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.

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.

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.

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.

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.

Velocity derivatives in turbulence at large Reynolds numbers exhibit probability densities with a large kurtosis. If the value of the kurtosis is about 40, as in many measurements of atmospheric turbulence, careful attention to the processing of such signals is required in order to obtain an acceptable level of accuracy. Signal-to-noise and integration-time limitations make measurements of moments higher than the fourth nearly impossible for signals of this kind.

The equilibrium configuration of a slowly rotating self-gravitating perfectly conducting inviscid liquid, in the presence of a small poloidal magnetic field, is considered for a case where the electric current is a simple function of the distance from the axis of rotation. Owing to the coupling of the magnetic field with the rotation the electric current may reverse direction. This could make the magnetic field zero on certain surfaces and impose restrictions on the parameters of the problem. A perturbation expansion of the nearly spherical surface of the liquid is constructed.

The primitive Kelvin wave (on a rotating, semi-infinite, plane sheet of water of uniform depth bounded by a vertical wall) is corrected for the effects of the Earth's curvature, the reduction in depth over the continental shelf, and bends in the coastline. The results are of interest for coastal propagation of the tides; numerical examples are given for the California coastline. It is found that the Earth's curvature reduces the wave speed south of Cape Mendocmo by 8–10% (the possible range for other coastlines is roughly ± 15%) and that the continental shelf reduces the wave speed by 2−8%. The off-shore mass transport (which vanishes identically for the primitive Kelvin wave) induced by curvature and/or the shelf also is calculated. The analysis of Packham & Williams (1968) for diffraction of a Kelvin wave by a corner is extended to obtain explicit results for the phase of the transmission coefficient. It is found that a sustained change in the direction of the coastline may induce a phase shift of the order of an hour (1·3 hours for the bend at Cape Mendocino), but that small distortions of the coastline without a sustained change in direction have negligible effects on the transmitted Kelvin wave at tidal frequencies.

A cylindrical MGD configuration comprising an electrically conducting, infinitely extended (cylindrical) jet which is both permeated and contained by stream-aligned magnetic fields is considered. Axisymmetric perturbations are then generated by exposure to a stationary, weak, azimuthal current source. Ignoring relativistic and dissipative effects, asymptotic solutions complying with a radiation condition are established for the steady state. Generally speaking, the disturbed system propagates a discrete superposition of wave functions, each of which is uniquely obtained for either the upstream or downstream region. In the non-trivial cases, there is no axial wave attenuation. However, transverse attenuations are experienced within the fluid-free enveloping field. The various jet-flow regimes are thoroughly examined. It is found, in particular, that a single stationary wave is produced upstream whenever M < 1, M2 + A2 [ges ] 2 and λβ < 1, while if M > 1 and A > 1 an infinity of discrete stationary waves occurs downstream. M and A being, respectively, the Mach and Alfvén numbers of the jet; λβ is a parameter involving M, A and A0 (an interface Alfvén number).

The water-wave transmission coefficient for a small slit in a thick vertical barrier is obtained theoretically and verified both experimentally and by comparison with an exact theory for the case of zero thickness. Similar shallow-water results are presented.

The growth of hypersonic boundary layers over both concave and convex surfaces is described, the strong-viscous-interaction equation due to Cheng et al. (1961) for curved surfaces with sharp leading edges being solved asymptotically for small and large arguments. Both the asymptotic solution for large arguments and a numerical integration predict an oscillatory behaviour of the boundary-layer thickness on concave surfaces. A modification of Cheng's theory, as suggested by Sullivan (1968) and Stollery (1970), is also examined and compared with experimental data reported here. The experiments were conducted in air using a hypersonic gun tunnel under cold wall conditions at M∞ = 12·25. They included measurement of surface pressure, heat-transfer distributions and schlieren studies for concave and convex models.