Let υ∈ be a sequence of DiPema-Majda approximate solutions to the 2-d incompressible Euler equations. We prove that if the vorticity sequence is weakly compact in the Hardy space H1 (R2) then a subsequence of υ∈ converges strongly in the energy norm to a solution of the Euler equations.

We consider the fluid motion induced when a circular cylinder performs small-amplitude oscillations about an axis parallel to a generator to which it is rigidly attached as in Fig l(a). In common with other fluid flows dominated by oscillatory motion, a time-independent, or steady streaming develops, and this is the focus of our attention. In particular we relate our results, qualitatively, to the observations that have been made in experiments.

An oncoming two-dimensional laminar boundary layer that develops an unstable inflection point and becomes three-dimensional is described by the Hall-Smith (1991) vortex/wave interaction equations. These equations are now examined in the neighbourhood of the position where the critical surface starts to form. A consistent structure is established in which an inviscid core flow is matched to a viscous buffer-layer solution where the appropriate jump condition on the transverse shear stress is satisfied. The final result is a bifurcation equation for the (constant) amplitude of the wave pressure. A representative classical velocity profile is considered to illustrate solutions of this equation for a range of values of the wave-numbers.

The flow induced by an oscillating circular cylinder which may perform transverse, torsional and axial vibrations is considered. The steady streaming associated with purely transverse vibrations of the cylinder may be significantly modified by the presence of, and interaction with, torsional oscillations. Similarly the interaction between the transverse and axial vibrations introduces a modification to the axial flow, which results in a steady streaming motion in the axial direction.

Almost seventy years ago Jeffery [1] showed that a finite velocity can result at infinity when the biharmonic equation is solved for the titled problem. Here, we extend his calculations to show that finite vorticity is the more general conclusion, and then indicate a resolution of the apparent paradox.

Görtler vortices are thought to be the cause of transition in many fluid flows of practical importance. In this paper a review of the different stages of vortex growth is given. In the linear regime nonparallel effects completely govern this growth and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Gortler vortices are given; in particular vortices induced by free-stream perturbations impinging on the leading edge of the wall are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of O(l) wavelengths are given and compared with experimental observations. The agreement between theory and experiment is shown to be excellent up to the point where unsteady effects become important. For small wavelength vortices the nonlinear regime is of particular interest since a strongly nonlinear theory can be developed there. Here the vortices can be large enough to drive the mean state which then adjusts itself to make all modes neutral. The breakdown of this nonlinear state into a three-dimensional time dependent flow is also discussed.

The nonlinear interactions that evolve between a planar or nearly planar Tollmien-Schlichting (TS) wave and the associated longitudinal vortices are considered theoretically, for a boundary layer at high Reynolds numbers. The vortex flow is either induced by the TS nonlinear forcing or is input upstream, and similarly for the nonlinear wave development. Three major kinds of nonlinear spatial evolution, Types I-III, are found. Each can start from secondary instability and then becomes nonlinear, Type I proving to be relatively benign but able to act as a pre-cursor to the Types II, III which turn out to be very powerful nonlinear interactions. Type II involves faster streamwise dependence and leads to a finite-distance blow-up in the amplitudes, which then triggers the full nonlinear three-dimensional triple-deck response, thus entirely altering the mean-flow profile locally. In contrast, Type III involves slower streamwise dependence but a faster spanwise response, with a small TS amplitude thereby causing an enhanced vortex effect which, again, is substantial enough to entirely alter the mean-flow profile, on a more global scale. Concentrated spanwise formations in which there is localized focussing of streamwise vorticity and/or wave amplitude can appear, and certain of the nonlinear features also suggest by-pass processes for transition and significant change in the flow structure downstream. The powerful nonlinear 3D interactions II, III seem potentially very relevant to experimental and computational findings in fully fledged transition; in particular, it is suggested in an appendix that the Type-Ill interaction can terminate in a form of 3D boundary-layer separation which appears possibly connected with the formation of lambda vortices in practice.

This theoretical work shows formally that unsteady interactive boundary layers can break up within a finite time by encountering a nonlinear localized singularity. The theory is an extension of, and is guided to a large extent by, Brotherton-Ratcliffe and Smith's (1987) work on a special case. Two major types of singularity are proposed, a “moderate” type yielding a singular pressure gradient and a “severe” type associated with a pressure discontinuity. Each type produces a singular response in the skin friction in the case of wall-bounded flows. The present finite-time singularity applies to any unsteady interactive flow, e.g., incompressible or compressible boundary layers, internal flows, wakes, in two or three dimensions; and the singularity and its associated change in flow structure have numerous repercussions, which are discussed, physically and theoretically, concerning boundary-layer transition in particular.

When a weak rotlet and a circular cylinder rotate together in a viscous fluid at low Reynolds number R, the Stokes' flow solution indicates a uniform stream as the radial distance r tends to infinity. It is shown here, when R is distinctly non-zero, that the flow is modified to form a spiral motion in the domain where R In r = O(l), but is not damped until the more distant domain R2 In r = O(l).

It is shown first that internal or external boundary-layer flow over obstacles or other surface distortions is susceptible to a novel kind of viscous-inviscid instability, involving growth rates much larger than those of traditional Tollmien-Schlichting and Gortler modes for instance. The same instabilities arise in liquid-layer flow at sub-critical Froude number, and they are associated with an interacting boundary-layer problem where the normalized pressure is equal to the normalized displacement decrement. Second, certain limiting linear and nonlinear disturbances are studied to shed more light on the overall instability process and each form of disturbance leads to a finite-time collapse, although different in each case. Thirdly, and in consequence, the work finds the significant feature that the whole interacting boundary layer can break down nonlinearly within a finite scaled time.

A study is made of Stokes flows in which a line rotlet or stokeslet is in the presence of a circular cylinder in a viscous fluid. In contrast to the Stokes Paradox for flow past an isolated cylinder, it is shown that if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity. A similar phenomenon can occur when two equal cylinders rotate with equal and opposite angular velocities, and the flow pattern is then such that there is a closed streamline enclosing both cylinders.

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 ( = 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

We consider the Falkner–Skan equation

in the special case β = – 1. It is known that this equation may be integrated twice to get the Riccati equation

Two spheres of different radii are approaching each other with equal and opposite velocities, the fluid flow around them being at low Reynolds number. The forces on the spheres can be calculated when they are very close by applying an asymptotic analysis — usually called lubrication theory — to the flow in the gap between the spheres. If the non-dimensional gap width is ε, the force is calculated here correct to O(ε In ε) for all ratios of the two spheres' radii. The analysis can be combined with earlier numerical calculations to find all the constants in the asymptotic expansion correct to O(ε).

The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross-section is discussed for large Reynolds numbers (R). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR, where h is a characteristic cross-sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O(h) and O(hR). When the aspect ratio is increased to O(R1/7), however, these two characteristic length scales both become O(R1/7h) and a match with plane channel flow instability is achieved. Thus the general cross-section produces temporal and spatial instability if the aspect ratio is O(R1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non-circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross-sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O(hR) scale modes in neutral stability for finite aspect ratios.

A higher-order, double boundary-layer theory is employed to investigate the mass transport velocity due to two-dimensional standing waves in a system consisting of two semi-infinite, homogeneous fluids of different densities and viscosities. For moderately large wave amplitudes, the leading correction to the tangential mass transport velocity near the interface is extremely significant and may typically contribute about 20% of the total velocity.

A study complementary to Part 1 (Smith 1979) is made of the linear stability characteristics, at high Reynolds number (R), of Poiseuille How through tubes with closed cross-sections. The first significant deviation of the upper branch of the neutral stability curve (Part 1 having described the lower bilanch) from that of plane Poiseuille flow arises when the aspect ratio is decreased from infinity to O(R1/11). The axial wavenumber α on the upper branch is then O(R-1/11). A further decrease of the aspect ratio, to a finite value, forces this α to fall sharply to O(R-1). A similar phenomenon occurs for the lower branch (Part 1). Thus the two branches are likely to meet only when the aspect ratio becomes finite, with the neutrally stable disturbances then having very large axial length scales.

The effects on a boundary layer of thickness O(LR−1/2) (where L is a typical streamwise lengthscale, and R is the Reynolds number) of a small unsteady hump at the wall is considered. The hump is of height O(LR−5/8) and length O(LR−3/8), and outside the boundary layer is potential flow. Three different regimes of unsteadiness parameter are considered, leading to a description of the flow over the complete spectrum for this size of excrescence.

In finding the resistance to the motion of a closely fitting plug along a tube filled with fluid, allowance must be made for the leakage of fluid through the gap between the plug and the tube. If there is no leakage, theoresistance is theoretically infinite. For a plug of length d in a tube of radius a and with a gap b between the plug and tube, the dominant term in the resistance comes from the flow in the gap, and is proportional to da/b. If, however, the plug is very short, so that it may be considered as a disk, the calculation of the flow near the disk shows that the resistance is now proportional to aln (a/b) and the presence of the disk increases the force on the tube by an amount equivalent to an increase in length of the tube by only a few radii.

The two inter-related aspects of this laminar flow study are, first, the effects of indentations of length O(a) and height O(aK-⅓) on an otherwise fully developed pipeflow and, second, the manner in which such a pipeflow adjusts ahead of any nonsymmetric distortion to the downstream conditions. Here K is the typical Reynolds number, assumed large, and a is the pipewidth. The flow structure produced by the particular slowly varying indentation, or by a suitable distribution of injection, comprises an inviscid core, effectively undisplaced, and a viscous wall-layer, where the swirl velocity attains values much greater than in the core and where the nonlinear governing equations involve the unknown pressure force. Linearized solutions for finite-length, unbounded or point indentations, and for finite blowing sections (which model the influence of a tube-branching), demonstrate the upstream influence inherent in the nonlinear problem, for steady or unsteady disturbances. It is suggested that the upstream interaction caused there provides the means for the upstream response in the general case where the indentation, say, produces a finite constriction of the tubewidth.