The effects of a uniform axial magnetic field directed towards an oscillating wall in a semi-infinite viscous fluid (or Stokes layer) is investigated. The linear stability and disturbance characteristics are determined using both Floquet theory and via direct numerical simulations. Neutral stability curves and critical parameters for instability are presented for a range of magnetic field strengths. Results indicate that a magnetic field directed towards the boundary wall is stabilizing, which is consistent with that found in many steady flows.

A pinned or free-floating rigid plate lying on the free surface of a thin film of viscous fluid, which itself lies on top of a horizontal substrate that is moving to the right at a constant speed is considered. The focus of the present work is to describe how the competing effects of the speed of the substrate, surface tension, viscosity, and, in the case of a pinned plate, the prescribed pressure in the reservoir of fluid at its upstream end, determine the possible equilibrium positions of the plate, the free surface, and the flow within the film. The present problems are of interest both in their own right as paradigms for a range of fluid–structure interaction problems in which viscosity and surface tension both play an important role, and as a first step towards the study of elastic effects.

The theory of two-layer hydraulics is extended to topography with co-varying width and height. When these variations of the non-dimensional width and total depth have a power law relationship, the solutions can still be presented in the Froude-number plane for both unidirectional and exchange flows. These differ from previous solutions, which were limited to treating width and height variations separately.

The subgrid-scale (SGS) physics in the convective atmospheric surface layer is studied using the SGS stress and SGS scalar flux. We derive the budget equations for the conditional mean SGS stress and SGS temperature flux and show that, for transport-equation-based SGS models, the budget terms must be correctly predicted by the SGS model in order for large-eddy simulation (LES) to reproduce the resolvable-scale velocity and temperature probability density functions. Field data from the Advection Horizontal Array Turbulence Study, which notably includes measurements of the fluctuating pressure and the advection of the velocity and temperature fields, are then used to analyse the budget equations. The results reveal the complex behaviour of the dynamics of the convective atmospheric surface layer. The budgets of the conditional mean SGS shear stress and SGS temperature flux are an approximate balance between the conditional mean production and pressure destruction, with the latter causing return to isotropy. The budgets of the normal SGS stress components are more complex. For strongly convective surface layers, energy is redistributed from the (smaller) vertical to the (larger) horizontal stress components during downdrafts, resulting in generation of anisotropy by the conditional mean SGS pressure–strain-rate correlation; wall pressure reflections can also enhance the anisotropy. The conditional mean SGS pressure transport, meanwhile, is a significant source of energy during updrafts as a result of the near-wall pressure minima. The vertical advection also plays a significant role in the transfer of SGS energy. For weakly convective surface layers, pressure transport is small while the SGS pressure–strain-rate correlation reverts to its usual role of causing return to isotropy. The results of the present study, particularly for the conditional mean SGS pressure–strain-rate correlation, provide new insights into the SGS physics first educed in a recent analysis of the mean SGS budgets by Nguyen et al. (J. Fluid Mech., vol. 729, 2013, pp. 388–422) and have important implications for near-wall models utilizing SGS transport equations in the convective atmospheric surface layer.

A hinged vane and a sensitive electrical system for recording the motion of the vane have been developed for the observation of fluctuating y-components of velocity in boundary layers. An approximate theory of the natural oscillations of such vanes is presented and experimentally verified. Using vanes as resonant detectors, meassurements have been made of oscillations injected into the laminar boundary layer on a flat plate in a wind tunnel with 0·3% free-stream turbulence. Points on the neutral Tollmien-Schlichting curve have thereby been obtained which lie close to the theoretical neutral curve.

The spin-up of a homogeneous rotating fluid bounded at the top and/or bottom by a permeable medium has been proposed by Bretherton & Spiegel (1968) as a model for the spin-up in natural flows where turbulent processes transmit the direct effect of the boundaries deeper into the fluid than does the laminar Ekman layer. The theoretical analysis for the spin-up of a laterally unbounded fluid bounded by a permeable medium below is presented here. In addition, an experimental study of the process is presented. Theory and experiment agree reasonably well with a maximum difference of about 8% in the predicted and measured spin-up times. The effects of the side-wall boundary have been studied theoretically by Howard (1969). Experimental observations in the side-wall boundary layer confirm qualitatively the results of Howard's theory.

The growth of an unbounded, density-stratified, turbulent shearing layer in the presence of a gravity field is studied using the postulate of marginal instability. It is found, for a similarity mean velocity and density distribution, that after a rapid initial growth rate the growth slows asymptotically to zero as the Richardson number approaches a value of 1/4. Furthermore, the theory predicts a constant dominant turbulent eddy scale in all but the initial stages of growth of the turbulent shearing layer.

Both the general growth characteristics and the constant dominant turbulent eddy scale predicted by the theory are confirmed by experimental data.

New variational principles are given for the two-dimensional interactions between gravity-driven water waves and a rotating and translating rectangular vessel dynamically coupled to its interior potential flow with uniform vorticity. The complete set of equations of motion for the exterior water waves, the exact nonlinear hydrodynamic equations of motion for the vessel in the roll/pitch, sway/surge and heave directions, and also the full set of equations of motion for the interior fluid of the vessel, relative to the body coordinate system attached to the rotating–translating vessel, are derived from two Lagrangian functionals.

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.

In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.

In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.

We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.

Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.

The capillary instability of a liquid thread containing a regular array of spherical particles along the centreline is considered with reference to microencapsulation. The thread interface may be clean or occupied by an insoluble surfactant. The main goal of the analysis is to illustrate the effect of the particle spacing on the growth rate of axisymmetric perturbations and identify the structure of the most unstable modes. A normal-mode linear stability analysis based on Fourier expansions for Stokes flow reveals that, at small particle separations, the interfacial profiles are nearly pure sinusoidal waves whose growth rate is nearly equal to that of a pure thread devoid of particles. Higher harmonics suddenly enter the normal modes for moderate and large particle separations, elevating the growth rates and yielding a stability diagram that consists of a sequence of superposed pure-thread lobes. A complementary numerical stability analysis based on the boundary integral formulation for Stokes flow reveals the strong stabilizing effect of particles whose radius is comparable to the thread radius. Numerical simulations of the finite-amplitude motion based on the boundary integral method demonstrate that thread breakup leads to particles coated with annular layers of different thicknesses.

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

A possible model of the flow in the neighbourhood of a point of reattachment of a supersonic laminar boundary layer consists of a three-tiered ‘triple-deck’ structure in which the basic problem reduces to that of solving the incompressible boundary-layer equations in the lower deck, a region of lateral and streamwise extent $O(R^{-\frac{5}{8}})$ and $O(R^{-\frac{3}{8}})$, where R is a representative Reynolds number of the flow (Daniels 1979). The present paper describes a scheme for the numerical solution of this problem which provides evidence in support of the proposed model and quantitative values for the O(R−¼) correction to the base pressure in the flow upstream and the O(R−⅝) correction to the position of reattachment relative to the point of intersection of the incoming shear layer and the wall. An important feature of the scheme is that it copes successfully with a flow which contains substantial reverse velocities.

A three-dimensional barotropic model of the wind-driven ocean circulation is examined and the flow near the east coast of the ocean is considered in detail. The model is linear and has constant coefficients of eddy viscosity. It is shown that a strong current may exist in the eastern boundary layer when upwelling or downwelling is present. No net northwards transport is produced as an equal deep counter current also occurs. A consequence of the downwelling is that the principal interior gyre is prevented from reaching the east coast and a secondary gyre is formed in this region.

Discrete particles released into turbulent carrier flows respond to that turbulence over periods of time which depend on the particle inertia and the initial state of the particle. Due to the turbulence, the velocity of the particles becomes random and they will become dispersed throughout the carrier flow. The level of randomness of the particle velocities can be quantified by the particle kinetic stress and the spreading rate is characterized by the dispersion coefficient. In the idealized case of stationary isotropic turbulence, the kinetic stress and dispersion coefficients approach limiting values long after release. Previous analysis by the author has provided development times for dispersion coefficients and kinetic stress in cases where particles were released from a point source: (i) from an initial state of rest, and (ii) with kinetic stress identical to that found in the steady state. This paper generalizes the analysis by allowing for arbitrary initial particle velocity and displacement distributions. As with the previous analysis, the primary focus is on high-inertia particles. Unlike the previous work, however, the present analysis is applicable to the common practice in direct numerical simulations of setting initial particle velocities equal to local fluid velocities. Several different estimators of the particle dispersion coefficient are considered. Whereas each estimator leads to the same long-time value, the time taken to approach this value can vary dramatically. Expressions for development times are given for the particle kinetic stress and dispersion coefficients. Consequences of the analysis for experimental and computational studies are discussed.

The effect of spatial variability of the hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for a sloped mean uniform flow in a randomly heterogeneous porous medium. The model also describes the interface between two fluids of differing densities, e.g. freshwater/saltwater and water/oil with the denser fluid at rest. We develop analytic solutions for the variance and integral scale of free-surface fluctuations and of specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of the head and the specific discharge beneath the free surface. Statistical moments are derived using a first-order approximation and then compared with their counterpart in an unbounded medium. The effect of anisotropy and angle of mean uniform flow on the statistical moments is analysed. The solutions can be used for solving more complex flows, slowly varying in the mean.