An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.

The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.

In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity ${\mathop\mu\limits^{*}}$. It is estimated from the available numerical data that for a suspension of identical rigid spherical particles \[ {\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2, \] the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.

In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.

The linear theory of rotating stratified fluids is applied to describe the steady axisymmetric motion of a stratified fluid in a rotating annulus for values of the stratification parameter $\overline{S} = \sigma N^2/(2\Omega)^2$ that are order one or larger: $\overline{S} \geqslant O(1)$. The motion is mechanically driven by either an applied velocity or by an applied stress at the top surface. The side walls are thermally insulated. Primary attention is given to a study of the meridional, or upwelling circulation. Simple analytical solutions are obtained with the aid of the narrow-gap approximation and a justifiable assumption of an infinite depth. In that case the meridional circulation is confined to a surface Ekman layer and to a Lineykin layer of thick-ness of $O(\overline{S}^{-\frac{1}{2}})$. It is shown how the solutions for the stress-driven upwelling flow in the annulus apply to a two-dimensional linear model of coastal upwelling in a stratified ocean.

An attempt has been made to identify the geometric character of the random fine-structure regions dispersed in fully turbulent fluid. The technique was measure-ment of two-position coincidence functions for the presence of velocity fine-structure. In this primitive approach, we tried to distinguish among only three possible categories for the random shapes: (a) ‘blobs’, (b) ‘rods’ and (c) 'slabs’, depending on whether three mean orthogonal dimensions of the domains were such that (a) all were of the same order, (b) one was an order larger than the other two, or (c) one was an order smaller than the other two.

Highly idealized paradigms for these three categories were studied analytically: the two-position coincidence functions were computed for the cases of (a) spheres, (b) circular cylinders and (c) plane slabs, each field containing randomly sized elements distributed randomly in space with a homogeneous and isotropic distribution. Comparison of the measured coincidence functions with these three paradigms suggests that the fine-structure regions are more nearly of ‘rod-like’ geometry than like either of the other two. No attempt was made to distinguish shapes which might be called 'strips’.

Part 3 of this study treats a prototype with insulating side walls at z = ± 1 for all x and insulating top and bottom walls a ty = ± a for x < 0 and at y = ± (a + bx) for x > 0, where the applied magnetic field is in the y direction and the flow is in the x direction. In the diverging portion (x > 0) of this duct, the entire mass flux is carried by high-velocity jets adjacent to the side walls, while the fluid elsewhere is stagnant. In the constant-area portion (x < 0), the fully developed flow is severely disturbed as it approaches the join at x = 0, and high-velocity jets occur even before the top and bottom walls begin to diverge. The analysis presented in Part 3 is not valid in the limit b → 0, and the object of this paper is to reconcile the stagnant core flow for b ≠ 0 with the fully developed flow for b = 0. Conditions are such that inertia forces are negligible.

The fist transitional stage occurs when 1 [Gt ] b [Gt ] M−½, where M is the (large) Hartmann number. The upstream disturbance disappears, and downstream each of the O(M−½) side-wall boundary layers splits into an O(b−lM−½) outer layer and an O(M−½) inner layer. The fluid outside these layers is still stagnant and an O(bM−½) velocity in the outer sublayers accounts for the mass flux. The viscous inner sublayers reduce the velocity in the outer sublayers to zero at the side walls.

The second transitional stage occurs when b = O(M−½). The outer sublayers spread across the entire duct so that none of the fluid is stagnant, and an O(1) core velocity accounts for the mass flux. This analysis is valid no matter how small b becomes, and as b → 0 the fully developed solution is recovered every-where.

The development of magnetohydrodynamic flow due to the passage of a uniform electric current past a sphere immersed in an incompressible viscous conducting fluid extending to infinity is considered. The flow field is the response of the fluid to the Lorentz force set up by the electric current and the associated magnetic field. The solution is based on the assumption that the flow field is weak and has a negligible effect on the electromagnetic variables. We also assume that the convection terms in the momentum equation are negligible. The solution, obtained by means of Laplace transforms, is analytic except for the evaluation of an integral which is done numerically. It is shown that the flow field spreads radially from the sphere into the fluid. The rate of development of the flow field increases with the ratio of the conductivity of the sphere to that of the fluid.

The behaviour of an initially plane, strong shock wave propagating into a conical convergence is investigated experimentally and theoretically. In the experiment a 10° half-angle cone is mounted on the end of a pressure-driven shock tube. Shock waves with initial Mach numbers varying from 6.0 to 10·2 are generated in argon a t a pressure of 1·5 Torr. During each run local shock velocities a t several positions along the cone axis are measured using a thin multi-crystal piezoelectric probe inserted from the vertex. This technique produces accurate velocity data for both the incident and reflected shock waves. In the corresponding analysis, a simplified characteristics method is used to obtain an approximate solution of the axisymmetric diffraction equations derived by Whitham (1959).

Both the shock velocity measurements and the axisymmetric diffraction solution confirm that the incident shock behaviour is dominated by cyclic diffraction processes which originate at the entrance of the cone. Each diffraction cycle is characterized by Mach reflexion on the cone wall followed by Mach reflexion on the axis, These cycles evidently persist until the shock reaches the cone vertex, where the measured velocity has increased by as much as a factor of three. Real-gas effects, enhanced in the experiment by increasing the initial Mach number and decreasing the pressure, apparently alter the shock wave behaviour only in the region near the vertex. Velocity measurements for the reflected shock within the cone show that the shock velocity is nearly constant throughout most of the convergence length.

It is shown that for low values of the ambient pressure the flow field for a steady spherically symmetric expansion can be divided into three parts termed the inviscied region, the intermediate layer and the shock layer. Analytic solutions are available in the first two regions and a complete integration of the equation is required in the third. Numerical solutions indicate that such a structure is achieved in the limit and the universality of the solutions in the individual regions is confirmed.

Experiments on fully developed turbulent flow in a channel which is rotating at a steady rate about a spanwise axis are described. The Coriolis force components in the region of two-dimensional mean flow affect both local and global stability. Three stability-related phenomena were observed or inferred: (i) the reduction (increase) of the rate of wall-layer streak bursting in locally stabilized (destabilized) wall layers; (ii) the total suppression of transition to turbulence in a stabilized layer; (iii) the development of large-scale roll cells on the de-stabilized side of the channel by growth of a Taylor-Gortler vortex instability.

An appropriate local stability parameter is the Richardson number formulated by Bradshaw (1969) for this case and the analogous cases of flow over curved walls and of shear-layer flow with density stratification. Local effects of rotational stabilization, such as reduction of the turbulent stress in wall layers, can be related to the local Richardson number in a simple way. This paper not only investigates this effect, but also, by methods of flow visualization, exposes some of the underlying structure changes caused by rotation.

The objectives of this study were to determine whether the addition of drag-reducing macromolecules alters the structure of the viscous sublayer and thereby modifies the production of kinetic energy in turbulent wall flows. This was accomplished by visualizing the near-wall region of a fully developed two-dimensional channel flow. Motion pictures were taken of dye injected into the near-wall region. Both water and a dilute drag-reducing polyethylene oxide-FRA solution were used as working fluids. The motion pictures were analysed to determine the spanwise spacing and the bursting rate of low-speed streaks that are characteristic of the viscous sublayer. The amount of drag reduction was established from pressure-drop measurements in pipe flows and a correlation that is independent of hydraulic diameter.

The data show that the time between bursts for an individual streak in a drag-reducing flow has the value for a water flow at the reduced wall shear. However, both the physical and the non-dimensional streak spacing is significantly increased in the drag-reducing flows and thus the spatially averaged bursting rate is decreased. This evidence strongly suggests that the dilute polymer solution decreases the production of turbulent kinetic energy by inhibiting the formation of low-speed streaks. A tentative explanation for this behaviour which is based upon the solution's high resistance to elongational strains and vortex stretching is offered.

According to classical boundary-layer theory, uniform flow past a semi-infinite wedge, inclined at a negative angle ½πβ to the direction of the free stream, does not separate unless β ≤−0·1988. It has been assumed, therefore, that, in the range −0·1988 < β < 0, the flow within the boundary layer is represented by the Falkner-Skan equation, which, as was shown by Stewartson (1954), has two admissible solutions. All such solutions for p < 0 appear to be somewhat unsatisfactory, however, because they require an adverse pressure gradient, which, by becoming infinite as the corner of the wedge is approached, could lead to separation even if, β > −0·1988. In addition, the structure of the high Reynolds number flow for β < −0·1988 has remained, to date, unresolved.

We present here a fundamentally different solution to this classical problem which eliminates the singularity in the potential region by allowing the flow to separate at the leading edge of the inclined surface. The associated flow field is then characterized by an essentially uniform free stream flowing over an inviscid and, to a high approximation, irrotational region of reverse flow in which the velocity is of O(R−½) in magnitude, R being the Reynolds number. Mixing of these two streams is confined to a free shear boundary layer, of O(R−½) in thickness, extending downstream from the leading edge and parallel to the direction of the undisturbed main flow. Finally, an additional boundary layer, of O(R−½) in thickness, is shown to exist between the separated region and the surface of the wedge. Owing to the absence of a characteristic length in the problem, similar solutions to the appropriate equations describing the flow in each region are obtained and are valid for all β < 0, provided that the Reynolds number is sufficiently large. The analysis is then extended to higher order in R to increase its range of validity and to demonstrate that the proposed structure of the flow field remains self-consistent. Although the solution is developed only for a semi-infinite wedge with β < 0, it is believed that certain of its features may be of value in the analysis of other problems involving high Reynolds number separated flows.

A small-scale apparatus was used to examine the effects of surface materials on the initiation of waves on water by the action of wind. Rigorous procedures were followed to ensure freedom from contamination of the initial water surface, and the effect of various forms of subsequent contamination were investigated, using a sensitive moire fringe wave slope observation technique. Provision was made in the design of the water container for insoluble films to circulate, avoiding the downwind buildup of contamination.

In measurements on pure water surfaces, no evidence of a ‘critical wind velocity’ was found, although slightly contaminated water was found to be appreciably less responsive to the wind excitation at wind speeds up to 4 m s−l. Measurements on monolayer-covered surfaces showed that, at these wind speeds, a compressed monolayer of relatively high surface tension and low dilational elasticity can have a greater stilling effect than one of low surface tension and high dilational elasticity. Measurements on a series of solutions of surface-active material, covering a range of surface tensions from 63 to 26 mN m−l at a concentration expected to reduce dilational elasticity by bulk-to-surface diffusion, indicated that the damping effect becomes steadily greater as the surface tension decreases.