The forced motion of a liquid heated non-uniformly from below and subject to a non-constant basic temperature profile is examined. The problem is formulated as a two-point boundary-value problem which is solved by a numerical method. In agreement with previous work, the presence of a positive vertical lapse rate of basic temperature tends to decrease the forced motion. It is found that constant heat generation within the fluid produces a parabolic basic temperature profile, and this tends to increase the forced velocity components.

Experiments are described which show the steady streaming of fluid under a flexible membrane, located near the bottom of a wave basin in shallow water. Circulation of fluid in a closed circuit was thus obtained, and measurements were made of the pressure head and total flow. The phenomenon might be used to extract power from waves at very low efficiency, but the theoretical basis is not yet understood.

The problem of the resonant response of a gas contained in a two-dimensional rectangular cavity to periodic (sinusoidal) velocity excitations at the walls of the cavity is investigated. It is found that in some neighbourhood of each resonant frequency there are discontinuous pressure disturbances (shock waves). The present theory is an extension of Chester's theory on resonances in closed tubes.

A model of Wilson Cloud formation following a low-altitude nuclear detonation is developed. It is shown that, for detonation yields between 10−3 kt and 100 kt, simple scaling laws characterize the evolution and physical properties of the Wilson Cloud.

Wave disturbances to baroclinic flows produce cyclones in the atmosphere and eddies in the oceans and have been extensively studied in laboratory experiments with differentially heated annuli of rotating fluid. Related analytical studies have concentrated mainly on the development of slowly growing waves on laterally uniform zonal flows. Neutral inviscid waves on such flows do not advect their own potential vorticity field whereas neutral waves on most laterally sheared baroclinic flows do. Scaling arguments suggest that on these laterally sheared flows the harmonics generated by the neutral waves play the dominant role in arresting the initial growth of weakly unstable waves. The arrest of a wave is chiefly accomplished by fully nonlinear advection within a critical layer centred on the wave's steering level whose depth is proportional to the wave's amplitude. Explicit numerical solutions illustrating these points are presented for a case in which the critical level is non-singular and the inviscid calculations comparatively straightforward. The stability of the solutions and the effects of diffusive fluxes on them are discussed. Potential vorticity diagnostics for a numerical simulation of a wave flow in a rotating annulus near the axisymmetric transition show that distortion of the wave's potential vorticity field is mainly confined to the vicinity of the steering level. Assumptions and approximations made in the explicit calculations which are of doubtful validity for this flow are highlighted.

If an unpolluted tributary joins a river less than about thirty channel breadths downstream of an effluent outlet, then the additional dilution can reduce the peak pollution level experienced at the shoreline. This paper identifies the optimal sites for steady discharges to take full advantage of the extra dilution.

The two-dimensional wave pattern produced in a homogeneous rotating fluid by a forcing effect oscillating with a frequency σ′0 and travelling with a uniform speed U along a line inclined to the axis of rotation at an arbitrary angle α is studied following Lighthill's technique. It is shown how the far field changes with α and σ′0.

For all σ′0 < 2Ω, except for σ′0 = 2Ω sin α (Ω being the angular velocity of the fluid), the forcing effect excites two systems of waves. When σ′0 → 2Ω sin α one of these systems spreads out, influencing the upstream side while the other shrinks in the downstream direction. This upstream influence is to the left or to the right of the line of motion of the forcing effect (the forcing line) according as σ′0 − 2Ω sin α[lg] 0 and increases as σ′0 − 2Ω sin α decreases. For σ′0 > 2Ω there is only a single system propagating downstream. As α varies these systems undergo a kind of rotation retaining the main features. α ≠ 0 or ½π makes the pattern asymmetric about the forcing line while a non-zero σ′0 splits the steady-case identical wave systems into two, which are otherwise coincident.

When σ′0 = 2Ω sin α the forcing effect excites straight unattenuated waves of fixed frequency travelling both ahead and behind in a ‘column’ parallel to the forcing line and enclosing it. Also there are two other systems, which propagate without penetrating into an upstream wedge. It is shown that this ‘column’ is the counterpart of the ‘Taylor column’.

Two impinging two-dimensional incompressible inviscid fluid jets of known widths and velocities produce two outgoing jets. The speeds of the outgoing jets are readily determined from the Bernoulli equation. Their two widths and two directions (four quantities) are related by conservation of mass and conservation of two components of momentum (three relations). Because these three conservation relations do not suffice to determine the four unknowns, Milne-Thomson (1968) states on p. 302 that ‘a unique solution is, in general, not possible’. He incorrectly attributes this indeterminateness to disregard of ‘the initial conditions from which this steady motion is supposed to arise’.

Most suspensions exhibit a rheological behaviour which cannot be represented by either Bingham's or Ostwald–De Waele's law. In studying such cases a very simple expression with only three parameters may be used. Starting with an intermediate law of this sort, this paper gives velocity profiles and head losses in laminar flow, which have been computed and plotted on diagrams in non-dimensional co-ordinates.

It has been found that transition flow rates in circular tubes for data taken from the literature and from experiments conducted on drilling muds at the Institut Français du Pétrole, are efficiently predicted by an empirical criterion (Ryan & Johnson 1959) which establishes a relation between a generalized Reynolds number and a generalized Hedström number.

The evolution of unsteady boundary layers on the plane of symmetry of a slender prolate spheroid in uniform motion at constant angle of attack after an impulsive start has been studied for the case of a prescribed pressure distribution. Calculated results have been obtained for angles of attack ranging from 30° to 50° and show, for example, that the unsteady-state solutions approach the steady-state solutions rapidly on the windward and leeward sides for α < αc (≈ 41°). This is also so on the windward side for α > αc. On the leeward side for α > αc, however, the unsteady boundary layer is initially unseparated, but develops a region of reversed flow with increasing time. Subsequently, the streamwise displacement thickness develops a pronounced peak, which leads to a singularity of the type observed by van Dommelen & Shen on a circular cylinder started impulsively from rest.

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.

A potential function/stream function formulation is introduced for the solution of the fully 3-D inverse potential ‘target pressure’ problem. In the companion paper (Part 1) it is seen that the general 3-D inverse problem is ill-posed but accepts as a particular solution elementary streamtubes with orthogonal cross-section. Under this simplification, a novel set of flow equations was derived and discussed. The purpose of the present paper is to present the computational techniques used for the numerical integration of the flow and geometry equations proposed in Part 1. The governing flow equations are discretized with centred finite difference schemes on a staggered grid and solved in their linearized form using the preconditioned GMRES algorithm. The geometry equations which form a set of first-order o.d.e.s are integrated numerically using a second-order-accurate space marching scheme. The resulting computational algorithm is applied to a double turning duct and a 3-D converging-diverging nozzle ‘reproduction’ test case.