A method is developed for determining any thin steady two-dimensional potential flow with free and/or rigid boundaries in the presence of gravity. The flow is divided into a number of parts and in each part the flow and its free boundaries are represented as asymptotic series in powers of the slenderness ratio of the stream. There are three basic flows, having two, one and no free boundaries and called jet flow, wall flow and channel flow, respectively. First the three expansions for these flows are found, extending results of Keller & Weitz (1952). They are called outer expansions to distinguish them from the inner expansions which apply near the ends of the stream or at the junction of two different types of flow. The inner and outer expansions must be matched at a junction to find how the emerging flow is related to the entering flow. This process can be continued to build up any complex flow involving thin streams. The method is illustrated in the case of a wall flow that leaves the wall to become a jet, which includes the case of a waterfall treated by Clarke (1965) in a similar way. In part 2 (to be published) other inner expansions are found and matched to outer expansions, providing the ingredients for the construction of the solutions of many flow problems.

The Stokes equations describing the creeping motion of two arbitrary-sized touching spheres are solved exactly through the use of tangent-sphere coordinates. For the case of a linear shear field at infinity, explicit results covering the entire range of size ratios are presented for: (a) the forces and torques on the aggregate; (b) the hydrodynamic forces on the individual spheres comprising a freely suspended aggregate, which are in general non-zero; (c) the contribution of the pair to the bulk stress of a dilute suspension; and (d) under free suspension conditions, the velocity of any material point relative to that of the undisturbed flow.

An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.

An experimental study is made of nonlinear interactions in a laminar free shear layer. Two disturbances (f1 and f2), excited by sound, amplify and grow independently for small amplitudes. At larger amplitudes the disturbances interact to generate fluctuations of sum and difference frequencies (f2 ± f1). Harmonics and subharmonics of f1 and f2 are also generated and all fluctuations interact to generate additional fluctuations of the form (nf2/m) ± (pf1/q); n, p = 1,2,3,…, m, q = 1,2. Nonlinear mode competition suppresses the growth of f1 or f2, depending on their relative amplitudes, and contributes to finite amplitude equilibration. An upper bound on the modal integral of total u′r.m.s.2 fluctuation energy is found. Fluctuation energy tends to be distributed among all possible frequency components, and its upper bound does not increase as the number of components increases.

The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. All these boundary conditions are modelled by two parameters: a length correction and a reflexion coefficient equivalent to the acoustic impedance.

The linear theory predicts large amplitudes near resonance and nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and wave form of the oscillation are predicted there.

In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.

An analysis is presented that is appropriate to three distinct phases in the temporal development of the flow past a semi-infinite vertical plate whose temperature is suddenly changed from that of the surrounding fluid. These are the initial stage when a one-dimensional solution describes the flow, then a local solution which describes the early stages of the departure from this, and finally an asymptotic solution which describes the manner in which the final steady state is achieved.

The results of Ackerberg (1970, 1971a), on the two-dimensional boundary-layer separation at a sharp trailing edge where a free streamline is attached, are extended to the axisymmetric case, with and without swirl. When the flow has swirl, the boundary-layer swirl velocity close to the wall may be opposite to that in the external flow; this may help explain the ‘bathtub vortex paradox’ observed by Sibulkin (1962). The solutions for the detached shear layers downstream of separation for two-dimensional and axisymmetric flows, with and without swirl, have been obtained. Some misprints in parts 1 and 2 are corrected in the appendix.

The behaviour of a conserved scalar field in the final period of an axisymmetric turbulent wake is investigated theoretically. Explicit formulae are derived for the behaviour, during and just prior to the scalar field final period, of quantities such as mean concentration, concentration correlations and velocity–scalar or vorticity–scalar cross-correlations. In particular, if τ is the time measured from a virtual origin it is shown that scalar intensity, which decays as τ−5/2, is more persistent asymptotically than turbulence intensity, which displays a τ−7/2 decay. By deriving the scalar field structure as perturbed by the Reynolds stresses acting on mean scalar gradients in the final period it is shown, e.g., that the mean field perturbation dies out at least as rapidly as τ−4, whereas the mean field itself decays as τ−1. Numerical results are presented to display the spatial structure of typical scalar correlations and velocity–scalar cross-correlations, which are compared where possible with non-asymptotic measurements in the wake of a heated sphere.

In deducing the viscous damping rate in surface waves confined by side walls, Ursell found in an example that two different calculations, one by energy dissipation within and the other by pressure working on the edge of the side-wall boundary layers, gave different answers. This discrepancy occurs in other examples also and is resolved here by examining the energy transfer in the neighbourhood of the free-surface meniscus. With due care near the meniscus a boundary-layer–Poincaré method is employed to give an alternative derivation for the rate of attenuation and to obtain in addition the frequency (or wave-number) shift due to viscosity. Surface tension is not considered.

The dispersion equation for waves on an infinite uniform jet column of elliptic cross-section is derived, and approximated for large eccentricity with the aid of new asymptotics for the modified Mathieu functions. It is shown that the effect of appreciable eccentricity on lateral disturbances is greatly to reduce their growth rates below those for a circular jet, regardless of whether the disturbance grows spatially or temporally. For ‘vertical’ disturbances it is shown that the behaviour of waves of general length is qualitatively similar to that of long waves on a two-dimensional jet. Thus the mode symmetric about the major axis has small growth rate whether the mode grows temporally or spatially, while the mode antisymmetric about the major axis has small growth rate if temporally growing, but large growth rate if spatially growing. Comments are made as to the relevance of these results to the mode of action of jet silencers which squash a round jet into a flat ‘fish-tail’ shape.

The Phillips (1960) convected wave equation is employed in this paper to study aerodynamic noise emission processes in subsonic and supersonic shear layers. The wave equation in three spatial dimensions is first reduced to an ordinary differential equation by Fourier transformation, then solved via the WKBJ method. Three typical solutions are required for discussions in this paper. The current results are different from the classical conclusions. The effects of refraction, convection, Mach-number dependence and temperature dependence of turbulent noise emission are analysed in the light of solutions to the Phillips equation. Owing to the inherent restrictions of the WKBJ transformation, the results of the present paper should be applied to wave radiation from shear layers whose thickness is no less than approximately one quarter of a wavelength. Such a condition is satisfied for turbulent round jets with an exit velocity greater than 0·6 times the ambient speed of sound.

We consider the flow in a differentially heated vertical slot filled with a stably stratified solution. The stability of the flow driven by the differential heating is investigated in the limits of small but finite amplitude disturbances and very large solute Rayleigh number RS = gβ(∂Sa/∂z)D4/KSv. If the Schmidt number H = KT/KS is of order 1, the growth of an initial perturbation at the neutral point is balanced by horizontal advection of solute and heat, and a steady equilibration amplitude is attained. The Nusselt number is independent of all fluid properties and is directly proportional to the Rayleigh number excess ε = (Ra – Rac)/Rac. If H is much greater than |RS|⅙, or if the disturbance wave-number is slightly less than the critical wavenumber, subcritical instabilities are possible. In particular a resonant instability is possible. These theoretical predictions are consistent with previous experimental results and with the laboratory results described in this paper. In the experiments we find that the mixing of the initial sugar gradient is accomplished by convection cells which undergo transitions to larger wavelengths. The breakdown of the interfaces between convection cells is described.

Delta wings with conically subsonic cones-bodies mounted on their compressive side are analysed in the hypersonic small disturbance regime. The weakly three-dimensional conditions associated with slender parabolic Mach cones are used to validate a linearized rotational approximation of the flow field. A combined integral–series representation is obtained for the pressure distribution between the wing-body and shock wave for arbitrary body cross-sections, and is specialized to give analytical formulae for arbitrary-order polynomial transversal contours. Numerical results are presented for wedge, parabolic and higher order cross-sections illustrating the dominant character of the cross-flow stagnation singularity associated with sharp wing-body intersections having a finite slope discontinuity. It is shown that the pressure has a logarithmic infinity at this secondary leading edge, as in corresponding Prandtl–Glauert irrotational flows. The relation of this finding to Lighthill's theorem on cross-stream vorticity is discussed. Other features of the pressure field are considered with particular emphasis on their relationship to a recently derived area rule for such configurations, and possibilities for favourable interference.

An analysis is made of the stability of an unsteady basic flow of a conducting fluid in the presence of a parallel magnetic field. The particular profile investigated is the classical Kelvin–Helmholtz profile modified by the addition of an oscillatory component. Two cases are considered in detail: that of a perfectly conducting fluid and that of a poorly conducting fluid. The investigation leads, in both cases, to an equation of the Hill type. It is concluded that the magnetic field has a stabilizing influence but is nevertheless unable to suppress the Kelvin–Helmholtz instability in an unsteady (basic) flow.

Expansions have been given in the past for steady Stokes waves at or near a largest wave with a 120° corner. It is shown here that the solution is more complicated than has been assumed: that the corner is not a regular singular point, and that waves of less than maximum amplitude have singularities of a different order.

A two-dimensional oil slick is examined as it spreads on water under the influence of gravitational and viscous forces. An exact analytical and numerical description of the flow is obtained in a certain rational asymptotic limit. The central result is an expression for the size of the slick as a function of time. This expression contains no free parameters to be empirically determined, and is in substantial agreement with experiment.

Burgers’ (1939) model equations for turbulence are considered analytically using a singular perturbation and nonlinear wave approach. The results indicate that there is an ultimate steady turbulent state. This is in agreement with the numerical results of Lee (1971) but not with Case & Chiu (1969): the last two papers start with a Fourier series approach.

A consequence of this model is that small disturbances ultimately grow into a single large domain of relatively smooth flow, accompanied by a vortex sheet in which strong vorticity is concentrated. This makes the results from the model different from those usually expected for turbulent flow fields. The model, as a result of its simplicity, has retained a degree of regularity which is not found in most forms of turbulence.

This paper deals with nonlinear streaming effects associated with oscillatory motion in a viscous fluid. A previous theory by Holtsmark et al. (1954) for the streaming near a circular cylinder in an incompressible fluid of infinite extent is reconsidered and used to obtain new numerical results, which are compared with earlier observations. The regime of validity of this theory is considered. The condition to be satisfied by the Reynolds number is found to be less stringent than was previously supposed.

The more recent theory by Wang (1968) based on the outer–inner expansion technique is discussed and corrected with the Stokes drift.

The case of an incompressible fluid enclosed between two coaxial cylinders, one of which is oscillating, is considered in detail. New theoretical and experimental results are given for various values of the parameters involved (Reynolds number, amplitude and cylinder radii).

The flow induced by impulsively starting the inner cylinder in a Couette flow apparatus is investigated by using a nonlinear analysis. Explicit finite-difference approximations are used to solve the Navier–Stokes equations for axisymmetric flows. Random small perturbations are distributed initially and periodic boundary conditions are applied in the axial direction over a length which, in general, is chosen to be the critical wavelength observed experimentally. Simultaneous occurrence of Taylor vortices is obtained at supercritical Reynolds numbers. The development of streamlines, perturbation velocity components and the kinetic energy of the perturbations is examined in detail. Many salient features of the physical flow are observed in the numerical experiments.

An analysis is presented of laminar fully developed flow in a curved tube of circular cross-section under the influence of a pressure gradient oscillating sinusoidally in time. The governing equations are linearized by an expansion valid for small values of the parameter (a/R) [Ka/ων]2, where a is the radius of the tube cross-section, R is the radius of curvature, ν is the kinematic viscosity of the fluid and K and ω are the amplitude and frequency, respectively, of the pressure gradient. A solution involving numerical evaluation of finite Hankel transforms is obtained for arbitrary values of the parameter α = a(ω/ν)½. In addition, closed-form analytic solutions are derived for both small and large values of α. The secondary flow is found to consist of a steady component and a component oscillatory at a frequency 2ω, while the axial velocity perturbation oscillates at ω and 3ω. The small-α flow field is similar to the low Dean number steady flow configuration, whereas the large-α flow field is altered and includes secondary flow directed towards the centre of curvature.