The three-dimensional marginal separation of a boundary layer along a line of symmetry is considered. The key equation governing the displacement function is derived, and found to be a nonlinear integral equation in two space variables. This is solved iteratively using a pseudospectral approach, based partly in double Fourier space, and partly in physical space. Qualitatively the results are similar to previously reported two-dimensional results (which are also computed to test the accuracy of the numerical scheme); however quantitatively the three-dimensional results are much different.

This paper describes numerical and asymptotic solutions of the steady two-dimensional boundary-layer equations governing buoyant flow on a horizontal, thermally insulated surface. The class of flows considered is one for which there is a uniform external stream at constant temperature but for which conditions upstream lead to a statically stable temperature field within the boundary layer. This has the effect of generating an adverse pressure gradient which, if sufficiently strong, causes the boundary-layer solution to terminate in a singularity. Results are obtained for a range of Prandtl numbers.

The rolling-up of the trailing vortex sheet produced by a wing of finite span is calculated as a series expansion in time. For a vorticity distribution corresponding to a wing with cusped tips, the shape of the sheet is found by summing the series using Padé approximants. The sheet remains analytic for some time but ultimately develops an exponential spiral at the tips. The centroid of vorticity is conserved to high accuracy.

The separation of the laminar boundary layer from a convex corner on a rigid body contour in transonic flow is studied based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number. It is shown that the flow in a small vicinity of the separation point is governed, as usual, by strong interaction between the boundary layer and the inviscid part of the flow. Outside the interaction region the Kármán–Guderley equation describing transonic inviscid flow admits a self-similar solution with the pressure on the body surface being proportional to the cubic root of the distance from the separation point. Analysis of the boundary layer driven by this pressure shows that as the interaction region is approached the boundary layer splits into two parts: the near-wall viscous sublayer and the main body of the boundary layer where the flow is locally inviscid. It is interesting that contrary to what happens in subsonic and supersonic flows, the displacement effect of the boundary layer is primarily due to the inviscid part. The contribution of the viscous sublayer proves to be negligible to the leading order. Consequently, the flow in the interaction region is governed by the inviscid–inviscid interaction. To describe this flow one needs to solve the Kármán–Guderley equation for the potential flow region outside the boundary layer; the solution in the main part of the boundary layer was found in an analytical form, thanks to which the interaction between the boundary layer and external flow can be expressed via the corresponding boundary condition for the Kármán–Guderley equation. Formulation of the interaction problem involves one similarity parameter which in essence is the Kármán–Guderley parameter suitably modified for the flow at hand. The solution of the interaction problem has been constructed numerically.

The present study concerns the Lagrangian dynamics of three-dimensional (3D) buoyancy-driven cavity flows under steady and laminar conditions due to a global temperature gradient imposed via an opposite hot and cold sidewall. This serves as the archetypal configuration for natural-convection flows in which (contrary to the well-known Rayleigh–Bénard flow) gravity is perpendicular (instead of parallel) to the global temperature gradient. Limited insight into the Lagrangian properties of this class of flows, despite its relevance to observed flow phenomena as well as scalar transport, motivates this study. The 3D Lagrangian dynamics are investigated in terms of the generic structure and associated transport properties of the global streamline pattern (‘Lagrangian flow topology’) by both theoretical and computational analyses. The Grashof number $Gr$ is the principal control parameter for the flow topology: limit $Gr=0$ yields a trivial state of closed streamlines; $Gr>0$ induces symmetry breaking by fluid inertia and buoyancy and thus causes formation of toroidal coherent structures (‘primary tori’) embedded in chaotic streamlines governed by Hamiltonian mechanisms. Fluid inertia prevails for ‘smaller’ $Gr$ and gives behaviour that is dynamically entirely analogous to 3D lid-driven cavity flows. Buoyancy-induced bifurcation of the flow topology occurs for ‘larger’ $Gr$ and underlies the emergence of ‘secondary rolls’ observed in the literature and to date unreported secondary tori for ‘larger’ Prandtl numbers $Pr$. Key to these dynamics are stagnation points and corresponding heteroclinic manifold interactions.

We consider a viscous drop, loaded with an insoluble surfactant, spreading over an inclined plane that is covered initially with a thin surfactant-free liquid film. Lubrication theory is employed to model the flow using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the late-time multi-region asymptotic structure of the spatially one-dimensional spreading flow. A simplified differential–algebraic equation model is derived for key variables characterising the spreading process, using which the late-time spreading and thinning rates are determined. Focusing on the neighbourhood of the drop’s leading-edge effective contact line, we then examine the stability of this region to small-amplitude disturbances with transverse variation. A dispersion relationship is described using long-wavelength asymptotics and numerical simulations, which reveals physical mechanisms and new scaling properties of the instability.

A new stability approach is developed for a wide class of strongly non-parallel axisymmetric flows of a viscous incompressible fluid. This approach encompasses all conical flows, and all steady and weakly unsteady disturbances, while prior studies were limited to specific flows and particular disturbances. A specially derived form of the Navier–Stokes equations allows the exact reduction of the linear stability problem to a system of ordinary differential equations. We found that disturbances originating at the boundaries of a similarity region cause a variety of steady bifurcations. Consideration of the still fluid allows disturbances to be classified into inner, outer and global modes, depending on the boundary conditions perturbed. Then we identify and study modes which cause bifurcation as the Reynolds number increases. The study provides improved understanding of (a) azimuthal symmetry breaking, (b) genesis of swirl, (c) onset of heat convection, (d) hydromagnetic dynamo, (e) hysteretic transitions, and (f) jump flow separation. We also discover and analyse two new bifurcations: (g) fold catastrophes and (h) appearance of radial oscillations in swirl-free jets. The stability analysis reveals that bifurcations (a), (c) and (f) are caused by inner perturbations, bifurcations (b), (d), (e) and (g) by outer perturbations, and bifurcation (h) by global perturbations. We deduce amplitude equations to describe the nonlinear spatiotemporal development of disturbances near the critical Reynolds numbers for (b) and (g). Disturbances switching between the basic and secondary steady states are found to grow monotonically with time.

The three-dimensional equations for a general inertial ocean current are transformed so that the temperature rather than the vertical co-ordinate appears as an independent variable. A downstream power series expansion is made of the equations and boundary conditions, which involves an expansion about the mean potential vorticity. A general first-order solution is obtained for a boundary current between two level surfaces, one of no motion and one of uniform temperature. The case of constant potential vorticity is treated for arbitrary inviscid boundary conditions; it is found that the current can exist as a boundary layer only if the open ocean geostrophic drift is westward everywhere in the depth interval. This result is extended to arbitrary potential vorticity distributions by an asymptotic analysis in physical space.

An approximate solution is presented to the problem of the propagation of a temperature wave through a stratified medium which both diffuses and radiates heat. The solution is a combination of two waves whose relative amplitudes vary with the distance from the lower boundary. Apparent diffusivities computed from phase lag and attenuation coefficients can differ greatly from each other and can vary with height even if the actual diffusivity does not.

An example, using parameters simulating the earth's atmosphere, suggests that analysis of the propagation of the diurnal wave upward from the earth's surface is likely to be inadequate if radiative effects are not considered.

A simplified example of this type of flow was examined in detail by developing two series, eventually matched, one about the nodal point and the other about the saddle point, and also by finite differences, marching from the nodal point to the saddle point. It was found that the results of marching the two series were in agreement with the finite difference method. The series solution near the saddle point is not unique, but numerical evidence indicates that the correct solution is that which has ‘exponential decay’ at infinity, and that this type of solution, if such exists, automatically emerges when the finite difference method is used.

The pressure-driven growth model is used to determine the shape of a foam front propagating into an oil reservoir. It is shown that the front, idealised as a curve separating surfactant solution downstream from gas upstream, can be subdivided into two regions: a lower region (approximately parabolic in shape and consisting primarily of material points which have been on the foam front continuously since time zero) and an upper region (consisting of material points which have been newly injected onto the foam front from the top boundary). Various conjectures are presented for the shape of the upper region. A formulation which assumes that the bottom of the upper region is oriented in the same direction as the top of the lower region is shown to fail, as (despite the orientations being aligned) there is a mismatch in location: the upper and lower regions fail to intersect. Alternative formulations are developed which allow the upper region to curve sufficiently so as to intersect the lower region. These formulations imply that the lower and upper regions (whilst individually being of a convex shape as seen from downstream) actually meet in a concave corner, contradicting the conventional hypothesis in the literature that the front is wholly convex. The shape of the upper region as predicted here and the presence of the concave corner are independently verified via numerical simulation data.

A three-dimensional direct numerical simulation (DNS) is applied to flows inside and outside an evaporating spherical water droplet in air, and the effect of relative humidity on the heat transfer between the droplet and ambient air is investigated. The initial air temperature is set to be 15 K higher than the initial droplet temperature. The results show that the local evaporation heat loss indicates the maximum on the front of the droplet and decreases on going from the front to the rear, and the reduction on the rear becomes marked for high-droplet Reynolds numbers. This is because the evaporation rate is suppressed on the rear surface by the presence of flow separations behind the droplet. The droplet temperature decreases in the low-humidity condition, whereas it increases in the high-humidity condition. This difference is caused by the heat balance of evaporation heat loss and convective heat gain from ambient air at the droplet surface.

It is shown that viscous Poiseuille flow sustains wave propagation in its centre region for waves whose phase speed is less than the maximum flow speed. When, moreover, there are critical levels, there exists a range of phase speeds for which over-reflection occurs and this range corresponds exactly to the range for which unstable eigenmodes exist. Consistent with the conjecture of Lindzen & Barker (1985), it is found that viscous boundary layers around the critical level and at the wall replace the exponential regions and wave sinks required for inviscid over-reflection.

Over-reflection, we find is confined to phase speeds for which these two boundary layers are in close proximity rather than widely separated or substantially overlapping.

Over-reflection is inevitably associated with a wave phase tilt opposite in direction to the shear at the critical level. All other cases yield a phase tilt in the direction of the shear. The former is consistent with the condition for the Orr mechanism to produce amplification (Boyd 1983).

Asymptotic methods are used to determine the dispersion equation for disturbances of rotating parallel flows in shallow water. From this equation the unstable modes and their growth rates are determined. The solution involves seven asymptotic expansions which are matched together. The results supplement and extend those which have been obtained previously using numerical methods by Griffiths, Killworth & Stern (1982) and by Hayashi & Young (1987).

Fast dynamo action in a chaotic time-periodic flow is investigated. Chaotic motion is created by perturbing a spatially periodic array of helical cells similar to Roberts’ cells, leading to an identifiable stretch–fold–shear fast dynamo mechanism. Using the stochastic Wiener bundle method to treat diffusion exactly, numerical results are presented suggesting fast dynamo action. A new numerical method for modelling the role of small magnetic diffusivity is introduced and results are compared with those calculated using the Wiener bundle method. Implications for the role of diffusion in the fast dynamo process are investigated. Finally the relation of the new method to a previously used ‘flux growth’ method are discussed.

The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of Reynolds & Potter (1967). In both the cases the least-stable centre-modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the Shanks (1955) method and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers α in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by Davey & Nguyen (1971).

This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman–Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable ‘inner’ asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general ‘global’, in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a ‘precursor’ of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side.

A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).

The Kelvin wave is the lowest eigenmode of Laplace's tidal equation and is widely observed in both the ocean and the atmosphere. In this work, we neglect mean currents and instead include the full effects of the Earth's sphericity and the wave dispersion it induces. Through a mix of perturbation theory and numerical computations using a Fourier/Newton iteration/continuation method, we show that for sufficiently small amplitude, there are Kelvin travelling waves (cnoidal waves). As the amplitude increases, the branch of travelling waves terminates in a so-called corner wave with a discontinuous first derivative. All waves larger than the corner wave evolve to fronts and break. The singularity is a point singularity in which only the longitudinal derivative is discontinuous. As we solve the nonlinear shallow water equations on the sphere, with increasing ε (‘Lamb's parameter’), dispersion weakens, the amplitude of the corner wave decreases rapidly, and the longitudinal profile of the corner wave narrows dramatically.

We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length $L$, a normalised thickness $0.01\leqslant \unicode[STIX]{x1D700}\leqslant 0.1$ and an apex angle $0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}$, and connected to inlet and outlet pipes of equal normalised radius $0.01\leqslant \unicode[STIX]{x1D6FF}\leqslant 0.1$ perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range $0.1\leqslant Re_{p}\leqslant 10$ and $0.1\leqslant Pe_{p}\leqslant 1000$, where $Re_{p}$ and $Pe_{p}$ are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet ($\text{D}$) or a Neumann ($\text{N}$) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when $\unicode[STIX]{x1D700}\geqslant \unicode[STIX]{x1D6FF}$ and a dominantly viscous straining flow when $\unicode[STIX]{x1D700}\ll \unicode[STIX]{x1D6FF}$. The thickness of the scalar boundary layer scales to the local Péclet number to the power $1/3$. The diffusive flux $j_{d}$ and the scalar $C_{s}$ at the wall where ($\text{D}$) or ($\text{N}$) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When $(\text{D})$ is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to $\unicode[STIX]{x1D6EC}_{\infty }=2.5694$. When $(\text{N})$ is imposed, $C_{s}$ varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and $j_{d}$ and $C_{s}$ are functions of the geometrical parameters $L$, $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$. In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.