The structure of the steady magnetohydrodynamic switch-on shock wave is investigated for several orderings of the four diffusivities involved in the problem. The various orderings are approximated to by allowing one or more of the appropriate diffusivities to approach zero, and approximate solutions that are uniformly valid to order unity are sought. In general, singular perturbation problems are encountered, the number occurring (from zero to a maximum of three) depending upon the ordering of the diffusivities and the magnitude of the downstream velocity normal to the shock relative to certain critical velocities downstream of the shock. Where necessary, the approximate solutions are rendered uniformly valid to first order by the insertion of boundary layers, for which the approximate equations are determined to first order. For most of the cases considered, the limiting forms of the integral curves are determined and they are sketched in appropriate three-dimensional phase spaces.

In this work we explore if it is possible to reproduce molecular-scale slip behaviour by using continuum equations. To that end it is noted that molecular-scale slip is affected by three factors: (i) near the wall, the fluid experiences a potential because of the wall; (ii) the fluid density responds to that potential, and hence, fluid compressibility is relevant; and (iii) the fluid can lose momentum to the wall. To incorporate these features we simulate shear flow of a compressible fluid between two walls in the presence of a potential. Compressibility effect is found to be important only in the near-wall region. The slip length is calculated from the mean velocity profile. The slip-length-versus-shear-rate trend is similar to that in molecular dynamic calculations. First, there is a constant value of the slip length at low shear rates. Then, the slip length increases beyond a critical shear rate. Lastly, the slip length reaches another constant value if the wall momentum loss parameter is non-zero. The scaling for the critical shear rate emerges from our results. The value of the slip length increases if the wall potential is less corrugated and if the momentum loss to the wall is low. An understanding of the overall force balance during various slip modes emerges from the governing equations.

An analytical solution of a biharmonic equation is presented in axisymmetric toroidal coordinates for Stokes flow due to surface tension gradient on the free surface of sessile drops. The stream function profiles exhibit clockwise and counter-clockwise toroidal volumes. The ring or dot formed by the downward dividing streamlines between these volumes predicts the experimentally deposited particle ring or dot well. This finding suggests that the downward dividing streamline can be taken to be a reasonable indicator of where deposition occurs. Different light patterns directed at different locations of the droplet can give rise to a single spot or ring. A relationship between the positions of the light intensity peak and possible locations of particle deposition is analysed to demonstrate that the streamlines can be generated on-demand to achieve particle deposition at predetermined locations on the substrate. Toroidal corner vortices called Moffatt eddies have appeared in other corner flows and develop in this optical Marangoni flow as well near the contact line.

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as ‘three-fluid’ flow in the same geometry. Assuming symmetry across the gap and based on the lubrication (‘equilibrium’) approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type.

Stokes flow is analysed for a combination body, consisting of a sphere attached to a slender body, translating along its axis in an infinite and otherwise un-disturbed fluid. The cross-section of the after-body, or tail, is circular; the radius, while not necessarily constant, is small compared with the radius of the spherical head. The tail is represented by a distribution of Stokeslets of strength per unit length F(z), located and directed along its axis. The interactive effect of head-tail attachment is manifested by the presence of image singularities located within the sphere. The image system for a single tail Stokeslet must be such that the no-slip condition is satisfied on the surface of the sphere. It is shown that this system consists of a Stokeslet, a Stokes doublet (stresslet only) and a source doublet located a t the image point. The strength F(z) is obtained by applying the no-slip condition to the combination body. The solution follows the lines of traditional slender-body theory, an expansion being performed in ascending powers of the reciprocal of the logarithm of the aspect ratio. The integral force parameters and F(z) are obtained to second order. The interactive effect is assessed, and the results are discussed in the context of a sedimenting micro-organism, such as a spermatozoon. The drag on the combination body is shown to be less by around 10% than the sum of the drags on an isolated sphere and tail. This drag, for a sperm-shaped body, is divided approximately equally between head and tail.

We examine the spatial evolution of an instability wave excited by an external source in a free, nearly non-dissipative, stably stratified shear flow with a small Richardson number Ri[Lt ]1. It turns out that at the nonlinear stage of evolution even so small a stratification modifies greatly the evolution behaviour compared with the case of a homogeneous flow which was studied in detail by Goldstein & Hultgren (1988).

We have investigated (analytically and numerically) different stages of evolution corresponding to different critical layer regimes, and determined the formation conditions and structure of a quasi-steady nonlinear critical layer.

It is shown that the stratification influence upon the nonlinear evolution is governed by the parameter (Pr−1)Ri/γ2L, where Pr is the Prandtl number and γL is the wave's linear growth rate (which is a measure of supercriticality), and this effect is important only when γL<Ri1/2, Pr≠1. The character of this influence radically depends on the sign of (Pr−1). Thus, when Pr<1 the amplitude in the course of the evolution varies in a limited range and either reaches saturation, when the supercriticality is small enough or, at higher supercriticality, performs quasi-periodic oscillations, whose structure becomes increasingly complicated with increasing γL. When Pr>1 stratification leads to the appearance of new evolutionary stages, namely the stage of explosive growth in the unsteady critical layer regime, and the stage of essentially unsteady evolution in the nonlinear critical layer regime, and to a modification of the power-law growth in the regime of a quasi-steady nonlinear critical layer.

We investigate instabilities that arise when the free surface of a liquid covered with an insoluble surfactant is vertically vibrated and inertial effects are negligible. In the absence of surfactants, the inertialess Newtonian system is found to be stable, in contrast to the case where inertia is present. Linear stability analysis and Floquet theory are applied to calculate the critical vibration amplitude needed to excite the instability and the corresponding wavenumber. A previously reported long-wavelength instability is found to persist to finite wavelengths, and the connection between the long-wavelength and finite-wavelength theories is explored in detail. The instability mechanism is also probed and requires the Marangoni flows to be sufficiently strong and in the appropriate phase with respect to the gravity modulation. For viscoelastic liquids, we find that instability can arise even in the absence of surfactants and inertia. Mathieu equations describing this are derived and these show that elasticity introduces an effective inertia into the system.

A statistical ensemble of large-eddy simulations (LES) is run simultaneously for the same flow. The information provided by the different large-scale velocity fields is used in an ensemble-averaged version of the dynamic model. This produces local model parameters that only depend on the statistical properties of the flow. An important property of the ensemble-averaged dynamic procedure is that it does not require any spatial averaging and can thus be used in fully inhomogeneous flows. Also, the ensemble of LES provides statistics of the large-scale velocity that can be used for building new models for the subgrid-scale stress tensor. The ensemble-averaged dynamic procedure has been implemented with various models for three flows: decaying isotropic turbulence, forced isotropic turbulence, and the time-developing plane wake. It is found that the results are almost independent of the number of LES in the statistical ensemble provided that the ensemble contains at least 16 realizations.

A solution is given for the sound waves generated when an infinite wedge passes at supersonic velocity through the plane interface between different media. The results are then applied to the particular case when the two media consist of the same perfect gas at different temperatures. It is observed that the flow can be deduced for any symmetrical aerofoil of infinite span by superposition, and the case of a double wedge is considered in detail.

We study inviscid Faraday waves in a brimful circular cylinder with pinned contact line. The pinning leads to a coupling of the Bessel modes and leads to an infinite system of coupled Mathieu equations. For large Bond numbers, even though the stability diagrams and the subharmonic and harmonic resonances for the free and pinned contact lines are similar, the free surface shapes can be quite different. With decreasing Bond number, not only are the harmonic and subharmonic resonances very different from the free contact line case but also interesting changes in the stability diagram occur with the appearance of combination resonance tongues. Points on these tongue boundaries correspond to almost-periodic states. These do not seem to have been reported in the literature.

This paper is concerned with the nonlinear interaction and development of a pair of oblique Tollmien–Schlichting waves which travel with equal but opposite angles to the free stream in a boundary layer. Our approach is based on high-Reynolds-number asymptotic methods. The so-called ‘upper-branch’ scaling is adopted so that there exists a well-defined critical layer, i.e. a thin region surrounding the level at which the basic flow velocity equals the phase velocity of the waves. We show that following the initial linear growth, the disturbance evolves through several distinct nonlinear stages. In the first of these, nonlinearity only affects the phase angle of the amplitude of the disturbance, causing rapid wavelength shortening, while the modulus of the amplitude still grows exponentially as in the linear regime. The second stage starts when the wavelength shortening produces a back reaction on the development of the modulus. The phase angle and the modulus then evolve on different spatial scales, and are governed by two coupled nonlinear equations. The solution to these equations develops a singularity at a finite distance downstream. As a result, the disturbance enters the third stage in which it evolves over a faster spatial scale, and the critical layer becomes both non-equilibrium and viscous in nature, in contrast to the two previous stages, where the critical layer is in equilibrium and purely viscosity dominated. In this stage, the development is governed by an amplitude equation with the same nonlinear term as that derived by Wu, Lee & Cowley (1993) for the interaction between a pair of Rayleigh waves. The solution develops a new singularity, leading to the fourth stage where the flow is governed by the fully nonlinear three-dimensional inviscid triple-deck equations. It is suggested that the stages of evolution revealed here may characterize the so-called ‘oblique breakdown’ in a boundary layer. A discussion of the extension of the analysis to include the resonant-triad interaction is given.

This investigation is concerned with the convective motions in a shallow layer of silicone oil on a plane, circular copper plate which is heated at the rim and cooled at the centre and at the same time rotated around a vertical axis. The oil is in touch with a glass lid, which is cooled uniformly. With sufficient heating axially symmetric concentric rings develop. The details of the motion can be described as a superposition of a circulation due to the horizontal temperature gradient and a circulation of opposite sense due to the centrifugal force, with the motions due to the vertical instability. There seems to be a conversion into rolls whose axes point radially, if the centrifugal circulation becomes too strong.

Laboratory experiments dealing with Reynolds stress gradients in shear-free turbulence inhomogeneous rotating fluids were conducted to better understand associated physicalphenomena. The study was motivated by possible applications to the oceanic environment wheresuch Reynolds stress gradients are ubiquitous (e.g. in the vicinity of the continental shelfbreak, where turbulence decays away from the boundary). The turbulence was generated byvertical oscillations of a circular shaft with O-ring surface roughness elements; the oscillation axis coincided with theaxis of symmetry of the cylindrical test cell.

In the absence of background rotation, the turbulence is strong in the immediate vicinityof the shaft surface and decays with the radial distance, r. The turbulencein the boundary layer is such that ur∼uθ∼w,where ur, uθ, w are the radial, azimuthal and vertical r.m.s. velocity components,respectively. These velocity components are found to be proportional to Sω,where S and ω are the stroke and frequency of the shaft oscillations,respectively, i.e. much the same as for the case of oscillating-grid turbulence, which hasbeen studied extensively.

When background rotation is present, the steady-state turbulent intensity close to theshaft is similar to that of the non-rotating experiments. Away from the shaft, in thecentral portion of the test cell, large-scale motions containing randomly distributedcyclonic and anticyclonic vortices are developed owing to small local Rossby numbers. In thevicinity of the shaft, a rectified anticyclonic flow Uθ isobserved. The magnitude of Uθ is found to be proportional to thecharacteristic r.m.s. turbulence velocity u, but independent of the rate ofbackground rotation.

Consideration of the equations of motion shows that mean flows should not be expected ifbackground rotation is absent. With rotation, however, the equations indicate that theturbulent stresses can initiate, further develop and then maintain a mean anticyclonic(rectified) flow around the cylinder; the azimuthal momentum equation is shown to play acritical role in the generation of the mean anticyclonic flow.

An important class of canonical problems that is employed in quantifying the slipperiness of microstructured superhydrophobic surfaces is concerned with the calculation of the hydrodynamic loads on adjacent solid bodies whose size is large relative to the microstructure period. The effect of superhydrophobicity is most pronounced when the latter period is comparable to the separation between the solid probe and the superhydrophobic surface. We address the above distinguished limit, considering a simple configuration where the superhydrophobic surface is formed by a periodically grooved array, in which air bubbles are trapped in a Cassie state, and the solid body is an infinite cylinder. In the present part, we consider the case where the grooves are aligned perpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinear motion perpendicular to the surface; rectilinear motion parallel to the surface, in the groove direction; and angular rotation about the cylinder axis. In this scenario, the flow is periodic in the direction parallel to the axis. Averaging over the small-scale periodicity yields a modified lubrication description where the small-scale details are encapsulated in two auxiliary two-dimensional cell problems which respectively describe pressure- and boundary-driven longitudinal flow through an asymmetric rectangular domain, bounded by a compound surface from the bottom and a no-slip surface from the top. Once the integral flux and averaged shear stress associated with each of these cell problems are calculated as a function of the slowly varying cell geometry, the hydrodynamic loads experienced by the cylinder are provided as quadratures of nonlinear functions of the latter distributions over a continuous sequence of cells.

A new time scale for turbulence modulation by particles is introduced. This time scale is inversely proportional to the number density and the radius of particles, and can be regarded as a counterpart of the phase relaxation time, an important time scale in cloud physics, which characterizes the interaction between turbulence and cloud droplets by condensation–evaporation. Scaling analysis and direct numerical simulations of dilute inertial particles in homogeneous isotropic turbulence suggest that turbulence modulation by particles with a fixed mass-loading parameter can be expressed as a function of the Damköhler number, which is defined as the ratio of the turbulence large-eddy turnover time to the new time scale.

A new theoretical framework, based on the analysis of Navier–Stokes solutions for the hypersonic laminar near wake of two-dimensional and axisymmetric blunt bodies, is presented. A semi-empirical relationship is derived between the free-stream Mach and Reynolds numbers and a characteristic wake Reynolds number. A control volume analysis was performed to assess the validity of some common assumptions used in the literature. Analysis of the momentum and vorticity equations is used to assess the dominant mechanisms of momentum transfer along and across the dividing streamline and centreline which enclose the near wake. An observed stagnation pressure gain along the dividing streamline is explained using the entropy transport equation, demonstrating an unbalance between entropy generation due to viscous dissipation and entropy diffusion. The rear-stagnation point flow is analysed using an analogy to a reversed flow jet which allows for the centreline Mach number to be solved. A new viscous–inviscid interaction theory is presented for the reattachment shock formation process for both planar and axisymmetric wakes. Finally, all of the sub-mechanisms are combined into an overall wake mechanism. The resulting equations constitute the first overall theoretical framework of the laminar near-wake mechanism including separation, reattachment, rear-stagnation point flow and dividing streamline stagnation pressure gain for both planar and axisymmetric near wakes. Scaling arguments are presented throughout the work for each of the key sub-mechanisms. Recommendations are made for how experimental and numerical results for the near wake should be presented. The equations and recommendations presented here are then used to perform a detailed disambiguation of laminar capsule studies in the literature.

We investigate magnetic Taylor–Couette flow in the presence of an imposed axial magnetic field. First we calculate nonlinear steady axisymmetric solutions and determine how their strength depends on the applied magnetic field. Then we perturb these solutions to find the critical Reynolds numbers for the appearance of wavy modes, and the related wave speeds, at increasing magnetic field strength. We find that values of imposed magnetic field which alter only slightly the transition from circular-Couette flow to Taylor-vortex flow, can shift the transition from Taylor-vortex flow to wavy modes by a substantial amount. The results are compared to those for onset in the absence of a magnetic field.