Results for three interrelated problems are obtained by making use of solutions of boundary-value problems obtained in a different context. The first one concerns a thin rigid circular disk rotating in a slow stream of viscous fluid, both when the fluid is conducting and when it is non-conducting. For the case of a conducting fluid formulae are given for both small and large Hartmann numbers. The second problem concerns a disk performing simple harmonic rotary oscillations about its axis of symmetry in a non-conducting viscous fluid which is at rest at infinity. The last problem is that of an arbitrary axisymmetric solid oscillating about its axis of symmetry in a bounded viscous fluid, and the solution is illustrated by the case of an oscillating disk.

Fluidic assembly provides solutions for assembling particles with sizes from nanometres to centimetres. Fluidic techniques based on patterned shapes of monolayers and capillary forces are widely used to assemble microfabrication devices. Usually, for self-assembly, the precondition is that the components must be mobile in a fluidic environment. In the present work, a shape-directed fluidic self-assembly of rod-like microstructures, such as an optical fibre on a wettable pad is demonstrated experimentally with submicrometre positioning precision. A model of the process is proposed, which accounts for the following two stages of the orientation of a fibre submerged in a sessile drop: (i) the drop melting and spreading over a wettable pad; (ii) fibre reorientation related to the surface-tension-driven shrinkage of the drop surface area. At the end of stage (ii), the fibre is oriented along the pad. The experimental results for the optical-fibre assembly by a solder joint have been compared to the modelling results, and a reasonable agreement has been found. The major outcome of the experiments and modelling is that surface tension forces on the fibre piercing a drop align the fibre rather than the flow owing to the spreading of the drop over the horizontal pad, i.e. stage (ii) mostly contributes to the alignment.

Exact and numerical similarity solutions for compressible perturbations to an incompressible, two-dimensional, axisymmetric vortex reference flow are presented. The reference flow consists of a set of two-dimensional, self-similar, incompressible vortices. Similarity variables, which give explicit expressions for the decay rates of the velocities and thermodynamic variables in the vortex flows, are used to reduce the governing partial differential equations to a set of ordinary differential equations. The ODEs are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are employed to study the dependences of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Additionally, several integral relations, which allow one to trace the energy transfer in a slightly compressible vortex, are derived.

Plane Poiseuille flow has long served as the simplest testing ground for Tollmien–Schlichting wave instability. In this paper, we provide a comprehensive comparison of equilibrium Tollmien–Schlichting wave solutions arising from new high-resolution Navier–Stokes calculations and the corresponding predictions of various large-Reynolds-number asymptotic theories developed in the last century, such as double-deck theory, viscous nonlinear critical layer theory and strongly nonlinear critical layer theory. In the relatively small to moderate amplitude regime, the theories excellently predict the behaviour of the numerical solutions at Reynolds numbers of order $10^{6}$ and above, whilst for larger amplitudes our computations suggest the need for further asymptotic theories to be developed.

Miscible liquids often come into contact with one another in natural and technological situations, commonly as a drop of one liquid present in a second, miscible liquid. The shape of a liquid droplet present in a miscible environment evolves spontaneously in time, in a distinctly different fashion than drops present in immiscible environments, which have been reported previously. We consider drops of two classical types, pendant and sessile, in building upon our prior work with miscible systems. Here we present experimental findings of the shape evolution of pendant drops along with an expanded study of the spreading of sessile drops in miscible environments. We develop scalings considering the diffusion of mass to group volumetric data of the evolving pendant drops and the diffusion of momentum to group leading-edge radial data of the spreading sessile drops. These treatments are effective in obtaining single responses for the measurements of each type of droplet, where the volume of a pendant drop diminishes exponentially in time and the leading-edge radius of a sessile drop grows following a power law of $t^{1/2}$ at long times. A complementary numerical approach to compute the concentration and velocity fields of these systems using a simplified set of governing equations is paired with our experimental findings.

The wave-action conservation equation for water waves is always derived from a Lagrangian for irrotational flow. This is quite satisfactory if the whole flow-field (i.e. waves and background current) is irrotational, but is inadequate for a background current with a large-scale (vertical) vorticity, even if the flow has negligible vorticity on the local scale of a few wavelengths. A wave-action conservation equation is derived for this case and equations governing the flow and the waves are given in a simple form closely parallel to the irrotational flow equations.

A linear stability analysis is conducted for the density-driven flow of variable viscosity miscible fluids in a vertically oriented capillary tube. The main goal is to assess the competition between the axisymmetric and the first azimuthal mode as a function of Rayleigh number, viscosity ratio and interfacial thickness parameter. In the absence of a net flow, the symmetry properties of the linearized set of equations indicate that the growth rates do not depend on which of the two fluids is the more viscous, although the shape of the eigenmodes does. For most parameter combinations, the first azimuthal mode is found to have larger growth rates than the axisymmetric mode. For thin interfaces and large Rayleigh numbers, however, the axisymmetric mode dominates above a certain viscosity ratio. An unexpected result is found regarding the influence of the interface thickness on the instability. For large viscosity ratios, intermediate interface thicknesses are found to be more unstable than either very thin or very thick interfaces. The reason for this behaviour is traced to a shift of the eigenfunctions towards the less viscous fluid, which allows the instability to grow in an overall less viscous environment. In the presence of a net axial flow, the upward and downward displacements of a more viscous fluid by a less viscous one are seen to result in the same growth rate. For large viscosity ratios, the axisymmetric mode becomes destabilized by the net flow, whereas the leading azimuthal mode is stabilized. This trend is in line with experimental observations.

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

A system of boundary integral equations is derived for flows in domains composed of a porous medium of permeability $k_{1}$, surrounded by another porous medium of different permeability, $k_{2}$. The incompressible Brinkman equation is used to describe the flow in the porous media. We first apply a boundary integral representation of the Brinkman flow on each side of the dividing interface, and impose continuity of the velocity at the interface to derive the final formulation in terms of the interfacial velocity and surface forces. We discuss relations between the surface stresses based on the additional conditions imposed at the interface that depend on the porosity and permeability of the media and the structural composition of the interface. We present simulated results for test problems and different interface stress conditions. The results show significant sensitivity to the choice of the interface conditions, especially when the permeability is large. Since the Brinkman equation approaches the Stokes equation when the permeability approaches infinity, our boundary integral formulation can also be used to model the flow in sub-categories of Stokes–Stokes and Stokes–Brinkman configurations by considering infinite permeability in the Stokes fluid domain.

We study the linear stability of thermocapillary-driven convection in a planar unbounded layer of an electrically conducting low-Prandtl-number liquid heated from the side and subjected to a transverse magnetic field. The thresholds of convective instability for both longitudinal and oblique disturbances are calculated numerically and also asymptotically by considering the Hartmann and Prandtl numbers as large and small parameters, respectively. The magnetic field has a stabilizing effect on the flow with the critical temperature gradient for the transition from steady to oscillatory convection increasing as square of the field strength, as also does the critical frequency, while the critical wavelength reduces inversely with field strength. These asymptotics develop in a strong enough magnetic field when the instability is entirely due to the jet of the base flow confined in the Hartmann layer at the free surface. In contrast to the base flow, the critical disturbances, having a long wavelength at small Prandtl numbers, extend from the free surface into the bulk of the liquid layer over a distance exceeding the thickness of the Hartmann layer by a factor O(Pr−1/2). For Ha [lsim ] Pr−1/2 the instability is influenced by the actual depth of the layer. For such moderate magnetic fields the instability threshold is sensitive to the thermal properties of the bottom of the layer and the dependences of the critical parameters on the field strength are more complicated. In the latter case, various instability modes are possible depending on the thermal boundary conditions and the relative magnitudes of Prandtl and Hartmann numbers.

A stability theory for the steady swirling flow above an infinite rotating disk immersed in an otherwise unbounded rigidly rotating fluid is developed in order to corroborate the various numerical computations considered for this problem. An analysis of the initial-value problem for linearized time-dependent perturbations on the steady-state similarity solutions shows that the disturbance equations have a stable continuum spectrum which, under certain conditions, exhibits only algebraic decay in time. In addition, a numerical analysis on the discrete spectrum shows that there are unstable eigenvalues for certain rotational rates of the disk relative to the fluid at infinity. The results obtained are in good agreement with the large-time behaviour of the corresponding solutions of the unsteady similarity equations.

When more than one wave is present in a system there exists the possibilty of a resonant interaction. Resonant modes become nonlinear at smaller amplitudes than nonresonant modes. If the nonlinearity causes increased growth rates then it may be that, for a time at least, the behaviour of the resonant modes will be the dominant feature. In shear layers resonant triads can be found where two oblique modes resonate with a plane wave and this case has received much attention in the literature. For a given plane wave, the resonance condition selects oblique modes of a certain wave angle and agreement has been found between predicted wave angles and those measured in experiments.

In this paper it is shown that resonance conditions can also be met between two planar waves in a Blasius boundary layer, where one of the waves is the usual unstable mode, and the other is a higher-order damped mode. The effects of wave modulation are modelled by performing a spatial analysis but allowing the frequency to become complex. It is found that for certain complex frequencies the strength of the nonlinear resonant interaction coefficients is greatly increased. Experiments have been performed in a low-turbulence wind tunnel in which disturbances with modulated and unmodulated sections were introduced into the boundary layer over a flat plate. It was found that disturbances with the frequency and modulation predicted by the theory do indeed show a much greater susceptibility to nonlinear breakdown than nonresonant disturbances.

The intradiurnal heating and cooling cycle of the mixed layer of a tropical ocean is investigated through the use of a ‘pseudo-two-dimensional’ numerical model. Particular emphasis is given to two-component diffusion resulting from dynamic instabilities in the water column. The conservation equations for salt and heat include the effects of solar heating, horizontal advection and turbulent fluxes at the sea surface, while wind mixing enters through the use of depth-dependent eddy diffusion coefficients resulting from the wave-orbital shear model of Kitaigorodskiy (1961). All inputs are treated as functions of time of day, or calculated via the bulk aerodynamic method.

The entrainment fluxes of salt and heat due to the mechanical stirring of the wind and the fluxes due to molecular diffusion are treated as separate, their respective contributions being added to form the diffusion coefficients used in an alternating-direction explicit scheme to integrate the heat and salt equations. Near the surface, in the absence of strong solar heating (i.e. during the night), these two fluxes alone are insufficient to remove the near-surface static instabilities; thus the presence of some additional process is suggested. A dynamic stability analysis is carried out, based on the temperature and salinity gradients. The resulting Rayleigh numbers indicate the possibility of double-diffusive convection, whereby the vertical transfers of salt and heat may proceed at rates far greater than can be accounted for by molecular diffusion alone. Therefore, the molecular diffusions in the model are increased by a factor roughly proportional to the one-third power of the ratio of the local effective Rayleigh number to a critical Rayleigh number. The modified molecular diffusivities are then added to the eddy diffusion coefficients due to the wind, to form the total diffusion coefficients used in the numerical integrations.

Comparisons are made between the model-generated profiles of temperature and the profiles observed in the ocean. The comparisons show reasonable agreement in the diurnal cycle of the heat wave at 1 m vertical resolution (except for the model-generated surface layer being too deep during the late afternoon hours). (Previous models typically predict only the temperature and thickness of a homogeneous layer.) The results obtained with the model are instructive in estimating the relative importance of the various mixing processes in the upper ocean.

Differences in experimental results for surface gravity wave phase speeds obtained by cross spectra and slope-height spectra are compared. It is shown that, for a simple two-dimensional model containing both dispersive and non-dispersive waves, the observed differences can be explained readily. The slope-height technique measures a weighted average of the wavenumber squared; for frequencies sufficiently large compared with the dominant frequency, the computed phase-speed variation with frequency is representative of the wave having the larger wavenumber - the dispersive wave. For the cross-spectral method, it is shown that the small-probe-separation assumption usually employed is not valid for frequencies much larger than the dominant frequency, except at the singular point where both waves have exactly equal spectral densities and the phase function represents an average of the two modes. For all other cases, the phase function approaches that of the wave having the larger spectral density, and essentially ignores the presence of the other wave, even if both modes are relatively close in their contributions to the total spectral density.

We show that an electric field parallel to an electrically neutral surface can generate flow of electrolytic mixtures in small channels. We term this solvo-osmotic flow, since the flow is induced by the asymmetric preferential solvation of ions at the liquid–solid interface. The generated flow is comparable in magnitude to the ubiquitous electro-osmotic flow at charged surfaces, but for a fixed surface charge density, it differs qualitatively in its dependence on ionic strength. Solvo-osmotic flow can also be sensitively controlled with temperature. We derive a modified Helmholtz–Smoluchowski equation that accounts for these effects.

Here, we study noisy transitional flows in imperfect millimetre-scale channels. For probing the flows, we use microcantilever sensors embedded in the channel walls. We perform experiments in two nominally identical channels. The different sets of imperfections in the two channels result in two random flows in which the high-order moments of the near-wall fluctuations differ by orders of magnitude. Surprisingly, however, the lowest-order statistics in both cases appear to be qualitatively similar and can be described by a proposed noisy Landau equation for a slow mode. The noise, regardless of its origin, regularizes the Landau singularity of the relaxation time and makes transitions driven by different noise sources appear similar.

A method of constructing plane potential flows past doubly periodic arrays of cylindrical obstacles is described. It is based on Rankine's well-known technique of determining by simple superposition the shape of an oval-shaped obstacle simultaneously with the flow round the obstacle. This paper deals with the use of such potentials in calculating effective medium conductivities for a class of doubly periodic two-phase materials which contain either non-conducting or perfectly conducting cylindrical inclusions embedded within a conducting matrix phase.

We present experiments, using an ultra-high-speed video camera, which indicate flow-induced buckling of a surfactant layer on the free surface of a deforming water drop. This results in the ‘puncturing’ of the liquid surface, sending out a narrow jet. The phenomenon is produced when a water drop falls onto and passes through a hemispherical soap film. The soap film wraps around the drop forming a closed pocket, with a thin layer of air separating the two. This occurs without initial contact between the soap film and the drop liquid, which becomes clear during the subsequent dynamics, as the original film unwraps fully from the drop surface. Under particular conditions the soap film makes contact with the top of the drop, producing a toroidal bubble, which is driven down the drop to form a hemisphere at its bottom. We propose that the resulting rapid reduction in surface area of the bubble leads to overpacking of the surfactant molecules and buckling of the surface, which allows the jet to emerge, from the bottom tip of the drop. Marangoni-driven boundary layers, converging at the bottom, may also play a role in forcing out the jet.