The potential problem for the flow at high Reynolds numbers R outside the boundary layer and wake of a thin flat plate at small incidence with allowance for displacement thickness is not fully denned unless the position of the wake is known in advance. The Kutta–Joukowski hypothesis does not provide a satisfactory first approximation to this because of the singularity in curvature of the streamline springing from the trailing edge in inviscid flow, which implies that the initial curvature of the wake in the real flow will be large enough to cause a modification to the potential flow. The net vorticity per unit length in a curved wake is approximately U∞ δ2∞dθ0/ds, where U∞, δ2∞ and dθ0/ds are respectively the undisturbed stream velocity, momentum thickness at infinity and curvature. The outer potential problem is set up with a vortex sheet of this strength to represent the wake, leading to a singular integro-differential equation for θ0(s). From the particular solution we obtain a proportionate correction – (CD/4π) (log4/CD) to the Kutta–Joukowski circulation, where CD is the drag coefficient. For laminar flow this is of order R−½ log (1/R). However, the solution also contains an arbitrary constant which cannot be settled without an examination of the near wake. The recent work of Brown & Stewartson (1970) suggests that this may lead to a term of the lower order $R^{-\frac{3}{8}}$, but depends on the assumption, not supported by the present analysis, that the pressure rise across the wake is o(R−¼).

This Part of J.F.M. has been planned as a salutation to Sydney Goldstein on the occasion of his retirement from the Gordon McKay Chair at Harvard. The readiness with which the editors of J.F.M. agreed to set aside this part in this way is a mark of the esteem in which S. G. is held throughout the world of fluid mechanics.

An analysis is presented of the flow field near a neutrally-buoyant rigid spherical particle immersed in an in compressible Newtonian fluid which, at large distances from the particle, is undergoing simple shear flow. Subject to conditions of continuity of stress at the particle surface and to conditions of zero net torque and zero net force on the sphere, the effect of fluid inertia on the velocity and pressure fields in the vicinity of the particle has been computed to $O(R^{\frac{3}{2}})$, where R = a2G/ν is a shear Reynolds number, a being the sphere radius, G the velocity gradient in the free stream (taken to be a positive number), and ν the kinematic viscosity.

Some streamlines have been computed and plotted. These illustrate how the fore–aft symmetry of the creeping-motion solution is destroyed when one includes inertial effects.

Knowledge of the velocity and pressure fields enables one to compute the effect of inertial forces in suspension rheology. The results include a correction to the Einstein viscosity law to $O(R^{\frac{3}{2}})$ for a dilute (non-interacting) suspension of spheres. In addition it is found that inertial effects give rise to a non-isotropic normal stress.

This paper deals with Stokes flow due to a stationary axially symmetric slender body in a uniform stream, which may be either parallel or perpendicular to the axis of the body. The effect of the body is represented by distributions of singularities along a segment of its axis of symmetry. Systems of linear integral equations for these distributions are obtained, and the first few terms of uniformly valid (in the Stokes region) asymptotic expansions in the slenderness ratio are discussed. The leading terms yield the expected result that the drag on the body in a transverse stream is double that in an axial stream. The second approximation to the ratio of these two drags is also independent of the body shape.

A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUε, where U is a measure of the undisturbed velocity and ε = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of ε2 are presented, together with iterative relations which allow their determination to the order of any power of ε. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of ε, and, for a cylindrical body, to the order of ε3.

The boundary-layer flow just upstream of the trailing edge of a flat plate is studied when a free streamline is attached to the edge. The separation at the edge occurs with an infinitely favourable pressure gradient and is characterized by a skin friction which is proportional to the inverse eighth power of the distance from the edge. The proportionality factor for the first-order term is independent of the upstream boundary-layer flow. The streamwise velocity profile at separation is non-analytic near the wall Y = 0, and starts with the term $Y^{\frac{2}{3}}$.

Turbulent boundary-layer wall-pressure measurements were made with ‘pinhole’ microphones three times smaller (relative to a boundary-layer displacement thickness) than microphones used in earlier work. The improved high-frequency resolution permitted examination of the influence of high-frequency eddies on smooth-wall pressure statistics. It was found that the space-time decay rate is considerably higher than previously reported. Measurements of cross-spectral density made with 5 Hz bandwidth filters disclosed low phase speeds at low frequency and small separation. Measurements were repeated on rough walls and parallels were drawn from knowledge of a smooth-wall boundary-layer structure to propose a structure for a rough-wall boundary layer. The effect of independently varying roughness height and separation on the large and small-scale turbulence structure was deduced from the measurements. It was found that roughness separation affected the very large-scale structure, whereas the roughness height influenced the medium and very small-scale turbulence.

A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted.

Photographs and simultaneous temperature measurements show that the thermal oscillations which occur in air between horizontal plates with unstable stratification are associated with lateral oscillations of wavy convection rolls. The phenomenon appears to consist of an oscillating motion nearly independent of depth superimposed upon the convective motions, and is best viewed from above. For large Prandtl number the photographic evidence suggests that the first unsteady motions to occur consist of a very similar oscillation, except that the features which oscillate are updraft and downdraft lines of short length. The periods of the oscillations obtained visually agree well with those reported previously from fixed temperature measurements, and a weak oscillation of doubled frequency is explained.

Thermal instabilities of a contained fluid are investigated for a fairly general class of problems in which the dynamics are dominated by the effects of rotation. In systems of constant depth in the direction of the axis of rotation the instability develops when the buoyancy forces suffice to overcome the stabilizing effects of thermal conduction and of viscous dissipation in the Ekman boundary layers. Owing to the Taylor–Proudman theorem, a slight gradient in depth exerts a strongly stabilizing influence. The theory is applied to describe the instability of the ‘lower symmetric régime’ in the rotating annulus experiments at high rotation rates. An example of geophysical relevance is the instability of a self-gravitating, internally heated, rotating fluid sphere. The results of the perturbation theory for this problem agree reasonably well with the results of an extension of the analysis by Roberts (1968).

The method of matched asymptotic expansions is used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number. This technique is restricted to problems for which the dimensionless frequency ε, defined as ωb/a0, is small, ω being the circular frequency, b the typical body dimension, and a0 the speed of sound. By combining Lorentz and Galilean transformations, the problem is transformed to a space in which the approximation appropriate to the inner region is found to be incompressible flow and that appropriate to the outer, classical acoustics. This approximation for the inner region is the unsteady counterpart of the Prandtl-Glauert transformation, but is not identical to use of that transformation in a straightforward quasi-steady manner. For wings in oscillatory motion, it is the same approximation as was given by Miles (1950).

To illustrate the technique, two examples are treated, one involving a pulsating cylinder in a stream, the other the impinging of plane sound waves upon an elliptical wing in a stream.

Observations were made of the appearance of hydrodynamic cavities behind a series of axisymmetric headforms. Among the phenomena investigated was the transition of the interfacial or separated boundary layer on the cavity surface. The first stage of this process, namely the spatial growth of instability waves could be distinguished by means of high-speed photography. Comparison is made with a theoretical instability analysis.

Experiments on fully developed cavity flows were carried out with the prime initial objective of investigating the effects of the addition of small quantities of ‘turbulent drag reducing additive’ upon the cavity-surface boundary-layer instability and transition reported in the previous paper (Brennen 1970). However, in most instances, the additives were found to cause an unforeseen instability in the wetted surface flow around the headform. Upon convection, the resulting disturbances dramatically disfigured the cavity surface, thus negating the original purpose. This new phenomenon warranted investigation and became the principal subject of this paper.

A numerical study has been made of steady laminar convection in an infinite horizontal layer of fluid, bounded above by a rigid plate held at constant temperature, and below by a thermal insulator. The fluid is heated by a uniform distribution of heat sources. Recent methods for the solution of the Navier-Stokes equations by finite differences are discussed, and the results of integrations in two and three space dimensions and time are presented in an attempt to determine the planform and other flow properties as functions of the Rayleigh and Prandtl numbers.

These results are compared with a preliminary study by Roberts (1967) using a mean-field equation approach and with experimental observations by Tritton & Zarraga (1967) and Hooper (private communication).

It is shown that the expansion assumed by Stewartson to describe the flow close to separation in a compressible boundary layer is incomplete. When the wall is cold an infinity of new terms involving log ξ, log log ξ and their products and quotients must be added at each algebraic stage. The skin friction then vanishes like x½ ln x where x is the distance to separation. None of the coefficients of the logarithmic terms are arbitrary and in particular the first two terms in the expansion of the skin friction are known if the heat transfer is given at separation. Convergence is so slow, however, that this is of no practical value.

The paper examines the stability of the uniform flow which approaches a two-dimensional stagnation region formed when a cylinder or a two-dimensional blunt body of finite curvature is immersed in a crossflow. It is shown that such a flow is unstable with respect to three-dimensional disturbances. This conclusion is reached on the basis of a mathematical analysis of a simplified form of the disturbance equation for the stream-wise component of the vorticity vector. The ultimate, or stable, flow pattern is governed by a singular Sturm–Liouville problem whose solution possesses a single eigenvalue. The resulting flow is one in which a regularly distributed system of counter-rotating vortices is super-imposed on the basic, Hiemenz-like pattern of streamlines. The spacing of the vortices is a unique function of the characteristics of the flow, and a theoretical estimate for it agrees well with experimental results. The analysis is extended heuristically to include the effect of free-stream turbulence on the spacing.

The problem is similar to the classical Görtler–Hämmerlin study of the stability of stagnation flow against an infinite flat plate, which revealed the existence of a spectrum of eigenvalues for the disturbance equation. The present analysis yields the same result when an infinite radius of curvature is assumed for the blunt body.

We obtain exact solutions of the linearized perturbation equations for the convective motions in a Boussinesq fluid contained in an infinite rectangular channel and heated from below. The top and bottom are assumed to be perfect heat conductors. The sides can either be conducting or insulating. We assume that the sidewalls are rigid, but allow the top and bottom to be free so that we can separate variables.

We find that the preferred modes of convection closely resemble transverse ‘finite rolls’ [as predicted by Davis (1967) for convection in a box] for channels with height to width ratios outside the range 0·1 to 1. Inside this range they show noticeable departures from roll form. We prove, however, that, except when the sidewalls are relaxed to infinity, ‘finite rolls’ are never exact solutions of the linearized equations, even though in most cases they are good approximations.

We also find that bringing the sidewalls closer together inhibits convection and, generally, produces thinner cells at the onset of convection.

A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small.

Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense.

Some exact solutions of the steady M.H.D. equations for an inviscid infinitely conducting fluid are discussed, and a generalization of the Prendergast model for an idealized magnetic star is obtained, in which there is a finite azimuthal swirl velocity of the fluid. The flow in all examples is bounded by a sphere and expressions are found for the distribution of energy between the fluid motion and magnetic field.

Initially, this study arose from questions concerning the scattering of tsunamis as they propagate over the irregular topography of the deep waters of the ocean. The mathematical problem to which this led is pertinent to many other phenomena, however, and we direct the analysis, here, to the propagation of gravity waves over an irregular bottom topography and to the lateral oscillations of an elastic string whose ends undergo random longitudinal displacements. Several facets of the mathematical problem are rather fascinating but the results do suggest that scattering is not the most important part of the tsunami propagation.