This paper is part of a series treating the flow in a variable-area, rectangular duct which is rotating about an axis that is perpendicular to the duct's centre-line and parallel to one pair of walls (sides). The speed of rotation is assumed to be sufficiently large that viscous effects are confined to boundary layers and free shear layers and that inertial effects are much smaller than viscous effects everywhere. Earlier papers present inertialess solutions for a prototype with parallel sides everywhere, parallel top and bottom upstream of a cross-section, and straight, symmetrically diverging top and bottom downstream of the same cross-section. The present paper presents inertial perturbations to the inertialess solutions for the prototype when the slope of the diverging top and bottom is small. This paper begins to bridge the gap between papers treating inertialess flows in ducts with arbitrary geometries and papers treating flows with significant inertial effects in ducts with restricted geometries.

When solving for three-dimensional laminar and turbulent boundary layers on smooth bodies of revolution at incidence, one has to contend with a difficulty near the nose where the usual formulations of the governing equations are singular. A transformation of the co-ordinate system is described which removes this singularity and enables the solution to be carried smoothly around the nose. A further difficulty arises if the body is slender and it is also shown how this may be overcome. As part of our continuing studies of this problem for both laminar and turbulent flows, we compute the laminar boundary layers on the line of symmetry for thin bodies taking the prolate spheroid as a paradigm. We show that if the angle of incidence α is less than 41°, separation never occurs at the nose no matter how thin the body. In contrast, the value of α which provokes separation at the leading edge of a two-dimensional airfoil tends to zero with the thickness ratio of the airfoil.

An experimental study of wind-generated capillary waves has been carried out in a laboratory wind-wave tunnel. In this tunnel it was possible to generate a stationary homogeneous wave field and to vary the wind over a range of speeds. A technique which makes use of a Preston tube has been used to measure the shear stress at the air-water boundary. Measurements of surface elevation and wave slope spectra in the capillary range of frequencies were obtained. Equations describing the wave spectra equilibrium range under the action of wind shear, surface tension and viscosity are derived. For frequencies beyond 15 hertz the measured spectra are in satisfactory agreement with the derived equations. Comparisons with other existing data have also been made.

Analytical and numerical analyses have been made of the physical behaviour of a collapsing bubble in a liquid. The mathematical formulation takes into account the effects of compressibility of the liquid, non-equilibrium condensation of the vapour, heat conduction and the temperature discontinuity at the phase interface. Numerical solutions for the collapse of the bubble are obtained beyond the time when the bubble reaches its minimum radius up to the stage when a pressure wave forms and propagates outward into the liquid. The numerical results indicate that evaporation and condensation strongly influence the dynamical behaviour of the bubble.

In addition, the propagation of the stress wave, both in a solid and a liquid, due to the collapse of the bubble has been observed by means of the dynamic photoelasticity. It is clearly demonstrated that the stress wave in a photoelastic specimen is caused by impact of the pressure wave radiated from the bubble.

Analyses are made of the mutual interactions between shock structure and the sidewall laminar boundary layer and their effects on the quasi-steady flat-plate laminar boundary layer in ionizing-argon shock-tube flows. The mutual interactions are studied using effective quasi-one-dimensional equations derived from an area-averaged-flow concept in a finite-area shock tube. The effects of mass, momentum and energy nonuniformities and the wall dissipations in the ionization and relaxation regions on the argon shock structure are discussed. The new results obtained for shock structure, shock-tube laminar side-wall and quasi-steady flat-plate boundary-layer flows are compared with dual-wavelength interferometric data obtained from the UTIAS 10 × 18 cm Hypervelocity Shock Tube. It is shown that the difference between the results obtained from the present method and those obtained by Enomoto based on Mirels’ perfect-gas boundary-layer solutions are significant for lower shock Mach numbers (Ms ∼ 13) where the relaxation lengths are large (∼ 10cm). In general, the present results agree better with our experimental data than our previous results for uncoupled ionizing flows.

A transonic-flow theory of thin, oblique wing of high aspect ratio is presented, which permits a delineation of the influence of wing sweep, the centre-line curvature, and other three-dimensional (3D) effects on the nonlinear mixed flow in the framework of small-disturbance theory. In the (parameter) domain of interest, the flow field far from the wing section pertains to a high subsonic, or linear sonic, outer flow, representable by a Prandtl–Glauert solution involving a swept, as well as curved, lifting line in the leading approximation.

Among the 3D effects is one arising from the compressibility correction to the velocity divergence, absent in classical works; this effect also leads to a correction in the outer flow in the form of an oblique line source. More important is the upwash corrections which includes the influence of both the near and far wakes, as well as the local curvature of the centre-line. For straight oblique wings, local similarities exist in the 3D flow structure, permitting the reduced equations to be solved once for all span stations. An analogy also exists between the oblique-wing problem and that of a 2D transonic flow which is weakly time-dependent; this provides an alternative method of solving numerically the inner airfoil problem.

Solutions to the reduced problem are demonstrated and compared with full-potential solutions for elliptic oblique wings involving high subcritical as well as slightly supercritical component flows.

In an ocean of uniform depth the propagation of small-amplitude plane waves is impeded by two vertical semi-infinite perfectly reflecting barriers extending from the bottom of the sea to above the free surface. The two screens do not lie in general in the same plane, and they are separated by a gap through which wave energy is transmitted from the open sea into the sheltered region. A transmission coefficient is established for small gap widths relative to the wavelength and the agreement with existing theoretical results for special cases is found to be very good.

Eulerian direct interaction is used to close Liouville's equation central to the transport of particles in a turbulent fluid where the dominant drag force is derived from the particle and local fluid velocities. The reliability of the equation is then tested by comparison of solutions with those of a computer simulation of particle motion with Stokes drag in a random velocity field. Using an empirical drag law accurate for particle Reynolds numbers up to 500, formulae are derived for the statistical moments central to particle dispersion for a weak drag force operating in a Gaussian isotropic stationary velocity field. These show for instance that the long time particle diffusion coefficient is in general greater than the equivalent value based on Stokes drag.

The fluid mechanics of self-propelling, slender uniflagellar micro-organisms is examined theoretically. The mathematical analysis of these motions is based upon the Stokes equations, and the body is represented by a continuous distribution of stokeslets and doublets of undetermined strength. Since the body is self-propelling, additional constraints on the total force and moment upon it are applied. A system of singular integral and auxiliary equations, in which the propulsive velocity and viscous force per unit length are the unknowns, is derived. The vector integral equation is decomposed into near- and far-field contributions, and the solution is determined by a straightforward iterative procedure. The flagella considered are of constant radius and are restricted to planar undulations. The analysis is applied to a small amplitude wave form of infinite length, and a third-order analytic solution is obtained. By means of numerical computation, the method is extended to large amplitude wave forms of both infinite and finite length. The validity and accuracy of the solution method, the effect of local curvature, and an approximate model for an attached cell body-proper are evaluated in light of alternative theories.

The solution method is systematically applied to a variety of wave-form shapes representative of actual flagella. For a sinusoidal wave form, the variations in propulsive velocity, power output and propulsive efficiency are examined as functions of the number of wavelengths on the flagellum, the amplitude and the flagellar radius. Wave forms of variable amplitude and variable wavelength are also considered. Among the significant results are the effect of the cell body on pitching, the significant differences between constant frequency and constant phase-speed undulations for variable wavelength wave forms, and comparisons with other pertinent theories.

The vorticity dynamics of a convective swirling boundary layer are studied from the viewpoint of steady, inviscid fluid-dynamics theory. Attention is confined to the region of flow lying directly below and within a circularly shaped updraft. Fluid enters the updraft region without vorticity save for that in the boundary layer upstream of the updraft radius. Solutions of the equation \[ r\eta = r^2\frac{dH}{d\psi}-\frac{d}{d\psi}\bigg(\frac{\Gamma^2}{2}\bigg) \] (e.g. Batchelor 1967, p. 545) are presented. By the nature of this approach it allows one to compute the ‘outer’ flow together with the outer boundary-layer structure and hence side-step the interaction problem. A drawback is that the inner viscous structure is not captured. These solutions are compared to some numerical solutions of the time-dependent, viscous axisymmetric Navier-Stokes equations which are reported elsewhere (Rotunno 1979). Although the agreement is not perfect, model results are close enough whereby a number of useful deductions concerning the effects of viscous diffusion and time-dependence may be made.