The effect of radiative transfer on the thermal stability of an ionized fluid, in the presence of uniform vertical magnetic field has been examined from hydromagnetic approximations. Two asymptotic cases (i) when the fluid is optically thin and (ii) when it is optically thick, have been examined in detail. The principle of exchange of stabilities and the concept of over-stability have been discussed and the physical conditions under which the former will hold, obtianed. Also the conditions as to which type of instability will arise first are derived and it has been shown that in the presence of large magnetic field over-stability arises earlier than convection under the first approximation.

This paper studies the impinging of two ideal fluid jets. The usual two-dimensional model of jet flow uses an ideal, incompressible, weightless fluid to describe these impinging jets, so that the problem becomes one of complex analysis which seems to have an infinite number of analytical solutions, except for direct jet impacts. The new approach presented here is based on the construction of a dividing line between the two jets. It gives an efficient procedure for solving this problem numerically when the jets flow in arbitrary directions and the solution obtained seems to be unique.

Laminar—turbulent transition in the decelerating boundary layer of Gad-el-Hak et al. (1984) is studied by solving the incompressible, time-dependent, three-dimensional Navier—Stokes equations numerically. The temporal and spatial evolution of the experimental flow structures is approximated with spatial periodicity and temporal evolution while maintaining mean-velocity profiles appropriate to the spatially-developing flow. Other than the mean flow, the initial flow fields include only small-amplitude white random noise. This and the large streamwise and spanwise dimensions of the numerical domain allow unstable waves to be selected by the dynamics, instead of being imposed arbitrarily as in previous numerical studies. In that sense, the transition is natural. In the early stages of transition, two-dimensional and slightly oblique waves grow rapidly owing to inflexional instability. Their subsequent nonlinear interactions trigger the breakdown and create a pattern of Λ vortices. The patterns of Λ vortices are more irregular than those found in ribbon-induced transition. The tips of the Λ vortices are rarely aligned with the flow direction, and they appear in local patches, consistent with the experimental visualizations of Gad-el-Hak et al. A simple model based on the interference of multiple modes of instability accounts for these features, but the specific pattern of Λ vortices depends on the random content of the initial flow field. A simulation of the later stages of transition is performed with higher numerical resolution, showing that the ‘naturally-born’ Λ vortices undergo breakdown processes similar to those of their ribbon-induced counterparts.

The wall jet with suction or injection is investigated; an analysis under conditions corresponding to similar flows is shown to reduce to an eigenvalue problem. Asymptotic solutions valid far from the surface are used to initiate the integration and circumvent the usual iteration associated with the two-point boundary-value problem. Typical solutions for various rates of suction and injection are obtained. It is found that the skin friction decreases with increasing rate of suction. Representative thermal solutions are obtained for Prandtl and Lewis numbers equal to one, under the special condition that the surface temperature is equal to the ambient temperature or that the enthalpy varies monotonically from the surface value to the ambient value.

The shape of liquid surfaces in regular N-pods in the absence of gravity is considered. A liquid volume in the vertex of a regular N-pod wets the adjacent faces if the sum of the liquid's contact angle γ with the faces and half the dihedral angle α between adjacent faces is smaller than π/2. A suggestion for why the surface shape in the wedge approaches its cylindrical shape at infinity exponentially is given. The range of this exponential decrease is related to the curvature of the meniscus and the angles α and γ: The decrement of the decrease generally shows a weak dependence on α+γ, predominantly depending on the liquid volume. Extremely close to the wetting limit, when α+γ approaches π/2, the decrement vanishes. The exponential meniscus shape leads to a similarity relation and allows small relative liquid volumes in polyhedrons to be split up into partial volumes ascribed to the corners and others ascribed to the wedges. The respective relations among volume, curvature, contact angle and corner geometry are obtained by numerical simulation and the limits of applicability are discussed. This greatly simplifies the calculation of liquid surfaces in the limit of small liquid volumes. The results obtained apply to liquid surfaces in a Space environment, e.g. to metallic melts in crucibles and to propellants and other technical fluids in tanks and reservoirs, as well as to liquid surfaces on Earth, e.g. to liquids trapped in polyhedral pores and to liquid foams, provided their characteristic length is sufficiently small compared to the capillary length.

In the fundamental configuration studied here, a steady hypersonic free stream flows over a thin sharp aligned airfoil or flat plate with a leading-edge shock wave, and the flow field in the shock layer (containing a viscous and an inviscid layer) is steady laminar and two-dimensional, for a perfect gas without real and high-temperature gas effects. The viscous and inviscid layers are analysed and computed simultaneously in the region from the leading edge to the trailing edge, including the upstream-influence effect present, to determine the interactive flow throughout the shock layer and the positions of the shock wave and the boundary-layer edge, where matching is required. Further theoretical analysis of the shock layer helps to explain the computational results, including the nonlinear breakdown possible when forward marching against enhanced upstream influence, for example as the wall enthalpy increases towards its insulated value. Then the viscous layer is computed by sweeping methods, for higher values of wall enthalpies, to prevent this nonlinear breakdown for airfoils including the flat plate. Thin airfoils in hypersonic viscous flow are treated, for higher values of the wall enthalpies and with the upstream-influence effect, as are hypersonic inviscid flows, by modifying the computational methods used for the flat plate. Also, the behaviour of the upstream influence for bodies of relatively large thickness, and under wall velocity slip and enthalpy jump for flat plates, is discussed briefly from a theoretical point of view.

Subsequent to the present work, computations based on the Navier–Stokes and on the parabolized Navier–Stokes equations have yielded excellent and good agreement respectively with the present predictions for large Mach and Reynolds numbers.

A crossing-shock-wave/turbulent-boundary-layer interaction is investigated using the k–ε turbulence model with a new low-Reynolds-number model based on the approach of Saffman (1970) and Speziale et al. (1990). The crossing shocks are generated by two wedge-shaped fins with wedge angles α1 and α2 attached normal to a flat plate on which an equilibrium supersonic turbulent boundary layer has developed. Two configurations, corresponding to the experiments of Zheltovodov et al. (1994, 1998a, b), are considered. The free-stream Mach number is 3.9, and the fin angles are (α1, α2) = (7°, 7°) and (7°, 11°). The computed surface pressure displays very good agreement with experiment. The computed surface skin friction lines are in close agreement with experiment for the initial separation, and are in qualitative agreement within the crossing shock interaction region. The computed heat transfer is in good agreement with experiment for the (α1, α2) = (7°, 7°) configuration. For the (α1, α2) = (7°, 11°) configuration, the heat transfer is significantly overpredicted within the three-dimensional interaction. The adiabatic wall temperature is accurately predicted for both configurations.

The source-sink flow within a rapidly rotating annular region, bounded by a pair of concentric circular cylinders and by horizontal end plates, is considered. Fluid is injected into the container at arbitrary locations on the inner radius and withdrawn at one or more locations on either the inner or outer radius, through axisymmetric slots. The upper end wall is maintained at a higher constant temperature than the bottom end plate, while the vertical side walls are thermally insulated; the entire apparatus is rapidly rotating about the common axis of the cylinders. An analysis of the flow and temperature distribution is carried out in the context of the Boussinesq approximation and by assuming that vertical buoyancy effects are negligible to leading order. A method for calculation of the temperature distribution within the container is developed for physical situations where the effects of the imposed thermal gradient and the source-sink flow are of comparable magnitudes; the procedure is applicable for an arbitrary distribution of sources and sinks on the side walls. The temperature problem is an unusual complicated boundary-value problem, and numerical solutions are obtained for a number of different cases. The results reveal a number of interesting flows and temperature fields within the container and indicate how the temperature field is influenced by the placement and temperature of the sources and sinks, as well as by the relative magnitudes of the imposed forced flow and vertical thermal gradient. The possible application of the present theory to centrifuges is indicated.

This paper considers the effect of an exponential variation in the background density field (as exists in compressible atmospheres) on the structure and dynamics of the quasi-geostrophic system, and compares the results with the corresponding Boussinesq limit in which background density variations are assumed small. The behaviour of the compressible system is understood via a closed-form analytic expression for the Green's function of the inversion operator relating potential vorticity and streamfunction. This expression makes explicit the anisotropy of the Green's function, inherited from the density profile, which has a slow, algebraic decay directly above the source and an exponential decay in all other directions. An immediate consequence for finite-volume vortices is a differential rotation of upper and lower levels that results in counterintuitive behaviour during the nonlinear evolution of ellipsoidal vortices, in which vortex destruction is confined to the lower vortex and wave activity is seen to propagate downwards. This is in contrast to the Boussinesq limit, which exhibits symmetric destruction of the upper and lower vortex, and in contrast to naive expectations based on a consideration of the mass distribution alone, which would lead to greater destruction of the upper vortex. Finally, the presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded case (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex, the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation, the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.

Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit $\unicode[STIX]{x1D719}\rightarrow 0$. The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The $O(1)$ slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the $O(1)$ slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.

The development of the boundary layer on the upper surface of a horizontal flat plate in a non-diffusive, stratified flow is described. It is shown that the flow can be characterized by two basic parameters, the Reynolds (RL) and Russell (RuL) numbers, and that, depending on the relative magnitude of these two parameters, three different régimes of flow can be defined. The delineation of these régimes and the description of the flow in each of them is obtained by deriving a uniformly valid first approximation to the Boussinesq equations of motion for a flow contained in the two-dimensional parameter space RuL > 0, RL > 1. The critical stratification for the self-blocking of a horizontal boundary layer is shown to be given by the condition RuL = O(RL½).

Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371–385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic, $\unicode[STIX]{x1D6FD}$, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.

New mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the axial direction), a multiple-timescale procedure is used to obtain explicitly parameterized high-order asymptotic approximations of solutions of the spatio-temporal evolution equation. Novel, hitherto-unexplained transitions from stability to instability are observed as inertia is increased. In particular, a critical Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_c$ is predicted at which occurs a supercritical pitchfork bifurcation in wave amplitude that is fully explained by the new asymptotic theory. For $\mathit{Re}<\mathit{Re}_c$, free-surface profiles converge algebraically-cum-exponentially to a steady state and, for $\mathit{Re}> \mathit{Re}_c$, stable temporally periodic solutions with leading-order amplitudes proportional to $(\mathit{Re}-\mathit{Re}_c)^{1/2}$ are found, i.e. in the régime in which previous related literature predicts exponentially divergent instability. For $\mathit{Re}=\mathit{Re}_c$, stable solutions are found that decay algebraically to a steady state. A model solution is proposed that not only captures qualitatively the interaction between fundamental and higher-order wave modes but also offers an explanation for the formation of the lobes observed in Moffatt’s original experiments. All asymptotic theory is convincingly corroborated by numerical integrations that are spectrally accurate in space and eighth/ninth-order accurate in time.

Parametrically forced gravity waves can give rise to high-velocity surface jets via the wave-depression cavity implosion. The present results have been obtained in circular cylindrical containers of 10 and 15 cm in diameter (Bond number of order $10^{3}$) in the large fluid depth limit. First, the phase diagrams of instability threshold and wave breaking conditions are determined for the working fluid used, here water with 1 % detergent added. The collapse of the wave-depression cavity is found to be self-similar. The exponent $\unicode[STIX]{x1D6FC}$ of the variation of the cavity radius $r_{m}$ with time $\unicode[STIX]{x1D70F}$, in the form $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6FC}}$, is close to 0.5, indicative of inertial collapse, followed by a viscous cut-off of $\unicode[STIX]{x1D6FC}\approx 1$. This supports a Froude number scaling of the surface jet velocity caused by cavity collapse. The dimensionless jet velocity scales with the cavity depth that is shown to be proportional to the last stable wave amplitude. It can be expressed by a power law or in terms of finite time singularity related to a singular wave amplitude that sets the transition from a non-pinching to pinch-off cavity collapse scenario. In terms of forcing amplitude, cavity collapse and jetting are found to occur in bands of events of non-pinching and pinching of a bubble at the cavity base. At large forcing amplitudes, incomplete cavity collapse and splashing can occur and, at even larger forcing amplitudes, wave growth is again stable up to the singular wave amplitude. When the cavity is formed, an impulse model shows the importance of the singular cavity diameter that determines the strength of the impulse.