Conjugate solutions of the equation of swirling cylindrical flow are considered in the case where the ‘primary’ flow has constant axial velocity. Numerical solutions are exhibited, which suggest a division of primary flows into four classes, and some conditions governing membership of these classes are derived. Finally, the case when the governing equation is slightly non-linear is considered.

The rate of mixing across a density interface between two layers of liquid has been measured in a laboratory experiment which allows a direct comparison between heat and salinity transports over the same range of density differences. Low Reynolds number turbulence was produced by stirring mechanically at a fixed distance from the interface, either in one or in both layers, and the results for these two sets of experiments are also compared. The measurements cover a factor of two in stirring rate and twenty in density. Over this range of conditions the ratio of entrainment velocity to stirring velocity can be expressed as functions of an overall Richardson number Ri, and in this form the results of the one and two stirred layer experiments are indistinguishable from one another. For density differences produced by heat alone, the functional dependence is close to Ri−1 except at small values of Ri where it approaches a finite limit. For experiments with a salinity difference across the interface, the mixing rate is the same as in the heat experiments at low values of Ri, but falls progressively below this as Ri is increased, with the approximate form $Ri^{\frac{3}{2}} $.

An interpretation of these results has been attempted, using a dimensional analysis and qualitative mechanistic arguments about the nature of the motion. The Ri−1 dependence implies a rate of change of potential energy proportional to the rate of working by the stirrer. The decreased mixing rates for salt have been attributed to a slower rate of incorporation of an entrained element into its surroundings by diffusion, which increases the tendency for it to return to the interface and dissipate energy in wave-like motions.

The unsteady laminar boundary-layer flow induced by the impulsive motion of a semi-infinite flat plate along its length is investigated. It is found that, unlike Stewartson's (1951, 1960) conclusion, a power-series solution is possible using the ‘correctly stretched’ variables in the analysis. The small-time solution, which is developed in powers of the time, shows a smooth transition from the initial Rayleigh flow to the final Blasius flow without an essential singularity and, furthermore, its validity extends to the whole time domain. The series solution developed for large times, however, seems divergent and merely asymptotic. No evidence is found for the existence of an essential singularity in the solution as described by Stewartson.

An experimental investigation of the instability of a laminar air flow over water shows two distinct modes of unstable oscillations as predicted by theory. The Tollmien–Schlichting waves instability could be excited by a ribbon vibrating in the air, and the neutral stability curve determined. The water wave instability mode could be excited by a ribbon vibrating in the water. The growth rates of these waves show only fair agreement with theoretical predictions.

This paper is an analysis of incompressible three-dimensional flows of electrically conducting fluids under the action of transverse magnetic fields which are assumed to be sufficiently strong that the interaction parameter N (= M2/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number. We also assume that R [Gt ] 1 and Rm (magnetic Reynolds number) [Lt ] 1, so that experimental verification of the theory may be possible.

The main results are: (i) when a thick body is placed in a parallel-sided channel with non-conducting walls the flow over it is highly dependent on the conductivity of the body, in a surprising way. If the body is non-conducting, there is no flow within that cylinder which circumscribes the body and is parallel to the magnetic field; outside the cylinder the flow is plane and potential and enters or leaves the surface shear layer of this cylinder at right angles. If the body is conducting, flow over it is possible and is of a different nature outside and inside the cylinder. (ii) When a non-conducting flat plate is placed in such a channel no blocking of the flow occurs. If the plate is elongated in the flow direction, the flow over it becomes identical to that calculated by Hasimoto (1960) and, if elongated at right angles to the flow, becomes identical to that calculated by Dix (1963).

Of particular interest in our analysis are the two types of layer which occur in these flows, the first being the Hartmann boundary layer, which is shown to have a controlling influence on the vorticity of the core flow in three-dimensional situations analogous to that of the Eckman layer in rotating-fluid flows. The second type, the free shear layer at the circumscribing cylinder, is of interest because of its internal structure and effect on the external flow.

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.

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).

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 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.

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.

A two-dimensional, semi-infinite, stratified shear flow in which the upstream dynamic pressure and density gradient are constant (Long's model) is considered. A complete set of lee-wave functions, each of which satisfies the condition of no upstream reflexion, is determined in polar co-ordinates. These functions are used to determine the lee-wave field produced by, and the consequent drag on, a semicircular obstacle as functions of the Froude number within the range of stable flow. The Green's function (point-source solution) for the half-space also is determined in polar co-ordinates.

The formation and transient dynamics of Stewartson layers are examined herein. The results are applied to a transient rotating fluid flow for which (i) Stewartson layers are presumed to be important, and (ii) the Ekman layers are found to be non-divergent, and hence unable to induce the crucial secondary circulation occurring in the spin-up process. The steady state is nevertheless reached after a spin-up time.