The drag on a cylindrical particle moving in a thin sheet of viscous fluid is calculated. It is supposed that the sheet is embedded in fluid of much lower viscosity. A finite steady drag is obtained, which depends logarithmically on the ratio of the viscosities. The Einstein relation is used to determine the diffusion coefficient for Brownian motion of the particle, with application to the movement of molecules in biological membranes. In addition, the Brownian motion is calculated using the Langevin equation, and a logarithmically time-dependent diffusivity is obtained for the case when the embedding fluid has zero viscosity.

A simple derivation is given of the parabolic flow first described by John (1953) in semi-Lagrangian form. It is shown that the scale of the flow decreases like t−3, and the free surface contracts about a point which lies one-third of the way from the vertex of the parabola to the focus.

The flow is an exact limiting form of either a Dirichlet ellipse or hyperbola, as the time t tends to infinity.

Two other self-similar flows, in three dimensions, are derived. In one, the free surface is a paraboloid of revolution, which contracts like t−2 about a point lying one-quarter the distance from the vertex to the focus. In the other, the flow is non-axisymmetric, and the free surface contracts like t−5.

The parabolic flow is shown to be one of a general class of self-similar flows in the plane, described by rational functions of degree n. The parabola corresponds to n = 2. When n = 3 there are two new flows. In one of these the scale varies as t12/7 and the free surface has the appearance of a trough filling up. In the other, the free surface resembles flow round the end of a rigid wall; the scale varies as t−4·17.

Inviscid methods of treating secondary flows due to streamline curvature are well known in the literature. The development of secondary motions at high Reynolds number in a bifurcation characteristic of the lung airways or the cardiovascular system is considered. In particular, the secondary motions in the initial section of a bifurcation, which can be considered a tube of slowly varying ellipticity, are calculated. Providing viscous effects are confined to regions near the walls, the calculated velocity field agrees with experimental observation. Thus the structure of the secondary motions can be considered an effect of stretching of vortex lines by the tube boundary.

In this paper we consider the flow field induced in an incompressible viscous conducting fluid in a hemispherical bowl by a symmetric discharge of electric current from a point source at the centre of the plane end of the hemisphere. This plane end is a free surface. We construct an analytic solution for the slow viscous flow and a numeriacl solution for the nonlinear problem. The streamlines in an axial cross-section form two sets of closed loops, one on either side of the axis. Our computations indicate that, for a given fluid, when the discharged current reaches a certain magnitude the velocity field breaks down. This breakdown probably originates at the vertex of the hemispherical container.

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.

Measurements are described of the fluid velocities close to rough beds oscillating in their own plane. The roughness with which most of the results were obtained consisted of smooth spheres closely packed in hexagonal formation. Some results are also given for beds of gravel. The beds were oscillated with simple harmonic motion in still air and the measurements were made with a hot-wire anemometer.

The measurements very close to the beds of smooth spheres show two maxima in the velocity profile during each half-cycle. One maximum corresponds to a component of velocity which varies nearly sinusoidally with time. The second forms quite a sharp peak and occurs close to ωt = 90°, 270°, where ω is the angular frequency of oscillation and t is time measured from the instant of maximum velocity of the plate. The phase at which this peak occurs shows little variation with distance from the bed. For values of βD > 3·0, where β = (ω/2ν)½, ν is the kinematic viscosity and D is the sphere diameter, the maximum velocity during each half-cycle is found at this peak over at least a certain range of distances from the bed. The variation with height of the nearly sinusoidal component of velocity is quite close to that given by Stokes’ (1851) solution for a flat plate. The peak at ωt = 90°, 270°, however, rises from zero at the bed to a maximum at a distance of about one-eighth of a sphere diameter above the crests and then falls off again.

The measurements with beds of gravel show a variation in velocity similar to that observed by Kalkanis (1957, 1964) and Sleath (1970). Because of the irregularity of the surface it is difficult to draw definite conclusions about the flow in the immediate vicinity of the bed.

A number of tests were carried out, with the beds of spheres, using a wire slanted at 45° to the bed in order to determine the velocity product uw.

An experiment was performed in a 3·5 by 3·5 m variable-height, closed convection box, with conditions ranging from a Rayleigh number of 4 × 104 up to 7 × 109, using air as the working fluid. Heat-flux measurements made at Rayleigh numbers up to 7 × 109 yielded a Nusselt number Nu = 0·13Ra0·30. Velocities and temperatures were measured up to Ra = 1·7 × 107, and Fourier spectra calculated to find the predominant horizontal scales of the motion midway between the boundaries. The predominant scale at Ra ∼ 105 was approximately four times the distance between plates, changing to six as Ra increased to 106. With side walls introduced so that the transverse aspect ratio was equal to five, Fourier spectra indicated considerable smaller scale motions, approximately equal to the layer depth. These motions decreased in size as Ra was increased. The results are discussed in relation to previous experimental and theoretical work.

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

The steady laminar motion of fluid through pipes of circular cross-section, the curvature of whose centre-line varies locally, is analysed theoretically. The flow in three kinds of pipes whose centre-lines are specified by \[ \hat{y} = a(1+\kappa^2\hat{x}^2)^{\frac{1}{2}},\quad\hat{y} = a\tan h\kappa\hat{x}\quad{\rm and}\quad\hat{y} = a\sin\kappa\hat{x} \] are treated as the examples of once-, twice- and periodically-curved pipes, respectively. The analysis is valid for any other two-dimensionally curved pipes, when centre-line curvature is small. At very small Reynolds number, the position of maximum axial velocity shifts towards the inner side of the pipe section; at large Reynolds number, on the contrary, it tends to the outer side, owing to centrifugal force. Furthermore, in the latter case, adaptation of the flow follows the change of mean-flow direction, with a phase lag.

The present paper and part 2 (adjacent) study the sound field produced by a convected point quadrupole embedded in and moving along the axis of a round plug-flow jet. Only subsonic eddy convection velocities are considered. We examine cold jets here and hot jets in part 2. A principal feature of the study is extensive comparison with jet-noise data. It appears that this simple model problem succeeds in explaining all the major interesting features of jet-noise data, on both hot and cold jets, for jet exit velocities in the low supersonic range. Particular success is achieved in explaining aspects of the data not explainable by the Lighthill acoustic-analogy approach. The picture of jet-noise generation that emerges (at least for jet velocities in the low supersonic regime) is in many respects a striking reaffirmation of the Lighthill point of view. It appears that there is an intrinsic or universal distribution of compact quadrupoles, whose strength and frequency distribution scale with the jet velocity and nozzle diameter as would be expected from simple dimensional reasoning, responsible for jet-noise generation. These quadrupoles are of course convected by the mean flow and satisfactory agreement with the data is obtained by assuming that they are devoid of any intrinsic directionality. There appears to be no significant jet Mach number (compressibility) or jet temperature effect on the scaling of this intrinsic distribution. The essential improvement over the Lighthill analysis is the incorporation of mean-flow shrouding effects on the radiation of the convected quadrupoles. It is perhaps no exaggeration to claim that, with the incorporation of such a shrouding effect, the problem of scaling jet noise with regard to the jet velocity, jet temperature, jet size and the angle from the jet axis appears to be completely resolved. (The ‘scaling’ principle cannot of course be very simply expressed and in fact needs calculations of the sort contained in the present paper to implement it.)

This paper continues the study of part 1 into the area of the noise of heated jets. First, this part of the study discusses how a convected wave equation approach based on Lilley's equation leads to additional dipole and simple source terms associated with the velocity fluctuations due to transverse gradients of the mean density of the flow. Once these source terms have been identified and roughly estimated, we revert to a plug-flow model of the jet flow (where now the jet temperature and jet density differ from the ambient values) to estimate the radiation of these singularities. Several novel physical aspects of hot-jet noise are uncovered by the analysis. Indeed the problem of hot-jet noise is the one where the greatest deviations from Lighthill's ideas on jet noise generation are evident. The results are applied to available data and a very satisfactory measure of agreement is obtained with respect to the various predictions of the theory. Mechanisms for ‘excess’ pure jet noise scaling on M6 and M4 are found to result from the density gradients of the mean flow. The satisfactory agreement with the data suggests a solution of the problem of scaling jet noise with regard to jet temperature effects. The ability to predict correctly the data also suggests that the jet temperature has very little effect on the turbulence source spectrum generating jet noise at least for jet exit velocities up to about 1·5 times the atmospheric speed of sound.