A simplified model is developed for an alveolar liquid lining undergoing cyclic stretching which mimics breathing motions. A thin, viscous film coats an extensible alveolar wall with small aspect ratio, $\varepsilon$. Scaling analysis and asymptotic theory are used to describe the interface profile and surfactant distribution during the oscillation cycle for either insoluble or soluble surfactants. The flow consists of two distinct regimes: an outer region away from the rigid endwalls where flow is near-parallel and a boundary-layer region at the rigid endwalls where flow is non-parallel. The system is solved asymptotically in the limit of $\varepsilon\ll 1$ and small strain amplitudes, $\Delta \ll 1$. For leading order in $\varepsilon$, steady streaming vortical flows are found at $O(\Delta^2)$ and their size, number and flow direction depend on the system parameter values. This preliminary model can be useful for understanding alveolar transport characteristics for slowly diffusing molecules with large Péclet number, such as endogenous surfactants and proteins as well as delivered surfactants, drugs and genetic material that may occur in various therapies or partial liquid ventilation. The flow pattern also provides a pathway for cell–cell signalling within the alveolus.

The problem of removing an unyielded visco-plastic material from the walls of a duct is considered. This forms a prototype problem for the mechanical removal of soft solids from the duct walls in both oilfield well construction and food processing.

We consider two-layer axial flows for which both fluids may be characterized as Herschel–Bulkley fluids, a class of fluids including Newtonian, Bingham and power-law models. If the yield stress of the displaced fluid is sufficiently large, it is possible for static layers of fluid to persist at the walls of the duct, as has been shown in Allouche, Frigaard & Sona (2000). Following Prager (1954) two variational principles are derived for these flows. These may be loosely interpreted as rate-of-strain minimization and stress maximization principles. The rate-of-strain minimization principle leads directly to existence and uniqueness results for this class of flows. The stress maximization principle leads to a number of qualitative results. An adaptation of the stress maximization principle also allows us to define the concept of a maximal static wall layer in terms of minimization of a certain functional over the space of admissible fluid–fluid interfaces, i.e. a shape optimization problem. We present a number of geometrically simple examples that demonstrate the validity of this method.

The laminar breakdown induced by purely travelling crossflow vortices in a three-dimensional flat-plate boundary-layer flow is investigated in detail by means of spatial direct numerical simulations. The base flow considered is generic for an infinite swept wing, with decreasing favourable chordwise pressure gradient and a sweep angle of $45^\circ$. First, the primary downstream growth and nonlinear saturation state of a single crossflow wave are simulated. Secondly, background disturbance pulses are added, and the subsequent mechanisms triggering transition to turbulence in this scenario are identified and analysed in detail. The saturated travelling crossflow vortex is found to give rise to a co-travelling secondary instability not unlike the instability in the much investigated steady crossflow-vortex case, but with characteristic differences. An analysis method with a spanwise Galilean transformation to travel with the primary wave and a consequently adapted timewise/spanwise Fourier decomposition of the disturbance flow enables unambiguous isolation of the various secondary disturbance modes. The resulting flow structures and their dynamics in physical space are visualized.

We describe the gravity-driven flow of a viscous fluid in a semi-infinite porous layer, $x>0$, from which fluid can drain freely at $x=0$. New experiments using a Hele-Shaw cell confirm that when the base of the layer is impermeable the motion of the current is self-similar and the dipole moment of the flow is conserved, as proposed theoretically by Barenblatt & Zel'dovich (1957). We extend the model to allow fluid to drain through the base of the porous layer into a thin horizontal layer of lower permeability. In this case we predict that the dipole moment of the current decays exponentially with time. At early times we find that the loss of fluid from the gravity current in the high-permeability layer is dominated by the draining at $x=0$, whereas at long times, the gravity-driven leakage into the underlying low-permeability layer is dominant. We successfully compare these analytic solutions for such draining currents with further laboratory experiments in which fluid drains from the end and through the base of a Hele-Shaw cell. We discuss the implications of these results for the dispersal of chemicals or pollutants injected into a layered porous rock.

We study by a fully nonlinear three-dimensional pseudospectral time-splitting simulation, the feedback control of a layer of fluid heated from below. The initial condition corresponds to a steady large-amplitude preferred convection state obtained at Prandtl number of 7.0 and Rayleigh number of 104, which is about six times the Rayleigh critical value. A robust controller based on the LQG (linear–quadratic–Gaussian) synthesis method is used. Both sensors and actuator are thermal-based, planar, andassumed to be continuously distributed. The simulated results show that large-amplitude steady-state convection rolls can be suppressed by the linear LQG controller action. The Green's function of the controller gives the shape of the control action corresponding to a point measurement. In addition, for Rayleigh numbers below the proportional feedback control stability limit, this controller appeared to be effective in damping out steady-state convection rolls as well. However, in a region very near the proportional control stability limit, proportional control action induces subcritical g-type hexagonal convection, which is obtained here through direct simulations. Note that well above this proportional control limit, the LQG still damps out all convection. The nonlinear plant model is validated by comparing check cases with published results.

The behaviour of water waves over periodic beds is considered in a two-dimensional context and using linear theory. Three cases are investigated: the scattering of waves by a finite section of periodic topography; the Bloch problem for infinite periodic topography; and sloshing motions over periodic topography confined between vertical boundaries. Connections are established between these problems.

A transfer matrix method incorporating evanescent modes is developed for the scattering problem, which reduces the computation to that required for a single period, without compromising full linear theory. The problem of the existence of Bloch waves can also be posed on a single period, leading to a close relationship between it and the scattering problem. Sloshing motions over periodic beds, which may be regarded as special cases of the Bloch problem, are also found to have a significant connection with wave scattering.

Integral equations methods allied to the Galerkin approximation are used to resolve the three problems numerically. In particular, the full linear solution for Bragg resonance is presented, allowing the accuracy of existing approximations to this phenomenon to be assessed. The selection of results given illustrates the main features of the work.

We describe a model for non-convecting diffusion-controlled solidification of a ternary (three-component) alloy cooled from below at a planar boundary. The modelling extends previous theory for binary alloy solidification by including a conservation equation for the additional solute componentand coupling the conservation equations for heat and species toequilibrium relations from the ternary phase diagram. We focus on growth conditions under which the solidification path (liquid line of descent) through the ternary phase diagram gives rise to two distinct mushy layers. A primary mushy layer, which corresponds to solidification along aliquidus surface in the ternary phase diagram, forms above a secondary (or cotectic) mushy layer, which corresponds to solidification along acotectic line in the ternary phase diagram. These two mushy layers are bounded above by a liquid layer and below by a eutectic solid layer. We obtain a one-dimensional similarity solution and investigate numerically the role of the control parameters in the growth characteristics. In the special case of zero solute diffusion and zero latent heat ananalytical solution can be obtained. We compare our predictions with previous experimental results and with theoretical results from a related model based on global conservation laws described in the Appendix. Finally, we discuss the potentially rich convective behaviour anticipated for other growth conditions.

An analysis of the vertical velocity field using the full generalized {omega} equation ($\omega$-equation) in a single mesoscale baroclinic oceanic gyre is carried out. The evolution of the gyre over 20 inertial periods is simulated using a new three-dimensional numerical model which directly integrates the horizontal ageostrophic vorticity, explicitly conserves the potential vorticity (PV) via contour advection on isopycnal surfaces, and inverts the nonlinear PV definition via the solution of a three-dimensional Monge–Ampère equation. In this framework the $\omega$-equation comes simply from the horizontal divergence of the horizontal ageostrophic vorticity prognostic equation. The ageostrophic vorticity is written as the Laplacian of a vector potential $\varphib$, from which both the velocity and the density fields are recovered, respectively, from the curl and divergence of $\varphib$. A new initialization technique based on the slow, progressive growth of the PV field during an initial time interval is used to avoid the generation of internal gravity waves during the initialization of the gyre. This method generates a nearly balanced baroclinic gyre for which the influence of internal gravity waves in the mesoscale vertical velocity field is negligible.

The numerical fields obtained are then used to carry out a first numerical analysis of the $\omega$-equation. The analysis shows that, for moderately high Rossby numbers, the local and the advective rates of change of the differential ageostrophic vertical vorticity $(\zeta'_z)$ are of the same order of magnitude as the three largest terms in the $\omega$-equation. There is, however, a large cancellation between these two terms, resulting in the approximate material conservation of $\zeta'_z$. This might explain the ‘over-applicability’ of the quasi-geostrophic (QG) $\omega$-equation for Rossby numbers larger than 0.1. The QG vertical velocity is only 22% smaller than the total vertical velocity for the case studied (having a Rossby number of −0.5).

This paper investigates the stability properties of a Blasius boundary layer perturbed by a small-amplitude steady three-dimensional distortion, which may be isolated or periodic along the spanwise direction. It is shown that once the strength of the distortion exceeds a threshold, the perturbed flow becomes inviscidly unstable. An isolated distortion which features a dominant low-speed streak induces a localized mode, while a periodic distortion supports spatially quasi-periodicmodes through a parametric resonance. The frequencies and growth rates of both types of mode are much higher than those of the viscous Tollmien–Schlichting (T–S) waves. For moderate distortions, the instability modes can be viewed as kinds of modified T–S waves, which amplify at rates in excess of the viscous instability. For a localized distortion, these modes do not reduce to the usual T–S wavesin the zero-distortion limit. The nonlinear development of the inviscid modes is also studied, and is foundto be governed by a slightly modified version of the evolution equation derived by Wu (1993). A numerical study suggests that nonlinearity has a strong destabilizing effect, and ultimately leads to an explosive growth in the form of a finite-distance singularity. The present theoretical model is found to capture qualitatively some key experimental observations.

This paper presents theoretical results on the instability of a Blasius boundary layer perturbed by Klebanoff modes (i.e. the low-frequency streaks known to be induced by free-stream turbulence). Herein, the Klebanoff distortions are modelled as the signature of a three-dimensional convected gust that may be either isolated or periodic along the spanwise direction. Relatively weak Klebanoff fluctuations can produce $O(1)$ changes tothe near-wall curvature of the base flow profile and, hence, fundamentally alter the nature of its instability characteristics.The perturbed flow is shown to support instabilities that are predominantly inviscid and have significantly larger growth rates and characteristic frequencies than the Tollmien–Schlichting(T–S) modes of an unperturbed Blasius flow. The spanwise mode shape of instabilities in the perturbed flow is determined by theSchrödinger equation, with a potential function that corresponds to the skin friction perturbation due to the Klebanoff distortion. The growth ratesof these modes are determined by the near-wall torsion of the perturbed flow.The unsteadiness of the Klebanoff distortion is shown to be a crucial element in determining the overall instability characteristics.

Laboratory experiments were conducted on a double-diffusing fluid stabilized by the faster diffuser ($T$) and destabilized by the slower diffuser ($S$), the arrangement that leads to salt fingering. The experiments were conducted in three tanks of depths 2 m, 1 m or 0.3 m, with initial linear profiles of $S$ and $T$ between two reservoirs at fixed or known values, $S_0 + \Delta S, T_0 + \Delta T$ in the top reservoir, $S_0, T_0$ in the bottom reservoir.

In the thermohaline staircase regime consisting of an alternating stack of convecting layers and finger layers, the overall fluxes $F_S$ and $F_T$ are clearly not depth independent. The overall Nusselt number, $N_S$ varies inversely with the number $n$ of finger layers plus convecting layers occurring in the experimental fluid. The largest Nusselt number occurs for $n = 1$. The number $n$ increases with increasing mean vertical gradient, up to a point. If the vertical gradients are large, $n$ is always equal to 1.

Internal measurements show that in the Rayleigh number range $|R_S|\sim 10^{15}$ and density ratio range $R_{\rho}\sim 1.1$ to 1.2, the finger flux $F^f_S\sim {(\delta S/h)}R_S^a R_{\rho}^b$ with $a = 0.18$ to 0.19 and $b = 1.8$ to 2.1. The data show dependence of $F^f_S$ on finger layer thickness $h$. In the convecting layers, $N^c_S \propto R^{0.21}$ for large enough aspect ratio.

The linear inviscid instability of an infinitely thin vortex sheet, periodically corrugated with finite amplitude along the spanwise direction, is investigated analytically. Two types of corrugations are studied, one of which includes the presence of an impermeable wall. Exact eigensolutions are found in the limits of very long and of very short wavelengths. The intermediate-wavenumber range is explored by means of a second-order asymptotic series and by limited numerical integration. The sheets are unstable to both sinuous and varicose disturbances. The former are generally found to be more unstable, although the difference only appears for finite wavelengths. The effect of the corrugation is shown to be stabilizing, although in the wall-bounded sheet the effect is partly compensated by the increase in the distance from the wall. The controlling parameter in that case appears to be the minimum separation from the sheet valley to the wall. The instability is traced to a pair of oblique Kelvin–Helmholtz waves in the flat-sheet limit, but the eigenfunctions change character both as the corrugation is made sharper and as the wall is approached, becoming localized near the crests and valleys of the corrugation. The study is motivated by the desire to understand the behaviour of lifted low-speed streaks in wall-bounded flows, and it is shown that the spatial structure of the fundamental sinuous eigenmode is remarkably similar to previously known three-dimensional nonlinear equilibrium solutions in both plane Couette and Poiseuille flows.