In this, the third, part we present a complete asymptotic analysis of the distribution of velocity and electric potential in an electrically conducting liquid between two circular electrodes of finite diameter, 2b, when a current is passed between them. The electrodes are set opposite to each other in insulating planes, a distance 2a apart, and a magnetic field is applied perpendicular to these planes.

The asymptotic solution is obtained under the restriction that the Hartmann number M satisfies both the conditions M [Gt ] 1 and M½l [Gt ] 1, where l = b/a. It enables us to calculate the distribution of velocity and electrical potential throughout the flow field, and provides an expansion for the resistance R between the electrodes in descending powers of M½l, which is correct provided terms of order M−1 and (lM½)−3 are neglected. Comparison of the theoretical results with experiment shows good agreement in the measurements of R and in the direct probe measurements of electrical potential within the fluid. This is one of the first experiments in which direct probe measurements of an MHD flow, as well as external measurements, have provided such a satisfactory confirmation of the theory. Direct measurements of the velocity, by means of a Pitot tube as in part 2, or by means of a hot-film anemometer undertaken by Malcolm (1968), agree less well with the theory.

Stability to infinitesimal disturbances, when a parallel magnetic field is imposed, is investigated for the free boundary-layer type flows, of low magnetic Reynolds number, between two unbounded parallel streams of a viscous, incompressible, electrically conducting fluid. Neutral stability curves are calculated for small wave-number making use of the limiting profile: previous results by another author are found to be incomplete. A qualitative neutral stability picture is conjectured for other values of the wave-number and, granted a certain part of this conjecture, the conclusion is that the critical Reynolds number remains zero until the parameter Q/R exceeds the value (Q/R)crit [eDot ] 0·0233. It is suggested that a sufficiently strong magnetic field can stabilize a flow of any finite Reynolds number.

A simple laboratory model for the wind-driven ocean circulation is re-studied experimentally and theoretically. Introduced by Pedlosky & Greenspan (1967), the model consists of a rotating cylinder with sloping bottom, the fluid inside being driven by the steady relative rotation of the cylinder's lid. A linear theory is developed to illustrate the modification in the interior and Stewartson boundary layers caused by variation of the bottom slope from 0 to O(1); Stommel's (1948) model is obtained when the bottom slope tan α [Lt ] E¼, and the Munk & Carrier (1950) model is obtained for E¼ [Lt ] tan α [Lt ] 1 (E is the Ekman number). Measurements of the interior cross-contour ‘Sverdrup’ velocity agree well with theory when the Ekman-layer Reynolds number RE is ≈ 1 or less. The western boundarylayer azimuthal velocity agrees reasonably well with theory, although the observed variation with depth and bottom slope were not predicted. The western boundary layer shows downstream intensification when RE is increased from ≈ 1 until topographic Rossby waves appear in the transition region between western boundary layer and interior. The motion becomes unstable when a critical value of RE is reached, independent of the bottom slope, and a low-frequency two-dimensional flow oscillation is observed. A brief comparison is made with previous wind-driven ocean circulation studies.

Bounds for the complex wave velocity c, determined by the Orr-Sommerfeld equation and the boundary conditions for channel flow, have been given by Joseph (1968a,b). In these notes it is shown how two of Joseph's theorems can be uniformly improved.

The main purpose of this work is to show how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Bénard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space may be 180° out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.

The paper describes calculations of the statistical properties of turbulence-induced plate vibration, the computational scheme making no reference to Fourier synthesis. The problem considered is that of a local turbulent field acting on an infinite thin elastic plate, and the statistics of the response field are described at large distances from the forcing region. Effects of mechanical dissipation in the plate are examined, along with a discussion of the relevance of the results to problems involving plates of finite extent.

This note presents a discussion of the roles of axial momentum flux, flow force, angular momentum flux and circulation in determining the strength and hence characterizing the structure of such narrow rotating axisymmetric core flows as swirling jets, vortex jets, sink vortices and vortex wakes. The salient (though sometimes neglected) features of these core flows are that perturbation pressure plays an essential role both in the coupling of axial and azimuthal velocity fields and in the transmission of force along the core, and that flux of angular momentum is invariant only along cores with zero gross circulation. A number of existing solutions are brought into relationship by the discussion, including Long's similarity solution for draining vortices and Reynolds’ dimensional treatment of swirling wakes.

This is a study of the magnetohydrodynamic flow of an incompressible viscous fluid between coaxial disks, with a uniform axial magnetic field B. The fluid has density ρ, viseosity η and electrical conductivity σ. The flow is assumed to be steady, and to be similar in the sense that the radial and tangential components of velocity increase linearly with radial distance from the axis of rotation. Most of the work is concerned with disks which are electrical insulators, one of which rotates while the other remains stationary. The imposed conditions can then be represented by the Reynolds number R = ρΩad2/η and the Hartmann number M2 = σB2d2/η, where Ωa is the angular velocity of the rotating disk and d is the gap between the disks. Asymptotic solutions are given for R [Lt ] M2, and numerical solutions are obtained for values of R and M2 up to 512. Experimental measurements are presented which are in general agreement with the theoretical flows, and the results for small values of the Hartmann number provide the first known experimental support for the purely hydrodynamic solutions in the range 100 < R < 800.

A conducting liquid with a free surface is subjected to a vertical magnetic force due to imposed, horizontal, magnetic and current fields in the liquid. Because the current field is modified differently by differently oriented surface waves, the propagation of gravity waves becomes strongly anisotropic. The cases of shallow and deep fluid are explored. The group velocity shows features that are reminiscent of magnetoacoustic waves. The need for stability of the surface sets limits on the magnetic force which may be imposed. The feasibility of experiments is discussed and the effect of ohmic damping and surface tension is found to be relatively unimportant under suitably chosen conditions.

An asymptotic solution is obtained to the problem of internal wave propagation in a horizontally stratified inhomogeneous fluid of non-uniform depth. It also applies to fluids which are not stratified, but in which the constant density surfaces have small slopes. The solution is valid when the wavelength is small compared to all horizontal scale lengths, such as the radius of curvature of a wavefront, the scale length of the bottom surface variations and the scale length of the horizontal density variations. The theory underlying the solution involves rays, a phase function satisfying the eiconal equation, and amplitude functions satisfying transport equations. All these equations are solved in terms of the rays and of the solution of the internal wave problem for a horizontally stratified fluid of constant depth. The theory is thus very similar to geometrical optics and its extensions. It can be used to treat problems of propagation, reflexion from vertical cliffs or from shorelines, refraction, determination of the frequencies and wave patterns of trapped waves, etc. For fluid of constant density, it reduces to the theory of Keller (1958). The theory is applied to waves in a fluid with an exponential density distribution on a uniformly sloping beach. The predicted wavelength is shown to agree well with the experimental result of Wunsch (1969). It is also applied to determine edge waves near a shoreline and trapped waves in a channel.

When the vertical boundaries of a container of stratified brine solution are heated, cells are observed to form in the brine with a layered structure. An explanation of this phenomenon has been given in terms of the successive growth of cells from the top and bottom of the container. An alternative explanation is given here which is based on an examination of the stability of the fluid layer in the neighbourhood of the vertical boundaries, a layer in which there exist horizontal gradients in both temperature and salinity, and the explanation is subjected to comparison with a carefully controlled laboratory experiment. The theoretical description is fairly general and it seems possible that the effects of horizontal gradients of salinity and temperature, which approximately compensate each other so as to leave no horizontal density gradient, may be important in the ocean, as suggested by Stern (1967).

A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.