The two-dimensional sail is considered in a free-streamline model to complement the oft-considered airfoil model which is limited to small angles of attack. The shape of the sail, the lift and drag coefficients, and the moment are obtained for various angles of attack and states of tension.

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.

Experiments are carried out to establish the relationship between the nature of the flow instability and the inclination angle of the plate. The angular dependence of the Rayleigh number characterizing the onset of instability is also determined. An electrochemical flow visualization technique is utilized to expose the patterns of fluid motion. It is found that for inclination angles of less than 14° (relative to the vertical), waves are the mode of instability. On the other hand, for inclination angles in excess of 17°, the instability is characterized by longitudinal vortices. The range between 14° and 17° is a zone of continuous transition, with the two modes of instability co-existing.

Finite-difference solutions of the equations of motion for steady incompressible flow around a circular cylinder have been obtained for a range of Reynolds numbers from R = 5 to R = 100. The object is to extend the Reynolds number range for reliable data on the steady flow, particularly with regard to the growth of the wake. The wake length is found to increase approximately linearly with R over the whole range from the value, just below R = 7, at which it first appears. Calculated values of the drag coefficient, the angle of separation, and the pressure and vorticity distributions over the cylinder surface are presented. The development of these properties with Reynolds number is consistent, but it does not seem possible to predict with any certainty their tendency as R → ∞. The first attempt to obtain the present results was made by integrating the time-dependent equations, but the approach to steady flow was so slow at higher Reynolds numbers that the method was abandoned.

The diagonal pattern of bed form which appears in sand-bed channels when the channel width to depth ratio is great and the flow is nearly critical is discussed from the theoretical viewpoint by using the method of characteristics. Some photographs illustrate the phenomenon.

The development of the boundary layer on the upper surface of a horizontal flat plate in a non-diffusive, stratified flow is described. It is shown that the flow can be characterized by two basic parameters, the Reynolds (RL) and Russell (RuL) numbers, and that, depending on the relative magnitude of these two parameters, three different régimes of flow can be defined. The delineation of these régimes and the description of the flow in each of them is obtained by deriving a uniformly valid first approximation to the Boussinesq equations of motion for a flow contained in the two-dimensional parameter space RuL > 0, RL > 1. The critical stratification for the self-blocking of a horizontal boundary layer is shown to be given by the condition RuL = O(RL½).

The boundary layer on the upper surface of a horizontal plane in a diffusive, stratified flow is examined. The analysis shows that density diffusion increases the role of the buoyancy forces and causes a significant change in the properties of the boundary layer when compared to the non-diffusive case. A uniformly valid first approximation for moderate Russell numbers is derived, and the effects of buoyancy and diffusion are evaluated by solving the resulting equations numerically.

Several generalizations of theorems of the types originally stated by Helmholtz concerning the dissipation of energy in slow viscous flow have been given recently by Keller, Rubenfeld & Molyneux (1967). These generalizations included cases in which the fluid contains one or more solid bodies and drops of another liquid assuming the drops do not change shape. Some further extensions are given herein which allow for drops which may be deformed by the flow and include the effect of surface tension. The admissible boundary conditions have also been extended and particular theorems applicable to infinite domains, spatially periodic flows and to flows in infinite cylindrical pipes are derived. Uniqueness theorems are also proved.

The three-dimensional steady flow of a shallow viscous liquid with non-uniform surface tension has been considered when the variation in surface tension results from the presence of an insoluble chemical contaminant on the surface. Similarly solutions for the particular problem of a channel flowing into a semi-infinite lake have been obtained, the depth and surface concentration at infinity being specified.

A study is made of the laminar flow in the neighbourhood of the trailing edge of an aerofoil at incidence. The aerofoil is replaced by a flat plate on the assumption that leading-edge stall has not taken place. It is shown that the critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is $\alpha^{*} = O(R^{\frac{1}{16}})$, where R is a representative Reynolds number, for incompressible flow, and α* = O(R−¼) for supersonic flow. The structure of the flow is determined by the incompressible boundary-layer equations but with unconventional boundary conditions. The complete solution of these fundamental equations requires a numerical investigation of considerable complexity which has not been undertaken. The only solutions available are asymptotic solutions valid at distances from the trailing edge that are large in terms of the scaled variable of order R−⅜, and a linearized solution for the boundary layer over the plate which gives the antisymmetric properties of the aerofoil at incidence. The value of α* for which separation occurs is the trailing-edge stall angle and an estimate is obtained from the asymptotic solutions. The linearized solution yields an estimate for the viscous correction to the circulation determined by the Kutta condition.

An analytical solution is obtained for the flow field due to the impinging of a plane shock wave of arbitrary strength by a thin wing moving in the opposite direction. The planform and the thickness distribution of the wing can be arbitrary and the speed of the wing can be either supersonic or subsonic relative to the undisturbed stream ahead of the shock or to that behind the shock. The solution is a generalization of the previous solution of Ting & Ludloff for the diffraction of shock wave by a two-dimensional stationary airfoil to a three-dimensional wing moving with supersonic or subsonic speed relative to the stream ahead of or behind the shock. The solution is employed for the analysis of the changes in aerodynamic forces when an airplane encounters a blast wave or a shock wave of another airplane. It is also used to study the diffraction of a shock wave or an N-wave advancing over flat terrains.

The equations governing the free-surface flow of a tubular film of liquid are derived from physical arguments, which throw some light on the formal process described in part 1. The solutions of the equations are discussed, in particular with reference to the film-blowing process for the manufacture of thin sheets of thermoplastic material. The qualitative adequacy of a model based on the dominance of viscous forces is demonstrated, and the effect of surface tension, air drag and non-isothermal flow is discussed briefly.

A stream with constant velocity U is impulsively started at time t = 0 past the trailing edge of a semi-infinite flat plate. According to boundary-layer theory, it is found that the flow at a distance x downstream from the trailing edge is unaware of the presence of the plate when x > Ut; at time t = x/U there is then a discontinuity in the velocity normal to the plate. It is the neglect of diffusion parallel to the axis of the plate that introduces the discontinuity, and when the complete Navier–Stokes equations are considered for t ≃ x/U, a solution is found that can be matched with that gained from boundary-layer arguments.

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.