We consider weakly nonlinear wave motions in a thermally stratified boundary layer. Attention is focused on the upper branch of the neutral stability curve, corresponding to small wavelengths and large Reynolds number. In this limit the motion is governed by a first harmonic/mean flow interaction theory in which the wave-induced mean flow is of the same order of magnitude as the wave component of the flow. We show that the flow is governed by a system of three coupled partial differential equations which admit finite-amplitude periodic solutions bifurcating from the linear, neutral points.

Taylor blast wave (TBW) theory and geometrical shock dynamics (GSD) theory describe a radially expanding shock wave front through an inert material, typically an ideal gas, in the strong blast wave limit and weak acoustic limit respectively. We simulate a radially expanding blast shock in air using a hydrodynamic simulation code and numerically describe the intermediate region between these two limits. We test our description of the intermediate shock phase through a two-dimensional simulation of the Bryson and Gross experiment. We then apply the principles of GSD to materials that follow the Mie–Gruneisen equation of state, such as plastics and metals, and derive an equation that accurately relates the acceleration, velocity and curvature of the shock through these materials. Along with detonation shock dynamics (DSD), which describes detonation shock propagation through high explosive fluids, we develop a hybrid DSD/GSD model for the simulation of heterogeneous explosives. This model enables computationally efficient simulation of the shock front in high explosive/inert mixtures consisting of simple or complex geometric configurations. We simulate an infinite two-dimensional slab consisting of one half explosive, PBXN-9, and one half aluminium and model the boundary angle conditions using shock polar analysis. We also simulate a series of high explosive unit cells embedded with aluminium spherical particles, and we compare the propagation of the detonation shock front with a direct numerical simulation performed with the ALE3D code.

The coupled dynamics of two conjugated liquid layers of disparate thicknesses, which coat a solid substrate and are subjected to a transverse temperature gradient, is investigated. The upper liquid layer evolves under the short-wavelength Bénard–Marangoni instability, whereas the lower, much thinner film undergoes a shear-driven long-wavelength deformation. Although the lubricating film should reduce the viscous stresses acting on the up to one hundred times thicker upper layer by only 10 %, it is found that the critical Marangoni number of marginal stability may be as low as if a stress-free boundary condition were applied at the bottom of the upper layer, i.e. much lower than the classical value of 79.6 known for a single film. Furthermore, it is experimentally verified that the deformation of the liquid–liquid interface, albeit small, has a non-negligible effect on the temperature distribution along the liquid–gas interface of the upper layer. This stabilizes the hexagonal pattern symmetry towards external disturbances and indicates a two-way coupling of the different layers. The experiments also demonstrate how convection patterns formed in a liquid film can be used to pattern a second conjugated film. The experimental findings are verified by a numerical model of the coupled layers.

In this paper we investigate the effect of vertical rotation on the linear stability of an unbounded region of vertically stratified fluid, which has compensating horizontal temperature and salinity gradients, so there is no overall horizontal density gradient. We find the most unstable perturbations for given linear vertical and horizontal gradients and show how the addition of rotation affects the results found for the non-rotating case by Holyer (1983), using molecular diffusivities. By using a transformation that, for the non-oscillatory instability, links the rotating case to the non-rotating case we show that the growth rate, the across-front slope and the wavenumber of the intrusion with the maximum growth rate is unchanged. The basic difference between the non-rotating and the rotating case, for the non-oscillatory instability, is that in the rotating case the interleaving layers slope both along and perpendicular to the direction of the horizontal temperature and salinity gradients and not just along them. The oscillatory instability has no simple transformation between the rotating and the non-rotating cases, and the addition of rotation changes the growth rate and the wavenumber of the instabilities.

In this paper the slip flow of viscous fluid at low Reynolds numbers past a flat plate aligned with the flow is studied theoretically on the basis of Oseen-Stokes equations of motion. An integral equation for the distribution of fundamental singularities representing the plate is derived and solved approximately in the vicinity of the edge and main portion of the plate. A formula for the local skin friction is obtained and discussed numerically. It is also shown that the slippage of the flow gives rise to reduction of the drag force on the plate by an amount O(K|ln K|), where K is the Knudsen number. The velocity change near the edge of the plate is of particular interest and is found to be logarithmically singular there.

A quantitative two-dimensional theoretical model is developed to describe the movement of water and salt along the long narrow extracellular channels which appear to be a common structural feature of all epithelial membranes. This study examines the transport behaviour of both open and closed membrane systems as a function of the geometric specialization of the channel and the active transport site location under the influence of three driving forces: trans-membrane osmotic and hydrodynamic pressure differentials and active transport. The previous one-dimensional hydrodynamic model of Diamond & Bossert (1967) and Segel(1970) was confined to closed channel systems such as the gall bladder in which the only mechanism for water movement is local osmosis due to active transport.

Approximate analytical solutions are presented for long constant-area open channels in which the active transport sites have been idealized as point solute sources. A streamwise co-ordinate straining technique has been used in these solutions to describe the nonlinear effects of convection over long distances. Closed-form solutions are also presented for the pressure and solute concentration distributions within simplified models of channel constrictions with varying degrees of occlusion.

Numerical results of the model have been compared with Cole's (1961, 1962) in vivo and in vitro experiments on the rabbit ciliary body. Satisfying agreement with the measured values of the solute and water fluxes has been obtained for both the living eye and the excised ciliary body. These results strongly suggest that the formation of aqueous humour in the rabbit is a pressure-dependent process in which local osmosis due to active transport accounts for only one-third of the total aqueous flow. The model has also been applied to the gall bladder epithelium using more general boundary conditions than allowed for in the model of Diamond & Bossert. New solutions yielding a vanishing diffusional flux at the channel exit were obtained. However, the model, like that of Diamond & Bossert, does not provide a rational explanation as to how the water in the cell interior is replenished.

We consider the stability of a long free film of liquid composed of two immiscible layers of differing viscosities, where each layer experiences a van der Waals force between its interfaces. We analyse the different ways in which the system can exhibit interfacial instability when the liquid layers are sufficiently thin. For an excess of surfactant on one gas–liquid interface, the coupling between the layers is relatively weak and the instability is manifested as temporally separated rupture events in each layer. Conversely, in the absence of surfactant, the coupling between the layers is much stronger and the instability is manifested as rupture of both layers simultaneously. These features are consistent with recent experimental observations.

The heat transfer across an inclined air layer in the longitudinal roll regime often falls below expected values. A companion paper (Ruth et al. 1980) presents the heat transfer measurement and flow visualization observations; the present paper discusses the mechanisms which are thought to be responsible for this anomalous behaviour and formulates a simple model which correlates the results of our previous study and that of Hart (1971). The observed suppression of heat transfer below values expected for pure longitudinal roll motion is explained in terms of a shear instability. A stabilizing mechansim, which would restore the longitudinal rolls at higher Rayleigh numbers, is described; the return of the heat transfer to expected levels at higher Rayleigh numbers is ascribed to this mechanism. Comparisons are made between the results of the present model and the analyses of Clever & Busse (1977).

Fluid particles can have a mean motion in time, even when the Eulerian mean flow disappears everywhere in space. In the present article, we demonstrate that this phenomenon, known as the Stokes drift, plays an essential role in the problem of magnetic field generation by fluctuation flows (kinematic dynamo) at high Rm. At leading order, the dynamo is generated by the Stokes drift that acts as if it were a mean flow. This result is derived from a mean-field dynamo theory in terms of time averages, which reveals how classical expressions for alpha and beta tensors actually recombine into a single Stokes drift contribution. In a test case, we find fluctuation flows that have a G. O. Roberts flow as Stokes drift and this allows to confront our model to direct integration of the induction equation. We find an excellent quantitative agreement between the prediction of our model and the results of our simulations. We finally apply our Stokes drift model to prove that a broad class of inertial waves in rapidly rotating flows cannot drive a dynamo.

Algebraic topology (homology) is used to characterize quantitatively non-Oberbeck–Boussinesq (NOB) effects in chaotic Rayleigh–Bénard convection patterns from laboratory experiments. For fixed parameter values, homology analysis yields a set of Betti numbers that can be assigned to hot upflow and, separately, to cold downflow in a convection pattern. An analysis of data acquired under a range of experimental conditions where NOB effects are systematically varied indicates that the difference between time-averaged Betti numbers for hot and cold flows can be used as an order parameter to measure the strength of NOB-induced pattern asymmetries. This homology-based measure not only reveals NOB effects that Fourier methods and measurements of pattern curvature fail to detect, but also permits distinguishing pattern changes caused by modified lateral boundary conditions from NOB pattern changes. These results suggest a new approach to characterizing data from either experiments or simulations where NOB effects are expected to play an important role.

This paper describes a theory for a piston-operated compressor in which shock waves are used to heat the gas being compressed. Detailed calculations are given showing the use of such a compressor to heat air for use in a hypersonic wind tunnel. The effect of real gas properties on the compression process is included in this discussion.

This paper concerns the effects of topographic changes upon the dispersion of contaminants in rivers. Exact analytical results are derived for the time-of-arrival, temporal variance and skewness far downstream of a sudden contaminant release. It is demonstrated that there can exist optimal discharge sites, such that at all positions far downstream the maximum concentrations are minimized.

Numerous experimental studies have documented that injecting low-salinity water into an oil reservoir can increase the amount of oil recovered. However, owing to the complexity of the chemical interactions involved in this process, there has been much debate over the dominant mechanism causing this effect. In order to further understand one proposed mechanism, multicomponent ionic exchange, we study the motion of an oil slug through a clay pore throat filled with saline water. The pore throat is modelled as a capillary tube connecting two bulk regions of water. We assume that the surfaces of the oil and the capillary are negatively charged and that, due to repulsion between these surfaces, the oil slug is separated from the capillary surface by a thin film of water. Ion interactions at the oil–water and clay–water interfaces are modelled using the law of mass action. By using lubrication theory to describe the thin-film flow in the water layer separating the oil from the clay surface, and the macroscopic flow through the capillary, we derive expressions for the thickness of the wetting film, and the velocity of the oil slug, given a pressure difference across the ends of the capillary. Numerical results show that the thickness of the water layer and the velocity of the oil slug increase as the salinity of the water is reduced, suggesting that this mechanism contributes to the low-salinity effect. An analytical solution is presented in the limit in which the applied pressure is small.

We investigate numerically the influence of polymer mixing on shear-driven turbulence. Of particular interest is the suppression of mixing that accompanies drag reduction with dilute polymer solutions. The simulations use the finite extensible nonlinear elastic model with the Peterlin closure (FENE-P) to describe the polymer stresses in the momentum equation, with polymer concentration allowed to vary in space and time. A thin slab of concentrated polymer was placed in an initially Newtonian homogeneous turbulent shear flow on a plane perpendicular to the mean velocity gradient, and allowed to mix in the gradient direction while actively altering the turbulence. The initially higher concentration of polymer near the centreplane suppressed production of turbulent kinetic energy and Reynolds stress in that region, while turbulence outside the polymer-rich region remained shear-dominated Newtonian turbulence. The rate of mixing in the shear direction was severely damped by the action of the polymer compared to a passive scalar in the corresponding Newtonian turbulent shear flow. This, in part, was a result of the same damping of vertical velocity fluctuations by the polymer that leads to the suppression of momentum flux. However, the cross-correlation between the polymer concentration and vertical velocity fluctuations was also suppressed, indicating that the explanation for the reduction in polymer mixing involves both the suppression of vertical velocity fluctuations and an alteration of turbulence structure by the polymer–turbulence interactions.

We consider the problem of three interacting resonant waves with arbitrary (non-conservative) nonlinear coupling. Such coupling arises naturally in the interaction of waves on shear flows, and in interactions between interfacial and gravity waves. We focus on the case where two modes are damped and have identical properties, and the third is linearly unstable. When the damping rates dominate the growth rate, the dynamics evolves on two disparate timescales and it is then possible to reduce the system to a multi-modal one-dimensional map, thus revealing clearly the complex sequence of bifurcations that occurs as the parameters are varied. We also investigate the effect on the equations of small additive noise; this can be simply modelled by a (deterministic) perturbation to the map. It is shown that even at very low levels, the effect of noise can be extremely important in determining the period and amplitude of the oscillations.

By using asymptotic analysis, an eigensolution technique has been developed for predicting the flow of gas contained in a pie-shaped cylinder of finite length rotating rapidly about its vertex. This problem has application to a conventional cylindrical gas centrifuge with radial walls. Three different types of boundary layers exist in the flow: Ekman layers on the top and bottom, buoyancy layers on the radial walls, and a cylindrical ‘pancake’ layer on the outer wall of the cylinder. A single sixth-order partial differential equation is obtained for the axial velocity in the cylindrical layer, and the other layers provide matching conditions. The problem is formulated for no-slip and prescribed temperature conditions on the solid surfaces and for adiabatic no shear stress with zero pressure at the inner free surface. Eigenvalues are computed for this problem and compared with those for the open cylinder, and solutions are presented for flows induced by mass throughput and by differential temperature conditions.

In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.

In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.