We consider the deformation of a thin thread of viscous liquid (viscida) as its ends are slowly moved together. Equations are deduced which are capable of describing the motion of the thread when the displacement of the axis from a straight line is either on the scale of the thread thickness (problem 1) or on the much larger scale of the thread length (problem 2). In the former case it is shown analytically that an arbitrary initial displacement evolves in such a way that, as the appropriately scaled time τ becomes large, the first mode of the disturbance emerges in a dominant role with an amplitude that is proportional to τ½ and independent of the initial amplitude. This provides the initial condition for problem 2, for which a numerical description is obtained.

In addition, we analyse the situation when the ends of the viscida are slowly pulled apart. In this case the high frequency end of the spectrum dominates as an arbitrary disturbance decays.

The velocity correlation function of a Brownian particle in a viscous compressible fluid is studied in the limit of very small compressibility. The main effect of compressibility is an initial rapid change of the particle mass from a real inertial mass to a virtual mass.

The nonlinear response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is re-examined here in part 1 for the case of strong steady forcing. Introduced by Pedlosky & Greenspan (1967), the model consists of a rapidly rotating right cylinder with a planar sloping bottom. The homogeneous contained fluid is driven by the slow rotation of the flat upper lid relative to the rest of the basin. Except in thin Ekman and Stewartson boundary layers on the solid surfaces of the basin, the horizontal flow in the interior and western boundary layer is constrained by the rapid rotation of the basin to be independent of depth. The model thus effectively simulates geophysical flows through the physical analogy between topographic vortex stretching in the laboratory model and the creation of relative vorticity in planetary flows by the β effect.

As the forcing is increased, the flow in both the sliced-cylinder laboratory and numerical models first exhibits downstream intensification in the western boundary layer. At greater forcing, separation of the western boundary current occurs with the development of stationary topographic Rossby waves in the western boundary-layer transition regions. The observed flow ultimately becomes unstable when a critical Ekman-layer Reynolds number is exceeded. We first review and compare the experimental and numerical descriptions of this low-frequency instability, then present a simple theoretical model which successfully explains this observed instability in terms of the local breakdown of the finite-amplitude topographic Rossby waves embedded in the western boundary current transition region. The inviscid stability analysis of Lorenz (1972) is extended to include viscous effects, with the consequence that dissipative processes in the sliced-cylinder problem (i.e. lateral and bottom friction) are shown to inhibit the onset of the instability until the topographic Rossby wave slope exceeds a finite critical value.

The response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is studied here in part 2 for the case of an oscillatory ‘wind’ stress. The model consists of a rapidly rotating right cylinder with a planar sloping bottom. This basin geometry contains no closed geostrophic contours, so that low frequency topographic Rossby wave modes possessing mean vorticity exist in the sliced-cylinder model by the physical analogy between topographic vortex stretching and the β effect for large-scale planetary flows. The interior flow in the laboratory model is driven by the time-dependent Ekman-layer suction produced by the periodic relative angular velocity of the upper lid. The frequency of the forcing is sufficiently small that the interior motion is quasi-geostrophic with the horizontal velocities being independent of depth. Simple two-dimensional analytic and numerical models are developed and compared very favourably with the laboratory results. The observed horizontal velocity field exhibits both (i) westward intensification and decreased horizontal scale when the forcing frequency is decreased, and (ii) a significant resonant magnification when the forcing frequency is tuned to the natural frequency of one of the lower inviscid topographic Rossby wave modes. The observed westward phase speed of the driven motion is accurately predicted and shows little dependence on the amplitude of the forcing. The instantaneous and mean Lagrangian fluid particle trajectories were measured in the laboratory model. The general derivation by Moore (1970) of the governing equations for the mean Lagrangian motion are extended to incorporate forcing and Ekman-layer dissipation. The results suggest that the mean Lagrangian flow should be significantly reduced near resonant frequencies, since the mean Eulerian motion is partially offset by the Stokes drift associated with the topographic Rossby wave modes. This result is consistent with the small observed amplitude of the mean Lagrangian motion. Also presented are the results for a laboratory experiment conducted using a combined steady and oscillatory ‘wind’ stress.

The stability of a compressible non-dissipative swirling flow to adiabatic infinitesimal disturbances of arbitrary orientation is considered. The resulting sufficient condition for stability is the general form of the effective Richardson criterion for swirling flows, first obtained, for axisymmetric modes only, by Howard. In addition, upper bounds to the growth rate of unstable modes are obtained and some extensions of the semicircle theorem to azimuthal disturbances are stated.

An experimental investigation of the instability and the acoustic radiation of the low Reynolds number axisymmetric supersonic jet has been performed. Hot-wire measurements in the flow field and microphone measurements in the acoustic field were obtained from different size jets at Mach numbers of about 2. The Reynolds number ranged from 8000 to 107000, which contrasts with a Reynolds number of 1·3 × 106 for similar jets exhausting into atmospheric pressure.

Hot-wire measurements indicate that the instability process in the perfectly expanded jet consists of numerous discrete frequency modes around a Strouhal number of 0·18. The waves grow almost exponentially and propagate downstream at a supersonic velocity with respect to the surrounding air. Measurements of the wavelength and wave speed of the St = 0·18 oscillation agree closely with Tam's theoretical predictions.

Microphone measurements have shown that the wavelength, wave orientation and frequency of the acoustic radiation generated by the dominant instability agree with the Mach wave concept. The sound pressure levels measured in the low Reynolds number jet extrapolate to values approaching the noise levels measured by other experimenters in high Reynolds number jets. These measurements provide more evidence that the dominant noise generation mechanism in high Reynolds number jets is the large-scale instability.

In a transonic nozzle flow in which the velocity is slightly supersonic in some neighbourhood of the nozzle throat, a shock wave may be present either very close to the throat or else somewhat further downstream. In the latter case, relatively simple series solutions in general provide an asymptotic description of the fluid motion except very close to the shock wave. These outer solutions are reviewed for symmetric two-dimensional flow, and it is shown that the shock-wave jump conditions are not satisfied. A correction is then derived in the form of an inner solution for a small region immediately behind the shock. The resulting solution exhibits the singularities in the pressure gradient, streamline curvature and shock-wave curvature which are expected to occur at the intersection of a normal shock wave and a curved wall. An extension to axisymmetric flow is also given.

The method of characteristics is used to calculate the supersonic flow past a wedge of small angle with non-equilibrium effects. The wave decay and development distances are presented in a concise similarity form which permits accurate extrapolation to very weak waves. The numerical solutions are compared with shock-tube flows of CO2 and N2O.

The method of characteristics is used to calculate the flow ahead of an impulsively started piston moving at constant velocity. Particular attention is paid to the development of weak shock waves which are either fully or partly dispersed at very large distances from the piston. It is found that the global features of the flows may be represented in similarity form, and the graphs obtained allow extrapolation to very weak waves.

The propagation of waves in an inviscid, electrically conducting fluid is considered. The fluid rotates with angular velocity Ω* and is permeated by a magnetic field b* which varies on the length scale L = Ql, where Q = Ω*l2/λ (l is the length scale of the waves, λ is the magnetic diffusivity) is assumed large (Q [Gt ] 1). A linearized theory is readily justified in the limit of zero Rossby number R0 (= U0/Ω*l, where U0 is a typical fluid velocity) and for this case it is shown that the total wave energy of a wave train is conserved and transported at the group velocity except for that which is lost by ohmic dissipation. The analysis is extended to encompass the propagation of a sea of random waves.

A hydromagnetic dynamo model is considered in which the fluid is confined between two horizontal planes perpendicular to the rotation axis a distance L0(=O(L)) apart. Waves of given low frequency Ω*0 (= O(R0Q½Ω*)) and horizontal wavenumber l−1 but random orientation are excited at the lower boundary, where the kinetic energy density is 2πρU20. The waves are absorbed perfectly at the upper boundary, so that there is no reflexion. The linear wave energy equation remains valid in the double limit 1 [Gt ] R0Q½[Gt ]Q−½, for which it is shown that dynamo action is possible provided $\Delta = L_0U^2_0/l^3\omega_0^{*2} > 1 $. When dynamo action maintains a weak magnetic field (Δ −1 [Lt ] 1) which only slightly modifies the inertial waves analytic solutions are obtained. In the case of a strong magnetic field (Δ [Gt ] 1) for which Coriolis and Lorentz forces are comparable solutions are obtained numerically. The latter class includes the more realistic case (Δ → ∞) in which the upper boundary is absent.

Computations of the steady viscous flow past a fixed spherical gas bubble are reported for Reynolds numbers in the range 0·1 [les ] R [les ] 200. Good agreement with Moore's (1963) asymptotic theory for the drag coefficient is obtained for R [les ] 40 and with the well-known small-R theory for R [les ] ½. The method of series truncation is used to reduce the problem to a nonlinear two-point boundary-value problem, which is then solved by an accurate and efficient finite-difference procedure.

The free shear layer stemming from a discontinuity in angular velocity at either of two parallel disks in almost rigid rotation with fluid between is re-examined. The sole discrepancy between theory and experiment is unaffected by higher-order approximation unless curvature effects are included, when it is reduced.

We consider steady non-axisymmetric source-sink flow of a perfect gas in a rapidly rotating circular cylinder for the case in which the sources and the sinks are distributed on the top and bottom. We apply a linearized analysis to a small perturbation from the state of rigid-body rotation. We show that the radial pressure gradient plays an important role in the determination of the axisymmetric part of the flow field and that the effects of viscosity and thermal conductivity govern the overall configuration of the non-axisymmetric part. We also show that the transport of gas in the main inner flow is axial and that radial transport is confined to the horizontal boundary layers.