After making the laboratory approximation of small magnetic Reynolds number, the steady, axisymmetric and purely azimuthal velocity profile that in principle can be generated in an incompressible viscous electrically conducting fluid contained in a fixed infinitely long circular cylinder by a magnetic field transverse to the cylinder axis and uniformly rotating with low frequency is subjected to infinitesimal axisymmetric perturbations. The principle of the exchange of stabilities is assumed to hold and the marginal-stability problem becomes a sixth-order eigenvalue problem involving the magnetic Taylor number and the axial wavenumber. An asymptotic analysis, based on the assumption that the magnetic Taylor number is large, and using solutions of the comparison equation d6y/dz6 = zy, is presented in order to obtain first approximations to the neutral-stability curves of the first and second eigenmodes, and compared with the results of direct numerical integration. It is found that at the onset of instability the secondary motions have a multi-cell structure, the motions in the region, near the cylinder wall, of adversely distributed angular momentum driving through weak viscous action the cells in the interior.

The perturbation of pre-existing surface gravity waves caused by the presence of an internal wave was studied both experimentally and analytically. An extensive series of experiments was performed, and quantitative results were obtained for the one-dimensional monochromatic interaction of internal waves and surface gravity waves. Internal wave-induced surface slope, amplitude and wavenumber modulations were measured for a wide range of interaction conditions. A complementary theoretical analysis, based on the conservation approach of Whitham (1962) and Longuet-Higgins & Stewart (1960,1961), was performed and a closed form solution obtained for the one-dimensional wave interaction. Both the theory and the experiment demonstrate that the effect increases with interaction distance. The maximum interaction effect is found to occur when the phase speed of the internal wave and the group velocity of the surface wave are matched. The phase of the internal wave at which maximum surface-wave modulation occurs is found to be a sensitive and continuous function of the relative wave speeds. The experimental data are in good agreement with the present theoretical analysis.

The linear stability of the developing flow of an incompressible fluid in the entrance region of a circular tube is investigated. The case of non-axisymmetric small disturbances is considered in the analysis. The main-flow velocity distribution used in the stability calculations is that from the solution of the linearized momentum equation. The eigenvalue problem consisting of the disturbance equations and the boundary conditions is solved by a direct numerical integration scheme along with an iteration procedure. An orthonormalization method is employed to remove the ‘parasitic errors’ inherent in the numerical integration of the coupled disturbance equations. The flow is found to be unstable to non-axisymmetric disturbances with an azimuthal wavenumber of one. Neutral-stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of these results is made with those for axisymmetric disturbances reported by Huang & Chen. It is found that the first instability of the flow is due to non-axisymmetric disturbances and occurs in the entrance region of the pipe with a minimum critical Reynolds number of 19 780.

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which \[ \sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0; \] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.

An extensive experimental study of the near wake of a 9° half-angle sharp cone at incidence at M∞, = 7 has been conducted. The model was suspended by six thin wires. The flow field in the plane of symmetry was surveyed from 0 to 2·5 diameters downstream of the base in the axial direction, and to the bow shock in the transverse direction. For three values of incidence (0° 5° and 15°))measurements were made of the static pressure, Pitot pressure and stagnation temperature, from which the streamline pattern was determined. In addition the u. = 0 line was found independently. The results of the measurements were also used to trace lines of constant Pitot pressure showing the shock waves, and lines of constant stagnation temperature in the viscous wake.

The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlehs, by which the rotational motion of a given body can be represented.

Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient CM = M/(8πμω0ab2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity ω0 about its axis in a fluid of viscosity μ) of various body shapes so far investigated is found to lie between $\frac{2}{3}$ and 1, usually very near unity for not extremely slender bodies.

For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.