An analytical study is made of non-equilibrium effects on hypersonic, inviscid flow over slender, axisymmetric bodies. Also, two-dimensional results are obtained for the purpose of comparison. The rate process under consideration is that of molecular vibration of the gas. The exact problem is solved by successive approximations based on a double-expansion scheme involving two small parameters: one represents the fact that the bodies considered are slender; the other represents the facts that the vibrational internal energy is small in comparison to the total enthalpy. The exact differential equations and boundary conditions are simplified to the hypersonic-small-disturbance-approximation form. The unknown quantities in this approximate problem are expanded into series of the small parameter, (γ − 1)/(γ + 1), which is ⅙ for a diatomic gas. In this formulation it is found that the classical hypersonic similitude can be extended by slight modifications to cover the added consideration of vibrational non-equilibrium. The modifications introduced are the normalization of all lengths by the characteristic relaxation length of the gas and the addition of a new dimensionless parameter, which is a measure of the excitation level of the vibrational internal energy in the flow field. Explicit, uniformly valid solutions are obtained for the specific problems of flow over a slender cone and of that over a thin wedge. The successive approximations are carried as far as necessary to show the non-equilibrium effect, which differs in order of magnitude for the various flow quantities. One interesting feature of the solutions is the non-monotonic behaviour in the relaxation of the surface pressure of both the cone and the wedge, in contrast to intuitive expectation. The result for a 20° cone in a free stream of oxygen at 300° K and a Mach number of 15 is displayed and compared with the numerical solution of the exact problem using the method of characteristics.

The solution is obtained in general terms for the velocity fluctuations generated in a laminar boundary layer by an oscillating disturbance on the boundry wall. The form of excitation is chosen to represent a vibrating ribbon of the type used by Schubauer to force disturbance in boundary layers. The forced wave system generated by the ribbon is shown to be a spatially growing one, which is described far downstream by an eigenmode of the system which has a complex wave-number.

The empirical description of turbulent wall layers across which the shear stress varies is considered. The description given by Townsend for zero-stress layers is found to be inapplicable to uniform pressure flows in pipes and two-dimensional channels, and to a boundary layer developing in the absence of a pressure gradient.

A general linear theory is developed to describe the manner in which rigid fluid rotation is established from a prescribed initial state of motion in a container of arbitrary shape. The container rotates with uniform angular velocity and is filled with a viscous incompressible fluid.

A new mean circulation theorem is proved and used to separate the flow into geostrophic motion and inertial oscillations. The basic eigenvalue problem is studied and important properties concerning spectrum, orthogonality and completeness are deduced. The effect of viscosity on the inviscid modes is calculated in a manner that maintains the solution uniformly valid through the spin-up time. All modes decay in this time scale which characterizes the entire transition to rigid rotation in all configurations.

An asymptotic solution to the Navier-Stokes equation is obtained for the incompressible flow of a viscous fluid past a semi-infinite flat plate when a slip boundary condition is applied at the plate. The results for the shear stress (and hence the slip velocity) on the plate differ basically from those obtained by previous authors who considered the same problem using some form of the Oseen equations.

We consider a very thin vortex filament in an unbounded, incompressible and inviscid fluid. The filament is not necessarily plane. Each portion of the filament moves with a velocity that can be approximated in terms of the local curvature of the filament. This approximation leads to a pair of intrinsic equations giving the curvature and the torsion of the filament, as functions of the time and the arc length along the filament. It is found that helicoidal vortex filaments are elementary solutions, and that they are unstable.

The intrisic equations also suggest a linear mechanism that tends to produce concentrated torsion and a non-linear mechanism tending to disperse such singularities.

The turbulent separated-flow region occurring at a compression corner under certain circumstances at supersonic speed has been examined in the light of recent improvements to base pressure theory (McDonald 1964). This base pressure theory is further extended from what could be termed a single-layer model of the re-attaching boundary layer to a two-layer model, thus enabling the inviscid shock configuration which occurs at the corner to be determined. Application of this analysis to some experimental results indicates a substantial measure of agreement.

While this analysis has been framed for estimating the scale of the corner interaction, the extension can of course be applied to increase the range of initial boundary-layer thicknesses to which McDonald's analysis is applicable. An example of such an application is shown to be in good agreement with experiment.

A theory describing boundary-layer surface-pressure fluctuations on a rigid surface is presented in a form that illustrates the main effect of compressibility. The most significant effect is that the correlation area is proportional to the square of mean-flow Mach number so it does not vanish in flow of finite compressibility. Modifications of the wave-number and frequency spectra by this effect are described, and the results applied to the computation of large plate response. That computation incorporates the effect of fluid loading, which enters the response equations as a dissipative term for components at supersonic phase velocity but merely as an added loading for subsonic components.

The interior of an insulating cylindrical container is supposed filled with an incompressible, electrically conducting, viscous fluid. An externally applied magnetic field is caused to rotate uniformly about an axis parallel to the cylinder generators (by applying two alternating components out of phase at right angles). Induced currents in the fluid give rise to a Lorentz force which drives a velocity field, which in general may have a steady and a fluctuating component. The particular case of a circular cylindrical container in a transverse magnetic field is studied in detail. Under certain reasonable assumptions, the resulting flow is shown to have only the steady component, and the distribution of this component is determined. Some conjectures are offered about the stability of this flow and about the corresponding flows in cavities of general shape.

The six simplest modes of motion are considered and three rotational effects investigated:

1. The effect of the rotation of the earth.

2. The effect of the rotation of the container.

3. The effect of the rotation of the liquid within the container.

The changes in frequency consequent on the rotation are derived in all three cases for all six modes. In the case of the earth's rotation the disposition and character of the amphidromic (nodal) points and the amphidromic waves that rotate round these points are investigated in detail. One mode is particularly interesting because it has four amphidromic points, the waves rotate in a positive sense around two of these and in a negative sense round the other two.

The investigation concerns the stability of an interface between two inviscid fluids of different densities which flow parallel to each other in an oscillatory manner. When the difference in the mean speeds is below the steady, critical speed for instability but is large compared to the amplitude of the fluctuations, parametric amplification of waves at the interface occurs, and the interface exhibits a resonance of a subharmonic nature. The paper concludes with a discussion of the stability of unsteady shear layers on the basis of the results.

The distortion of a uniform magnetic field, aligned with the flow at infinity, by the potential flow of an inviscid conductor about a circular cylinder is determined. Potential flow of the fluid occurs when the interaction parameter is small; this is the case studied here. In the flow-potential and stream-function plane the problem may be formulated as a singular integral equation. Solutions of this equation show that for small fluid conductivities the magnetic field lines are distorted in the sense of being dragged along by the motion of the fluid. This process continues as the conductivity increases, with fewer and fewer of the magnetic field lines entering the body. For large conductivity this reduced flux of field lines enters over most of the body surface and exits in the neighbourhood of the rear stagnation point; behind the body there is a jet-like structure of magnetic field lines.

The instability of a viscous fluid between two infinite vertical plates and heated from below in the presence of a magnetic field perpendicular to the plates is investigated, and the most critical stability boundary in the space of the Rayleigh number R, Hartmann number M, and the horizontal wave number a is determined. It is found that the most unstable mode is a symmetric mode with zero wave-number, and that for any M the fluid is unstable for any non-zero R, however small.

A series expansion is derived for the three-dimensional boundary-layer flow over a flat plate, arising from a general main-stream flow over the plate. The series involved are calculated as far as terms of order ξ2, where ξ is a non-dimensional parameter defining distance measured from the leading edge of the plate. The results are applied to an example in which the main stream arises from the disturbance of a uniform stream by a circular cylinder mounted downstream from the leading edge of the plate, the axis of the cylinder being normal to the plate. Calculations are made for shear stress components on the plate, and for the deviation of direction of the limiting streamlines from those in the main stream.

The transient pressure field produced behind a plane shock wave of arbitrary strength, which encounters a slender supersonic cone head on, is theoretically predicted. The analysis is restricted to small cone angles (in order that Mach-reflexion will occur at the surface) and to inviscid flow.

A conical-flow technique is utilized to transform the small-perturbation equations of time-dependent rotational motion of the fluid behind the deflected plane shock into time-independent equations. From these equations, a boundary-value problem for pressure alone, in the Mach-reflexion region, is derived which is solved numerically. Results for the small perturbation pressure field of the cone are contrasted with Smyrl's (1963) results, which hold for a moving wedge, for a range of values of cone and plane shock speeds.