In this paper time dependent expansions of monatomic gases with spherical symmetry are discussed. For the particular case of Maxwellian molecules closed expressions for the moments up to second order are obtained in regions of the flow where the inviscid solution is no longer valid. These solutions are derived in a general form using the particle path function as a parameter. The structure of the inviscid solution is such that this simplification can be made. The novelty of the present approach is that solutions already derived in previous papers can be obtained from the general solution in various limits; both the results for steady flow and the expansion of a fixed mass of gas can be derived in this manner. Finally, a particular example is constructed in order to illustrate the general theory.

The inviscid compressible flow stability problem is mathematically similar to that of sound propagation in a sheared flow field. This similarity has been exploited by applying an inner expansion technique to study the effect of finite shear gradients on free parallel flow instabilities. This technique had previously been used to investigate the effect of thin boundary layers on sound propagation in ducts. The expansion, which is applicable to flow profiles involving thin, but finite, shear layers separating regions of uniform flow, offers a significant computational advantage over the numerical methods commonly employed to determine the stability of continuous mean flow profiles. Although equally applicable to three-dimensional and to spatially growing hydrodynamic instabilities, the procedure is demonstrated by application to the eigenvalue problem for temporal instabilities of shear layers and jets in plane inviscid compressible flow.

For the case of vanishingly thin shear layers, the eigenvalue equations derived here reduce to those obtained by Miles (1958) for parallel flows bounded by vortex sheets. The series solution of Graham & Graham (1969), valid for linear shear-layer profiles of arbitrary thickness, provides a basis of comparison for the expansion-method results. Unstable-mode eigenvalues obtained using the two methods are found to be in good agreement for a significant range of values of the ratio of shear-layer thickness to axial wavelength.

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

A remarkably stable type of revolving hollow water jet was generated in a vertical pipe, fixed in the base of a large tank, by first establishing Borda free flow, in which the jet springs clear of the pipe wall. Swirl was then imparted to the oncoming water, and an air core formed in the jet which was of varicose shape with alternate swellings and contractions of both its inner and outer surfaces. The observed wavelengths are compared with the theory, in which inertia and surface tension but not viscosity and gravity are taken into account.

A workshop was held at MIT in February 1986 for specialists performing theoretical research on the interactions of water waves with floating or submerged bodies. The principal applications of this field are related to ship hydrodynamics and to wave loadings on offshore platform. In addition to the traditional approach based on linearization of the wave and body motions, substantial progress has been made on nonlinear problems including both analytical and numerical studies. Subsequent workshops are planned on an annual basis.

We examine disturbances leading to optimal energy growth in a spatially developing, zero-pressure-gradient turbulent boundary layer. The slow development of the turbulent mean flow in the streamwise direction is modelled through a parabolized formulation to enable a spatial marching procedure. In the present framework, conventional spatial optimal disturbances arise naturally as the homogeneous solution to the linearized equations subject to a turbulent forcing at particular wavenumber combinations. A wave-like decomposition for the disturbance is considered to incorporate both conventional stationary modes as well as propagating modes formed by non-zero frequency/streamwise wavenumber and representative of convective structures naturally observed in wall turbulence. The optimal streamwise wavenumber, which varies with the spatial development of the turbulent mean flow, is computed locally via an auxiliary optimization constraint. The present approach can then be considered, in part, as an extension of the resolvent-based analyses for slowly developing flows. Optimization results reveal highly amplified disturbances for both stationary and propagating modes. Stationary modes identify peak amplification of structures residing near the centre of the logarithmic layer of the turbulent mean flow. Inner-scaled disturbances reminiscent of near wall streaks, and amplified over short streamwise distances, are identified in the computed streamwise energy spectra. In all cases, however, propagating modes surpass their stationary counterpart in both energy amplification and relative contribution to total fluctuation energy. We identify two classes of large-scale energetic modes associated with the logarithmic and wake layers of the turbulent mean flow. The outer-scaled wake modes agree well with the large-scale motions that populate the wake layer. For high Reynolds numbers, the log modes increasingly dominate the energy spectra with the predicted streamwise and wall-normal scales in agreement with superstructures observed in turbulent boundary layers.

The period $T$ of the Helmholtz mode in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir is determined in terms of elliptic integrals. It is shown that $T$ decreases monotonically with increasing amplitude $A$ and is within 1% (10%) of the linear limit $T_0$ for $A{/}h_0\,{<}\,0.4(1.0)$, where $h_0$ is the ambient depth.

The rise speed of supercritical carbon dioxide spheres in aqueous surfactant solution was investigated experimentally. Decanonic acid was added to water as a surfactant, and then the rise speed of the CO$_{2}$ spheres was measured at pressures 5 MPa to 10 MPa and at temperatures 28 $^{\circ}$C to 40 $^{\circ}$C generating different phases of spheres, namely gas, liquid, and supercritical. The results revealed that gas bubbles and liquid droplets showed the same behaviour as a rising rigid sphere due to the adsorption of surfactant onto the surface of the spheres. In contrast, due to the absorption of surfactant into the spheres, the drag coefficient of spheres of supercritical fluid was lower than that of a rigid sphere and was similar to that of a fluid sphere with a moving boundary. This behaviour mainly occurred in the rising process of the spheres composed of supercritical CO$_{2}$.

Understanding particle transport and localisation in porous channels, especially at moderate Reynolds numbers, is relevant for many applications ranging from water reclamation to biological studies. Recently, researchers experimentally demonstrated that the interplay between axial and permeate flow in a porous microchannel results in a wide range of focusing positions of finite-sized particles (Garcia & Pennathur, Phys. Rev. Fluids, vol. 2 (4), 2017, 042201). We numerically explore this interplay by computing the lateral forces on a neutrally buoyant spherical particle that is subject to both inertial and permeate forces over a range of experimentally relevant particle sizes and channel Reynolds numbers. Interestingly, we show that the lateral forces on the particle are well represented using a linearised model across a range of permeate-to-axial flow rate ratios. Specifically, our model linearises the effects of the permeate flow, which suggests that the interplay between axial and permeate flow on the lateral force on a particle can be represented as a superposition between the lateral (inertial) forces in pure axial flow and the viscous forces in pure permeate flow. We experimentally validate this observation for a range of flow conditions. The linearised behaviour observed significantly reduces the complexity and time required to predict the migration of inertial particles in permeate channels.

This paper aims to narrow the gap between the Lagrangian and Eulerian descriptions of rotating stratified fluids. To this end, without loss of generality the primitive Lagrangian equations with arbitrary oriented time-dependent rotation and arbitrary stable stratification have been simplified and made more amenable for analysis. The bulk of the work is concerned with developing in parallel exact Lagrangian and Eulerian descriptions of a particular interesting class of motions of rotating stratified incompressible Boussinesq fluids: the vertically uniform columnar motions. The Lagrangian description is confined to ideal fluids, while the Eulerian one includes viscosity and diffusivity. Assuming the rotation axis to be parallel to gravity, with the rotation rate being an arbitrary function of time, and the buoyancy frequency to be constant, it is found that for vertically uniform motions there is always an exact split into horizontal and vertical subsystems. Evolution of the horizontal velocities and displacements is governed by the classical equations of two-dimensional incompressible hydrodynamics, only slightly modified by accounting for the variable rotation rate. These equations are independent of stratification and vertical motions. The Coriolis term is potential and can be incorporated into pressure. The vertical motions represent a manifestation of packets of inertia–gravity waves with strictly horizontal wavevectors, and are exactly described by linear equations independently of the wave amplitudes. They do not depend on rotation, either constant or variable. The wavepackets do not interact with each other or with horizontal motions. For ideal fluids or those with Rayleigh friction there are explicit solutions describing these motions for arbitrary initial conditions. The Cauchy problem for the columnar motions in ideal fluids is found to be well posed. Thus there is a natural extension of well-studied two-dimensional incompressible hydrodynamics which retains the property of the absence of vortex stretching: all two-dimensional flows could be ‘dressed up’ by adding appropriate vertical motions of a rotating stratified fluid. All the columnar motions could be described in such a way. The examined columnar motions exist under arbitrary relations between the parameters of rotation and stratification and, in particular, without rotation. In the limit of strong rotation one recovers the results known in the literature, in particular, under additional assumptions of small amplitude and steadiness of motions the solutions describe the classical Taylor–Proudman columns.

Under suitable conditions, an immersed granular bed can be destabilized by local thermal forcing and the induced buoyant force. The destabilization is evident from the triggering and establishment of a dense fluid-like granular plume. Varying the initial granular layer average height $h$, a time series of the free layer surface is extracted, allowing us to dynamically compute the underlying volume of the granular layer. Different observed phenomena, namely the initial interface deformation, the lowering of the average granular interface (i.e. decrease of the granular layer volume) and the emission of a plume, are analysed. We show that the phenomenon is mainly driven by heat transfer, for large $h$ and also involves a variable height thermal boundary condition and Darcy flow triggering, for small $h$. Simple modelling with no adjustable parameters not only allows us to capture the observed scaling power laws but is also in quantitative agreement with the obtained experimental data.

We investigate mechanisms through which energy cascades from the mesoscale, $O(100~\text{km})$, to the submesoscale, $O(10~\text{km})$, for oceanic fronts in a reduced gravity shallow water model using two different profiles. The first idealization of an ocean front has an interfacial depth that is a smooth hyperbolic tangent profile and is an extension of the piecewise constant potential vorticity profile studied in Boss et al. (J. Fluid Mech., vol. 315, 1996, pp. 65–84). By considering a range of minimum depths, all of which have the same velocity profile, we are better able to isolate the effect of vanishing layer depths. We find that the most unstable mode exists in a one-layer model and does not need two layers, as previously speculated. Moreover, we find that even without a vanishing layer depth there are other modes that appear at both larger and smaller length scales that have a gravity wave structure. The second profile is the parabolic double front from Scherer & Zeitlin (J. Fluid Mech., vol. 613, 2008, pp. 309–327). We find more unstable modes than previously presented and also categorize them based on the mode number. In particular, we find there are pairs of unstable modes that have equal growth rates. We also study the nonlinear evolution of these oceanic fronts. It is determined that vanishing layer depths have significant effects on the unstable dynamics that arise. First, stronger gravity wave fields are generated. Second, cyclonic fluid that moves into the deeper waters is stretched preferentially more in comparison to the deep water scenario and destabilizes more easily. This results in smaller scale vortices, both cyclones and anticyclones, that have length scales in the submesoscale regime. Our results suggest that the nonlinear dynamics of a front can be very efficient at generating submesoscale motions.

The approach of Visbeck, Marshall & Jones (1996) is used to explain the primary results of a recent laboratory study of isolated convection in a two-layer fluid reported by Narimousa (1996).