Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425–432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr–Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.

We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh–Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck–Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared with one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh–Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation of the streamlines is explained by means of the Darcy equation which is used to describe the transport of momentum. In addition to the properties of the rolls, their stability against arbitrary infinitesimal perturbations is investigated. We compute stability balloons for the pure fluid case as well as for a wide parameter range of Lewis numbers and separation ratios that are typical for binary gas and fluid mixtures. The stability regions of rolls are found to be restricted by a crossroll, a zigzag and a new type of oscillatory instability mechanism, which can be related to the crossroll mechanism.

The force F(x) and torque G(x) acting on a conducting body or particle suspended at position x in a magnetic field ${\rm Re} (\hat{\boldmath B}_0({\boldmath x}){\bf e}^{-{\rm i}\omega t})$ are determined to leading order in the ratio of the scale a of the particle to the scale L of the field. When the particle is spherical, F decomposes naturally into an irrotational ‘lift’ force FL and a solenoidal ‘drag’ force FD related to G by \[ F^{\rm D} + {\textstyle\frac{1}{2}}\nabla \wedge G = 0. \] This relationship is important when the action of the field on a suspension of such spheres is considered, because it implies that the net effective force per unit volume acting to generate bulk flow is zero in any region where the concentration c is uniform. However, non-uniformity is generated by the force ingredient FL and bulk flow is then generated through interaction of G and ∇c. These effects are demonstrated for two examples involving rotating and travelling fields. Interactions of a finite number N of spheres are also considered, and in particular it is shown that when the field is a uniform rotating one, the governing dynamical system is at leading order Hamiltonian with four independent integral invariants. When N [ges ] 4, the system in general exhibits chaos.

Simultaneous measurements of the longitudinal and spanwise components sx and sz of the fluctuating velocity gradient at the wall of a pipe and the fluctuating velocity u and w at various distances from the wall have been analysed in order to provide information on the origin of flow-oriented wall eddies. Special conditional-averaging techniques, which use multiprobe measurements, are developed to capture the wall structures as they evolve in both space and time. It is found that wall eddies of wavelength λ+ ≈ 100, detected from measured patterns of sz(z), are closely related to spanwise variations of the velocity at y+ = 20 and 40. Analysis of measurements from probes arranged in the direction of the mean flow supports the notion that the wall structures result from the propagation of spanwise disturbances of the velocity at some distance from the wall, both downstream and towards the wall. A flow model, different from vortex models hypothesized by previous investigators, is proposed to account for the relation of flow phenomena in the viscous wall region to the low-speed wall streaks.

The conditions for hydraulic control to occur in a continuously stratified fluid are discussed, using density as a vertical coordinate in place of height. A suitable definition of Froude number, which varies with depth, is given. Three conditions for control emerge. One is that the flow be everywhere well-behaved; another is that control occurs when the local long-wave speed vanishes. These are shown to be equivalent. The location of the control is determined indirectly by the Froude number, which occurs as the coefficient in an ordinary differential equation; the Froude number must be somewhere less than a critical value for control to occur. The third condition requires the coalescence of two different solutions for the same boundary conditions at the point of control. It is shown that this requirement is non-trivial: examples given include a simple control by topography, a virtual control, and a control by a constriction. A direct connection with layered theory is produced. Brief discussions of bidirectional flow (where the isopycnal surface of zero velocity must be flat) and weak shocks are given.

Following Wallis, the relation between non-viscous bubbly flow and conduction in composites is examined. The bubbles are treated as incompressible and correspond to non-conducting inclusions. A simple relation is found between the effective conductivity and the energy coefficient which is agreement with previous calculations. It is shown that the energy coefficient is frame dependent and, in the frame of zero volumetric flux, is equal to the virtual mass density. Zuber's virtual mass density corresponds to the conductivity of the Hashin–Shtrikman coated-sphere geometry. This connection is exploited to extend Zuber's result to ellipsoidal bubbles. The hyperbolicity of effective equations derived from a variational principle is analysed for various bubble configurations. Without bubble clustering the equations are ill-posed (unstable). However, when the bubbles group into ellipsoidal clusters the resulting effective equations are well-posed for a wide range of parameter values.

The stability criterion for almost cylindrical dielectric liquid bridges subjected to axial electric fields in the presence of residual axial gravity is obtained. In its absence, a perfectly cylindrical equilibrium solution is allowed for all values of the relevant parameters, which are the slenderness of the liquid bridge, the electrical Bond number and the relative permittivity between the outer and inner media. This basic solution is unstable beyond a critical slenderness which varies with the electrical parameters (González et al. 1989). The destabilization takes place axisymmetrically. The inclusion of the gravitational Bond number as a new, small parameter may be treated by means of the Lyapunov-Schmidt Method, a well-known projection technique that gives the local bifurcation diagram relating the admissible equilibrium amplitudes for the liquid bridge and the aforementioned parameters. As in the absence of applied electric field, the gravitational Bond number breaks the pitchfork diagram into two isolated branches of axisymmetric equilibrium solutions. The stable one has a turning point whose location determines the new stability criterion. Quantitative results are presented after solving the resulting set of linear recursive problems by means of the method of lines.

We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters α−1 = (ν/Ωa02)½ and ε = U0/a0Ω, where ν is the kinematic viscosity, Ω is the oscillation frequency, a0 is the undisturbed semi-major axis and U0 is a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, Rs = ε2α2.

In this paper we examine the linear stability of an annular film surrounding a dielectric-fluid core in a tube in the presence of double layers of charges at the film core and at the film–tube interfaces, when the fluid-fluid interface is of low tension. In the absence of electrostatic forces, the surface tension force arising from the circumferential curvature destabilizes, and that from the axial curvature stabilizes the system. The competition is such that waves larger than the unperturbed interface circumference are unstable and those shorter are stable. For charged layers in the film, two cases are examined: (i) double-layer repulsion where the volume charge density is everywhere of the same sign and (ii) double-layer attraction where the diffusive layers next to the film interfaces are of opposite signs. In the first case, double-layer repulsion and surface tension lowering stabilize the destabilizing action of the circumferential component of the surface tension force, and a window of stability can exist. In the case of double layers of opposite signs, double-layer attraction destabilizes the system, and growth rates larger than those caused by pure capillarity can arise. Finally, for the case of a core bounded by an infinite electrolyte, surface tension lowering stabilizes the destabilizing action of the circumferential component of the surface tension force and destabilizes the longitudinal one, although the magnitudes of these effects may differ. As a result the thread can become unstable to waves shorter than the interface circumference.

The propagation of hydromagnetic planetary waves in a rotating thin shell of fluid in regions of magnetic and velocity shears is studied using the β-plane approximation. In slowly varying shear the use of the WKBJ approximation makes it possible to construct the various types of ray trajectory that can occur and consequently the conditions that give rise to critical-latitude phenomena and trapping are deduced.

The opposite extreme to the WKBJ limit, namely reflexion and refraction of waves by a current-vortex sheet, is also analysed. In this case the conditions that lead to wave amplification (or over-reflexion) are investigated. Qualitatively, it is found that reflected Alfvén modes are amplified if the jump in the flow speed across the sheet lies between two speeds which are respectively greater and less than the sum of the Alfvén speeds on either side of the sheet. Also, Rossby waves incident upon a sufficiently strong easterly flow can suffer over-reflexion.

The general case of reflexion and refraction at a finite double (magnetic and velocity) shear layer is discussed. In analogy with the invariance of ‘wave action’ of gravity waves in a shear flow we construct a quantity [Ascr ] which is invariant except at critical latitudes, where it is discontinuous. By using the asymptotic solutions near these critical latitudes and by adopting the proper matching procedure for the solutions on either side of these latitudes it is possible to relate the two constant values of [Ascr ] on either side of each critical latitude. These general results are then applied to various profiles of shear flow and magnetic field so as to elucidate the manner in which an incident wave is reflected from and transmitted through a double layer.

In this paper, we study the transition from regular to Mach reflection (RR → MR) in the dual solution domain due to the influence of an upstream disturbance, by considering the transition as an evolutionary rather than an abrupt process. From numerical simulation, we observe for the early stage of transition a multiple interaction structure, composed of a triple-shock structure, a type VI shock interaction and a shock/slipline interaction. In the end, we observe a pure unsteady MR structure. Under self-similar assumption of the triple point for the first stage and including additional Mach waves over the slipline for the last stage, we develop an idealized unsteady model to obtain the evolution of the Mach stem height and the time taken for the Mach stem to stabilize. The triple point is found to move at a nearly constant speed in the multiple interaction stage which occupies about one quarter of the transition time. In the pure unsteady MR stage, which occupies the rest of transition, the speed of the triple point drops nonlinearly until the Mach stem stabilizes.

There are a number of papers devoted to the construction of the exact solitary wave solution using a series. Power series in amplitude or Fourier series have usually been used. In the present paper we accomplish the exact summation of the Witting (1975) series and show that this series describes other flows, not solitary waves. One such flow is fluid suction under a curvilinear roof. The left half of it is similar to the left half of a maximal-amplitude solitary wave flow.

The evolution of three-dimensional disturbances in an incompressible three-dimensional stagnation-point flow in an inviscid fluid is investigated. Since it is not possible to apply classical normal-mode analysis to the disturbance equations for the fully three-dimensional stagnation-point flow to obtain solutions, an initial-value problem is solved instead. The evolution of the disturbances provides the necessary information to determine stability and indeed the complete transient as well. It is found that when considering the disturbance energy, the planar stagnation-point flow, which is independent of one of the transverse coordinates, represents a neutrally stable flow whereas the fully three-dimensional flow is either stable or unstable, depending on whether the flow is away from or towards the stagnation point in the transverse direction that is neglected in the planar stagnation point.

The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the ‘imperfect’ bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625; J. Fluid Mech., vol. 778, 2015, pp. 451–484) is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by Daniels (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 173–197) and Hall & Walton (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 199–221; J. Fluid Mech., vol. 90, 1979, pp. 377–395) in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.

An idealised model is presented to study the formation of sorted bed forms generated by a wind-driven along-shore current. The study employs a linear stability analysis to describe the time development of perturbations of both bottom composition and bed elevation, superimposed on a flat bed composed of a sediment mixture homogeneously distributed in space. The model considers both bed and suspended loads and takes into account the averaged influence of waves on the flow field and the transport of sediment. The results show that the positive coupling between waves, along-shore current and the erodible heterogeneous bed leads to the amplification of two modes, which exhibit distinct characteristics. A first mode is found to be dominant when moderate hydrodynamic conditions are considered and is primarily amplified by the convergence of sediment transport induced by the changes in the bed elevation. This mode has wavelengths of the order of hundred metres and has coarse (fine) sediments in its troughs (crests). By increasing the height of the waves and/or the strength of the steady current, the second mode can become dominant. This mode is characterised by shorter wavelengths and results from the interaction between the convergence of sediment transport related to changes in the bottom composition and that induced by perturbations of the bed elevation. These bed features can have an up-current or a down-current shift between the centre of the coarse-grained bands and the trough of the bottom wave. Typical growth times of the amplified features are of the order of hundreds of days and the migration rates, in the direction of the along-shore current, range between 0.1 and 10 m per day. A qualitative comparison of the model results with field observations indicates that the generation of two distinct modes provides a possible explanation for the broad range of characteristics of the natural bed features.

This study considers the natural convection flow in a water body subjected to heating by solar radiation. The investigation into this type of natural convection flow has been motivated by the fact that it is known to play a crucial role in the daytime heat and mass transfer in shallow regions of natural water reservoirs and lakes, with a resultant impact on biological activity. An analytical solution for temperature in such an internally heated system shows that the temperature stratification consists of an upper stable stratification and a lower unstable stratification. One of the important consequences of such a nonlinear temperature stratification is the limitation of the mixing driven by rising thermal plumes with the penetration length scale of the plumes determining the lower mixed layer thickness. A theoretical analysis conducted in the present study suggests that in relatively deep waters, the lower mixed layer thickness is equal to the attenuation length of the radiation, which has important implications for water quality, including the transport of pollutants and nutrients in the water body. Scalings are also obtained for the quasi-steady boundary layer. The theoretical analysis is validated against numerical simulations.

Numerical simulations of geostrophic turbulence above topography are used to compare (a) nonlinear generation of system entropy, S, (b) selective damping of enstrophy and (c) development of vorticity–topography correlation. In the damped cases, S initially increases, approaching a quasi-equilibrium (maximum S subject to the instantaneous, though decaying, energy and enstrophy). When strongly scale-selective damping is applied, onset of the vorticity–topography correlation follows the timescales for enstrophy decay. During the period of decay, it is shown that nonlinear interaction continues to generate S, offsetting in part the loss of S to explicit damping.

Experiments were carried out to investigate the drop formation behaviour of a hydrophobic hydrate-forming liquid, HCFC-141b (CH3CCl2F), at a single nozzle in a water stream under hydrate-formable thermodynamic conditions. Attention was focused on the relation between the clathrate-hydrate formation and the drop formation. It was observed that two discrete hydrate crusts grow along the liquid–liquid interface; one forms a frontal cap and the other forms a cylindrical root on each growing drop before its detachment from the nozzle. Most of the latter crust remains at the tip of the nozzle after the detachment of the drop so that it grows into a bell-shaped or nearly cylindrical funnel composed of hydrate deposits in the course of successive growth/detachment of drops. The size of these drops is dependent on the instantaneous diameter of the hydrate-funnel tip rather than the diameter of the nozzle itself. Thus, the size of the drops successively released into the water stream generally varies synchronously with quasi-periodical alternation of growth and breaking of the hydrate funnels. The growth and breaking of the hydrate funnels and the resultant drop-size variation are significantly dependent on the system temperature (or the system subcooling from the liquid/liquid/hydrate equilibrium temperature), the nozzle diameter, and the velocity of the drop-forming liquid through the nozzle.

This study investigates the influence of the condensation coefficient of vapour on the collapse of a bubble composed of condensable gas (vapour) and non-condensable gas (NC gas). We simulated vapour and NC gas flow inside a bubble based on the molecular gas dynamics analysis in order to replicate the phase change (viz., evaporation and condensation) precisely, by changing the initial number density ratio of the NC gas and vapour, the initial bubble radius and the value of the condensation coefficient. The results show that the motion of the bubble is unaffected by the value of the condensation coefficient when that value is larger than approximately 0.4. We also discuss NC gas drift at the bubble wall during the final stage of the bubble collapse and its influence on the condensation coefficient. We conclude that vapour molecules can behave as NC gas molecules when the bubble collapses, owing to the large concentration of NC gas molecules at the gas–liquid interface. That is, the condensation coefficient reaches almost zero when the bubble collapses violently.

A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some ‘from deep to shore’ computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in Chamberlain & Porter (1995). When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1.

It is also shown that the dominant asymptotic analysis, responsible for the new formula, is valid for all slope angles $\alpha{<}\pi/2$ and not just the special angles considered by Friedrichs.