The adiabatic approximation for a solitary wave in a channel of gradually varying breadth b and uniform depth is tested by experiment and by numerical solution of the generalized Korteweg-de Vries (KdV) equation. The results for a linearly diverging channel show good agreement with the prediction α (dimensionless wave amplitude)$\propto b^{-\frac{2}{3}}$. The experiments and numerical solutions for the linearly converging channel show that the wave growth is well approximated by α ∞ b−½. The discrepancy between the diverging and converging channels is shown to be due to nonlinear effects associated with the choice of the spatial variable as the slow variable in the generalized KdV equation. The measured and computed profiles display the predicted ‘shelves’ of elevation and depression in the converging and diverging channels, respectively.

For a particle with two or three perpendicular planes of symmetry rotating at low Reynolds number in a Couette flow field, there are three orbits in which the motion is simply periodic. The three scalars Bi involved in Bretherton's shape tensor are found experimentally from periods of rotation in these orbits. Experiments on right parallelepipeds, ranging from thin platelets through cubes to rectangular section rods, are described. Each of the Bi is found to depend on the aspect ratio of one of the particle cross-sections, with only slight influence from its third dimension. Results are expressed in terms of a relationship between this aspect ratio and that of an equivalent ellipse, incorporating a weak function of the third particle dimension. The equations of motion governing a general doubly periodic motion, and incorporating experimental Bi values, are integrated numerically and compared satisfactorily with experimental observations.

When heat is applied at the bottom of a stable salinity gradient a series of layers with uniform temperature and salinity is formed. The evolution of this system is investigated in the laboratory and a numerical model of the process is developed. New layers are formed sequentially at the top of a growing convection region while lower down adjacent layers merge. For given fluid properties the convection depends upon one parameter Q, which is proportional to the (suitably non-dimensionalized) ratio of the salinity gradient to the heat flux. We find that the depth of the top of the convecting region and the number of layers present increase like the square root of time over the range of Q examined. This permits the definition of an effective conductivity, KT, for the total series of layers which is directly proportional to κT, the molecular thermal diffusivity, and inversely proportional to Q. The vertical growth of the layers is thus retarded by increasing Q. The average thickness of the layers decreases with increasing salinity gradient and appears to be independent of the applied heat flux.

A model equation is derived to study trapped nonlinear waves with a turning effect, occurring in disturbances induced on a two-dimensional steady flow. Only unimodal disturbances under the short wave assumption are considered, when the wave front of the induced disturbance is plane. In the neighbourhood of certain special points of sonic-type singularity, the disturbances are governed by a single first-order partial differential equation in two independent variables. The equation depends on the steady flow through three parameters, which are determined by the variations of velocity and depth, for example (in the case of long surface water waves), along and perpendicular to the wave front. These parameters help us to examine various relative effects. The presence of shocks in a continuously accelerating or decelerating flow has been studied in detail.

Natural convection flows induced by localized heating of the base of a thermally stratified enclosure are examined. The enclosure is a vertical circular cylinder with height equal to radius. The wall temperature increases linearly with height, and a small heat source is centrally located on the floor. Parameters of the problem are the ambient stratification rate (Γ−1), the Prandtl number (Pr), and a Grashof number (Gr) based on the temperature and the diameter of the heated spot. Over wide ranges of Γ and Gr, vertically layered convection cells are observed in the upper part of the enclosure in both laboratory experiments and numerical calculations. For the case of strong buoyancy and weak stratification, plume-like flows exist immediately above the heat source. Streak photographs are in qualitative accord with the numerical calculations, except for a range of Gr when an azimuthal rotation of the laboratory plume is observed. All flows are otherwise laminar. The theoretical results reveal a √Gr scaling at large Gr for the velocities and heat transfer rates, and a correlation of strongly stratified, viscous flows with the group Gr Pr Γ−1.

Numerical solutions have been obtained for the problem of steady, incompressible, viscous flow between two infinite concentric cylinders, the cross-sections of the inner and outer cylinders being circular and square respectively. The square cylinder is fixed and the flow is driven by the rotation of the circular cylinder. Solutions are given for Reynolds number in the range 1–1400 and for several values of the parameter B, defined as the ratio of the side of the square to the diameter of the circle.

The turbulent velocity fluctuations at eight positions on sections of a plane wake and a boundary layer have been sampled simultaneously and recorded in digital form on magnetic tape for subsequent numerical analysis. Two configurations have been used (lines of equally-spaced sensors in planes normal to the flow, and arrays with three rows) with sensors responsive both to streamwise and cross-stream components of the fluctuations. To the extent that the Taylor approximation of ‘frozen’ flow is valid, the recorded fluctuations may be interpreted as instantaneous values at grid points in the volume swept out by the array.

The records have been examined, (a) to find evidence for flow patterns with marked periodicity in one direction, and (b) to select dimensions and orientations for simple eddy flow patterns whose random superposition would lead to correlation functions with a close resemblance to those calculated from the recorded data. In the wake, clear evidence was found for periodic flow patterns that resemble the eddies of a von Kármán street, but, although the spacing of eddy centres in each group was uniform, it varied considerably from one group to another, suggesting that groups are being observed in different stages of development.

Two kinds of correlation were calculated from the records, (i) simple mean values of velocity products, and (ii) mean values of the products weighted by the total intensity or Reynolds stress in the effective volume swept by the array of sensors. For both kinds, the correlations are well described by simple inclined roller-type eddies, but the correspondence is greatly improved by weighting in favour of intensity or Reynolds stress. It appears that the eddies contributing most to intensity or Reynolds stress are less variable in form than all the eddies together, and that those contributing most to Reynolds stress are significantly different in shape and in orientation from those contributing most to turbulent energy.

The motion of two immiscible fluids with arbitrary viscosities flowing through a capillary with an almost flat fluid-fluid interface is investigated in the limit of small capillary and Reynolds numbers. A complete description of the dynamics of the fluids is presented. It is shown that the motion of the fluid away from the moving contact line can be completely determined in terms of one material parameter, and how the capillary can be viewed as a device for measuring it. The dynamic behaviour of various contact angles, measured by others, is calculated. It is shown that they all depend on the radius of the capillary; hence, they do not represent properties of only the materials of the system.

A plane horizontal layer of a fluid with depth d is considered into which heat is introduced. Within the Boussinesq approximation an exact expression is obtained for the efficiency of convection γ in transforming the rate of heat supplied into the generation of kinetic energy. It agrees with results of numerical and laboratory experiments whose data can be used to calculate this value. In laboratory experiments γ is usually of order 10−8 to 10−6. Using this quantity estimates are obtained for the r.m.s. velocity of convective motions $\overline{u}$ and their time scale $\tau = d/\overline{u}$ for a regime where viscosity is important. These estimates agree with the results of a number of previous numerical experiments over a wide range of Rayleigh, R, and Prandtl numbers. Dimensionless convection equations normalized by these scales show the existence of thermal boundary layers and of almost isothermal regions within the bulk of the fluid. From this, the main regimes of heat transfer follow immediately: the Nusselt number N ∼ (R − Rcr)¼ for moderate R and $N \sim R^{\frac{1}{3}}$ for sufficiently large R.

A number of simple experiments have been carried out to measure $\overline{u}$ and τ for convection in water. Their results confirm the theoretical dependences of $\overline{u}$ and τ on external parameters and show that a smooth transition region exists from the viscous regime of convection to the more fully-developed turbulent one. The latter regime is considered by a scaling analysis. The results are compared with the author's measurements and other experimental data.

Similarly density convection is considered which arises by the separation of a medium into light and heavy fractions. Differences and analogies with the thermal convection are established. Elementary experiments confirm qualitatively the predicted dependences for $\overline{u}$ and τ. Applications of the results obtained are briefly discussed for studies of heat and mass transfer in the ocean and of convection in the Earth's mantle.

In the last section some general properties are considered for various forced flows, convection, turbulence and some types of atmospheric circulation, that allow one to formulate a ‘rule’ of the fastest response, which asserts that the kinetic energy of a fluid system is of order of the supplied power times the fastest relaxation time which the system possesses.