We show that the rippling often observed on small progressive gravity waves can be explained in terms of a nearly resonant harmonic nonlinear interaction. The resonance condition is that the phase speeds of the two waves must be nearly identical. The in viscid analysis is generalized to any order in a small parameter proportional to the wave steepness. Wave tank measurements provide experimental evidence for most of the predicted results. The phenomenon of resonant rippling is further shown to be not just peculiar to capillary-gravity waves, but in fact possible for any weakly nonlinear dispersive wave system whose dispersion relation has discrete pairs of solutions nearly satisfying the resonance conditions.

Section 1 is in the nature of a general survey of fluid flows within the human body : including the lungs (airflow in the airways and the special characteristics of the pulmonary blood circulation), the general systemic circulation of the blood, and the urinary tract. Problems of the microcirculation, including blood flow in the narrower capillaries, gas exchange with the terminal airways (alveoli), and exchange of gas and nutrients with peripheral tissue, are treated in the more specialized second section, which describes in some detail modern views concerning peripheral resistance.

The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.

A simple model is suggested to explain the flow mechanism at the upstream edge of a square-edged broad-crested weir. The separation bubble that may be seen to occur at this point is treated as an area of constant static head, while the main flow outside the bubble is deemed to be irrotational and divided from the bubble by a free streamline. If this model is accepted, the flow pattern will be such that energy requirements will be met along the free surface and the streamline bounding the bubble while, within these boundaries, the Laplace equation will be satisfied at every point. Accordingly, a solution satisfying these conditions is established by the use of a relaxation technique.

In practice, it is likely that the cavity flow within the bubble will be bounded not by a single streamline but by a turbulent mixing zone and that there will be some increase in pressure near the point of re-attachment. Nevertheless, the surface profile and flow pattern observed in experiments show fair agreement with those predicted using the simple model. Whilst acknowledging, then, that the bubble and the zone bounding it are in fact of a more complex character, we may say that this simplified treatment affords a sound model of the main flow and so permits a better understanding of the action of the square-edged broad-crested weir.

It is shown that for a given body and a given orientation g there is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid with g vertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.

Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

We define a generalized vortex to have azimuthal velocity proportional to a power of radius r−n. The properties of the steady laminar boundary layer generated by such a vortex over a fixed coaxial disk of radius a are examined. Though the boundary-layer thickness is zero a t the edge of the disk, reversals of the radial component of velocity u must occur, so that an extra boundary condition is needed at any interior boundary radius rE to make the structure unique. Numerical integrations of the unsteady governing equations were carried out for n = − 1, 0, ½ and 1. When n = 0 and − 1 solutions of the self-similar equations are known for an infinite disk. Assuming terminal similarity to fix the boundary conditions at r = rE when ur > 0, a consistent solution was found which agrees with those of the self-similar equations when rE is small. However, if n = ½ and 1, no similarity solutions are known, although the terminal structure for n = 1 was deduced earlier by the present authors. From the numerical integration for n = ½, we are able to deduce the limit structure for r → 0 by using a combination of analytic and numerical techniques with the proviso of a consistent self-similar form as rE → 0. The structure is then analogous to a ladder consisting of an infinite number of regions where viscosity may be neglected, each separated by much thinner viscous transitional regions playing the role of the rungs. This structure appears to be characteristic of all generalized vortices for which 0.1217 < n < 1.

A statically stable stratified free shear layer was formed within the test section of a wind tunnel by merging two uniform streams of air after uniformly heating the top stream. The two streams were accelerated side by side in a contraction section. The resulting sheared thermocline thickened gradually as a result of molecular diffusion and was characterized by nearly self-similar temperature (odd), velocity (odd) and Richardson number (even) profiles. The minimum Richardson number J0 could be adjusted over the range 0·07 ≥ J0 ≥ 0·76; the Reynolds number Re varied between 30 and 70. Small periodic disturbances were introduced upstream of the test section by a fine wire oscillating in the thermocline. The wire generated a narrow horizontal beam of internal waves, which propagated downstream and remained confined within the thermocline. The growth or decay of these waves was observed in the test section. The results confirm the existence of a critical Richardson number the value of which is in plausible agreement with theoretical predictions (J0 ≅ 0·22 for the Reynolds number of the experiment). The growth rate is a function of the wavenumber and is somewhat different from that computed for the same Reynolds and Richardson numbers, but the calculation assumed velocity and density profiles which were also somewhat different.

Existing experiments on the low-speed flow downstream of steps and fences, and some new measurements downstream of a backward-facing step, are used to demonstrate the complicated nature of the flow in the reattachment region and its effect on the slow non-monotonic return of the shear layer to the ordinary boundary-layer state. A key feature of the flow is found to be the splitting of the shear layer at reattachment, where part of the flow is deflected upstream into the recirculating flow region to supply the entrainment; the part of the flow that continues downstream suffers a pronounced decrease in eddy length scale, evidently because the larger eddies are torn in two. This phenomenon will occur in all cases where a shear layer reattaches after a prolonged region of separation, either at low speed or in supersonic flow. For simplicity, the discussion in the present paper is confined to low-speed flows.

The theory of Stokes flows about slender particles is considered for the case in which the particle cross-section is arbitrary and non-uniform along the axis of the particle, subject to certain smoothness assumptions. As distinct from previous works on slender particles in which the particle is represented approximately by a line distribution of Stokeslets, the representation adopted here is the exact one of a surface distribution of Stokeslets. This representation may then be used to recover the familiar one-dimensional integral equation for the equivalent line density of Stokeslets, together with estimates for the range of particle shapes for which the equation is valid. Within this range it is found that there is a class of particles for which the established perturbation scheme, which is used to obtain a solution to the integral equation, is singular. This class of particle shapes is illustrated by the example of a uniformly twisted particle whose pitch of twist is large enough compared with the cross-sectional dimension to ensure the validity of the equation, but small enough to make the usual method of solution singular. It is shown how the equation may be transformed so that an approximate solution can be found by means of a regular perturbation scheme. The results indicate that, for each of the axial and transverse components of motion, there is an equivalent particle of circular cross-section for which the total force and couple are the same as for the original particle. However, the radii of the equivalent cylinders are different for each component, the transverse component being affected by the twist while the axial component is not.

The development and evaluation of a hot-film probe, suitable for use within arteries and operated with a commercial constant-temperature anemometer and linearizcr, is described. The performance of the system in the recording of arterial velocity wave forms is described, and instantaneous and time-averaged velocity profiles constructed from measurements in the thoracic aorta of dogs are presented. The profiles were blunt, with boundary layers estimated to be less than 2 mm thick throughout the cycle, and significant skews were observed, the explanation for which appears to lie in the influence of local geometry on the flow. A preliminary study of flow disturbances in the aorta based on visual observation of instantaneous velocity wave forms and frequency spectrum analysis is reported. The occurrence of flow disturbances and turbulence is shown to be related to peak Reynolds number and the frequency parameter α. The possible roles of free-stream disturbances and boundary-layer transition in generating these disturbances are discussed.

A non-dissipative fluid rotates uniformly in the annular region between two infinitely long cylinders and is permeated by a magnetic field varying with distance from the axis of rotation. The hydromagnetic stability of this system is examined theoretically. When the magnetic field is azimuthal the system can always be rendered stable to axisymmetric disturbances by sufficiently rapid rotation (Michael 1954). Unless the magnetic field everywhere decreases with radius, however, the system may be unstable to non-axisymmetric disturbances even when the rotation speed exceeds a typical Alfvén speed by many orders of magnitude. ‘Slow’ hydromagnetic waves, akin to those invoked in a recent theory of the geomagnetic secular variation (Hide 1966), may then be generated by the spatial variations of the magnetic field. All unstable waves so generated propagate against the basic rotation, i.e. ‘westward’, when the field is azimuthal, and this property is in fact remarkably insensitive to variations in both magnitude and direction of the imposed field.

To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.

Velocity distributions within the boundary layer of a swirling flow of incompressible fluid in a convergent conical nozzle have been investigated. Theoretical calculations with boundary conditions more appropriate to physically existent situations discounted the existence of 'super-velocities’ within the boundary layer. Parallel experimental investigations demonstrated an interdependence of core and boundary-layer flows which precluded the maintenance of the flow conditions required by the analysis.

The effect, of high intensity large-scale free-stream turbulence on the flow past a rigid circular cylinder has been studied experimentally a t subcritical Reynolds numbers. Grids were used to produce homogeneous turbulence fields with longitudinal scales ranging from 0·36 to 4·40 cylinder diameters and with longitudinal intensities greater than 10%. Power and cross-spectra of the turbulence components (the ‘system input’) have been measured in order to carefully define the turbulence characteristics.

In the response experiments, a special model measured arbitrary two-point pressure correlations. Subsequent integrations yielded the specbral properties of the unsteady lift and drag. Measurements of mean drag and Strouhal frequency indicate that to some extent even severe large-scale turbulence can be considered to be qualitatively equivalent to an increase in the effective Reynolds number. Vortex shedding is not seriously disrupted by severe turbulence, but is affected more by low than by high frequencies. The unsteady lift response is still dominated by the vortex shedding, whereas the unsteady drag becomes primarily a response to turbulence. The cross-spectra of the drag forces for the one turbulence case examined overlay well when plotted against lateral separation divided by wavelength. This has enabled a ‘describing function’ for the drag response to turbulence to be derived. This describing function is the central element needed for the calculation of the structural response of such cylinders in the drag direction.

Quasi-steady creeping flow in models of small airway units of the lung is investigated. A respiratory unit of the lung is modelled by a sphere, an oblate and a prolate ellipsoid of revolution, and a circular cylinder of finite length. The solution of the Stokes equations for each of these geometries is indicated for general axi-symmetric boundary conditions. For particular cases consistent with the models, streamlines are plotted and some velocity profiles are shown. It is suggested that bulk 00w in the ha1 generations of the lung is significant for gas transport even though diffusion is the predominant mechanism there.

The steady-state interaction between surface waves and long internal waves is investigated theoretically using the radiation stress concepts derived by Longuet-Higgins & Stewart (1964) (or Phillips 1966). It is shown that, over internal wave crests, those surface waves for which cg0cosϕ0 > ci experience a change in direction of propagation towards the line of propagation of the internal waves and their amplitudes are increased. Here cg0 is the surface-wave group speed at U = 0, ϕ0 is the angle between the propagation direction of the surface waves at U = 0 and the propagation direction of the internal waves, and ci is the phase speed of the internal waves. If cg0cos ϕ0 < ci the direction of the surface waves is turned away and their amplitudes are decreased. Over troughs the opposite effects occur.

At positions where the local velocity of surface-wave energy transmission measured relative to the internal wave phase velocity is zero, i.e. cg + U − ci = 0, there is a singularity in the energy of the surface waves with resulting infinite amplitudes. It is shown that at these critical positions two wavenumbers which were real and distinct on one side coalesce and become complex on the other. The critical positions are thus shown to be barriers to the propagation of those wave-numbers. It is also shown that there is a critical position representing the coalescence of three wavenumbers. Surface-wave crest configurations are shown for three numerical examples. The frequency and direction of propagation of surface waves that exhibit critical positions somewhere in an internal wave field are shown as a function of the maximum horizontal surface current. This is compared with measurements of wind waves that have been reported elsewhere.

The shock propagation theory of Brinkley & Kirkwood (1947) is extended to provide a uniformly valid analytic solution of point-explosion problems both when the undisturbed medium is uniform and when it is stratified. This is achieved mainly by selecting the parameter expressing a similarity restraint in this theory such that initially it gives precisely the Taylor–Sedov solution, while asymptotically, in the weak regime, still retaining the well-known Landau–Whitham–Sedov form of the solution for shock overpressure. The shock overpressure, as calculated by the present method for spherical and cylindrical blast waves in the entire regime from the point of explosion to where they have become very weak, shows excellent agreement with that from the exact numerical solutions of Lutzky & Lehto (1968) and Plooster (1970). The solution for a spherical shock propagating in an exponential atmosphere stratified by a constant acceleration due to gravity also shows a good agreement with the exact numerical solution of Lutzky & Lehto.

An accelerating liquid drop, under the action of surface tension, is shown to be unstable to small disturbances above a first critical value of the Bond number. Both numerical and second-order asymptotic methods are employed in order to characterize the normal-mode response and the neutral-stable modes at larger values of the Bond number. The transient response of an initially spherical drop that is accelerated by the flow of an external gas is studied as an initial-value problem. A unified theory, that includes acceleration as well as aerodynamic effects, is presented in order to account for the complete dynamic range of Weber and Bond numbers. The results are compared with experimental observations that range from continuous vibration to irreversible aerodynamic distortion and unstable shattering.