This paper considers steady, slowly varying water waves propagating over a gently sloping bed on a steady current. The current varies linearly with depth, and so has constant vorticity ω. The analysis is two-dimensional and dissipation is neglected. Definitions, and expressions correct to second order in the amplitude, are given for the radiation stress, wave energy density E and total energy flux. An average La-grangian [Lscr ], obtained by heuristic arguments from Clebsch potentials, leads to the result that for this particular problem E equals the wave action [Lscr ]ω times the angular frequency ωrm relative to a frame of reference moving with the average-over-depth current velocity Um. This determines the variation of the amplitude with distance explicitly. An analytical expression for the height of the mean water surface is found by a heuristic argument which compares the conservation equations for total energy and wave action. All the results have been checked directly by substitution back into the basic equations. Graphs illustrate the effect of the vorticity ω on the wavelength, amplitude and set-down.

This paper extends two earlier papers in which DiPrima & Stuart calculated first (1972b) the critical Taylor number to order ε2, where the eccentricity ε is proportional to the displacement of the axes of the circular cylinders, and second (1975) the torque and load to order ε associated with nonlinear effects of Taylor vortices. In the latter paper, it was shown that to order ε the torque arising from the Taylor vortices is identical with that for the concentric problem, which was first calculated, by a perturbation method, by Davey (1962). This deficiency is remedied in the present paper, where the calculation is taken to order ε2. It is found that, as ε rises, the torque associated with the Taylor vortices falls slightly when we keep constant the percentage elevation of the Taylor number above the ε-dependent critical value. This result is in accordance with experimental observations by Vohr (1967, 1968). In addition, results of calculations of the pressure field developed by the Taylor-vortex flow in association with the eccentric geometry are presented; this is larger than in the concentric case owing to a Reynolds lubrication effect. Also given are the associated components of the load on the inner cylinder, but only for Taylor numbers close to the critical value.

One additional observation by Vohr, for cylinders with a mean ratio of the gap to the inner radius of 0·099, was that the maximum Taylor-vortex strength with ε = 0·475 occurred some 50° downstream of the maximum gap for a 20% elevation of the Taylor number above the critical value. Calculations in the two earlier papers (1972b, 1975) gave 90 and 76°, respectively, for that angle. Note that in the 1975 paper a geometrical correction of order ε was included. Here, with an additional modification of order ε due to the flow, this result is improved to 49° by the extended analysis presented, although the ‘small’ parameters are somewhat outside the range for which perturbation theory is expected to be valid.

Details are given of an implicit six-point finite-difference scheme for solving two-temperature, laminar, boundary-layer flows not in chemical equilibrium in ionizing argon. The analysis extends previous work by considering the radiation-energy loss and the chemical reactions due to atom-atom and electron-atom collisions in the ionizing boundary-layer and free-stream flow. Also included are variations in transport properties based on known elastic-scattering cross-sections, effects of chemical reactions, radiation-energy loss and the electric-sheath wall boundary conditions. The results are compared with dual-wavelength interferometric boundary-layer data obtained by using a Mach-Zehnder interferometer 23 cm in diameter with the UTIAS 10 × 18 cm Hypervelocity Shock Tube for shocks of initial Mach numbers Ms ∼ 13 and 16 moving into argon at a pressure p0 ∼ 5 torr and temperature T0 ∼ 297 °K. Considering the difficulties involved in solving such complex plasma flows, satisfactory agreement was obtained between analytic and experimental total-density profiles and electron-number-density profiles for the case Ms ∼ 16 and good agreement for Ms ∼ 13.

The steady flow of a suspension of closely fitting, neutrally buoyant, incompressible and elastic spheres through a circular cylindrical tube is investigated under the assumption that lubrication theory is valid in the fluid region. A series solution giving the displacement field of an elastic incompressible sphere under axisymmetrically distributed surface tractions is developed. It is found that, for closely fitting particles, flow properties of the suspension are strongly dependent on the shear modulus of the elastic material and the velocity of the particle.

A model of the diffusive interface in double-diffusive convection at high Rayleigh number is proposed. The interface is assumed to have a double structure: two marginally stable boundary layers from which blobs or thermals arise on the outer edges of the interface, separated by a diffusive core across which all transport takes place by molecular diffusion. The model is time-independent and comparison is made with unsteady ‘run-down’ experiments on the assumption that the experiments run down through a sequence of equilibrium states each of which can be considered separately. The model predicts a constant ratio of the buoyancy fluxes of the two components at a value equal to the square root of the ratio of their molecular diffusivities, and individual fluxes in reasonable agreement with the available experimental data. Some time-dependent features of the model are also examined.

Laser-Doppler velocity measurements in water are reported for the flow in the plane of symmetry of a turbulent spot. The unsteady mean flow, defined as an ensemble average, is fitted to a conical growth law by using data at three streamwise stations to determine the virtual origin in x and t. The two-dimensional unsteady stream function is expressed as $\psi = U^2_{\infty}tg(\xi, \eta) $ in conical similarity co-ordinates ζ = x/U∞t and η = y/U∞t. In these co-ordinates, the equations for the unsteady particle displacements reduce to an autonomous system. This system is integrated graphically to obtain particle trajectories in invariant form. Strong entrainment is found to occur along the outer part of the rear interface and also in front of the spot near the wall. The outer part of the forward interface is passive. In terms of particle trajectories in conical co-ordinates, the main vortex in the spot appears as a stable focus with celerity 0·77U∞. A second stable focus with celerity 0·64U∞ also appears near the wall at the rear of the spot.

Some results obtained by flow visualization with a dense, nearly opaque suspension of aluminium flakes are also reported. Photographs of the sublayer flow viewed through a glass wall show the expected longitudinal streaks. These are tentatively interpreted as longitudinal vortices caused by an instability of Taylor-Görtler type in the sublayer.

The three-soliton solution of the two-dimensional Korteweg-de Vries equation is analysed to show that the structure of the interaction can be represented in terms of the motion of two-soliton resonant interactions (resonant triads) as described by Miles (1977). The schematic development of the interaction with time is obtained and shown to approximate closely to computer calculations of the analytic solution. Similar results follow for interactions of more solitons and other equations.

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

Three-dimensional cellular convection concentrates magnetic flux into ropes when the magnetic Reynolds number is large. Amplification of the magnetic field is limited by the Lorentz force and the maximum field in a flux rope can be estimated. Boundary-layer analysis yields a completely self-consistent solution for a model of convection driven by imposed horizontal temperature gradients, and the transition from a kinematic to a dynamic regime can be followed in detail. The maximum value of the amplified field is proportional to the square root of the ratio of the viscous to the magnetic diffusivity.

Two-dimensional time-periodic water waves on a gently sloping bottom are investigated under the classical long-wave assumptions that ε = h′0/λ′ and δ = H′/h′0 are small parameters (H′ being the wave height, h′0 a characteristic water depth and δ′ the horizontal scale for the oscillatory motion) and the assumption that δ/ε2 = O(1) as δ and ε tend to zero.

It is shown that for a bottom slope hx for which hx = o(ε3) the governing KdV equation with slowly varying coefficients (derived by Johnson 1973a) has a time-periodic solution which in the first approximation is a slowly varying cnoidal wave. The second-order approximation in an asymptotic expansion with respect to the bottom slope represents the deformation of this wave due to the sloping bottom. For hx [Gt ] ε5 this deformation is larger than the second-order contribution from the basic expansion with respect to wave amplitude which underlies the KdV equation itself; the calculation is then a consistent approximation to physical reality.

Numerical results for the deformation are given. Also, the wave profiles are compared with experiments on a plane of slope $h_x = \frac{1}{35}$ and show good agreement even for the large values of H’/h’ appropriate to waves rather close to breaking.

A theoretical study is made of the motion of a small sphere in an incompressible viscous fluid bounded by two plane walls. The analysis is based on the Stokes equations of motion and the method of reflexions is employed to obtain the lowest-order wall effect on the sphere, which is allowed either to rotate or not as it translates in an arbitrary direction. The effect of two plane walls on the force and torque (or angular velocity) experienced by the sphere is not accurately estimated by superposing the contributions from the two individual walls.

A small drop of liquid 1 falls through a less dense liquid 2 and approaches the horizontal interface between liquid 2 and an underlying layer of liquid 1. After a short time the drop will be brought to rest (or nearly) in a hollow in the interface. Before the drop can coalesce with its bulk phase, the thin film of liquid 2 trapped between them must be squeezed out, and become sufficiently thin that rupture can occur. This is the film drainage problem. Early calculations, based on simple lubrication theory, fail to take proper account of two effects which are investigated here and shown to be decisive. They are the circulation induced in the drop and in the lower bulk fluid, which tends to speed up drainage, and the constriction in the film thickness at its periphery, which tends to slow it down. This constriction has been observed and some existing theories have attempted to model it in an ad hoc manner. We give here a physical explanation and calculate the minimum thickness explicitly. The effect of circulation in the adjacent fluids is also calculated.

Convection in a porous cavity driven by heating in the horizontal is analysed by a number of different techniques which yield a fairly complete description of the two-dimensional solutions. The solutions are governed by two dimensionless parameters: the Darcy-Rayleigh number R and the cavity aspect ratio A. We first find solutions valid for shallow cavities, A → 0, by using matched asymptotic expansions. These solutions are given up to O(A6R4). For A fixed, we find regular expansions in R by semi-numerical techniques, up to O(R30) in some cases. Series-improvement techniques then enable us to cover the range 0 ≤ R ≤ ∞. A limited result regarding bifurcations is noted. Finally, for R → ∞ with A fixed, we propose a self-consistent boundary-layer theory which extends previous approximate work. The results obtained by these different methods of solution are in good agreement with each other and with experiments.

Euromech 98 was held in Eindhoven inNovember 1977 with the authors as chairmen. The Colloquium was attended by 48 participants from a number of European countries. Bubbles play an important role in many areas of technology: propeller-induced cavitation in ship building, cavitation in fluid machinery, nucleate boiling in reactorsand similar devices, and many processes (centrifuges, mixers) in the chemical process industry. A wide variety of research, both of fundamental and of more applied nature, is going on in universities and in industrial laboratories. Representatives from all theabove areas were included among the participants and consequently the topics raised in the presentations and discussions covered a very broad field.

The conservation law for the flux of axial momentum in a turbulent jet is examined. The examination discloses that for a plane jet out of a wall the momentum flux is reduced appreciably because the induced flow towards the jet has a component in the direction opposite to the main jet flow and because of the pressure field generated in the ambient fluid. Existing experimental results confirm this conclusion.

Unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point for both cold and hot walls has been studied for the case when the velocity of the incident stream varies arbitrarily with time. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) an accelerating stream and (ii) a fluctuating stream. The results indicate that the variation of the density-viscosity product across the boundary layer, the wall temperature and the nature of stagnation point significantly affect the skin friction and heat transfer.

The supersonic flow field past a circular cone of semi-apex angle 7·5° at incidences up to 26° has been investigated experimentally at a free-stream Mach number of 2·94. The experiments were made using a five-hole conical-head directional probe. Since the flow past the cone was found to be conical, the flow phenomena may be described in a plane perpendicular to the cone axis. This paper gives the direction of the conical streamlines, the conical sonic line and the conical Mach number in such a plane, all of which may be deduced from the measurements. At incidences exceeding twice the cone semi-apex angle a separated flow regime was observed which consisted of two main vortices on either side of the leeward plane of symmetry, and probably also two secondary vortices close to the cone surface. From incidences of 17° onwards, an embedded region of conical supersonic flow terminated by a shock wave was revealed. At 22°, approximately, a second embedded shock wave was measured close to the cone surface and extended slightly across the leeward plane of symmetry. This shock wave may have been generated by the vortex system, which induced supersonic cross-flow components towards the cone.

An analysis of the stability of thin viscous liquid sheets, such as those emitted from industrial spraying nozzles, is presented. These sheets are in the form of a circular sector whose thickness reduces as the distance from the nozzle increases.

The Kelvin-Helmholtz type of instability usually observed causes the breakup and atomization of the sheet, as required in most industrial spraying processes. Waviness, like that of a flapping flag, is produced and increasing amplitudes finally cause breakup.

An analytical solution in the form of hypergeometric functions for the shape of the sheet and the waves is obtained. This solution includes, as special cases, analyses existing in the literature, in addition to establishing the possibility of a new type of instability dependent on the distance from the nozzle. Also, the classical type of instability, in which the waves increase with time, is examined and relations for unstable waves as a function of parameters such as the fluid viscosity, surface tension and sheet velocity are obtained. It is shown that there is no single wave that has a maximum growth rate, but that the wavenumber for maximum instability increases with the distance from the nozzle orifice.

The onset of convection in shear flow driven by lateral heating and also uniformly heated from below is investigated numerically by Galerkin's method. Stress-free as well as rigid, perfectly conducting boundaries are considered. The analysis is valid for small and moderate Prandtl numbers. The magnitude of the lateral basic temperature gradient may be expressed by a dimensionless Grashof number G, while the uniform heating from below is represented by a Rayleigh number Ra. Depending on the values of G, Ra and the Prandtl number Pr, a variety of interesting situations arise. In particular it is demonstrated that the form of the most unstable mode, i.e. whether it is a roll with axis aligned along the basic flow (a longitudinal roll) or one with axis normal to the basic flow (a transverse roll), depends on the value of the Prandtl number. For small values of G, the marginally stable disturbances are found to be steady, while for larger values of G, oscillatory instability occurs. For all values of G considered here (G [lsim ] 3000), computations of the energy balance for the marginally stable disturbances show that the main instability mechanism is of thermal origin, while the effect of shear may be important in selecting the preferred mode of disturbance.