The unidirectional flow of an incompressible, electrically conducting, viscous fluid along cylindrical pipes is considered. An external magnetic field, B0, which lies in the plane transverse to the flow is applied. It is shown that the governing equations, written in the co-ordinate system traced out by B0, are mathematically very similar to those for a uniform field.

The paper deals mainly with ducts whose walls are insulators. Though exact solutions (valid for all values of the Hartmann number) are derived, the limit of high Hartmann number is taken for detailed discussion. Transition layers (or, loosely, ‘wakes’) can arise which are centred on curved field lines. In some cases, reversed flow occurs in part of the core (‘radial-type’ fields). Situations also arise where the magnitude (and sign) of the velocity remains the same as for B0 = 0, whatever the strength of the applied, transverse (azimuthal) magnetic field.

Linear stability theory of Bénard convection in a rotating fluid (Chandrasekhar 1961) has shown that fluids with large ([ges ] 1) Prandtl number, σ, exhibit behaviour markedly different from that of fluids with σ [les ] 1. This difference in behaviour extends also into the finite-amplitude range (Veronis 1959, 1966I). In this paper we report a numerical study of two-dimensional Be´nard convection in a rotating fluid confined between free boundaries, with σ = 6·8 and σ = 0·2 for the range of Taylor number 0 [les ] [Fscr ]2 [Lt ] 105 and for Rayleigh numbers, R, extending an order of magnitude from the critical value of linear stability theory. The behaviour of water (σ = 6·8) is dominated by the rotational constraint even for relatively moderate values (∼ 103) of [Fscr ]2. A study of the resultant velocity and temperature fields shows how rotation controls the system, with the principal behaviour reflected by the thermal wind balance; i.e. the horizontal temperature gradient is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. A fluid with a small Prandtl number (σ = 0·2) becomes unstable to finite-amplitude disturbances at values of the Rayleigh number significantly below the critical value of linear stability theory. The subsequent steady vorticity and temperature fields exhibit a structure which is quite different from that of fluids with large σ. The rotational constraint is balanced primarily by non-linear processes in a limited range of Taylor number ([Fscr ]2 [les ] 103·6). For larger values of [Fscr ]2 the system first becomes unstable to infinitesimal oscillatory disturbances but a steady, finite-amplitude flow is established at supercritical values of R which are none the less smaller than the values that one would expect from linear theory. The ranges of Taylor number in which the above phenomena occur are different from those which were estimated on the basis of an earlier study (Veronis 1966 I) which made use of a minimal representation of the finite-amplitude velocity and temperature fields. No subcritical, finite-amplitude oscillatory motions were found in the present study. Comparison with some of the experimental features observed and reported by Rossby (1966) is also discussed and it is pointed out that some of the differences between theory and experiment may be traced to the restrictive conditions (two-dimensionality and free boundaries) of the present study.

Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 √2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.

A conference on ‘Stratified Fluids’ was held in Ann Arbor, Michigan, 11–14 April 1967, under the sponsorship of the National Science Foundation. The meeting was organized by Professor C.-S. Yih (University of Michigan) with the assistance of the other members of the scientific committee: T. B. Benjamin, W. R. Debler, L. N. Howard, R. R. Long, J. W. Miles, W. H. Munk and O. M. Phillips. Thirty papers were delivered on subjects involving the application of stratified fluid problems to geophysical phenomena, waves in stratified fluids, experimental and observational investigations, and stability. We give here a brief account of the proceedings. The papers delivered will not be published in a formal volume; references to where they can be found are given at the end of this article.

A uniform liquid film on a horizontal flat plate may be unstable to small disturbances when an air stream flows over the liquid surface. The stability of such films is examined for cases where the film is contaminated by an insoluble surface-active agent.

Two approximate analyses are given, which are applicable to liquid films at moderately large Reynolds numbers and at fairly small Reynolds numbers, respectively. These supplement previous work on uncontaminated films by Miles (1960), Cohen & Hanratty (1965) and Craik (1966).

At large liquid Reynolds numbers, the presence of surface contamination enhances stability due to increased dissipation in the viscous layer just within the liquid surface; but, at small liquid Reynolds numbers, there exists a class of disturbances for which surface elasticity may be destabilizing.

This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.

The magnetohydrodynamic flow between two parallel conducting planes is investigated for the case when the flow is driven by the current produced by electrodes placed one in each plane, the applied magnetic field being perpendicular to the planes. An exact solution is presented for the case of line electrodes placed opposite each other and an approximate method of determining the flow in this case for large Hartman number (M) is also given. It is shown that for large M the asymptotic form of the exact solution agrees with that obtained by the approximate method. The results are generalized to cover the case of line electrodes displaced relative to each other. A similar investigation is carried out for the case of two point electrodes opposite each other.

A penetrative convection problem which depends on the density maximum of water near 4 °C is solved numerically. When a layer of water has its lower boundary maintained at 0 °C and its upper boundary at some temperature above 4 °C, the layer will be divided into a lower convectively unstable region and an upper convectively stable region. Steady-state finite-amplitude solutions to this problem are obtained using the mean field approximation and free boundaries.

Convective mixing alters the temperature structure of the layer so that the temperature of a large fraction of the layer is slightly below the temperature of maximum density, in agreement with laboratory measurements. The largest motions are found in a principal convective cell which extends from the lower boundary to a temperature of 7° or 8 °C. Above the principal cell one or more counter cells may form, depending on the temperature of the upper boundary. The velocities in the counter cells are substantially less than those in the principal cell and fall off rapidly going upward. When the temperature of the upper boundary is 7 °C or higher, convection first takes place at a finite amplitude and at a Rayleigh number less than that predicted by the linear theory. When the temperature of the upper boundary is 10 °C or higher, the upper boundary is no longer important dynamically to the system.

As the Rayleigh number is increased above critical stability the velocities in the principal cell, heat transport and the distortion of the temperature field all increase. In addition, the principal cell becomes more slender and fills a greater fraction of the layer. Also, as the Rayleigh number increases the counter cells become more flattened, fill a smaller fraction of the layer, and the velocities in the counter cells decrease relative to those in the principal cell.

The most important penetration of convective motions takes place in the form of nearly horizontal motions in the lowest part of the stable region, corresponding to the upper part of the principal cell. The velocities of these motions are a large fraction of the largest vertical velocities in the unstable region.

The Navier–Stokes equations are used to study the unsteady structure of a weak shock wave reflecting from a plane wall. Both an adiabatic and an isothermal wall are considered. Incident and reflected shock structures are found by expanding the dependent variables in asymptotic series in the shock strength; the first-order terms are shown to satisfy an equation analogous to Burgers equation. The structure of the wave during reflexion is obtained from an expansion in which the first-order terms satisfy the acoustic equations. The isothermal wall boundary condition requires the introduction of a thermal layer adjacent to the wall. In this case viscosity and convection play a role secondary to the wall temperature boundary condition in determining the structure of the reflected wave. The presentation is simplified by introducing a generalized Burgers equation that gives the same first-order results as the Navier–Stokes equations. Correct second-order results are obtained from this equation simply by applying a correction to the result for the temperature.

The problem of an unsteady axisymmetric expansion of a monatomic gas into a vacuum is considered in the limit of small source Knudsen number. It is shown that a solution of the Boltzmann equation for Maxwell molecules valid for large time can be constructed, which matches with the known equilibrium solution for an inviscid expansion of a fixed mass of gas into a vacuum provided that the region near the zero density front is excluded. This solution is formally the same as that obtained for the similar problem of steady spherical expansion into a vacuum—the variations along each particle path of the unsteady flow being the same as that in the steady flow.

Near the front, the expansion procedure breaks down and the equations require a different scaling. A modified form of the Boltzmann equation is obtained which leads to a corresponding set of moment equations. Unfortunately, the set of moment equations is no longer closed and no essential simplification has been made.

An investigation of the interaction of a jet of air impinging normally on a liquid surface has been carried out. The regimes of stable, oscillating, and dispersing indentions were established as a function of velocity and surface tension.

For the stable region, energy and force balances were formulated. The force balance, when integrated, described the indention profile well and will have further use in predicting the point of onset of instability.

A linear relationship between the square of the instability velocity and the surface tension and between the square of the critical dispersion velocity and the surface tension has been determined experimentally.

This study was begun as an attempt to either confirm or disprove conflicting results of previous research upon a classical problem in potential theory. An integral equation resulting from a surface distribution of vorticity is used to solve numerically for the flow through a circular orifice. Free-surface profiles and contraction coefficients are determined for four different ratios of orifice area to pipe area, and a comparison is made between the numerical and experimental results of both this study and previous studies.

Riegels (1938) investigated the breakdown of Hele-Shaw flow in a Hele-Shaw cell with unusually large separation distance 2h* between the walls. A theoretical outer expansion for the velocity was constructed in the case where the obstacle is a circular cylinder, using an intuitive inner boundary condition that seems to be correct in the limit h* → 0, but without explicit matching with the inner expansion.

An inner expansion has now been found, and it shows that the solution in the inner layer forces terms into the outer expansion that are larger than those found by Riegels whenever h* is finite and not zero.

The paper considers the effect on the steady flow past a sphere of a uniform upstream distribution of dust particles having a small relaxation time. Using a potential solution as an upstream model of the gas flow at large Reynolds numbers R, an equation for the concentration of dust near the sphere is derived and solved numerically. It is shown that in this inviscid model there exists a dustfree layer adjacent to the sphere. A drag force is computed, and it is also shown that particles do not collide with the sphere until the Stokes number σ is greater than $\frac{1}{12}$ if we assume the gas flow unchanged by the presence of dust particles, which is in agreement with previous work of Langmuir & Blodgett (1946). The paper concludes with a discussion of the effect of a viscous boundary layer in which it is suggested that the dust-free layer is preserved when σR½ [Gt ] 1, but is prevented from forming by the viscous boundary layer when σR½ [Lt ] 1.

The rate of damping of perturbations on a shock wave reflected from a perturbed flat wall was measured in a shock tube. Incident shock wave Mach numbers of 1·45 and 1·09 in air together with sinusoidal and Gaussian wall perturbations were employed. These measurements were compared with a modified form of a linearized theory due to Zaidel (1960). The linearization was performed about the basic solution of a plane shock wave reflected normally from a flat wall.

The rate of decay and the frequency and phase of oscillations agreed very well with the theoretical predictions; the amplitudes of the oscillations were some-what larger than predicted. The reflected shock shape was initially in good agreement with theory, but higher frequency perturbations on the reflected shock front caused deviations from the predicted shape after the shock front had travelled about one wall-wavelength away from the wall.

The generally satisfactory agreement between theory and experiment supports the use of linearized analysis in predicting shock wave stability.

A layer of viscous liquid with a free surface is set in motion by the lower boundary moving simple-harmonically in its own plane. The stability of this motion is investigated. Since the primary flow is time-dependent, the time variable cannot be separated from at least one space variable, and a new approach must be used to investigate the problem. In this paper the stability of long waves is studied by a perturbation method which has not been applied before to problems of stability of unsteady flows, and it is found that the flow under consideration can be unstable for long waves.

In a uniform rotating liquid of uniform depth, Kelvin waves may be propagated in one direction along a straight boundary of the liquid. The Wiener-Hopf technique is used to obtain the wave field due to Kelvin waves incident at a rightangle corner. An asymptotic solution is obtained for the field far from the corner. It is shown that for low frequencies the Kelvin waves are propagated round the corner without change of amplitude, but that for high frequencies cylindrical waves of the ‘Poincaré’ type are generated at the corner, so that the amplitudes of the Kelvin waves propagated round the corner are reduced.

Starting from Head's semi-empirical method for incompressible flow, two approaches to the prediction of turbulent boundary-layer development in compressible flow are explored. The first uses Head's incompressible method in conjunction with a compressibility transformation similar to Stewartson's transformation for laminar flow; the second carries over Head's physical arguments to treat the compressible flow directly. Measurements in supersonic flow, both on flat plates and downstream of an abrupt pressure rise, show broad agreement with the predictions of the second method but do not support the compressibility transformation. In particular, measurements on flat plates reveal that as Mach number increases the entrainment rate decreases to a lesser extent than the skin-friction coefficient. Whilst this result is consistent with the second treatment in this paper, it is difficult to reconcile with any of the compressibility transformations discussed, and the validity of these transformations in turbulent flow is therefore questioned.

The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio ε across the shock is very small, (ii) the pressure at a point P of the disturbed flow field is not very small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal through P. Terms O(ε) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.

Measurements of the distributions of pressure on a bluff flat plate (fence) have been correlated with the characteristics of the smooth-wall boundary layer in which it is immersed. For zero pressure-gradient flows, correlations are obtained for the variation of form drag with plate height h which are analogous in form to the ‘law of the wall’ and the ‘velocity-defect law’ for the boundary-layer velocity profile. The data for adverse pressure-gradient flows is suggestive of a ‘law of the wake’ type correlation. Pressures on the upstream face of the bluff-plate are determined by a wall-similarity law, even for h/δ > 1, and are independent of the pressure-gradient history of the flow; the separation induced upstream is apparently of the Stratford-Townsend type. The effects of the history of the boundary layer are manifested only in the flow in the rear separation bubble, and then only for h/δ > ½. The base pressure is also sensitive to free-stream pressure gradients downstream of the bluff-plate. The relative extent of upstream influence of the bluff-plate on the boundary layer is found to increase rapidly as h/δ decreases. One set of measurements of the mean flow field is also presented.