Nonlinear instability of a zonal flow of slightly viscous Boussinesq fluid in a rapidly rotating frame is studied mathematically by the method of normal mode cascade, the flow being along a rectangular channel with horizontal and vertical rigid walls. Viscosity is represented approximately by supposing that its only effects occur in Ekman layers near the top and bottom walls of the channel, after the linear model of Barcilon. Self-interaction of one slightly unstable mode is found to lead to equilibration with supercritical instability. Also, interactions of two slightly unstable modes plausibly lead to equilibration. These results are related to the literature of experiments on differentially heated, rotating annuli.

Inviscid bubbles confined by the slow axisymmetric straining motion of a very viscous fluid are considered for the case when the surface tension is weak. The shape of the bubbles is determined using slender-body theory, and it is found that these bubbles have pointed ends, in agreement with well-established experimental results. The description obtained is invalid within exponentially small neighbourhoods of the ends and a local analysis suggests that the tips are cusp-like. In both the description of the major portion of the bubble and of the ends, there is an apparent non-uniqueness because a certain parameter can take on a countably infinite number of values. This non-uniqueness is not resolved.

Improved models for the movement of fluid by cilia are presented. A theory which models the cilia of an organism by an array of flexible long slender bodies distributed over and attached at one end to a plane surface is developed. The slender bodies are constrained to move in similar patterns to the cilia of the microorganisms Opalina, Paramecium and Pleurobrachia.

The velocity field is represented by a distribution of force singularities (Stokes flow) along the centre-line of each slender body. Contributions to the velocity field from all the cilia distributed over the plane are summed, to give a streaming effect which in turn implies propulsion of the organism. From this we have been able to model the mean velocity field through the cilia sublayer for the three organisms. We find that, in a frame of reference situated in the organism, the velocity near the surface of the organism is very small – up to one half the length of the cilium – but it increases rapidly to near the velocity of propulsion from then on. This is because of the beating pattern of the cilia; they beat in a near rigid-body rotation during the effective (‘power’) stroke, but during the recovery stroke move close to the wall. Backflow (‘reflux’) is found to occur in the organisms exhibiting antiplectic metachronism (i.e. Paramecium and Pleurobrachia). The occurrence of gradient reversal, but not backflow, has recently been confirmed experimentally (Sleigh & Aiello 1971).

Other important physical values that are obtained from this analysis are the force, bending moment about the base of a cilium and the rate of working. It is found, for antiplectic metachronism, that the force exerted by a cilium in the direction of propulsion is large and positive during the effective stroke whereas it is small and negative during the recovery stroke. However, the duration of the recovery stroke is longer than the effective stroke so the force exerted over one cycle of a ciliary beat is very small. The bending moment follows a similar pattern to the component of force in the direction of propulsion, being larger in the effective stroke for antiplectic metachronism. In symplectic metachronism (i.e. Opalina) the force and bending moment are largest in magnitude when the bending wave is propagated along the cilium. The rate of working indicates that more energy is consumed in the effective stroke for Paramecium and Pleurobrachia than in the recovery stroke, whereas in Opalina it is found to be large during the propagation of the bending wave.

This paper describes a theoretical investigation into (i) the response of a spherical particle to a one-dimensional fluid flow, (ii) the motion of a spherical particle in a uniform two-dimensional fluid flow about a circular cylinder and (iii) the motion of a particle about a lifting aerofoil section. In all three cases the drag of the particle is allowed to vary with (instantaneous) Reynolds number by using an analytical approximation to the standard experimental drag-Reynolds-number relationship for spherical particles.

Measurements of velocity and temperature fluctuations have been made in flows of air over a strongly heated horizontal surface in conditions of large negative Richardson number and approximating to horizontally homogeneous turbulent flow with constant shear stress and upwards flux of total heat. Within a layer that includes most of the total variation of mean temperature and velocity, the effect of the mean flow on the thermal structure is nearly confined to a change in the length scale which measures the thickness of the viscous-convective layer at the surface but which is also the length scale for the fully turbulent region. Measurements of the mean velocity and the mean-square fluctuation in the stream direction were made for a flow velocity of about 0·70 ms−1 and showed that the profiles of mean temperature and mean velocity are closely similar, implying proportionality at each height of the transfer coefficients for heat and for momentum. The ratio is estimated to be 1·4. The measurements were carried out in conditions such that the Monin-Obukhov length was in the range 8–60 mm and the maximum height of observation was 80 mm.

Previous experiments on a small-scale rectangular water-channel with slotted walls revealed that under certain circumstances the flow could become unsteady. Although these oscillations were suppressed at the time, the mechanism of the instability was not fully understood. Theoretical work, on a similar form of instability in a circular slotted-wall wind tunnel, has been extended to describe the oscillations in the water channel. Further experiments have been performed, and in some of these the length of the outer chamber surrounding the working section was altered. Good agreement between the theoretical and experimental results was obtained. Consideration has also been given to the occurrence of the self-induced oscillations in flumes of different scale and form.

According to classical boundary-layer theory, when two uniform parallel streams are brought into contact at large Reynolds number (R) the location of the dividing streamline remains indeterminate to O(R−½) if both streams are subsonic and semi-infinite in extent. It is demonstrated here that this indeterminacy is a fundamental property of such a system which cannot be resolved, as Ting (1959) proposed, by balancing the pressure across the viscous mixing region to higher order in R.

A theoretical and experimental study is made of the second-order resonant interaction between triads of linearly damped waves, one common member of which is continuously forced. In the case of a single triad, if the forced wave exceeds a critical amplitude defined by properties of the triad members, energy proceeds irreversibly to the other two waves. A stable limit state is reached where all power in excess of that required to sustain a critical amplitude in the forced wave is transferred to the other waves, which also reach steady terminal amplitudes.

It is shown that when two or more triads are simultaneously at resonance the only stable limit state is one wherein the forced wave has fallen to the lowest critical amplitude, and the only other two waves remaining are those of the triad possessing this critical amplitude. Regardless of their initial amplitudes, all other waves not externally forced ultimately disappear.

The theory is applied to the interaction of standing internal gravity waves in a linearly stratified liquid. The experiments described here quantitatively confirm the major predictions.

An instability mechanism, leading to the generation of cross-waves in a closed channel, was examined recently by Garrett (1970). His theory is not applicable to long channels where the wavemaker produces a primary field which is a progressive wave train. In such cases, the heaving of the mean surface, of considerable significance in the instability mechanism, is confined to the nonpropagating field near the wavemaker. Here the theory of resonant interactions is extended to describe the energy transfer from this forced localized field to the cross-wave field. There are close analogies between the present results and Garrett's, although the resonant bandwidth estimated here is an order of magnitude smaller. The theory indicates that nonlinear effects may control the decay of cross-waves down the channel.

The von Kármán vortex streets formed in the wakes of vibrating smooth cylinders and cables were studied using a hot-wire anemometer and flow visualization by fog injection in a wind tunnel. All the experiments took place in the flow regime where the vibration and vortex-shedding frequencies lock together, or synchronize, to control the formation of the wake. Since the flow in the vortex formation region is fundamental to further understanding of the interaction between a vibrating bluff obstacle and its wake, detailed measurements were made of the formation-region flow for Reynolds numbers between 120 and 350. The formationregion length is shown to be a fundamental parameter for the wake, and is dependent on a shedding parameter St* related to the natureally occurring Strouhal number for the von Kármán street. The effects of vibration amplitude and frequency on the mean and fluctuating velocity fields in the wake become apparent when the downstream displacement is scaled with the formation length. The von Kármán vortex street behind a vibrating cylinder is divided into three predominant flow regimes: the formation, stable and unstable regions. Fundamental differences exist in the vortex streets generated behind stationary and vibrating cylinders, but many classical characteristics, including the manner of vortex breakdown in the unstable region, are shared by the two systems.

The incipient effect of buoyancy on the heat loss from a fine hot wire in two-dimensional steady incompressible flow is considered. The wire is taken to be horizontal, the mainstream is taken to be normal to the wire and to be directed upwards at an acute angle α to the vertical, and the product Nε = NG/R3 is assumed to be small, where N, G and R denote respectively the Nusselt, Grashof and Reynolds numbers (defined, in the usual way, in §§2 and 3). Three parts of the flow are distinguished and discussed in turn. These are (i) the wire's thermal wake, (ii) an outer irrotational flow induced as a small perturbation of the mainstream by the thermal wake, and (iii) an inner flow, near the wire, in which diffusion either dominates or balances convection. It is shown that the change caused by buoyancy in the wire's heat loss is due largely to the irrotational flow induced by the wire's thermal wake. When log Nε is significantly large, the change caused by buoyancy in the wire's heat loss is almost entirely due to the irrotational flow. The increment caused by buoyancy in the Nusselt number is then approximately the same as would be produced, in the absence of buoyancy, if the mainstream speed were increased by a factor 1– 2σ−1Nεlog(Nε) cos α, where σ is the Prandtl number.

The solutions of the incompressible quasi-cylindrical momentum-integral equations describing the flow in the viscous core of a wing-tip vortex are obtained in a closed form and are shown to have two distinct branches. The discontinuities of these solutions have infinite axial gradients and therefore, following Hall, are assumed to signal the inception of the vortex breakdown. Benjamin's finite transition, with its excess flow force dissipated, is shown to give results equivalent to a sudden cross-over, upstream of the discontinuity, from one branch solution to another. The critical point of such a cross-over is downstream from the cross-over, at the discontinuity. Sarpkaya's experimental data, and the nature of the solutions ahead of the discontinuity, suggest that the physical manifestation of the discontinuity is the spiral breakdown, whereas the cross-over seems to be related to the rapidly expanding and subsequently contracting axisymmetric bubble. This therefore implies that the beginning of the spiral breakdown is the all important disturbance which triggers off not only the downstream asymmetric departure of the flow from its quasi-cylindrical form but also the formation of the upstream axisymmetric cross-over bubble. Solutions for the turbulent flow downstream from the spiral breakdown indicate that the wing-tip vortex breakdown can result in an appreciable reduction of the maximum circumferential velocity and should thus lessen the danger that trailing vortices present to following aircraft.

An idealized two-dimensional flow due to a point source of x momentum is discussed. In the far field the flow is modelled by a jet region of large vorticity outside which the flow is potential. After use of the transformation \[ \zeta^3 = (\xi + i\eta)^3 = x + iy, \] the equations suggest naively obvious asymptotic expansions for the stream function in these two regions, namely \[ \sum_{n=0}^{\infty}\xi^{1-n}f_n(\eta)\quad {\rm and}\quad\sum_{n=0}^{\infty}\xi^{1-n}F_n(\eta/\xi) \] respectively. Consistency in matching these expansions is achieved by including logarithmic terms associated with the occurrence of eigensolutions. Fn is easy to find and Jn can be found in closed form so the inner and outer eigensolutions may be fully determined along with the complete structure of the expansions.

In this paper experiments in which cross-waves were generated in front of a wave-maker at one end of a long channel are described. The primary field produced was a progressive wave train, but, at certain frequencies, a standing wave developed in front of the wavemaker. This wave, whose crests were at right angles to the wavemaker and which had frequency half that of the wavemaker, is known as a cross-wave.

An instability mechanism for the formation of cross-waves in a long channel has been presented in Part 1 (Mahony 1972). In the present paper we describe some measurements concerning the formation of the waves. The initial growth rate of the waves has been measured and the curves of neutral stability determined for two of the possible modes. The results are in good agreement with Mahony's theory. After the early stages in the development of the waves, there is an increase in their growth rate. This new rate was found to be about twice the initial growth rate. Also reported are some measurements of the amplitude of the cross-wave field along the channel.

The deformation of initially spherical drops of radius r0 subjected to an external flow of velocity u is experimentally examined for a large range of Weber and Bond numbers. Observations of the changes in the response are compared with recent analytical predictions. The data show that beyond a critical Weber number the response ceases to be vibratory and becomes monotonic with time. Subsequently, it is found that, although the response is unstable, the deformation imposed by the external aerodynamic pressure distribution remains the dominant factor. Measurements of the drag coefficient yield a mean value ofCD = 2·5 over a large Reynolds-number range. The time at which Taylor instability occurs is shown to be inversely proportional to Bond number to the one-quarter power. There is little evidence of the instability occurring until a normalized time $t^{*} = (\epsilon^{\frac{1}{2}}\bar{t}u/r_0)$ is approximately unity; here ε is the gas/liquid density ratio (ρ/ρ′) and $\bar{t}$ is the real time.

This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

An approximate solution is developed for the system of equations describing flow and ion transport in a diffuse electrical double layer slightly perturbed from equilibrium. The approximation is valid only when the potential difference across the diffuse layer is small, less than about 25 mV. When the approximate solution is used to study wave motion of low-tension interfaces, it is found chat ion transport in diffuse layers slows down interfacial motion in both stable and unstable situations. Although the slowing effect is relatively small (a few per cent or less) when the small potential approximation applies, the form of the solution suggests that the effect could be significant for potential differences in the 50–100 mV range, which exist in many systems of interest. There are also indications that the slowing effect can significantly influence wave motion of thin liquid films with diffuse layers, e.g. soap films, although a detailed analysis of the thin-film situation is not carried out.

The paper explores the whole class of self-similar solutions for blast waves bounded by Chapman-Jouguet detonations that propagate into a uniform, quiescent, zero counter-pressure atmosphere of a perfect gas with constant specific heats. Since such conditions can be approached quite closely by some actual chemical systems at N.T.P., this raises the interesting possibility of the existence of Chapman-Jouguet detonations of variable velocity. The principal virtue of the results presented here is, however, more of theoretical significance. They represent the limiting case for all the self-similar blast waves headed by gasdynamic discontinuities associated with a deposition of finite amounts of energy, and they exhibit some unique features owing to the singular nature of the Chapman-Jouguet condition.

The problem considered here is to determine the shape of a symmetric two-dimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical attempts to solve these equations prove to be unsuccessful, it is shown that the optimal plate shapes must have blunt noses. This problem is next formulated by a method using finite Fourier series expansions, and optimal shapes are obtained for various ratios of plate arc-length to plate width.

Experimental studies of the flow field near the wall in a turbulent boundary layer using hot-wire probes are reported. Measurements of the product uv are studied using the technique of conditional sampling with a large digital computer to single out special events (bursting) when large contributions to turbulent energy and Reynolds stress occur. The criterion used to determine when the product uv is sampled is that the streamwise velocity at the edge of the sublayer should have attained a certain value. With this simple criterion we find that 60% of the contribution to $\overline{uv}$ is produced when the sublayer velocity is lower than the mean. This result is true at both low, Rθ = 4230, and high, Rθ = 38 000, Reynolds numbers. With a more strict sampling criterion, that the filtered sublayer velocity at two side-by-side points should be simultaneously low and decreasing, individual contributions to $\overline{uv}$ as large as 62 $\overline{uv}$ have been identified. Additional measurements using correlations between truncated u and v signals reveal that the largest contributions to the Reynolds stress and turbulent energy occur when u < 0, v > 0 during an intense bursting process and the remainder of the contributions occur during a less intense recovery process. Thus, contributions to the turbulent energy production and Reynolds stress at a point near the wall are of relatively large magnitude, short duration and occur intermittently. A rough measure of the intermittency factor for uv at a point near the wall is 0·55 since 99% of the contribution to $\overline{uv}$ is made during only 55% of the total time.