The initial value problem presented by mixing and chemical reaction in the wake of a flat plate is solved using the boundarylayer approximation. When a cool combustible mixture and its hot combustion products are separated by a finite, perfectly insulating flat plate, the velocity, temperature, and combustible concentration are determined in the vicinity of the trailing edge.

The mixing problem without chemical reaction is solved in terms of a ‘universal solution’ for a given initial temperature ratio and Prandtl number from which the solution for arbitrary temperature ratios can be obtained.

The mixing problem with chemical reaction is solved in terms of a ‘universal solution’ for the first two terms of an assumed series solution for the temperature. In this case the ‘universality’ is with respect to a parameter B characterizing the chemical and hydrodynamic initial conditions.

The axial distance from the trailing edge to the first local temperature maximum is given in terms of the initial conditions and is shown to be greatly shortened by the presence of the viscous wake as compared with non-viscous mixing.

The paper is devoted to the creation of an original model of propagation of weak but finite-amplitude waves initiating an exothermic process connected with chemical reaction or relaxation of a non-equilibrium medium. The medium is single phase (a fluid), or a homogeneous two-phase medium (liquid with gas bubbles). A nonlinear differential model describing wave-kinetic interaction and wave evolution is derived. The linear dispersion and dissipative features of these systems are investigated both analytically and numerically. Attention is paid to explanation of the physical mechanisms resulting in the formation of a self-sustained solitary wave, which in terms of synergetics could be called a ‘dissipative structure’.

The question is addressed of how much energy is reflected when a Kelvin wave propagating along a straight channel hits a bend. The solution is expressed as a truncated series of Kelvin waves and several evanescent cross-channel Poincaré modes. The bend acts as a diffraction grating – for bends of certain angles there is complete transmission and between these angles there are lobes of reflection. The width of the lobes of the diffraction pattern is directly proportional to the wavelength of the incident Kelvin wave, as in optics, electromagnetism, etc. The effect of changing the inside radius of the bend is also examined. The reflection of energy is generally small unless the Poincaré modes are nearly propagating.

The Rayleigh-Janzen expansion method is extended to plane and steady flows which contain one or more point vortices interacting with a smooth or sharp-edged obstacle. A uniformly valid approximate solution of the compressible-flow equations is deduced by applying a perturbation method and by using matched asymptotic expansions to solve the resulting singular perturbation problem. The method yields compressibility corrections for the vortex positions and for the velocities. Results are presented for the flow past a circle and a pair of symmetric vortices (Föppl's flow). They show that the compressibility effects are substantial and are consistent with experimental data.

Observations of inertial waves generated by uniform horizontal flow over ridges and truncated axisymmetric obstacles in a homogeneous fluid rotating about a vertical axis are discussed and compared with linear theory. The dependence of the flow on obstacle shape, Ro, H, E and E is investigated. Here Ro = U/2ΩL is the Rossby number, H = Ro(D/L), E = v/2ΩL2 is the Ekman number, and ε = h/L is the non-dimensional height of the obstacle, where U is the basic velocity, Ω is the angular frequency, L is a streamwise length, D is the depth of the fluid, h is the height of the obstacle, and v is the kinematic viscosity. Previous linear analysis of this problem has been for the limit H fixed, Ro→0, referred to here as the small-Ro limit. However, it is shown that certain linear terms neglected in the small-Ro limit can be important for finite Ro, and are included in the analysis given here. The observed flow is then well described by linear theory for H/ε [Gt ] 1, particularly in the case of two-dimensional flow over a ridge. However, for H/ε [Gt ] 1 the flow field is dominated by a vertical columnar motion, which is not adequately described by the analysis.

In this paper we consider the problem of the concentration jump of a vapour in the vicinity of a plane wall, which consists of the condensed phase of the vapour, in a rarefied gas mixture of that vapour (A) and another ‘inert’ gas (B). The general formulation of the problem of determining the concentration-jump coefficient CdA is given. In the Knudsen layer the simplest model of Boley-Yip theory is used to simplify the Boltzmann equations for the binary gas mixture. The numerical calculation of the concentration jump coefficient CdA for various values of evaporation coefficient αA is illustrated for the case of nB [Gt ] nA for which experimental data are available.

In this paper we investigate, using theory and direct numerical simulations (DNS), the forward in time (FIT) and backward in time (BIT) probability density functions (PDFs) of the separation of inertial particle pairs in isotropic turbulence. In agreement with our earlier study (Bragg et al., Phys. Fluids, vol. 28, 2016, 013305), where we compared the FIT and BIT mean-square separations, we find that inertial particles separate much faster BIT than FIT, with the strength of the irreversibility depending upon the final/initial separation of the particle pair and their Stokes number $St$. However, we also find that the irreversibility shows up in subtle ways in the behaviour of the full PDF that it does not in the mean-square separation. In the theory, we derive new predictions, including a prediction for the BIT/FIT PDF for $St\geqslant O(1)$, and for final/initial separations in the dissipation regime. The prediction shows how caustics in the particle relative velocities in the dissipation range affect the scaling of the pair-separation PDF, leading to a PDF with an algebraically decaying tail. The predicted functional behaviour of the PDFs is universal, in that it does not depend upon the level of intermittency in the underlying turbulence. We also analyse the pair-separation PDFs for fluid particles at short times, and construct theoretical predictions using the multifractal formalism to describe the fluid relative velocity distributions. The theoretical and numerical results both suggest that the extreme events in the inertial particle-pair dispersion at the small scales are dominated by their non-local interaction with the turbulent velocity field, rather than due to the strong dissipation range intermittency of the turbulence itself. In fact, our theoretical results predict that for final/initial separations in the dissipation range, when $St\gtrsim 1$, the tails of the pair-separation PDFs decay faster as the Taylor Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is increased, the opposite of what would be expected for fluid particles.

Hydromagnetic Taylor-vortex flows of liquid metals, subjected to an axially applied magnetic field, are considered. The supercritical state of flow is treated by means of perturbation method with multiple scale expansions in time and space. Results show that the supercritical state of flow is significantly affected by the imposed magnetic field. Calculations on finite equilibrium amplitude of Taylor vortices are compared with experiment. In the case of Taylor–Couette flows between insulating cylinders, good agreement is found between theory and experiment, while, in the case of electro-magnetically driven flows between steady conducting cylinders, disagreement occurs once magnetic field effects become significant. These results are discussed.

The leading-edge contamination (LEC) problem of an infinite swept wing is shown here as vortex-induced instability. The governing equation for receptivity is presented for LEC in terms of disturbance energy based on the Navier–Stokes equation. The unperturbed shear layer given by the swept Hiemenz boundary-layer solution is two-dimensional and an exact solution of incompressible the Navier–Stokes equation. Thus, the LEC problem is solved numerically by solving the full two-dimensional Navier–Stokes equation. The contamination at the attachment-line is shown by solving a receptivity to a convecting vortex moving outside the attachment-line boundary layer, which triggers subcritical spatio-temporal instability.

The mechanism of LEC is shown to be due essentially to a convecting counter-clockwise rotating vortex, whereas a clockwise rotating vortex displays much weaker receptivity. These results are consistent with experimental results for the bypass mechanism.

The role of linear and nonlinear mechanisms in the contamination problem is discussed as interactions between vorticity and velocity terms of the developed receptivity equation. The computed temporal growth rates reveal pattern formation during such instabilities. Proper orthogonal decomposition (POD) of the numerical solution shows the structure of the leading eigenvector as the coherent eddy excited during the bypass transition.

The reflection of a straight-crested gravity wave by a non-secular perturbation h1(x) in depth relative to an otherwise flat bottom of depth h0 is calculated through an expansion in ε∝h1/h0. Explicit results are developed up to second order for the sinusoidal patch h1=−bsin(mπx/l), 0<x<l, and reduced for Bragg resonance. Trapped modes are absent at first order but appear at second order and contribute O(ε2)/O(ε3) to the maximum (Bragg-resonant) reflection coefficient for odd/even m. A third-order approximation that includes the dominant contributions of the third-order components of the resonant peak of the reflection coefficient for large m, but neglects the trapped modes, predicts resonant peaks that agree with the values measured by Davies & Heathershaw (1984).

In a recent study by Yeung & Parkinson (1997), a wake width was proposed which allowed the bluff-body potential-flow model by Parkinson & Jandali (1970) to be extended to include the flow around an oblique flat plate. By incorporating this wake width in the momentum equation originally derived by Eppler (1954) for separated flow, the drag of the plate is related to its inclination and base pressure through a simple analytical condition. It allows the base pressure, which is usually treated as an empirical input, to be determined theoretically and thus the model becomes self-contained. Predictions of the base pressure, drag and width of wake are found to be in reasonable agreement with the experimental data. When applied to the symmetrical flow around a wedge of arbitrary vertex angle, similar agreement with experimental measurements is obtained as well. It is also demonstrated that this condition is compatible with the free-streamline models by Wu (1962) and Wu & Wang (1964) such that the corresponding predictions are in good agreement with experiment.

An obstacle in westerly flow on a periodic β-plane can generate resonant Rossby waves and cause large perturbations to the flow field. The pattern and strength of the flow can vary markedly in response to relatively small changes in the forces driving the system. The aim of this paper is to develop a simple theory valid in these circumstances. Such a theory has application to the dynamics of planetary scale quasi-steady perturbations in the atmosphere.

A series of models for barotropic quasi-geostrophic flow in an annulus is presented. An implicit quasi-linear model, with zonally averaged flow parametrized in terms of topography shape and net zonal mass transport, gives good agreement with nonlinear calculations. In one case stable multiple equilibria are predicted and confirmed, but the multiplicity regime is small.

Numerical solutions, obtained via the boundary-integral technique, are used to consider the effect of a linear axisymmetric straining flow on the existence of steady-state configurations in which a neutrally buoyant spherical particle straddles a gas–liquid interface. The problem is directly applicable to predictions of the stability of particle capture in flotation processes, and is also of interest in the context of contact angle and surface tension measurements. A primary goal of the present study is a determination of the critical capillary number, Cac, beyond which an initially captured particle is pulled from the interface by the flow, and the dependence of Cac on the equilibrium contact angle θc. We also present equilibrium configurations for a wide range of contact angles and subcritical capillary numbers.

We show that a recent publication by Liron & Barta (1992) concerning a single-layer boundary integral equation for the tractions is mathematically equivalent to Karrila & Kim's (1989) Riesz method. In actual computational schemes, the second viewpoint is preferable since the integral operator has a spectral radius less than one and even large problems can be solved by fast iterative methods.

Linearized solutions for the flow field of a rotating blade row in an infinitely long annular duct are reviewed. An isolated rotor is assumed to operate in a uniform axial flow so that the disturbance field is steady in a blade fixed co-ordinate system. Both three-dimensional and compressibility effects are included, but attention is confined to subsonic flows. Previously published source-flow solutions omitted a term which affected the thickness part of the rotor flow field constructed from them. Corrected source and rotor-thickness solutions are given, and then the source or monopole solution is used to form a pressure dipole solution. The rotor-loading contribution to the flow field is found by superposition of the revised dipole solutions. The present version of the dipole representation of the steady-loading field is shown to be equivalent to an existing vortex representation, but different from an existing dipole representation. The behaviour of the blade-surface pressure and velocity distributions is described for both the thickness and loading cases. Sample numerical evaluations of the surface quantities are presented.

The channel induction furnace is an electrically efficient device for the heating and stirring of liquid metals. In this paper an axisymmetric model for the channel flow is proposed, in which the fluid is confined to the inside of a torus. An exact solution for the magnetic field is found in terms of toroidal harmonic functions. Finite-difference methods are used to calculate the primary, cross-channel motions under the assumptions of a small skin depth, a constant eddy viscosity and no thermal dependence. Non-axisymmetric perturbations to the channel shape are considered and the perturbed field calculated. The secondary circulation along the channel is discussed.