The quasi-geostrophic flow over an obstacle placed on the lower of two horizontal planes in rapid rotation about the vertical z axis is considered. The flow field is calculated in the limit of vanishing viscosity after assuming the flow far upstream of the obstacle to be uniform, of magnitude V. The effects that occur in homogeneous flow are compared with those that occur in stratified flow.

If the flow is homogeneous, there is a region of closed streamlines if \[ h_0R^{-1} > \min_r\left[r\left/\int_0^r xh(x)dx\right. \right], \] where the obstacle is assumed to be cylindrically symmetric and given by z = Hh0h(r/L), H is the distance between the planes and R is the Rossby number V/(fL). For any obstacle the right-hand side of (1) is greater than zero and hence h0 must be positive for a closed-streamline region to occur. It is argued, and illustrated by a particular example, that because (1) involves an integral of h(x) a representative flow pattern can be obtained for obstacles of less than critical height by considering the special case of a flat-topped obstacle, as is done by Ingersoll (1969).

If the flow is stratified with constant Brunt-Väisälä frequency N, the condition for the existence of closed streamlines is shown to be \[ h_0R^{-1} > \min_r \left[B\int_0^{\infty}\int_0^{\infty}xt\cot h(Bt)h(x)J_0(tx)J_1(tr)\,dt\,dx \right]^{-1}, \] where B = NH/fL. In contrast to the homogeneous situation, the right-hand side of (2) can be zero and is so if the obstacle is somewhere vertical. Such obstacles will produce a closed-streamline region no matter now small their height and will hence not lead to patterns representative of smooth obstacles. This is because a stratified column of fluid cannot be stretched or compressed over an infinitesimal distance. Instead, the column bends markedly and the fluid flows around the obstacle. The critical conditions (1) and (2) for a number of specific obstacles are calculated and discussed.

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.

These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

This paper examines the effect of steady flow on the generation of sound by aerodynamic sources located in the neighbourhood of a scattering body. The analysis is facilitated by the use of a reverse-flow reciprocal theorem which is valid for a certain class of inhomogeneous flow problems. In the case of acoustic ‘bremsstrahlung’, in which sound is generated during the passage of a silent fluid inhomogeneity, such as an entropy spot, past an obstacle in the flow, an elegant discussion of the interaction can be given in terms of a renormalized Green's function. Here the effect of the obstacle is equivalent to a local distortion of the geometry of the three-dimensional space occupied by the medium, the extent of which governs the properties of the acoustic radiation. This is illustrated by means of a detailed analysis of the sound generated during the unsteady convection of a source of constant strength through a neck in a duct.

A small perturbation analysis is made of the flow inside a spherical explosion created in a gas absorbing energy from an array of lasers with a common focus. The influence of a small counterpressure outside the leading shock wave is investigated for power deposition constant in time. The departure from self-similarity is described by a power series in the variable y, the inverse square of a Mach number based on the wave-front velocity. To first order, the flow equations are solved numerically, the shock velocity is obtained and the radial profiles of velocity, density and pressure are presented graphically. The flow singularity at the radiation focus is discussed.

The effect of gravity acting perpendicularly to the stratification induced by rotation in a compressible fluid is investigated. The γ = 1 approximation used by Gans (1974) in an investigation of the free modes of a rotating gas is used to find an inviscid solution. The boundary layers are calculated, and their suctions are shown to lead to a resonance. The amplitude of the resonance mode is calculated and seen to be of the same order as the simple inviscid solution, so that a corrected solution can be obtained.

The stability of the system is discussed in a simple fashion. It seems likely that a turbulent core will form in a cylinder. The interaction of such a core with the remainder of the fluid is beyond the scope of this paper. Finally a procedure by which a zonal flow field can be found is given.

The general problem of potential flow past a slender body of revolution is considered. The flow incident on the body is described by an arbitrary potential function and hence the results presented here extend those obtained by Handels-man & Keller (1967 α). The part of the potential due to the presence of the body is represented as a superposition of potentials due to point singularities (sources, dipoles and higher-order singularities) distributed along a segment of the axis of the body inside the body. The boundary condition on the body leads to a linear integral equation for the density of the singularities. The complete uniform asymptotic expansion of the solution of this equation, as well as the extent of the distribution, is obtained using the method of Handelsman & Keller. The special case of transverse incident flow is considered in detail. Complete expansions for the dipole moment of the distribution and the virtual mass of the body are obtained. Some general comments on the method of Handelsman & Keller are given, and may be useful to others wishing to use their method.