The stability of a steady, plane, one-dimensional trans-Alfvénic shock to small transverse disturbances in the velocity and magnetic field is discussed. In the undisturbed flow the magnetic field and velocity are normal to the plane of the shock which is treated as a discontinuity in an inviscid gas of zero thermal conductivity. However, the electrical conductivity of the ambient gas is taken to be finite, i.e. the magnetic diffusivity is very much larger than both the viscous diffusivity and the thermal diffusivity.

A small uniform transverse perturbation of the magnetic field is imposed at zero time. A specific super-Alfvénic normal shock is chosen. Computed values of the magnetic field, in and near the shock, are given for various later times. Two diffusing Alfvén waves of the amplitudes predicted by infinite conductivity theory are shown to propagate away from the shock region, leaving behind the expected steady-state shock profile.

Similar computations are carried out for a specific trans-Alfvénic normal shock. An analytic asymptotic solution, valid for large times, is also obtained. This result agrees with the computations carried out. The shock profile of the transverse quantities is one which grows linearly with time. Outside this magneto-hydrodynamic shock region which surrounds the trans-Alfvénic hydrodynamic shock (discontinuity), steady states are reached in each of which the transverse velocity and transverse magnetic field are uniform. An incident Alfvén wave, consisting of a weak diffusing current sheet, produces these same effects. The resolution of an arbitrary transverse fluctuation, in which both the magnetic field and the velocity have limiting values at large distances from the shock, is discussed. The solution for large times is found.

It is shown that the trans-Alfvénic normal shock and its two limiting cases, the null switch-on and null switch-off shocks, are unstable to general transverse disturbances although there exist particular disturbances of this kind which will not destroy them. An integral condition is obtained which, together with the relevant boundary conditions, determines the profiles of the transverse quantities in the trans-Alfvénic normal shock whenever a steady state is reached. This removes the puzzling arbitrariness of these profiles.

Numerical integration of the boundary-layer equations associated with flow past a semi-infinite flat plate in the presence of an aligned magnetic field has shown that the solutions are not unique if ε < 1 for certain values of β < 1, where ε is an intrinsic property of the fluid and β a property of conditions at infinity. An analytic explanation of this phenomenon is given here. The main properties as β → 1 of the unique solutions when ε > 1 are elucidated. Further, the equations associated with flow past a solid boundary in which the magnetic field is zero are solved numerically. The solutions appear to be unique but, on the other hand, the maximum value β0, of β, for which they can be found, tends to zero with ε.

In an earlier paper (Gallagher & Mercer 1962) the numerical results for the first eigenvalue of the problem of plane couette flow were given. The higher eigen-values are now examined and are found to be in agreement with those of Southwell & Chitty (1930), but to disagree with those of Grohne (1954).

The paper is discussion of measurements of the statistical properties of the pressure field at the wall of turbulent attached shear flows. These measurements have been made only in part by the author. A preliminary discussion is given of the important limitations imposed by the imperfect space resolution of contemporary pressure transducers. There follows a discussion of the appropriate scales of the pressure field. It is shown that measurements of the longitudinal cross-spectral densities lead to similarity variables for the space-time covariance of the pressure and for the corresponding spectra. The existence of these similarity variables may be due to the dispersion of the sources of pressure by the mean velocity gradient. Such a mechanism is illustrated by a simple model. Lateral cross-spectral densities also lead approximately to similarity variables.

Computations based directly upon detailed pressure-velocity correlation measurements by Wooldridge & Willmarth reveal that an important part of the pressure at the wall of a boundary layer is contributed by source terms which are quadratic in the turbulent velocity fluctuations; the interaction of the mean strain rate with normal velocity fluctuations, being in effect limited to a region very near the wall, supplies a dominant contribution only at high frequencies and its scales, downstream convective speed and convective memory are markedly smaller than those of the observed wall pressure.

The inner part of the Law of the Wall region (y* [les ] 100) seems to be substantially free of pressure sources and within that region (a) the pressure can be given in terms of its boundary value, and (b) the local velocity field is dependent upon but unbale to affect appreciably the turbulent pressures.

Hypersonic flow past axisymmetric power-law bodies $(y \propto x^m; m = 0, { \frac {1}{10}},{ \frac {1}{4}},{ \frac {1}{3}},{ \frac {1}{2}},{ \frac {3}{4}}\; and \; 1)$ has been studied experimentally and a detailed analysis of the dependence of shock-wave shape on body shape made.

Results indicate a continuous relationship between the exponent of the shock-wave shape and that of body shape over the range considered.

The flow field due to the oscillations of a disk is governed largely by diffusion near the boundary, but the inertia forces cannot be neglected at large distances. The solution is obtained as a single expansion for the case when the disk is performing small-amplitude sinusoidal oscillations.

A study has been made of the factors influencing the break-down of sheets formed by the impingement of two liquid jets. It is shown that disintegration generally results from the formation of unstable waves of aerodynamic or hydrodynamic origin. While the characteristics of the former waves are fairly well understood, little is known about the latter. The results of this study indicate that hydrodynamic (or ‘impact’) waves are generated when the Weber number of each jet (ρlDV2 sin 2θ/γ, where ρl is the liquid density, D the jet diameter, V the mean jet velocity, θ the half-angle of impingement, and γ is the surface tension) is above a critical value, and that their formation is independent of the Reynolds number. Drop sizes have been measured and are shown to be critically dependent upon the mechanism of disintegration.

The order of the dispersion relation for the propagation of hydromagnetic waves along a magnetized cylindrical plasma falls by unity when the plasma resistivity, σ−1, tends to zero. A consequence of this is that the two boundary conditions necessary on an insulating wall are reduced to a single condition, a reduction brought about by the development of a current sheet. If the ratio, $\Omega \equiv \omega|\omega_{ci}$ of the wave frequency to the ion cyclotron frequency is also assumed to be vanishingly small, then the nature of the single boundary condition to be adopted in the limit σ−1 → 0 depends, for the slow hydromagnetic wave, on the limiting value of ρ½Ω2 . Similarly, if Ω [Gt ] 1, and the fast hydromagnetic wave is being considered, then the relevant boundary condition is found to depend on the limiting value of Ωσ−½.

The ‘resistive’ waves that are found to accompany the fast and slow waves, in order to satisfy the boundary conditions for small but finite values of σ−1, are studied in some detail and their contribution to the wave damping is determined.

In this paper we examine the steady, two-dimensional convective motion which occurs in a horizontal circular cylinder whose wall is non-uniformly heated. One observes several qualitatively different physical phenomena depending on the wall temperature distribution and the value of the Rayleigh number. The low-Rayleigh-number behaviour for the single convective cell heated from below is related to the classical Rayleigh stability problem. The critical Rayleigh number for the single circular cell is approximately five times the value for Rayleigh's multi-celluar configuration. The flow which exhibits a nearly parabolic velocity profile near the critical Rayleigh number, gradually changes to a rigidly-rotating-core behaviour as the Rayleigh number increases. The speed of core rotation is a function of the Prandtl number, whereas the boundary-layer thickness is primarily a function of the Rayleigh number. When the heating is from side to side, the solution shows that as the Rayleigh number increases the core motion is progressively arrested leaving a narrow circulating band of fluid adjacent to the wall. An oblique heating produces a hybrid phenomenon, a low-Rayleigh-number behaviour which is characteristic of the sideways heating case and a high-Rayleigh-number interior motion characteristic of the bottom heating case. To determine the core motion in the high-Rayleigh-number limit, Batchelor's work concerning the uniqueness of incompressible, exactly steady, closed streamline flows with small viscosity is extended to include flows with small thermal conductivity.

The transition Reynolds number and the turbulent Reynolds number induced by a sinusoidally fluctuating free stream have been determined experimentally. The oscillating flow was produced in a closed-circuit wind tunnel by means of a rotating shutter valve which had a range of frequencies from 4 to 125 c/s corresponding to a range of the dimensionless frequency parameter, $\omega v |U^2_\infty$, of 2·29 × 10−6 to 4·49 × 10−5. The dimensionless amplitude parameter, $\Delta U|U_\infty$, could be adjusted by means of shutter blades of various widths from a value of 0.0775 to 0.667.

Flows in both the free stream and the boundary layer were monitored simultaneously by means of two transistorized constant-temperature hot-wire anemometers. The transition Reynolds number, the turbulent Reynolds number and turbulent intermittency factor, γ, were determined from the velocity-time traces recorded on a dual-channel oscilloscope.

It was found that the transition Reynolds number depends only on the amplitude of the oscillations and that the dimensionless transition length is a function only of the frequency. The time-space distribution of turbulent bursts in the transition region indicates that the location as well as duration of bursts is quite regular and closely tied to the fluctuations of free-stream velocity, confirming the analysis of Greenspan & Benney.

This paper considers the question of how large the magnetic energy can be in stationary homogeneous turbulence at large Reynolds number of an incompressible conducting fluid for which the magnetic diffusivity λ is much less than the kinematic viscosity ν. An approximate equation, that takes account of the effect of the Lorentz forces on the turbulence, is proposed for the spectrum function of the magnetic energy for wave-numbers lying in the equilibrium range. This equation is used to determine the magnetic spectrum function and the level of magnetic energy for the case when a statistically steady magnetic field is maintained by a relatively small input of magnetic energy, by a weak applied field say, on the scale of the energy-containing eddies; it being supposed as suggested by earlier work that the magnetic energy would eventually die away in the absence of external electromotive forces. The results are complicated, there being essentially four different régimes depending in a fairly involved way on the relative values of λ/ν, the turbulent Reynolds number, and the ratio of the energy of the applied field to the energy of the turbulence. The main conclusions about the amplification factor are shown diagrammatically in figure 4. The wavenumber at which the magnetic spectrum has its maximum tends to decrease, as the applied field is increased, from the conduction cut-off wave-number $(\epsilon |v \lambda ^2)^{\frac{1}{2}}$ to values lying in the inertial subrange, much less than $(\epsilon |v ^3)^{\frac{1}{4}}$.

The case of a turbulent dynamo is also examined, and it is concluded that, if it exists, the equilibrium magnetic energy would be given by Batchelor's criterion of equipartition of energy between the magnetic field and the small, energy-dissipating eddies. The magnetic energy in the dynamo is found to lie mainly in Fourier components of wave-number about $(\epsilon |v \lambda ^2)^{\frac{1}{2}}$.

The motion at low Reynolds number of a drop in a quiescent unbounded fluid is investigated theoretically by means of a singular-perturbation solution of the axisymmetric equations of motion. Special attention is paid to the deformation of the drop. It is shown that for small values of the Weber number We the drop will first deform exactly into an oblate spheroid and then, with a further increase in We, into a geometry approaching that of a spherical cap. These results are quite insensitive to the ratio of the viscosities of the two fluid phases. The first-order effect of the deformation on the drag of the drop is also included in the analysis.