This paper is concerned with steady flow in collapsible tubes, such as veins, at fairly low Reynolds number. Lubrication theory is used to calculate the velocity and pressure distribution in an elliptical tube whose cross-sectional area and eccentricity vary slowly and in a given way with longitudinal distance x. The transverse velocity field and the effect of inertia on the primary velocity and pressure distributions are calculated to first order in the relevant small parameter. The results of these calculations are combined with a relationship between transmural pressure and the cross-sectional area at any x which is close to that measured in (large) veins, and are used to predict the pressure and flow in a collapsible tube when a given distribution of external pressure is applied. Different relationships between the tube perimeter and cross-sectional area are examined. The theory is applied to an experiment in which a segment of collapsible tube is supported between two rigid segments, and squeezed; predictions of the relationship between the pressure drop and flow rate are made for various experimental conditions. In particular, when the resistance of the downstream rigid segment is held constant, a range of flow rates is found in which the pressure drop falls as the flow rate is raised; this agrees with experiment.

A photogrammetrical technique has been used to study the interaction of two identical explosively produced spherical shock waves and to compare this interaction with the reflexion of one of the spherical shocks from the ground. It is postulated that there was no energy loss in the interaction of the two shock waves and that the interaction therefore simulated the reflexion of a spherical shock from an ideal non-energyabsorbing surface. The ‘ideal’ reflexions were compared with real reflexions from two types of ground surface: one smooth and the other rough. Experiments were carried out with the centres of the spherical shocks at two separations so that observations could be made of the interaction of shocks of different strength. Significant differences were shown to exist in both the strengths of the Mach shocks and in the triple-point trajectories over the different surfaces. The results are intended to aid in the evaluation of computer codes being developed to simulate spherical-shock reflexions from real surfaces.

High-speed motion pictures (2000 frames/s) of saltating spherical glass microbeads (of diameter 350–710 μm and density 2·5 g/cm3) were taken in an environmental wind tunnel to simulate the planetary boundary layer. Analysis of the experimental particle trajectories show the presence of a substantial lifting force in the intermediate stages of the trajectories. Numerical integration of the equations of motion including a Magnus lifting force produced good agreement with experiment. Typical spin rates were of the order of several hundred revolutions per second and some limited experimental proof of this is presented. Average values and frequency distributions for liftoff and impact angles are also presented. The average lift-off and impact angles for the experiments were 50° and 14° respectively. A semi-empirical procedure for determining the average trajectory associated with given conditions is developed.

The stability of longitudinal rolls in an inclined convection layer is investigated for various angles of inclination. Three types of instability are responsible for the transition from longitudinal rolls to three-dimensional forms of convection in different regimes of the parameter space. The role of the wavy instability is emphasized since it does not correspond to a transition in the case of a horizontal layer. The analysis emphasizes the cases of air and water as convective media. Comparison of the theoretical results with experimental data indicates that the stability analysis based on infinitesimal disturbances correctly describes the observed instabilities.

A finite-difference procedure for three-dimensional parabolic flows is used to predict the development of Taylor vortices in the flow between concentric rotating cylinders, resulting from the growth of small disturbances of a Couette flow. Predictions of such flows are presented in the developing and fully developed region. A precise calculation of the wavelength of the vortices has been possible by employing a periodic boundary condition on the pressure field. The predicted torque coefficient compares well with experimental data. The critical Taylor number has also been predicted with good accuracy.

A linear stability analysis is applied to a system consisting of a horizontal fluid layer overlying a layer of a porous medium saturated with the same fluid, with uniform heating from below. Surface-tension effects at a deformable upper surface are allowed for. The solution is obtained for constant-flux thermal boundary conditions.

The collapse of a composite circular viscous tube owing to external pressure and surface tension is considered. It is shown that, for small surface tension, the collapse time, at which the tube closes, is very sensitive to the viscosity of the inner tube.

A subgrid model is presented for a passive scalar advected by a randomly prescribed frozen velocity field. In addition to an eddy diffusivity it has a term which describes the injection of noise from the subgrid scales, and a term which implicitly describes the coupling between the large eddies and the eddies just below the limit of resolution of the finite-difference grid. The construction of the model is based upon an iterative procedure which models away first the dissipation-range eddies and then the eddies of a slightly larger size, etc.

This paper is the first of a pair that describe two-point velocity measurements made at various radial positions in water in fully developed pipe flow. Axial velocity fluctuations were measured with hot-film anemometers at two points sufficiently close together that the turbulence structure remained essentially unchanged while passing between them. Phases of the cross-spectra of these velocities were then determined and interpreted in terms of a wave model of the turbulence structure. The model assigns an axial velocity and streamwise inclination to the lines of equal phase of each frequency component of the spectra.

In general, the lines of equal phase for each frequency component are inclined to the wall in the flow direction, the lower frequencies being more inclined than the higher frequencies, though all lines of equal phase at points in the central region of the pipe tend towards the perpendicular. For points near the wall the inclinations are very pronounced.

In the central region, phase velocities of lower frequency components are lower than those for higher frequencies. All phase velocities could be normalized with respect to position by the local mean velocity. The group velocity of small-scale (large wavenumber) disturbances in the core region appears to be approximately constant and of the order of the local mean velocity. This leads to a modified form of Taylor's hypothesis.

The variance in all the measurements increases rapidly in the region y+ < 26. This may be due to the intermittent nature of the flow near the wall (which is discussed in part 2) or to a rotation of the ‘frozen’ pattern by the mean shear field between the two sensors. The magnitude of the latter effect is estimated in this paper and is significant very near the wall. The results in the central region are not affected.

Developing laminar flow in a liquid film issuing from a full slot and descending along a vertical wall is analysed for the limiting cases of low flow rate and high surface tension. Neither limiting case is approached uniformly and the singular perturbation method with matched asymptotic expansions is employed. The location of a free boundary, the curved meniscus, is unknown a priori, and the limiting meniscus profiles are obtained. An interesting viscous boundary layer is found at low flow rates.

The unsteady creeping motion of a thin sheet of viscous liquid as it advances over a gently sloping dry bed is examined. Attention is focused on the motion of the leading edge under various influences and four problems are discussed. In the first problem the fluid is travelling down an open channel formed by two straight parallel retaining walls placed perpendicular to an inclined plane. When the channel axis is parallel to the fall line there is a progressive-wave solution with a straight leading edge, but inclination of the axis generates distortions and these are calculated. In the second problem a sheet with a straight leading edge travelling over an inclined plane penetrates a region where the bed is uneven, and the subsequent deformation of the leading edge is followed. The third problem considers the flow down an open channel of circular cross-section (a partially filled pipe) and the time-dependent shape of the leading edge is calculated. The fourth problem is that of flow down an inclined plane with a single curved retaining wall. These problems are all analysed by assuming that a length characteristic of the geometry is large compared with the fluid depth divided by the bed slope, and all the solutions display extreme sensitivity to the data.

Recent work (Hunter & Riahi 1975) on nonlinear convection in a rotating fluid is extended to a multi-modal regime. The schematic multi-boundary-layer method of Busse (1969) and the upper-bound technique of Howard (1963) are used to obtain upper bounds on the Nusselt number N. It is shown that there are infinitely many modes in the range $Ta \ll R^{\frac{3}{2}}$, where Ta is the Taylor number and R is the Rayleigh number, and different types of mode optimize N in different regions of the parameter space (R, Ta). While the optimal N is independent of Ta for Ta [Lt ] R, it is found that it increases with Ta in $R \ll Ta \ll (R \log R)^{\frac{4}{3}}$ and decreases as Ta increases in $(R \log R)^{\frac{4}{3}} \ll Ta \ll R^{\frac{3}{2}}$, and that the functional dependence of the optimal N on R and Ta is continuous (within a logarithmic term) throughout the region of R, Ta space.

The flow of an ostensibly two-dimensional wall jet over a logarithmic spiral has been studied both experimentally and theoretically. It is established that, if the skin friction is effectively constant, the flow may be self-preserving, and this is confirmed experimentally for the two spirals studied ($x/R = \frac{2}{3}$ and x/R = 1). The rate of growth has been predicted using the integral momentum equation and the integral equation for the combined mean and turbulent energy. Important assumptions in this theory are that the turbulence structure parameter $\overline{u^{\prime}v^{\prime}}/\overline{q^{\prime 2}}$ and the normalized mean position of the superlayer are invariant with curvature, and the experiments show that this is nearly true. The growth is constant for each spiral and increases with curvature. Using the measured rate of growth, the integral energy equation gives a satisfactory prediction of the turbulent shear stress, but the two-dimensional integral momentum does not. The turbulence is very intense in these flows and the Reynolds stresses were corrected using correlations of up to fourth order. However, the corrections may still have been too small, which would account for some of the difference between the calculated and measured shear stress. The outer flow of a wall jet strongly influences the inner boundary layer and this effect is found to increase with curvature. The conventional logarithmic law of the wall ceases to apply for $x/R >\frac{2}{3}$.

This experimental investigation determines the effect of the various parameters of a subsonic axisymmetric jet upon a fast chemical reaction between ozone contained dilutely in the primary jet and diluted nitric oxide being entrained from the ambient fluid. Radial profiles of ozone densities are determined from absorption measurements of a thin beam of ultraviolet light which is probed across the jet. Data have been obtained for five jet Reynolds numbers between 4000 and 32 000 to reflect variations in both gas density and nozzle velocity; the data confirm Reynolds number independence of reactant concentration profiles to within experimental accuracy. When the non-dimensional chemical reaction rate exceeds a determined value, the ozone profiles are observed to be independent of further increases in the reaction speed. The measured ozone concentration profiles are related directly to the molecular-scale mixing in the jet and to mixing intermittency in the mixing zone in particular.

The flows induced by the presence of an insulating sloping boundary in a doublediffusive system are examined. In the diffusive case, when the component with the larger diffusivity is unstably distributed, it is known that under certain circumstances horizontal motions are induced near the slope, and that a series of horizontal layers forms. We investigate the formation and properties of the layers, in particular their vertical scale and its dependence on the stratification and the slope angle. The scale of the layers is found to be a strong function of Gρ, the ratio of the vertical density gradient of the unstably distributed component to that of the stably distributed component. At low values of Gρ, no layering was observed; at larger values of Gρ layers were formed, and their scale increased as Gρ → 1. A weak dependence of scale on slope angle was also observed with the scale diminishing as the angle of the slope to the horizontal increased.

A new form of layering has been observed when the basic stratification is in the finger sense. At high enough values of Gρ the basic stratification is unstable to finger motions and these exist throughout the fluid. When a slope is introduced, horizontal motions are set up near the slope which cause the fingers to break down and layers are produced. There is considerable horizontal motion in these layers as well as convective motions driven by the fingers in the interfaces between the layers. The formation of these layers and some of their properties are documented.

The generation, and thence the dissipation, propagation and radiation, of waves in a compressible fluid subjected to a magnetic field is studied as an extension of the ‘acoustic analogy’ (Lighthill 1952) to magneto-acoustics. A formal theory of magneto-acoustic waves introduces (i) a single differential operator describing propagation, (ii) a dynamic and a magnetic tensor modelling generation and (iii) a dissipation tensor to complete the wave equation. The interpretation of these tensors indicates the magnitude of the physical processes of wave generation, by turbulence and inhomogeneities, and of wave dissipation, by viscous and electrical resistance and heat conduction. The total quadrupole components are classified according to the mode of emission. If the magnetic field or compressibility is neglected we obtain, respectively, ‘aerodynamic acoustics’ and a corresponding theory for Alfvén waves. These hydrodynamic and hydromagnetic results contrast with magnetodynamics, when the magnetic field is dominant. The magneto-acoustic far field implies a law of directivity and intensity of radiation. The main results have been collected in three summary tables (see appendix).

Three-dimensional homogeneous isotropic turbulence at very high Reynolds number R is studied using a variant of the Markovian eddy-damped quasi-normal theory. In the case without helicity, numerical calculations indicate the development of a $k^{-\frac{5}{3}}$ inertial range in the energy spectrum and an onset of significant energy dissipation at a time t* which appears to be independent of the viscosity v as v → 0; analytical arguments having a bearing on this behaviour, described as an ‘energy catastrophe’, are also discussed. The skewness factor (for t > t*), which increases with R, tends to 0·495 when R → ∞. When helicity is present, the existence of simultaneous energy and helicity cascades is demonstrated numerically. It is also shown that the helicity cascade inhibits the energy transfer towards large wavenumbers, in agreement with preliminary low Reynolds number results of Herring and with the conclusion of Kraichnan (1973) based on analysis of the interaction between two helicity waves. This inhibition implies a delay of the onset of energy dissipation at zero viscosity. It is shown that, whatever the relative rate of helicity and energy injection, a regime is attained at large wavenumbers k where the relative helicity tends to zero (with increasing k) and helicity is carried along locally and linearly by the energy cascade like a passive scalar. In practice, the linear regime is attained when the relative helicity is less than about 10%. The Kolmogorov constants of energy and helicity in the inertial range are determined. The impossibility of pure helicity cascades of a type conjectured by Brissaud et al. (1973a) is demonstrated. Finally it is shown that, because of dissipation and non-positive-definiteness of the helicity spectrum, non-zero total helicity may appear in the decay of unforced turbulence with zero total initial helicity, if the helicity spectrum is not initially identically zero.

The paper generalizes an earlier problem of Grundy (1972) by considering the expansion of a (uniform) initially contained gas into a low-density non-uniform ambient atmosphere of density ρ0r−k, where k > 0 and r is a non-dimensional radial co-ordinate. Regarding the flow as a perturbation of the perfect-vacuum expansion, we set up a boundary-value problem with boundary conditions on the contact front separating the two gases and on the strong shock which propagates into the ambient atmosphere. A large time solution to the problem can be developed by constructing an outer expansion valid near the contact front and an inner expansion valid near the shock. The matching process encounters two kinds of difficulty both of which imply that the large time solution is indeterminate from an asymptotic analysis alone.

The asymptotic analysis does show however that the shock velocity tends to a constant only for restricted values of k. For the remaining values the shock has a k-dependent power-law behaviour. The paper examines the location of the transition and determines the asymptotic power-law dependence of the shock velocity.

In a slow linear shear flow over a cylinder in contact with a plane, there is an infinite set of eddies within the cusps at the point of contact. If the cylinder is not in contact with the plane, there is a flux of fluid between the cylinder and the plane, no matter how small the gap. When the gap is approximately 0·685 times the cylinder radius or less, the flow separates from the boundaries. Single eddies form alternately on the plane and the cylinder. These interlace as the cylinder approaches the plane and force the fluid which flows through the gap to take a tortuous path. The expressions for the force and torque acting on the cylinder are also given.

Experiments were conducted with two layers of stably stratified fluid in a wind-wave tank. The slope of the density interface was measured and was related to the wind stress, the density difference between the two fluids and the depth of the interface.