Two parallel circular jets, in inviscid incompressible flow, with uniform axial velocityof the same magnitude and direction are placed near to one another, resulting in a stronglycoupled field. A given (small) wavenumber in the axial direction is taken and a dispersionrelation is found relating the frequency and wavenumber for a given disturbance mode, alongwith the velocity potentials within and exterior to the jets. The problem is tackledanalytically using bipolar coordinates and asymptotic forms for the dispersion relation arefound in the small-separation and large-separation limits. Results are then compared withthe corresponding two-dimensional problem for plane jets. It is concluded thatclose-proximity interactions greatly destabilize the varicose mode of the coupled jet, andgreatly stabilize the sinuous mode.

The Navier-Stokes equations for flow in a rotating circular pipe are solved numerically, subject to imposing helical symmetry on the velocity field v = v(r, θ + αz,t). The helical symmetry is exploited by writing the equations of motion in helical variables, reducing the problem to two dimensions. A limited study of the pipe flow is made in the parameter space of the wavenumber α, and the axial and azimuthal Reynolds numbers. The steadily rotating waves previously studied by Toplosky & Akylas (1988), which arise from the linear instability of the basic steady flow, are found to undergo a series of bifurcations, through periodic to aperiodic time dependence. The relevance of these results to the mechanism of laminar-turbulent transition in a stationary pipe is discussed.

We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds $Ca=U\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of $-Ca$. The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

Potential flow analysis, including unsteady effects, has been applied to live swimming squid, Loligo pealei. Squid were modelled as slender, axisymmetric bodies. High-speed video records, recorded at frame rates of 125 to 250 Hz, provided time-varying body outlines which were digitized automatically. Axisymmetric renderings of these body outlines and the real motion of the squid were used as the input of the potential flow analysis. Axial and lateral inviscid fluid forces simply due to the flow past the squid body were calculated from pressures coefficients obtained from the unsteady Bernoulli equation. Lateral forces were found to play virtually no role in determining muscle stresses in squid jet propulsion. Axial pressure forces were also found to be small in comparison to both net force (based on the observed whole body kinematics) and estimations of skin friction. These findings demonstrate the effects of the highly adapted shape of squid with regard to hydrodynamics. The work suggests that skin friction and working fluid intake are the most significant sources of drag on a swimming squid.

Motions in the solid mantle of silicate planets are predominantly driven by internal heat sources and occur in laminar regimes that have not been systematically investigated. Using high-resolution numerical simulations conducted in three dimensions for a large range of Rayleigh–Roberts numbers ( $5\times 10^{3}\leqslant Ra_{H}\leqslant 10^{9}$ ), we have determined the characteristics of flow in internally heated fluid layers with both rigid and free slip boundaries. Superficial planforms evolve with increasing $Ra_{H}$ from a steady-state tessellation of hexagonal cells with axial downwellings to time-dependent clusters of thin linear downwellings within large areas of nearly isothermal fluid. The transition between the two types of planforms occurs as a remarkable flow polarity reversal over a small $Ra_{H}$ range, such that downwellings go from isolated cylindrical structures encircled by upwellings to thin interconnected linear segments outlining polygonal cells. In time-dependent regimes at large values of $Ra_{H}$ , linear downwellings dominate the flow field at shallow depth but split and merge at intermediate depths into nearly cylindrical plume-like structures that go through the whole layer. With increasing $Ra_{H}$ , the number of plumes per unit area and their velocities increase whilst the amplitude of thermal anomalies decreases. Scaling laws for the main flow characteristics are derived for $Ra_{H}$ values in a $10^{6}$ – $10^{9}$ range. For given $Ra_{H}$ , plumes are significantly colder, narrower and wider apart beneath free boundaries than beneath rigid ones. From the perspective of planetary studies, these results alert to the dramatic changes of convective planform that can occur along secular cooling.

The source mechanism of inertia–gravity waves (IGWs) observed in numerical simulations of the differentially heated rotating annulus experiment is investigated. The focus is on the wave generation from the balanced part of the flow, a process presumably contributing significantly to the atmospheric IGW field. Direct numerical simulations are performed for an atmosphere-like configuration of the annulus and possible regions of IGW activity are characterised by a Hilbert-transform algorithm. In addition, the flow is separated into a balanced and unbalanced part, assuming the limit of a small Rossby number, and the forcing of IGWs by the balanced part of the flow is derived rigorously. Tangent-linear simulations are then used to identify the part of the IGW signal that is rather due to radiation by the internal balanced flow than to boundary-layer instabilities at the side walls. An idealised fluid set-up without rigid horizontal boundaries is considered as well, to investigate the effect of the identified balanced forcing unmasked by boundary-layer effects. The direct simulations of the realistic and idealised fluid set-ups show a clear baroclinic-wave structure exhibiting a jet–front system similar to its atmospheric counterparts, superimposed by four distinct IGW packets. The subsequent tangent-linear analysis indicates that three wave packets are radiated from the internal flow and a fourth one is probably caused by boundary-layer instabilities. The forcing by the balanced part of the flow is found to play a significant role in the generation of IGWs, so it supplements boundary-layer instabilities as a key factor in the IGW emission in the differentially heated rotating annulus.

We treat the problem of the stability of an infinite horizontal array of cylinders, spaced periodically, by a direct numerical simulation of the Navier–Stokes equations for steady flow at Reynolds numbers less than or equal to 100. We find that the only stable configuration for the array is one with equal spacing between cylinders and all cylinders lying on a line perpendicular to the flow. The array is found to be stable under displacements of the cylinders perpendicular and parallel to the array. We say a perturbation is stable when it gives rise to a force which acts to restore the original stable configuration. Our results are consistent with experiments in which spheres were confined by the sidewalls of a fluidized bed to move in two dimensions. As a secondary issue we consider the variation with parameters of the length and width of wakes behind cylinders.

Grid is an important factor in numerical simulation of hypersonic aerothermodynamics. This paper introduces three criteria for determining grid size in the transition flow regime when using the computational fluid dynamics (CFD) method or the direct simulation Monte Carlo (DSMC) method. The numerical relationship between these three criteria sizes is deduced according to the one-dimensional fluid theory. Then, the relationship is verified using the CFD method to simulate the flow around a two-dimensional cylinder. At the same time, the dependence of simulation accuracy on grid size in the CFD and DSMC methods is studied and the mechanism is given. The result shows that the simulation accuracy of heat flux especially depends on the normal grid size next to surfaces, where the $Re_{\mathit{cell},w}$ criterion and the $\unicode[STIX]{x1D706}_{w}$ criterion based on local parameters are applicable and equivalent, while the $Re_{\mathit{cell},\infty }$ criterion based on the free-stream parameter is only applicable under the assumption of constant viscosity coefficient and constant temperature wall conditions. On the other hand, the trend of the heat flux changing with grid size obtained by CFD and DSMC is exactly the opposite. Therefore, the grid size must be strictly satisfied with the grid criteria when comparing CFD with DSMC and even the hybrid DSMC with Navier–Stokes method.

The linear stability of Couette–Poiseuille flow of two superposed fluid layers in a horizontal channel is considered. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier–Stokes equations in each fluid layer, convection–diffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim of this study is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics on the instability. A linear stability analysis is performed and the growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using long-wave approximations. It is found that the system is stable when the surfactant is sufficiently soluble in the bulk and if the fluid viscosity ratio $m$ and thickness ratio $n$ satisfy the condition $m<n^{2}$. On the other hand, the effect of surfactant solubility is found to be destabilising if $m\geqslant n^{2}$. Both of the aforementioned results are manifested for low bulk concentrations below the critical micelle concentration; however, when the equilibrium bulk concentration is sufficiently high (and above the critical micelle concentration) so that micelles are formed in the bulk fluid, the system is stable if $m<n^{2}$ in all cases examined.

The dynamics of the orientations of prolate ellipsoids in general linear shear flow is considered. The motivation behind this work is to gain a better understanding of the motion and the orientation probability distribution of ice particles in clouds in order to improve the modelling of their collision. The evolution of the orientations is governed by the Jeffery equation. It is shown that the possible attractors of this equation are fixed points, limit cycles and an infinite set of periodic solutions, named Jeffery orbits, in the case of simple shear. Linear stability analysis shows that the existence and the stability of the attractors are determined by the eigenvalues of the linear part of the equation. If the eigenvalues possess a non-vanishing real part, then there always exists either a stable fixed point or a stable limit cycle. Pure imaginary eigenvalues lead to Jeffery orbits. The convergence to a stable fixed point or to a stable limit cycle may either be monotonic or may be retarded due to the occurrence of non-normal growth. If non-normal growth occurs the convergence rate may be much slower compared with the characteristic time scale of the shear. Expressions for the characteristic time scale of convergence to the stable solutions are derived. In the case of non-normal growth, expressions are derived for the delay in the convergence. The orientation probability distribution function (p.d.f.) is computed via the solution of the Fokker–Planck equation. The p.d.f. is either periodic, in the case of simple shear (pure imaginary eigenvalues), or it converges to singular points or strips in the orientation space (fixed points and limit cycles) on which it grows to infinity. Time-independent p.d.f.s exist only for imaginary eigenvalues. Unlike the case where Brownian diffusion is present, the steady solutions are not unique and depend on the initial conditions.

The heating from above of an initially homogeneous layer of solid crystals, saturated liquid and glass ballotini (an inert matrix filler) is considered both experimentally and theoretically. The heat flux causes crystals at the top of the layer to dissolve, forming liquid which, being more concentrated and dense than the interstitial liquid below, drives convection in the lower layer. Mixing of this high-concentration liquid into the lower layer leads to precipitation, thereby releasing latent heat which raises the temperature of the lower layer. Dissolution of solid crystals from the top leaves behind a closely packed layer of glass ballotini overlain by a layer of clear liquid, both of which deepen with time. The initially homogeneous porous medium thus develops into a three-layer stratified system of (from the top): clear liquid; clear liquid with close-packed ballotini; and the evolved initial assemblage of solid crystals, ballotini and saturated liquid. Data from laboratory experiments compare well with analytical and numerical results from a one-dimensional theoretical model. The model is based on the concept that the heat supplied from above is used entirely for the dissolution of solid crystals at the upper boundary of the lower layer. The resulting compositional convection redistributes the dissolved salt uniformly throughout the lower layer, where it partly recrystallizes to restore chemical equilibrium. The crystallization and associated release of latent heat leads to a gradual and uniform increase of both the solid fraction and temperature of the lower layer. Some geological consequences of the model are presented in the concluding section.

Measurements of subsonic air jets show that the peak noise usually occurs when observations are made at small angles to the jet axis. In this paper, we develop further understanding of the mathematical properties of this peak noise by analysing the properties of the overall sound pressure level with an acoustic analogy using isotropy as a paradigm for the turbulence. The analogy is based upon the hyperbolic conservation form of the Euler equations derived by Goldstein (Intl J. Aeroacoust., vol. 1, 2002, p. 1). The mean flow and the turbulence properties are defined by a Reynolds-averaged Navier–Stokes calculation, and we use Green's function based upon a parallel mean flow approximation. Our analysis in this paper shows that the jet noise spectrum can, in fact, be thought of as being composed of two terms, one that is significant at large observation angles and a second term that is especially dominant at small observation angles to the jet axis. This second term can account for the experimentally observed peak jet noise (Lush, J. Fluid Mech., vol. 46, 1971, p. 477) and was first identified by Goldstein (J. Fluid Mech., vol. 70, 1975, p. 595). We discuss the low-frequency asymptotic properties of this second term in order to understand its directional behaviour; we show, for example, that the sound power of this term is proportional to the square of the mean velocity gradient. We also show that this small-angle shear term does not exist if the instantaneous Reynolds stress source strength in the momentum equation itself is assumed to be isotropic for any value of time (as was done previously by Morris & Farrasat, AIAA J., vol. 40, 2002, p. 356). However, it will be significant if the auto-covariance of the Reynolds stress source, when integrated over the vector separation, is taken to be isotropic in all of its tensor suffixes. Although the analysis shows that the sound pressure of this small-angle shear term is sensitive to the statistical properties of the turbulence, this work provides a foundation for a mathematical description of the two-source model of jet noise.

A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi-two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number ($\hbox{\it Ha} \,{\gg}\, 1$). This is analogous to the case of low-Ekman-number rotating flows, with Ekman layers replacing Hartmann layers. This has been at the origin of the homogeneous model for flows in a rotating frame of reference, with its rich structure: geostrophic contours and shear layers. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii (Isv. Akad. Nauk SSSR Mekh. Zhidk Gaza, vol. 3, 1968, p. 3) play a role similar to that of the geostrophic contours. However, a general theory for quasi-two-dimensional magnetohydrodynamics is lacking. In this paper, a model is proposed which provides a general framework. Not only can this model account for otherwise disconnected past results, but it is also used to predict a new type of shear layer, of typical thickness $\hbox{\it Ha}^{-1/4}$. Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd & Walker (J. Fluid Mech. vol. 84, 1978, p. 471), and the problem of a rectangular cross-section duct flow in a slowly varying transverse magnetic field. For the first problem, the existence of thick shear layers of dimensionless thickness of order of magnitude $\hbox{\it Ha}^{-1/4}$ explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past Walker & Ludford (J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.

It is shown that for low values of the ambient pressure the flow field for a steady spherically symmetric expansion can be divided into three parts termed the inviscied region, the intermediate layer and the shock layer. Analytic solutions are available in the first two regions and a complete integration of the equation is required in the third. Numerical solutions indicate that such a structure is achieved in the limit and the universality of the solutions in the individual regions is confirmed.

The effects of surface corrugation with small amplitude on the growth of Tollmien–Schlichting (T–S) waves were examined experimentally in a zero-pressure-gradient boundary layer. Two- and three-dimensional corrugations of sinusoidal geometry with wavelengths of the same order as that of the two-dimensional T–S wave were considered. The corrugation amplitudes were one order of magnitude smaller than the boundary-layer displacement thickness. Streamwise growth of T–S waves on the corrugated walls was compared with that in the boundary layer on the smooth surface. A distinct difference was found in the destabilizing effect between the two- and three-dimensional corrugations. The two-dimensional corrugation significantly enhanced the growth of two-dimensional T–S waves even when the corrugation amplitude was only ∼10% of the displacement thickness. On decreasing the corrugation amplitude, the growth rate of two-dimensional T–S waves asymptotically approached that in the smooth-wall case. On the other hand, the three-dimensional corrugation had only a small influence on the growth of two-dimensional T–S waves even when the corrugation amplitude was as large as 20% of the displacement thickness. For three-dimensional corrugations, however, a pair of oblique waves was generated and developed by an interaction between the two-dimensional T–S wave and the corrugation-induced mean-flow distortion for the corrugation wavelength considered. On increasing the corrugation amplitude, the oblique waves generated were increased in amplitude and thus significantly influenced the secondary instability process.

A particle with a density slightly larger than that of the fluid in which it is immersed will sediment. However, if the particle's temperature is higher than that of the fluid, the terminal velocity of sedimentation will be smaller and can even change sign. When the terminal velocity is zero we say there is thermal levitation. Thermal levitation can also occur when the density and temperature of the particle are smaller than those of the fluid. Using a two-component thermal lattice Boltzmann equation method, we study this phenomenon and show it can be stable or unstable.