A flat plate x = 0, |y| < L is initially at rest in an electrically conducting, inviscid, incompressible fluid permeated by a uniform magnetic field (B0, 0, 0). The plate is impulsively accelerated to a small velocity (— U, 0, 0) which is then kept constant. It is assumed that LV/λ 1 and U/V [Lt ] 1, where V is the Alfven velocity, and λ is the magnetic diffusivity.

Four stages in the development of the flow are distinguished, the last three being:

1. (ii) L [Lt ] Vt [Lt ] (λ)½. During this stage the initial potential flow is being disturbed by propagation of electric current and vorticity from the plate. The initial discontinuity on the plate has only propagated a small distance away compared to L, but a large distance compared to the length scale of diffusion (λt)½. Exact solutions to the flow are found in the neighbourhood of y = — L and y = 0 (x = 0).

2. (iii) Vt [Lt ] L [Lt ] (λt)½. The asymptotic behaviour of the electric current and vorticity on the plate are determined showing that a column of fluid of length Vt moves with the plate, aligned to the magnetic field. The transverse diffusion of the current sheets bounding the column accelerates the fluid in layers of thickness O(λt)½.

3. (iv) (λt)½ [Lt ] L. Unlike stages (ii) and (iii), where the motion is dominated by the propagation of vorticity and electric current as Alfven waves from the plate, diffusive mechanisms completely dominate the motion. ‘Slug’ flow is maintained. The nature of the flow, including the structure of the layers bounding the column are determined for |x| [Lt ] (λt)½.

The conditions under which stationary waves may exist on a stream of water of finite depth are investigated theoretically for the case of a current which is uniform except for a constant defect in velocity in a region near the free surface. The analysis is extended to provide a two-dimensional theory for the surface profile induced by a simplified model of a hovering craft. The relevance of this work to the use of high speed flumes is discussed, and an example demonstrates the importance of the velocity distribution near the free surface.

A simple model is suggested to describe flow in a Hele-Shaw cell when the Hele-Shaw parameter Λ is not necessarily small. The averaged flow is potential with a conservative body force proportional to the local velocity. The elementary ramifications of this are deduced and comparisons made with experiment. In particular no separation is predicted if Λ is less than O(1), in agreement with experiment. Furthermore, the separation cavities occurring for large Λ are completely stagnant. The theory predicts attached viscous shear layers in the wake of a lifting body, reminiscent of certain MHD problems. These tails were not observed experimentally.

This paper presents an experimental investigation of the conditions of instability and post-critical flow in an electrically driven shear layer which has been previously studied both theoretically and experimentally. The shear layer forms as a primarily azimuthal flow in an electrically conducting fluid between the edges of two parallel and coaxial circular electrodes, mounted in insulating planes, when a current is passed between them in the presence of a strong, axially applied magnetic field. The aximuthal velocity at any radial position in the layer is maximum near the electrode planes and decreases to zero at the centreplane between the electrodes. Radial and axial secondary velocity components apparently exist, though smaller in magnitude than the azimuthal velocity throughout most of the layer.

The experiments employ a miniature hot-film sensor as an instantaneous velocity indicator in mercury. The following experimental results are reported. \begin{eqnarray*} & R_c = 2420,\quad 150 \leqslant M \leqslant 350,\\ & R_c = 345\,M^{\frac{1}{3}},\quad 350\leqslant M \leqslant 650, \end{eqnarray*} where Rc is the critical Reynolds number of the shear layer, based upon the maximum velocity in the layer and the layer thickness, and M is the Hartmann number based upon the distance from the electrodes to the centreplane. For values of M below about 300, oscillograms of the hot-film signal show that the layer goes unstable in a chaotic fashion. For M ≥ 300, however, the instabilities grow to produce a wave-like post-critical flow of steady amplitude. The waves are apparently parallel to the applied magnetic field and drift in the azimuthal direction at a constant proportion (10–15%) of the critical velocity. The maximum intensity of the peak to peak fluctuations from the measured mean azimuthal velocity appears to be about 0·05. These fluctuations decrease to zero at the centreplane. By an approximate measurement of wavelength it is determined that the integral number of waves in the post-critical flow increases with M.

The physical mechanism governing the onset of instability has not been established. The manner in which hydrodynamic and electromagnetic forces interact with the destabilizing disturbances cannot be fully understood without further theoretical analysis.

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.

In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

The instability of a plane layer of non-conducting fluid which is in hydrostatic equilibrium between two semi-infinite conducting fluids with surface charges is discussed for both inviscid and viscous fluid models. It is shown that for both the inviscid and viscous fluid cases, the criteria for instability are the same. Consideration is given to the relevance of the results in explaining the mechanism by which the presence of an electric field promotes more readily the coalescence of water droplets on a water surface by viewing the onset of disruption of the air film as the instability of the air film under the action of the electrostatic field produced by the surface charges on the water surfaces.

A cylindrical tank, full of fluid, is oscillating with frequency ω and rotating with angular velocity Ω about its axis of symmetry. It is assumed that the amplitude of oscillation, δ, is small and the viscosity is low such that boundary layers exist. Analysis shows that the unsteady boundary layer is of thickness [ε/(1 − 2Ω/ω)]½ on the top and bottom plates and of thickness ε½ on the side walls, where ε = ν/2ω. The interior unsteady flow shows source-like behaviour at the corners. The steady flow field is caused by the steady component of the non-linear centrifugal forces coupled with an induced steady rotation of the interior. This rotation, of order δ2ω, is prograde when Ω/ω < 0·118 and retrograde otherwise. Maximum retrograde rotation occurs at Ω/ω = 0·5. A steady boundary layer of thickness [ε/(1 − 2Ω/ω)]½ exists on the top and bottom plates, and of thicknesses \[ \epsilon^{\frac{1}{2}},\quad (\nu/L^2\Omega)^{\frac{1}{3}},\quad (\nu/L^2\Omega)^{\frac{1}{4}} \] on the side walls. Experimental measurements of the interior induced steady rotation compare well with theory.

We apply the Cameron—Martin—Wiener (formerly ‘Wiener—Hermite’) expansion of a random velocity field to the analytical study of turbulence. The kernels of this expansion contain all statistical information about the ensemble. Complete expressions are derived for constructing statistical quantities in terms of the kernels, and for the equations of motion of the kernels. We rigorously prove the Gaussian trend of the velocity field of the Navier—Stokes equation in the very late stage when the non-linear term is neglected. The n-dependence (n is the order of derivative) of the flatness factor, minus three for derivatives of the velocity field, shows a rapid increase with n in this stage.

The late decay problem of the Burgers model of turbulence is studied analytically with a view to obtaining suggestive guidelines for fitting the non-linear aspects of the model turbulence. We can divide the energy spectrum density into two parts, the larger of which is a kind of steady solution, which we call the ‘equilibrium state’, which remains self-similar in time in terms of an appropriate variable. The deviation from this ‘equilibrium solution’ satisfies the Kármán—Howarth equation. As initial velocity field, we take two particular cases: (a) a pure Gaussian, and (b) a non-Gaussian velocity field. With these two cases a detailed spectral analysis has been obtained. The energy spectrum deviation from ‘equilibrium’ declines exponentially to zero for all wave-numbers. The Gaussian case shows that the flatness factor minus three increases rapidly with n, while the non-Gaussian case does not show any marked dependence on n.

The hydrodynamic force experienced by a small solid sphere of radius ap resting on a solid plane wall in axisymmetric stagnation flow, ${\bf v}_{\infty} = \Omega(- z^2{\bf i}_z + z\tilde{\omega}{\bf i}_{\tilde{\omega}})$, or in planar stagnation flow, v∞ = Ω(−z2iz + 2zxix), is computed on the basis of Stokes’ creeping flow equations. In both cases, as well as for any flow whose z component of velocity is −Ωz2, this force is found to be Fz = − 60·87μΩap3, where μ is the viscosity of the fluid. The uniform flow parallel to the line of centres of two touching spheres of arbitrary radii is also solved.

Theoretical solutions exist for the flow about aerofoils in various modes of unsteady motion. This is also true for aerofoils whose lift is augmented by a jet flap. A common feature of these theories is the assumption of small amplitude displacements from the mean path and the generation of a continuous vortex sheet wake. However, in practice when the motion is oscillatory and of finite amplitude, an alternating trail of individual vortices is often found. The present visual investigation includes the measurement of some vortex street parameters associated with heaving aerofoils. The results are compared with those of conventional thin aerofoil theory in order to assess the significance of this departure from the conventional small displacement flow model.

A problem with possible physiological applications concerns the escape of a viscous fluid through a small hole in a wall. The solution presented here is for a line source of sinusoidally pulsating strength located at the origin x = y = 0, where the plane y = 0 is a rigid wall and the fluid is at rest at y = + ∞. The linearized Navier—Stokes equations are solved, and results in the form of streamline plots are shown and discussed.

It is the purpose of this note to correct Smyrl's (1963) results for the perturbation pressure in the interaction field produced behind a plane shock which is being diffracted by a supersonically moving thin yawed wedge.

The refractive influence on tsunami run-up of the offshore bottom topography of islands is analyzed. Shallow water theory is used to treat problems in which the geometry resembles that of individual Hawaiian islands and in which the incident wave is plane and monochromatic.

Mathematically, the differential equation for long-wave propagation is converted into an integral equation to which numerical methods are applied. Results of practical importance include the run-up on island coastal areas. The results are used in conjunction with earlier one-dimensional analyses to estimate the total tsunami run-up.

The steady two-cell viscous vortex solution of Sullivan (1959) is extended to yield unsteady two-cell viscous vortex solutions which behave asymptotically as certain analogous unsteady one-cell solutions of Rott (1958). The radial flux is a parameter of the solution, and the effect of the radial flow on the circumferential velocity, is analyzed. The work suggests an explanation for the eventual dissipation of meteorological flow systems such as tornadoes.