Experiments are presented in which two-dimensional internal waves were generated in a stably stratified fluid by the movement of a long horizontal circular cylinder. The cylinder was moved with a constant velocity at Reynolds numbers based on the diameter of the cylinder of between 6 and 100. Under these conditions the internal wave system is stationary with respect to the moving cylinder and the phase configuration of the waves compares well with Lighthill's theory for waves in dispersive systems.

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived.

A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in (a) a shear flow, and (b) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section.

The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt & Mason 1961; Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.

When a fluid of low conductivity flows parallel to a plane wall from behind which projects a magnetic field, an electric potential field is established throughout the fluid. In this paper the potential field is obtained explicitly in terms of the velocity field when the latter is unidirectional and depends only on the coordinate normal to the wall. Experiments with a variety of velocity profiles are described, and the agreement with the theory is satisfactory. The effect of slow variation of the profiles in the direction of the flow is considered, and is shown to be unimportant under the conditions of the experiments.

Experiments described here show that the rate of heat transfer in a supersonic, turbulent, boundary layer on a concave wall is increased by the streamwise curvature of the wall. For the arrangement investigated, the pressure is kept constant along the wall, and the increase of about 20% is therefore only due to the wall curvature. For a convex wall, a comparable decrease was found, also with constant pressure along the wall.

It is likely that this change in heat-transfer rate is mainly due to an increase or decrease of turbulent mixing by the effect of the curvature. The increase on a concave wall can in principle also be explained by large-scale vortices with axes in the flow direction (Görtler vortices). However, disturbances of this type cannot explain the decrease observed on the convex wall. They cannot therefore be the only cause.

A simple criterion indicating the effect of curvature on the turbulent motion normal to the wall is given. It is derived from an inviscid-flow analysis. The criterion shows that, for most Mach numbers and wall temperatures of practical interest, the change in mixing depends mainly on the velocity gradient normal to the wall. For high supersonic and hypersonic Mach numbers, however, there is a layer near the outer edge of the boundary layer where the change depends mainly on the temperature (density) gradient.

A curved screen placed across a two-dimensional channel causes the streamlines to be deflected on passing through because of the variation in pressure drop across the section and the refraction effect at the screen. Uniform flows far upstream and far downstream are required by the boundary conditions. An analytical description is based on the separation of the field into two regions distant from the screen in which viscosity and molecular diffusion are negligible, plus a thin layer along the screen in which energy loss and streamline deflexion are concentrated. These are described by empirical relationships. For linear velocity and density distribution upstream of the screen, equations can be simplified so that algebraic relationships between the variables at the screen surface are obtained. These have been solved numerically for the shape of screen required to produce a specified velocity distribution. An approximate solution is also obtained for general velocity profiles and the screen shape which produces uniform shear is derived. Experimental verification of the analysis is obtained from measurements of the velocity and temperature distributions downstream of the derived screen shapes mounted in a wind tunnel 45.6 cm square.

It is also shown that the boundary layers along a tunnel wall are accelerated or retarded by the screen depending on the loss coefficient. This effect is evident in all observations.

The case of homogeneous fluid is described by a simplified version of the analysis and several examples of velocity distributions are produced. These are verified by experiment and compared with those predicted by Elder (1959).

Lift can be generated on a circular cylinder with its axis normal to an air flow by blowing a sheet of air tangentially round the upper surface from a narrow slot or slots. This lift force may be estimated by matching the external inviscid flow field with separation points calculated by Spalding's unified boundary-layer theory. The theory reproduces experimental results reasonably well, except in certain special conditions fully discussed.

An analysis is presented for the stability of a viscoelastic liquid film flowing down an inclined plane with respect to three-dimensional disturbances. It is shown that under certain circumstances, these disturbances are more unstable than the two-dimensional ones, contrary to Squire's theorem.

The steady two-dimensional flow of an inviscid incompressible fluid of variable density is considered in a long channel, bounded above by a rigid horizontal plane and below by an obstacle. For certain variations with height of the speed aud density in the incident stream, the governing equation is the reduced wave equation. Drazin & Moore (1967) have recently used this fact to develop a waveguide analogy. In this note the wave-guide analogy is further developed and several uniqueness theorems obtained. When the obstacle satisfies a certain convexity condition it is shown that the upstream conditions and the obstacle uniquely determine the flow; that is, there is no critical internal Froude number or obstacle height for which the problem fails to be well posed.

This investigation is concerned with the convective motions in a shallow layer of silicone oil on a plane, circular copper plate which is heated at the rim and cooled at the centre and at the same time rotated around a vertical axis. The oil is in touch with a glass lid, which is cooled uniformly. With sufficient heating axially symmetric concentric rings develop. The details of the motion can be described as a superposition of a circulation due to the horizontal temperature gradient and a circulation of opposite sense due to the centrifugal force, with the motions due to the vertical instability. There seems to be a conversion into rolls whose axes point radially, if the centrifugal circulation becomes too strong.

This paper shows how electrochemical techniques can be used in studies of flow around solid objects to measure the velocity gradient at the solid boundary. The method holds the advantages that it is not necessary to calibrate and that the test element is easy to fabricate. A study of the distribution of the wall velocity gradient around a cylinder of 1 in. diameter indicates that boundary-layer theory correctly predicts the measurements between the front stagnation point and the separation for Reynolds number, R, greater than 150. The wall velocity gradients in the wake are much smaller than in the front part of the cylinder and they reveal a minimum which is quite close to the separation point.

Wave propagation in one-dimensional erodible-bed channels is discussed by using the shallow-water approximation for the fluid and a continuity equation for the bed. In addition to gravity waves, a third wave, which gives the velocity of propagation of a bed disturbance, is found. An appropriate dimensional analysis yields the quasi-steady approximation for the complete shallow-water equations.

The well-known linear stability analysis of free-surface flows is extended to include the erodibility of the bed. The critical Froude number Fc above which the free-surface of the fluid may become unstable is obtained. It is shown that erodibility increases the stability of the free surface, in qualitative agreement with previous experiments if qb > qs, qb and qs being respectively the contact-bed discharge and suspended-material discharge. The stability theory is also used to discuss coupled beds and surface waves. From it, five different configurations have been obtained: a sinusoidal wave pattern moving downstream, a transition zone and antidunes moving upstream, moving downstream and stationary. These bed forms are in agreement with experimental results; hence shallow-water theory seems to give a reasonable explanation of the boundary instability.

It is shown that the quasi-steady approximation and Kennedy's (1963) stability analysis will be in agreement if (kh)2 [Lt ] 1, where k is the wave number, and h is the depth of the water. When the phase shift δ is introduced in the quasi-steady approximation, the five bed patterns derived from the full equations are found again.

The linear boundary-layer analysis by Stewartson & Roberts (1963) and by Roberts & Stewartson (1965) for the motion of a viscous fluid inside the spheroidal cavity of a precessing rigid body is extended to include effects due to the nonlinear terms in the boundary-layer equation. The most significant consequence is a differential rotation super-imposed on the constant vorticity flow given by the linear theory. In addition it is shown that a tidal bulge of the cavity forces a fluid motion similar to that caused by the precessional torque. The relevance of both effects for the liquid core of the earth is briefly discussed.

An investigation is made of the stability of time-periodic azimuthal flows between coaxial, circular cylinders. The disturbance equations are linearized and consideration is limited to the effects of axisymmetric disturbances in a fluid with zero viscosity. It is found convenient to examine separately the cases of flows with zero time-mean and non-zero time-mean respectively. Some remarks are made concerning the definition of stability in relation to such flows and their general stability characteristics are evaluated and discussed.

A solution is offered for the flow induced by a two-dimensional line sink in a saturated, density-stratified porous medium. It is found that fluid is selectively withdrawn from a thin layer at the elevation of the line sink and not from the entire medium. The velocity distributions predicted by the theory are checked by experiments in a Hele–Shaw cell and good agreement found.

An operational method is used to obtain an exact solution for the heat transfer from a surface on which the temperature is prescribed to a stream in which the velocity profile is given by \[ u = (\mu_{e}\rho_w)^{-1}\left[\tau_w\eta + \left(\frac{1}{2\rho_w}\frac{dp}{dx}\right)\eta^2\right], \] where \[ \eta = \int_0^y \rho dy \] and other symbols have their usual meanings. The solution is expanded for small and large values of a dimensionless parameter proportional to $\sigma^{\frac{1}{4}}\tau_w/(dp/dx)^{\frac{2}{3}}$: the leading term when this is large is precisely Lighthill's (1950) expression for heat transfer at high Prandtl number, and that when it is small corresponds to Liepmann's (1958) expression for heat transfer at a separation point.

Particular attention is given to the case of heat transfer from a small element of length l maintained at a constant temperature ΔT above that of the surrounding adiabatic wall: this represents the gauge for skin friction measurement described by Bellhouse & Schultz (1966) and Brown (1967 a). It is shown that the Nusselt number for heat transfer from such an element is of the form $\alpha^{\frac{1}{3}}f(\beta/\alpha^{\frac{4}{3}}) $, where \[ \alpha = \sigma\rho_w\tau_w\frac{l^2}{\mu^2_w},\quad\beta = \bigg(\frac{\sigma\rho_w}{\mu^2_w}\bigg)\frac{dp}{dx}l^3, \] and expansions for small and large values of $\beta/\alpha^{\frac{4}{3}}$ are given. Over the whole range both are adequately represented for experimental purposes by the equation \[ c_1\alpha + \frac{c_2\beta}{Nu} = Nu^3, \] which has a form suggested by consideration of the integral approximation of Curle (1962). The experimental application of the results to both laminar and turbulent flows is discussed.

The boundary layer on an axisymmetric surface above which the flow is rotating about the axis of symmetry is considered. Transformations of the governing equations which permit the generalizations of a known solution for one meridian shape in incompressible flow to a family of meridian shapes are shown to exist. For compressible flow, a transformation of the Stewartson-Illingworth type was found which reduces a compressible flow problem to an incompressible case. Also, remarks are made concerning the invariance of the turbulent boundary-layer integral equations assuming particular semi-empirical shear laws.

Power spectral density measurements were made of the water surface displacement in a wind-generated wave train in the Stanford wind, water-wave research facility. Growth rates of different components were determined from the measured spectra. The resulting values were compared with those predicted by the viscous Reynolds stress mechanism of energy transfer from wind to wave which was proposed by Miles. Where the growth was exponential the theory could be made to predict growth rates successfully at wave frequencies less than 3·5 c/s. At higher frequencies the theory predicted values an order of magnitude larger than those measured. Limited regions of linear growth were found at the lowest wind speed for components with frequencies less than 2·5 c/s. The scatter in the data did not permit a quantitative comparison with theory to be made for this range.

In this part we first extend the theory of part 1 to analyse the distribution of velocity and electric current in an electrically conducting liquid between two circular electrodes of finite diameter, when a current is passed between them. The electrodes are set opposite to each other in insulating planes and a magnetic field is applied perpendicular to these planes. When the Hartmann number M [Gt ] 1 we find that the current is confined to the cylinder of fluid joining the electrodes. This effect is accounted for by the velocity which is induced in thin layers of thickness O(M−½), at the circumference of the cylinder. In our analysis we concentrate on these interesting layers and, amongst other results, we find that in the limit M → ∞ the resistance of the fluid between the electrodes becomes that of the cylinder of fluid joining the electrodes.

We then describe some experiments to test the validity of this theory. In these experiments we measured, as a function of the magnetic field, (a) the potential difference between the copper electrodes, the fluid being mercury, (b) the electric potential distribution in the fluid between the disks and in the thin layers between the electrode edges, by means of an electric potential probe, and (c) the velocities induced in the layers using a Pilot tube. Our conclusions were: (i) the overall predictions of the theory were correct; (ii) the results of the two probes approximately correlated with each other, despite the theory still having some limitations and the behaviour of these probes still being somewhat uncertain.

It is pointed out that recent theoretical results for the spin-up of a continuously stratified fluid with insulating boundary conditions are at variance with experiment. However, it is then shown that the theoretical analysis contains an inconsistent scaling assumption, so that the apparent contradiction is not a real one. The nature of the spin-up as shown by the experiments is briefly described.

Hydromagnetic surface waves propagating on the surface of a finitely conducting fluid are studied experimentally and theoretically. An alternating magnetic field is used, so that the field is largely excluded from the bulk of the fluid. Thus, the magnetic damping of the waves is negligible, and the perfectly conducting d.c. case is simulated. The effects of a finite electromagnetic skin depth, δ, are included in the calculation of the theoretical wave dispersion relation. It is predicted that as δ is increased and becomes comparable to the wavelength of the surface wave, the effect of the magnetic field on the fluid motion is decreased. This prediction is confirmed experimentally.