The problem of capillary–gravity waves generated by certain moving oscillatory surface-pressure distributions is investigated. The main difficulty of the problem lies in finding the real roots of the modified frequency equations. This is dealt with by the use of certain geometric considerations. The critical condition that results from the formation of double roots of the modified frequency equations is represented as a surface. This surface divides the whole space into several distinct regions. For points in different regions the propagation of waves is different. The waves are determined in all cases.

A class of ‘exact’ steady and unsteady solutions of the Navier—Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating in the (x, y)-plane with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x, — y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large Kármán's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation-point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Kármán's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite-time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial-value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

Stably stratified and non-stratified flows past a touching pair of vortices with continuous velocity are considered. An asymptotic solution for the very long eddies is determined. Numerical results cover the whole range of subcritical stratification and eddy length.

According to linear theory and assuming the liquids to be inviscid and the bubbles to remain spherical, bubbles set in oscillation attract or repel each other with a force that is proportional to the product of their amplitude of volume pulsations and inversely proportional to the square of their distance apart. This force is attractive, if the forcing frequency lies outside the range of eigenfrequencies for volume oscillation of the two bubbles. Here we study the nonlinear interaction of two deformable bubbles set in oscillation in water by a step change in the ambient pressure, by solving the Navier–Stokes equations numerically. As in typical experiments, the bubble radii are in the range 1–1000 μm. We find that the smaller bubbles (~5 μm) deform only slightly, especially when they are close to each other initially. Increasing the bubble size decreases the capillary force and increases bubble acceleration towards each other, leading to oblate or spherical cap or even globally deformed shapes. These deformations may develop primarily in the rear side of the bubbles because of a combination of their translation and harmonic or subharmonic resonance between the breathing mode and the surface harmonics. Bubble deformation is also promoted when they are further apart or when the disturbance amplitude decreases. The attractive force depends on the Ohnesorge number and the ambient pressure to capillary forces ratio, linearly on the radius of each bubble and inversely on the square of their separation. Additional damping either because of liquid compressibility or heat transfer in the bubble is also examined.

The spectral eddy viscosity (SEV) concept is a handy tool for the derivation of large-eddy simulation (LES) turbulence models and for the evaluation of their performance in predicting the spectral energy transfer. We compute this quantity by filtering and truncating fully resolved turbulence data from direct numerical simulations (DNS) of neutrally and stably stratified homogeneous turbulence. The results qualitatively confirm the plateau–cusp shape, which is often assumed to be universal, but show a strong dependence on the test filter size. Increasing stable stratification not only breaks the isotropy of the SEV but also modifies its basic shape, which poses a great challenge for implicit and explicit LES methods. We find indications that for stably stratified turbulence it is necessary to use different subgrid-scale (SGS) models for the horizontal and vertical velocity components. Our data disprove models that assume a constant positive effective turbulent Prandtl number.

The classical problem of nonlinear capillary waves on two-dimensional fluid sheets is reconsidered. The problem is formulated in terms of a complex potential, and solutions are sought using Fourier series expansions. A collocation technique combined with Newton's method is used to compute the Fourier coefficients numerically. Using this procedure, the exact solutions of Kinnersley (1976) are recomputed and various symmetric and antisymmetric wave profiles are presented, including the limiting configurations which exhibit trapped bubbles of air. Most important, three new solution branches which bifurcate nonlinearly from the symmetric Kinnersley solution branch are identified. The wave profiles along these new branches do not possess the symmetry or antisymmetry of the Kinnersley solutions, although their limiting configurations also display trapped air bubbles. No bifurcations are found along the antisymmetric Kinnersley solution branch.

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

Consider Stokes flow in the semi-infinite wedge bounded by the sidewalls ϕ = ±α and the endwall z = 0. Viscous fluid fills the region 0 < r < ∞, 0 < z < ∞ bounded by these planes; the motion of the fluid is driven by boundary data given on the endwall z = 0. A consequence of the linearity of the problem is that one can treat the velocity field q(r, ϕ, z) as the sum of a field qa(r, ϕ, z) antisymmetric in ϕ and one symmetric in it, qs(r, ϕ, z). It is shown in each of these cases that there exists a real vector eigenfunction sequence vn(r, ϕ, z) and a complex vector eigenfunction sequence un(r, ϕ, z), each member of which satisfies the sidewall no-slip condition and has a z-behaviour of the form e−kz. It is then shown that one can, for each case, write down a formal representation for the velocity field as an infinite integral over k of the sums of the real and complex eigenfunctions, each multiplied by unknown real and complex scalar functions bn(k) and an(k), respectively, that have to be determined from the endwall boundary conditions. A method of doing this using Laguerre functions and least squares is developed. Flow fields deduced by this method for given boundary data show interesting vortical structures. Assuming that the set of eigenfunctions is complete and that the relevant series are convergent and that they converge to the boundary data, it is shown that, in general, there is an infinite sequence of corner eddies in the neighbourhood of the edge r = 0 in the antisymmetric case but not in the symmetric case. The same conclusion was reached earlier for the infinite wedge by Sano & Hasimoto (1980) and Moffatt & Mak (1999). A difficulty in the symmetric case when 2α = π/2, caused by the merger of two real eigenfunctions, has yet to be resolved.

The injection of a two-dimensional jet into a uniform stream is considered, the fluids being assumed inviscid and incompressible. When the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. Inner and outer expansions are developed for the jet and the external flow. The first-order outer solution in the jet is the usual thin jet approximation which fails in the neighbourhood of the jet exit except for 90° injection, when it is uniformly valid. The basic non-linearity introduced by the pressure condition along the vortex sheet separating the jet from the external flow appears as a non-linear boundary condition for the first-order outer solution in the external flow. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. The first-order fluid speed and streamline deflection angle are shown to be given correctly to O(1) uniformly in the external flow (for all injection angles) by the first-order outer solution.

Experimental studies of the effect of a transverse velocity gradient on the mass transfer between a fixed solid spherical particle and liquid and of the mass-transfer intensity between solid sphere and liquid in a uniform flow have been carried out. Both types of flow were formed between two rotating circular cylinders parallel to each other and separated by a distance of 10mm. The following experimental procedures were applied : a method of stroboscopic visualization to measure velocity profiles, and an electrochemical method to measure the mass-transfer intensity between sphere and liquid. Results were obtained for a wide range of the relevant dimensionless parameters, viz the Reynolds and PBclet numbers specified for each case appropriately. In the case of simple shear flow these ranges are: 0.03 < Re* < 1, 150 < Pe* < 13000, and for the mass-transfer intensity 5 < Nu < 24. It is shown that the experimental data agree fairly well with the approximation formula $Nu = 1.134 Pe^{*\frac{1}{3}}$ obtained theoretically by Batchelor (1979) for an ambient pure straining motion with the assumptions that Re* [Lt ] 1 and Pe* [Gt ] 1. In the case of a uniform flow the ranges of the dimensionless parameters are 0.4 < Re < 20, 3000 < Pe < 40000; and 13 < Nu < 35. The experimental data are here in good agreement with theoretical dependences of the type Nu ∝ Pe1/3 obtained by Aksel'rud (1953) and Acrivos & Taylor (1962).

The generation of a standing wave of vortices in thin channels has been experimentally observed and discussed in the literature for the last several years. The specific cause of the wave and its response to various conditions remains largely unexplored. In this paper we model pulsatile flow through thin channels with inserted deflectors to generate the vortex wave, and we examine various measures to quantify its effects. We focus on the numerical solution of the transient vortex wave phenomenon and its response to a superimposed bulk flow, variations of pulsation, deflector spacing and shape as well as transverse suction. The quantifying measures are mapped over a Reynolds’ number–Strouhal number domain.

The different modifications of the models of two-phase multicomponent filtration (Collins 1961; Nikolaevsky et al. 1968) enable one to study the dynamics of filtration flows, taking into account phase transitions. The equations of multicomponent filtration are quite complicated and only in a few individual cases do they allow for an exact solution. The most frequently used of these appears to be the solution of the stationary problem of the flow of a multicomponent mixture toward a well or a system of wells (Khristianovich 1941). In the present paper we show that at certain values of pressure, temperature and composition of the multicomponent mixture a stationary solution of the problem may not exist. The absence of a stationary solution is related to the possibility of a spatially homogeneous solution losing its stability under a perturbation (Mitlin 1986a, 1987b). We obtain an analytical criterion for instability. As an illustration, we present the results of the numerical solution of the planar linear problem of the evolution of a multicomponent system whose pressure and composition are perturbed with respect to their constant values, which are equal at both ends. We have done a numerical analysis of the plane-radial problem of the operation of a gas–condensate well with different mass fluxes, applying the conditions of a real deposit. There are several ranges of flux where the flow becomes pulsating. It is shown that the time within which the stationary solution sets in is a non-monotonic function of flux and on approaching the stability limit diverges in inverse proportion to the undercriticality of debit. We have analysed the connection between the observed instabilities and the thermodynamics of two-phase multicomponent mixtures. It is shown that the instabilities are associated with the system entering the region of retrograde condensation. We discuss the relation of retrograde phenomena to the effect of negative volume of heavy components and, as a consequence, to the negative compressibility of an individual volume of a two-phase mixture moving in a porous medium. It is shown that the observed autowave modes are relaxation oscillations in a distributed system. By using the method of perturbations in the interphase equilibrium time, we have analysed the loss of stability in a more general – non-equilibrium – model. We show that the instabilities are generated according to the Landau–Hopf scenario and calculate the period of auto-oscillations. The one-mode approximation of the theory leads to the Van der Pol equation. In conclusion we present an experimental confirmation of the theory.

The classical quasi-geostrophic model in an active layer with an arbitrary vertical structure is modified by adding a boundary condition at the interface with a passive (motionless) lower layer: the difference between isopycnal and interface elevations is a Lagrangian constant, so that a particle in this boundary remains there and conserves its density. The new model has the appropriate integrals of motion: in particular, a free energy quadratic and positive definite in the deviation from a state with a uniform flow, made up of the internal and ‘external’ potential energies (due to the displacement of the isopycnals and the interface) and the kinetic energy.

Eady's model of baroclinic instability is extended with the present system, i.e. including the effect of the free lower boundary. The integrals of motion give instability conditions that are both necessary and sufficient. If the geostrophic slope of the interface is such that density increases in opposite directions at the top and bottom boundaries, then the basic flow is nonlinearly stable. For very weak internal stratification (as compared with the density jump at the interface) normal modes instability is similar to that of a simpler model, with a rigid but sloping bottom. For stronger stratification, though, the deformation of the lower boundary by the perturbation field also plays an important role, as shown in the dispersion relation, the structure of growing perturbations, and the energetics of the instability. The energy of long growing perturbations is mostly internal potential, whereas short ones have an important fraction of kinetic energy and, for strong enough stratification, external potential.

Hot-wire measurements were made simultaneously in two homogeneous ‘horizontal’ planes in the far-wake region of a cylinder. A technique developed using hot-wire data to identify the spatial characteristics of the large-scale bulges at the interface between the internal turbulent motions and the external irrotational flow was used to unambiguously relate these outer intermittent bulges to the inner coherent structures. It was found that a turbulent bulge is made up of a combination of a horseshoe vortex (whose legs form one double-roller eddy) and the straining region present just upstream of this structure. The approach also allowed the evaluation of the two most prominent phenomenological models for the entrainment mechanism in the far-wake region: the Kelvin–Helmholtz instability and Townsend's growth–decay cycle. It was found that the decaying and re-forming of the bulges and entrainment structures is not likely to occur. Rather, the evidence is that the large-scale bulges remain coherent for long streamwise distances in equilibrium with the overall similarity of the flow.

The response of an electromagnetic flowmeter to the flow rate through it should ideally be independent of the velocity distribution of the liquid within it. In general such flowmeters do not exhibit this ideal behaviour and the variation in response with different velocity distributions, for a given flow rate, can be very large. The sensitivity of a flowmeter to velocity distribution is here described in terms of a ‘worst flow’, which is a particular velocity distribution peculiar to a given flowmeter. The ‘worst flow’ is described mathematically in terms of functions which depend upon the design parameters of the flowmeter. The flowmeter response to its ‘worst flow’, for a given kinetic energy of motion, is formulated. This enables a criterion (ε) to be defined which describes the sensitivity of a given flowmeter to velocity profile effects and which permits different flowmeters to be compared in this respect. Methods of evaluating ε are discussed and its value is calculated for a conventional flowmeter, which employs small electrodes and an approximately uniform magnetic field. It is shown that it is possible actually to generate this worst flow in a given flowmeter, by operating the flowmeter in the role of an electromagnetic pump. This could lead to the direct measurement of ε The possibility is discussed of other definitions of ε based on different boundary conditions and constraints and which might lead to a less severe criterion.