A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi-two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number ($\hbox{\it Ha} \,{\gg}\, 1$). This is analogous to the case of low-Ekman-number rotating flows, with Ekman layers replacing Hartmann layers. This has been at the origin of the homogeneous model for flows in a rotating frame of reference, with its rich structure: geostrophic contours and shear layers. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii (Isv. Akad. Nauk SSSR Mekh. Zhidk Gaza, vol. 3, 1968, p. 3) play a role similar to that of the geostrophic contours. However, a general theory for quasi-two-dimensional magnetohydrodynamics is lacking. In this paper, a model is proposed which provides a general framework. Not only can this model account for otherwise disconnected past results, but it is also used to predict a new type of shear layer, of typical thickness $\hbox{\it Ha}^{-1/4}$. Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd & Walker (J. Fluid Mech. vol. 84, 1978, p. 471), and the problem of a rectangular cross-section duct flow in a slowly varying transverse magnetic field. For the first problem, the existence of thick shear layers of dimensionless thickness of order of magnitude $\hbox{\it Ha}^{-1/4}$ explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past Walker & Ludford (J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.

It is shown that for low values of the ambient pressure the flow field for a steady spherically symmetric expansion can be divided into three parts termed the inviscied region, the intermediate layer and the shock layer. Analytic solutions are available in the first two regions and a complete integration of the equation is required in the third. Numerical solutions indicate that such a structure is achieved in the limit and the universality of the solutions in the individual regions is confirmed.

The effects of surface corrugation with small amplitude on the growth of Tollmien–Schlichting (T–S) waves were examined experimentally in a zero-pressure-gradient boundary layer. Two- and three-dimensional corrugations of sinusoidal geometry with wavelengths of the same order as that of the two-dimensional T–S wave were considered. The corrugation amplitudes were one order of magnitude smaller than the boundary-layer displacement thickness. Streamwise growth of T–S waves on the corrugated walls was compared with that in the boundary layer on the smooth surface. A distinct difference was found in the destabilizing effect between the two- and three-dimensional corrugations. The two-dimensional corrugation significantly enhanced the growth of two-dimensional T–S waves even when the corrugation amplitude was only ∼10% of the displacement thickness. On decreasing the corrugation amplitude, the growth rate of two-dimensional T–S waves asymptotically approached that in the smooth-wall case. On the other hand, the three-dimensional corrugation had only a small influence on the growth of two-dimensional T–S waves even when the corrugation amplitude was as large as 20% of the displacement thickness. For three-dimensional corrugations, however, a pair of oblique waves was generated and developed by an interaction between the two-dimensional T–S wave and the corrugation-induced mean-flow distortion for the corrugation wavelength considered. On increasing the corrugation amplitude, the oblique waves generated were increased in amplitude and thus significantly influenced the secondary instability process.

A particle with a density slightly larger than that of the fluid in which it is immersed will sediment. However, if the particle's temperature is higher than that of the fluid, the terminal velocity of sedimentation will be smaller and can even change sign. When the terminal velocity is zero we say there is thermal levitation. Thermal levitation can also occur when the density and temperature of the particle are smaller than those of the fluid. Using a two-component thermal lattice Boltzmann equation method, we study this phenomenon and show it can be stable or unstable.

We report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.

We discuss a model for the evolution of a mushy zone which forms during the solidification of a binary alloy cooled from below in a tank with finite height. Our focus is on behaviours of the system that do not appear when either a semi-infinite domain or negligible solute diffusion is assumed. The problem is simplified through an assumption of negligible latent heat, and we develop a numerical scheme that will permit insights that are critical for developing a more general procedure. We demonstrate that a mushy zone initially grows rapidly, then slows down and eventually retreats slowly. The mushy zone vanishes after a long time, as it is overtaken by a slowly growing solid region at the base of the tank.

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

Macroscopic models based on moment equations are developed to describe the transport of mass and energy near the phase boundary between a liquid and its rarefied vapour due to evaporation and hence, in this study, condensation. For evaporation from a spherical droplet, analytic solutions are obtained to the linearised equations from the Navier–Stokes–Fourier, regularised 13-moment and regularised 26-moment frameworks. Results are shown to approach computational solutions to the Boltzmann equation as the number of moments are increased, with good agreement for Knudsen number ${\lesssim}1$, whilst providing clear insight into non-equilibrium phenomena occurring adjacent to the interface.

We present laboratory experiments that show that fingering patterns can emerge when circular interfaces of strain-rate-softening fluids displace less viscous fluids in extensionally dominated flows. The fingers were separated by regions in which the fluid appeared to be torn apart. Initially, the interface had a large dominant wavenumber, but some of the fingers progressively merged so that the number of fingers gradually declined in time. We find that the transition rate to a lower wavenumber during this cascade is faster the larger is the discharge flux of the displacing fluid. At late times, depending on the discharge flux, the pattern either converged into an integer wavenumber or varied stochastically within a finite range of wavenumbers, implying convergence to a fractional wavenumber. In that stage of the evolution we find that the average wavenumber declines with the discharge flux of the displacing fluid.

We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds $Ca=U\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of $-Ca$. The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

The dynamics of the orientations of prolate ellipsoids in general linear shear flow is considered. The motivation behind this work is to gain a better understanding of the motion and the orientation probability distribution of ice particles in clouds in order to improve the modelling of their collision. The evolution of the orientations is governed by the Jeffery equation. It is shown that the possible attractors of this equation are fixed points, limit cycles and an infinite set of periodic solutions, named Jeffery orbits, in the case of simple shear. Linear stability analysis shows that the existence and the stability of the attractors are determined by the eigenvalues of the linear part of the equation. If the eigenvalues possess a non-vanishing real part, then there always exists either a stable fixed point or a stable limit cycle. Pure imaginary eigenvalues lead to Jeffery orbits. The convergence to a stable fixed point or to a stable limit cycle may either be monotonic or may be retarded due to the occurrence of non-normal growth. If non-normal growth occurs the convergence rate may be much slower compared with the characteristic time scale of the shear. Expressions for the characteristic time scale of convergence to the stable solutions are derived. In the case of non-normal growth, expressions are derived for the delay in the convergence. The orientation probability distribution function (p.d.f.) is computed via the solution of the Fokker–Planck equation. The p.d.f. is either periodic, in the case of simple shear (pure imaginary eigenvalues), or it converges to singular points or strips in the orientation space (fixed points and limit cycles) on which it grows to infinity. Time-independent p.d.f.s exist only for imaginary eigenvalues. Unlike the case where Brownian diffusion is present, the steady solutions are not unique and depend on the initial conditions.

Rapid shallow granular flows over inclined planes are often seen in nature in the form of avalanches, landslides and pyroclastic flows. In these situations the flow develops an inversely graded (large at the top) particle-size distribution perpendicular to the plane. As the surface velocity of such flows is larger than the mean velocity, the larger material is transported to the flow front. This causes size segregation in the downstream direction, resulting in a flow front composed of large particles. Since the large particles are often more frictional than the small, the mobility of the flow front is reduced, resulting in a so-called bulbous head. This study focuses on the formation and evolution of this bulbous head, which we show to emerge in both a depth-averaged continuum framework and discrete particle simulations. Furthermore, our numerical solutions of the continuum model converge to a travelling wave solution, which allows for a very efficient computation of the long-time behaviour of the flow. We use small-scale periodic discrete particle simulations to calibrate (close) our continuum framework, and validate the simple one-dimensional (1-D) model with full-scale 3-D discrete particle simulations. The comparison shows that there are conditions under which the model works surprisingly well given the strong approximations made; for example, instantaneous vertical segregation.

Laboratory experiments are used to investigate the processes governing steady convectively driven circulation in a basin that communicates with a large external reservoir over a shallow sill. The motion is maintained by a steady loss of buoyancy distributed over the surface of the basin. Turbulent convection associated with the forcing produces a horizontal buoyancy gradient across the sill and the resulting mean flow consists of a layer directed into the basin near the surface with a dense counter flow below.

To first order, the magnitude of the exchange flow over the sill is determined by the horizontal momentum balance within the basin. Measurements of the mean and turbulent flow fields are used to show that inertia, buoyancy and friction may each contribute significantly to the balance. The interior flow produces a horizontal pressure gradient near the surface which must also contribute to the momentum balance. The density of the lower layer at the sill reflects the cumulative effect of interior processes, such as mixing, and these in turn influence the hydraulically controlled exchange flow over the sill. The basin dynamics are therefore coupled in a nonlinear fashion with the submaximal sill exchange. This coupling is investigated first by showing how interior processes are affected by changes in the magnitude of the forcing, and then by observing the associated variation of the flow state at the sill. The flow state is defined in terms of its relative proximity to the theoretical maximal exchange limit. Results show that the exchange flows are submaximal with flow rate approximately 85% of the maximal limit. This state appears to change very little in response to increasing forcing.

For a stratified basin, which exhibits a deep stagnant layer under the convectively driven near-surface exchange flow, the possibility of basin ventilation or erosion of deep fluid exists in the long term. This process and its dependence on external parameters is also explored.

The heating from above of an initially homogeneous layer of solid crystals, saturated liquid and glass ballotini (an inert matrix filler) is considered both experimentally and theoretically. The heat flux causes crystals at the top of the layer to dissolve, forming liquid which, being more concentrated and dense than the interstitial liquid below, drives convection in the lower layer. Mixing of this high-concentration liquid into the lower layer leads to precipitation, thereby releasing latent heat which raises the temperature of the lower layer. Dissolution of solid crystals from the top leaves behind a closely packed layer of glass ballotini overlain by a layer of clear liquid, both of which deepen with time. The initially homogeneous porous medium thus develops into a three-layer stratified system of (from the top): clear liquid; clear liquid with close-packed ballotini; and the evolved initial assemblage of solid crystals, ballotini and saturated liquid. Data from laboratory experiments compare well with analytical and numerical results from a one-dimensional theoretical model. The model is based on the concept that the heat supplied from above is used entirely for the dissolution of solid crystals at the upper boundary of the lower layer. The resulting compositional convection redistributes the dissolved salt uniformly throughout the lower layer, where it partly recrystallizes to restore chemical equilibrium. The crystallization and associated release of latent heat leads to a gradual and uniform increase of both the solid fraction and temperature of the lower layer. Some geological consequences of the model are presented in the concluding section.

The fluid flow and mass transfer characteristics in two-dimensional hydrodynamic focusing are theoretically investigated by considering different physical properties for the sample and sheath flows. Adopting a single-domain formulation, which assigns the region variable physical properties, three-dimensional analytical solutions are obtained for species transport under hydrodynamically fully developed conditions. In addition, simplified analytical solutions are derived assuming a uniform velocity field appropriate to electrokinetic focusing. The results show that the normalized overall mean velocity is an increasing function of the height to width ratio and a decreasing function of the sheath to sample viscosity ratio. The dependence of this normalized mean velocity on the sheath to sample flow-rate ratio is, however, non-monotonic: it grows with the flow-rate ratio when the sample fluid is more viscous than the sheath fluid, whereas the opposite is true when the sheath fluid is more viscous. Moreover, although an increase in either the viscosity or flow-rate ratios results in creating a smaller value of the normalized focused width, varying the channel aspect ratio may lead to either thinner or thicker focused regions, depending on the viscosity ratio. The inspection of the mass transport characteristics reveals that only the viscosity ratio and the Péclet number can significantly alter the mixing length. Surprisingly, the minimum mixing length in the presence of significant axial diffusion is achieved for a single-phase flow. Finally, the dimensionless mixing length is reduced by increasing the Péclet number, although the changes are negligible when this parameter is above 10. This threshold is only true when the Péclet number is calculated based on the higher of the sample and sheath diffusion coefficients.

The equations for three-dimensional hydrostatic Boussinesq dynamics are equivalent to a variational principle that is closely analogous to the variational principle for classical electrodynamics. Inertia–gravity waves are analogous to electromagnetic waves, and available potential vorticity (i.e. the amount by which the potential vorticity exceeds the potential vorticity of the rest state) is analogous to electric charge. The Lagrangian can be expressed as the sum of three parts. The first part corresponds to quasi-geostrophic dynamics in the absence of inertia–gravity waves. The second part corresponds to inertia–gravity waves in the absence of quasi-geostrophic flow. The third part represents a coupling between the inertia–gravity waves and quasi-geostrophic motion. This formulation provides the basis for a general theory of inertia–gravity waves interacting with a quasi-geostrophic mean flow.