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.

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.

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.

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.

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.

Turbulent temperature and velocity fluctuations in air were measured at a height of 4 m over a tidal mud flat. Particular attention was focused on the high-wavenumber, small-scale region of the spectra of these fluctuations. The measurements of the velocity fluctuations were made with a constant-temperature hot-wire anemometer; the hot wire consisted of a platinum wire 5 μm in diameter and approximately 1 mm in length. Temperature fluctuations were measured with a platinum resistance thermometer which consisted of a platinum wire 0·25 μm in diameter and about 0·30 mm in length.

The velocity spectra results agree well with the classical results of Grant, Stewart & Moilliet (1962) and Pond, Stewart & Burling (1963). In addition, they extend the velocity spectrum in air to slightly higher wavenumbers. The one-dimensional Kolmogorov constant K’ estimated from these data was 0·51.

The temperature spectra clearly show the shape of the one-dimensional temperature spectrum in air beyond the $-\frac{5}{3}$ region. In air temperature and velocity spectra are very similar. The value of the scalar constant K’0, which appears in the scalar $-\frac{5}{3}$ law, computed from these data was 0·81. Direct measurement was made of all parameters that enter into the calculation of it.

An approximate numerical method for calculating flow profiles in arteries is developed. The theory takes into account the nonlinear terms of the Navier-Stokes equations as well as the nonlinear behaviour and large deformations of the arterial wall. Through the locally measured values of the pressure, pressure gradient and pressure–radius function the velocity distribution and wall shear at a given location along the artery can be determined. The computed results agree well with the corresponding experimental data.

Linear wave theory is extensively used in research on the design of ship hull forms. Difficulty is being encountered, however, because of substantial differences between the calculated and measured phase geometry of the wave patterns generated. It seems likely that these differences may be at least partly due to nonlinear effects on phase velocity, and a nonlinear analysis of the Kelvin pattern has been undertaken as a basis for estimating the possible magnitude of such effects. It is noted that the Kelvin pattern due to a source in a finite tank contains a set of discrete free wave modes. An analysis of the nonlinear interactions in the general case of a steady multidirectional pattern of discrete cosine wave modes is undertaken, special attention being paid to the distortion of the phase anatomy, and the resulting theory is applied to the case of the Kelvin pattern in a tank. Sample computations using this analysis are discussed.

Attention is drawn to a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly. The flows are related to the ellipsoids of Dirichlet (1860).

Depending on a parameter P, the cross-section may take the form of a variable ellipse (P < 0), a hyperbola (P > 0) or a pair of parallel lines (P = 0). The elliptical case was investigated both theoretically and experimentally by Taylor (1960). The hyperbolic case (P > 0) is remarkable in that the flow develops a singularity when the angle between the asymptotes approaches a right-angle. It is suggested that this solution represents a possible instability near the crest of a standing gravity wave of large amplitude.

In the intermediate case (P = 0) the solution describes an open-channel flow in which the fluid filaments are stretched uniformly in a horizontal direction. The latter flow is demonstrated experimentally.

The equations governing two-dimensional turbulence are written as an infinite system of ordinary differential equations, in which the dependent variables are the coefficients in the expansion of the vorticity field in a double Fourier series. The variables are sorted into sets which correspond to consecutive bands in the wavenumber spectrum; within each set it is supposed that the separate variables will exhibit statistically similar behaviour. A low order model is then constructed by retaining only a few variables within each set. Multiplicative factors are introduced into the equations to compensate for the reduced number of terms in the summations. Like the original equations, the low order equations conserve kinetic energy and enstrophy, apart from the effects of external forcing and viscous dissipation.

A special case is presented in which the bands are half octaves and there is effectively only one dependent variable per set. Solutions of these equations are compared with conventional numerical simulations of turbulence, and agree reasonably well, although the nonlinear effects are somewhat underestimated.

The boundary layers forming on the walls of an aligned cylinder in a rotating fluid in axial motion are studied theoretically. The analysis shows that the side-wall boundary layer is of the Blasius type when the Rossby number exceeds the inverse square root of the Reynolds number and is transformed to the Stewartson $\frac{1}{3}$-layer when the Rossby number is less than this value. A second thicker boundary layer is superimposed on the $\frac{1}{3}$-layer whenever the difference between the azimuthal velocities of the ambient fluid and the boundary exceeds the axial velocity. Its thickness varies according to the relative magnitudes of these velocities and yields the Stewartson ¼-layer thickness only when the ratio of the azimuthal velocity difference to the axial velocity is of order E⅙, where E is the Ekman number. A uniformly valid solution is obtained for the first case when the boundary layer is of the Blasius type.