are derived for various orders of the ratio of the characteristic time associated with a temporal flow disturbance to the time taken by a sonic disturbance to traverse the transonicregime. The case where this ratio is large compared to one is studied in detail. A similarity solution involving an arbitrary function of time is found and it is shown that this solution corresponds to unsteady chimel flows with either stationary or time-varying wall shapes. Numerical computations are presented showing the temporal changes in flow structure as a disturbance dies out exponentially for the following typical nozzle flows: simple accelerating (Meyer) flow and flow with supersonic pockets (Taylor and limiting Taylor flow).

A thermally driven steady axisymmetric flow of gas of small diffusivity in a vertical circular cylinder rotating rapidly about its axis of symmetry is studied. The side wall is a thermal insulator and the horizontal end plates are perfect conductors. The temperature of the top end plate is kept slightly higher than that of the bottom one.

The boundary-layer method is applied to solve the linearized basic equations and the following results are obtained.

1. The axial velocity in the inner core is fully controlled by the Ekman suction on the horizontal plates and is the same as that in the case of a perfectly conducting side wall.

2. The closed circulation in the side-wall Stewartson E½ layer is strongly suppressed compared with the case of a perfectly conducting side wall.

This situation is reflected in the inner temperature field, which deviates from that in the case of a perfectly conducting side wall. The critical parameter governing the solution is found to be (γ − 1) PrG0E−1/3/4γ, where Pr is the Prandtl number, γ the ratio of specific heats, E the Ekman number and G0 the square of the Mach number based on the peripheral speed of the cylinder.

The equations which govern the flow at high Reynolds number in the vicinity of the trailing edge of a finite flat plate at incidence to a uniform supersonic stream are solved numerically using a finite-difference procedure. The critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is α* = O(R−¼) (Brown & Stewartson 1970), where R is a representative Reynolds number for the flow, and results are computed for three such values of α* which characterize the possible behaviour of the flow above the plate. The final set of computations leads to a numerical value for the trailing-edge stall angle α*s, the angle of incidence which just causes the flow to separate at the trailing edge of the plate. Analytic solutions are available in the form of asymptotic expansions near the trailing edge in terms of the scaled variable of order R−⅜. A multi-layer-type of expansion which occurs in the case α* = αs* is presented in detail for comparison with the computed solution.

The low-speed combustion of initially unmixed gaseous reactants under an irreversible two-step chain reaction is examined. Both equilibrium burning, in which two spatially separated flames of zero thickness arise, and near-equilibrium burning, in which two spatially separated flames of small but finite thickness arise, are studied by limit-process expansion techniques. Two time-dependent flows are examined: the first is (one-dimensional) transient mixing flow; and the second is (two-dimensional) transient counterflow. The latter flow, in which there is an impressed finite strain parallel to the flame, such that the flame itself is longitudinally stretched, is discussed as elucidating the characteristics of combustion in non-equilibrium turbulent shear flow.