The solution given by Gerstner for water waves of finite amplitude, which is valid for a semi-infinite liquid of arbitrary stratification, is reconsidered. By a transformation of co-ordinates, it is shown to represent edge waves propagating along a sloping shore.

The evolution of plane switch-on and switch-off shocks in the presence of small normal disturbances is examined. (Normal disturbances are those in which the perturbed quantities are functions only of time and the distance from the plane of the shock wave.) In the situation under discussion the magnetic diffusivity of the ambient gas is much greater than each of the viscous diffusivities and the thermal diffusivity.

It is shown that small normal disturbances can eventually cause appreciable changes in the flow pattern around either switch-on or switch-off shocks. Numerical computations are carried out for both a specific null switch-on and a specific null switch-off shock, in the presence of a known disturbance. The results of each of these computations, for large times, are in agreement with analytic, asymptotic solutions which are obtained.

A mechanism is suggested whereby switch-on and switch-off shocks, across which an appreciable deflexion of the direction of the magnetic field lines takes place, can adjust themselves to small normal disturbances.

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

A solution for the weak bore is found in which the mean profile is dominated by viscosity, so that the velocity variation is given essentially by a quasi-uniform Poiseuille flow. It is found that such a transition between flows of different depths is possible provided the Froude number is less than 1·58. The possibility of superposing an inviscid perturbation on such a flow is then investigated. Under favourable circumstances the effect of this perturbation is to add to the profile of the free surface a term which decays exponentially in front of the bore, but is oscillatory behind it.

The application of a heated thin metallic film to the measurement of mean skin friction in laminar and turbulent flow on a flat plate, circular cylinder and in an annular tunnel is described. The manner in which transition and separation are detected with this instrument is illustrated by reference to tests on a circular cylinder. The influence of the ambient air temperature and the gauge temperature on the behaviour of the instrument is analysed and it is shown that a reliable element can be constructed, capable of being moved from one location to another, while retaining its calibration. An element, calibrated on an oscillating flat plate, is used to obtain the spectral density of skin friction in a turbulent boundary layer.

The stability of a two-dimensional Couette-Poiseuille flow is investigated. The primary unidirectional flow is between two infinite parallel plates, one of which moves relative to the other. The results for the case of Poiseuille flow agree with Lin's results and all flows for which the plate velocity exceeds 70% of the maximum velocity of the Poiseuille component of flow are found to be stable for all finite Reynolds numbers. Results of intermediate cases are also obtained.

The attenuation of progressive, dispersive, oscillatory wave systems by dissipation within the viscous boundary layer in a long laboratory channel was studied both experimentally and theoretically, for the conditions of uniform water depth and over uniform impermeable slopes. In general, the observed dissipation exceeded that predicted by linear, small-amplitude theory for the boundary layer on the sides and bottom of the channel. On the hypothesis that the additional dissipation might be due to the presence of a similar boundary layer at the free water surface, the theory is extended to include the surface effect. The experimental results essentially confirm the theory—even in the markedly nonlinear region near shore.

The linearized stability problem for steady, cellular convection resulting from gradients in surface tension is examined in some detail. Earlier work by Pearson (1958) and Sternling & Scriven (1959, 1964) has been extended by considering the effect of gravity waves. In order to avoid the use of an assumed coupling mechanism at the interface, the relevant dynamical equations were retained for both phases. It is shown that the existence of a critical Marangoni number is assured, and that for many situations this critical value is essentially that which is appropriate to the case of a non-deformable interface. Usually, surface waves are important only at very small wave-numbers, but they are dominant for unusually thin layers of very viscous liquids.

The von Kármán (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.

The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.

The present paper proposes a theory for the mechanical behaviour of a fluid with a rigid microstructure. The microstructure is described by a director frame of three vectors and a second-order tensor W and its gradient are proposed as measures of the kinematics of this frame. When the frame is spinning without deforming, W reduces to the director spin velocity. Postulating the existence of a couple stress in addition to the classical Cauchy stress, the linear constitutive equations for such a structured fluid are derived and then specialized to the case of transverse isotropy.

These equations are used to study rectilinear shearing flow. When [dtri ]W = 0, the condition for a non-interacting substructure, the results of the theory are shown to be in agreement with the work of Jeffery and of Ericksen. For mutually interacting substructure particles, [dtri ]W ≠ 0, a geometric analysis of the non-linear differential equations is performed in order to exhibit the effects of particle concentration on the flow kinematics.

Numerical solutions of the equations of steady free convection in a vertical slot have been obtained by a procedure which relies solely on an iterative solution of Poisson's equation. With a mesh spacing of $\frac{1}{40}$ the solutions are stable up to a Rayleigh number of order 105. The numerical experiments are comparable to the author's laboratory experiments (Elder 1965) and to the experiments of Eckert & Carlson (1961). The growth of the boundary layers and the uniform vertical temperature gradient are well described. It is tentatively suggested that certain unstable motions found prior to the divergence of the numerical solution are comparable to the secondary flows found by the author.