High-frequency plane sound waves in ideal gases with internal dissipation are discussed. Particular applications to dissociating diatomic gases and gases displaying vibrational relaxation are considered. A criterion in the form of an inequality is derived for the validity of the high-frequency approximation and an asymptotic analysis is developed.

An examination of the isotropic vorticity theory was made in an adverse pressure gradient flow based on experimental data obtained in a conical diffuser having a total divergence angle of 8° and an area ratio of 4:1 with fully-developed pipe flow at entry. The results showed that the rates and the ratio of production and dissipation of the turbulent vorticity were constant in the core region of the diffuser but increase significantly in the wall layer. The overall vorticity balance was essentially the same at all axial stations. The analysis of Batchelor & Townsend (1947) for isotropic vorticity was found to be valid in the core region of the diffuser for an order-of-magnitude higher Rλ (200 [les ] Rλ 600) than in grid turbulence. The magnitude of the skewness of ∂u1/∂t was constant in the core region and comparable to that for grid turbulence. Also, this region of constant skewness extended over a larger portion of the flow cross-section compared to pipe flow. On the basis of these results, it was concluded that assumptions of isotropy in the fine structure are valid in the core region of the diffuser.

Davis has shown by means of a lengthy calculation that, for two-dimensional oscillations in a canal of width 2a, the mth eigenvalue has the form \[ N_m = {\textstyle\frac{1}{2}}m\pi - \frac{\lambda_1+\lambda_2}{4m\pi}+o\bigg(\frac{1}{m}\bigg), \] where λ1/a and λ2/a are the curvatures of the bounding cross-sectional curve C at its vertical intersections with the free surface. Here the same result is obtained more simply.

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush–liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.

The linear stability of a stably stratified fluid-filled cavity subject to vertical oscillations is determined via Floquet analysis. Retaining the viscous and diffusion terms in the Navier–Stokes–Boussinesq equations, with no-slip velocity boundary conditions, no-flux temperature conditions on the sidewalls and constant temperatures on the top and bottom walls, we find that the instabilities are primarily subharmonic (as is typical in many parametrically forced systems), except for in a few low-forcing-frequency ranges where the instabilities are synchronous. When the viscosity is small, the Floquet modes resemble the inviscid eigenmodes of the unforced problem, except in boundary layers. We establish scaling laws quantifying how viscosity regularizes the degeneracy associated with the inviscid idealization, and how it scales the thickness and intensity of the boundary layers. The product of boundary layer thickness and intensity remains constant with decreasing viscosity, leading to a delta distribution of vorticity on the walls in the limit of zero viscosity. This is in contrast to the zero wall vorticity in the inviscid case.

A modified version of Bateman's variational formulation of the incompressible Navier-Stokes equations and boundary conditions (see Dryden, Murnaghan & Bateman 1956) is introduced. This is employed to examine a particular nonlinear problem of hydrodynamic stability which was treated previously, using a ‘direct’ approach, by Craik (1971). This problem concerns the resonant interaction at second order of a triad of wave modes in a parallel shear flow.

The present method is conceptually attractive; it also has the major advantage over the ‘direct’ method of a substantial reduction in algebraic complexity, which allows results to be derived far more readily. Also, some further improvements are made upon Craik's previous analysis. Such a variational approach may often be simpler than present conventional methods of tackling nonlinear viscous-flow problems. The present paper shows how other problems of nonlinear stability and wave interactions may be tackled in this way.

Low-frequency waves that develop in a shallow layer of fluid, contained in a channel with linearly slopping bottom and rotating with uniform angular speed are investigated theoretically and experimentally. Exact numerical solutions of the eigenvalue problem, obtained from the linearized shallow water equations on the f-plane, show that the waves are trapped near the channel's shallow wall and propagate along it with the shallow side on their right in the Northern hemisphere. The phase speed of the waves is slower compared with that of the harmonic theory in which bottom slope is treated inconsistently. A first-order approximation of the cross-channel dependence of the coefficient in the eigenvalue equation yields an approximation of the cross-channel velocity eigenfunction as an Airy function, which, for sufficiently wide channels, yields an explicit expression for the wave's dispersion relation. The analytic solutions of the eigenvalue problem agree with the numerical solutions in both the wave trapping and the reduced phase speed. For narrow channels, our theory yields an estimate of the channel width below which the harmonic theory provides a more accurate approximation. Laboratory experiments were conducted on a 13 m diameter turntable at LEGI-Coriolis (France) into which a linearly sloping bottom of 10 % incline was installed. A wavemaker generated waves of known frequency at one end of the turntable and the wavenumbers of these waves were measured at the opposite end using a particle imaging velocimetry technique. The experimental results regarding the phase speed and the radial structure of the amplitude are in very good agreement with our theoretical non-harmonic predictions, which support the present modification of the harmonic theory in wide channels.

The relationship between numerical finite-amplitude equilibrium solutions of the full Navier–Stokes equations and nonlinear solutions arising from a high-Reynolds-number asymptotic analysis is discussed for a Tollmien–Schlichting wave-type two-dimensional vortical flow structure. The specific flow chosen for this purpose is that which arises from the mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travelling-wave solutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol. 678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number $R= 1{0}^{8} $ and confirm that the behaviour of its so-called lower branch solutions, which typically produce a smaller modification to the laminar state than the other solution branches, quantitatively agrees with the axisymmetric asymptotic theory developed in this paper. We further find that this asymptotic structure breaks down when the disturbance wavelength is comparable with $R$. The new structure which replaces it is investigated and the governing equations are derived and solved. The flow visualization of the resultant solutions reveals that they possess a streamwise localized structure, with the trend agreeing qualitatively with full Navier–Stokes solutions for relatively long-wavelength disturbances.

This paper examines the low-Reynolds-number migration of an insulating and rigid particle that is freely suspended in a viscous liquid metal and subject to uniform ambient electric and magnetic fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ Under the same physical assumptions as Part 1, a whole boundary formulation of the problem is established. It allows the determination of the particle rigid-body motion without calculating the modified electric field and the flow induced by the Lorentz body force in the fluid domain. The advocated boundary approach, well-adapted for future numerical implementation, makes it possible to obtain an analytical expression for the translational velocity of any ellipsoidal particle (the simplest case of non-spherical orthotropic particles). The behaviour of a spheroid is carefully investigated and discussed both without and with gravity. The migration of this simple non-spherical particle is found to depend on both its nature (prolate or oblate) and the ambient uniform fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ The spheroid translates without rotation, and not necessarily parallel to $\hbox{\bf\itshape E}\wedge \hbox{\bf\itshape B}.$ For adequately selected fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B},$ the spheroid may either migrate parallel or anti-parallel to a sphere and even be motionless.

Two inequalities are proposed for the purpose of bounding from below the mean shear U′o(z) in turbulent channel flow. The first of these inequalities pertains to the energetics of the boundary layer, and the second pertains to the logarithmic form of the asymptote to Uo. These inequalities imply a maximum value of von Kármán's constant, the numerical value of which lies between the measurements of Laufer (1951) and the value obtained from bulk discharge measurements in a pipe. The formalism, which contains no adjustable parameters, is then applied to the turbulent thermal convection problem, and a lower bound for the mean temperature $\overline{T}_o(z)$ is obtained. The minimum value of the latter at large distances from the boundary is in fair agreement with Townsend's (1959) measurements. Although the proposed inequalities have not been deduced from the equations of motion, they provide facts which may be useful in the search for new variational formulations of theturbulent transport problem.

We examine nonlinear geostrophic adjustment in a rapidly rotating (small Rossby number Ro) stably neutrally stratified (SNS) fluid consisting of a stratified upper layer with $N>f$ ($N$ is the buoyancy frequency, $f$ the Coriolis parameter) and a homogeneous lower layer, the density and other fields being continuous at the interface between the layers. The angular speed of rotation is non-parallel to gravity; the traditional and hydrostatic approximations are not used. The wave spectrum in the model consists of internal and gyroscopic waves. During the adjustment an arbitrary long-wave perturbation is split in a unique way into slow quasi-geostrophic (QG) and fast ageostrophic components with typical time scales $(Ro\, f)^{-1}$ and $f^{-1}$, respectively. The QG flow is governed by two coupled nonlinear equations of conservation of QG potential vorticity (PV) in the layers. The fast component is a sum of internal waves and inertial oscillations (long gyroscopic waves) confined to the homogeneous layer and modulated by an amplitude depending on coordinates and slow time. On times $t\sim (\, f\, Ro)^{-1}$ the slow component is not influenced by the fast one but the inertial oscillations amplitude is coupled to the QG flow and obeys an equation practically coinciding with that in the barotropic case (Reznik, J. Fluid Mech., vol. 743, 2014, pp. 585–605). A non-stationary boundary layer with large vertical gradients of horizontal velocity develops in the stratified layer near the interface to prevent penetration of the inertial oscillations into the stratified fluid; an analogous weaker boundary layer arises near the upper rigid lid. At large times the internal waves gradually decay because of dispersion and the resulting motion consists of the slow QG component and inertial oscillations confined to the barotropic lower layer.

The global linear stability of steady axisymmetric flow through a model fusiform aneurysm is studied numerically. The aneurysm is modelled as a Gaussian-shaped inflation on a vessel of circular cross-section. The fluid is assumed to be Newtonian, and the flow far upstream and downstream of the inflation is a Hagen–Poiseuille flow. The model aneurysm is characterized by a maximum height $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ and width $W$, non-dimensionalized by the upstream vessel diameter, and the steady flow is characterized by the Reynolds number of the upstream flow. The base flow through the model aneurysms is determined for non-dimensional heights and widths in the physiologically relevant ranges $0.1 \leq H \leq 1.0$ and $0.25 \leq W \leq 2.0$, and for Reynolds numbers up to 7000, corresponding to peak values recorded during pulsatile flows under physiological conditions. It is found that the base flow consists of a core of relatively fast-moving fluid, surrounded by a slowly recirculating fluid that fills the inflation; for larger values of the ratio $H/W$, a secondary recirculation region is observed. The wall shear stress (WSS) in the inflation is vanishingly small compared to the WSS in the straight vessels. The global linear stability of the base flows is analysed by determining the eigenfrequencies of a modal representation of small-amplitude perturbations and by looking at the energy transfer between the base flow and the perturbations. Relatively shallow aneurysms (of relatively large width) become unstable by the lift-up mechanism and have a perturbation flow which is characterized by stationary, growing modes. More localized aneurysms (with relatively small width) become unstable at larger Reynolds numbers, presumably by an elliptic instability mechanism; in this case the perturbation flow is characterized by oscillatory modes.