Steady streaming confined between three-dimensional wavy surfaces

We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σ h where σ ∼ O(1), but long-wavelength variations λ, with h/λ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel’s wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number

Generalized Lagrangian Coordinates for Transport and Two-Phase Flows in Heterogeneous Anisotropic Porous Media

We show how Lagrangian coordinates provide an effective representation of how difficult non-linear, hyperbolic transport problems in porous media can be dealt with. Recalling Lagrangian description first, we then derive some basic but remarkable properties useful for the numerical com- putation of projected transport operators. We furthermore introduce new generalized Lagrangian coordinates with their application to the Darcy–Muskat two-phase flow models. We show how these generalized Lagrangian coordinates can be constructed from the global mass conservation, and that they are related to the existence of a global pressure previously defined in the literature about the subject. The whole representation is developed in two or three dimensions for numerical purposes, for isotropic or anisotropic heterogeneous porous media

Weak-inertial flow between two rough surfaces

“Oseen–Poiseuille” equations are developed from an asymptotic formulation of the three-dimensional Navier–Stokes equations in order to study the influence of weak inertia on flows between rough surfaces. The impact of the first correction on macroscopic flow due to inertia has been determined by solving these equations numerically. From the numerical convergence of the asymptotic expansion to the three-dimensional Navier–Stokes flows, it is shown that, at the macroscopic scale, the quadratic correction to the Reynolds equation in the weak-inertial regime vanishes generalizing a similar result in porous media

Magnetohydrodynamic convectons

Numerical continuation is used to compute branches of spatially localized structures in convection in an imposed vertical magnetic field. In periodic domains with finite spatial period, these branches exhibit slanted snaking and consist of localized states of even and odd parity. The properties of these states are analysed and related to existing asymptotic approaches valid either at small amplitude (Cox and Matthews, Physica D, vol. 149, 2001, p. 210), or in the limit of small magnetic diffusivity (Dawes, J. Fluid Mech., vol. 570, 2007, p. 385). The transition to standard snaking with increasing domain size is explored

Spatially localized magnetoconvection

Numerical continuation is used to compute branches of time-independent, spatially localized convectons in an imposed vertical magnetic field focusing on values of the Chandrasekhar number Q in the range 10 < Q < 103. The calculations reveal that convectons initially grow by nucleating additional cells on either side, but with the build-up of field outside owing to flux expulsion, the convectons are able to transport more heat only by expanding the constituent cells. Thus, at large Q and large Rayleigh numbers, convectons consist of a small number of broad cells

Spatially localized binary fluid convection in a porous medium

The origin and properties of time-independent spatially localized binary fluid convection in a layer of porous material heated from below are studied. Different types of single and multipulse states are computed using numer- ical continuation and the results related to the presence of homoclinic snaking of single and multipulse states

États spatialement localisés dans la convection de double diffusion

Une méthode numérique de continuation est utilisée pour étudier les états stationnaires spatialement localisés dans la convection de double diffusion au sein d'une couche bi-dimensionnelle d'un fluide binaire confiné entre deux parois horizontales munies de conditions aux limites de non-glissement. La concentration du composant le plus lourd est maintenue supérieure à la paroi inférieure et la convection induite par une différence de température imposée entre le haut et le bas. Dans une certaine gamme de paramètres, des états spatialement localisés appelés convectons se forment. L'origine et les propriétés de ces états sont étudiées et reliées au phénomène d'homoclinic snaking

Un nouveau préconditionneur pour les problèmes elliptiques à coefficients variables

On présente dans cette Note un nouveau préconditionneur pour l’inversion du système algébrique issu de la discrétisation par méthode spectrale d’un problème elliptique du second ordre à coefficients variables et non séparables. Ce préconditionneur est construit en discrétisant un problème similaire à l’original et obtenu par moyenne des coefficients. L’inversion du préconditionneur utilise une méthode directe connue sous le nom de diagonalisation successive

Nonlinear Marangoni convection in circular and elliptical cylinders

The spatial organization of single-fluid Marangoni convection in vertical cylinders with circular or elliptical horizontal cross section is described. The convection is driven by an imposed heat flux from above through Marangoni stresses at a free but undeformed surface due to temperature-dependent surface tension. The solutions and their stability characteristics are obtained using branch-following techniques together with direct numerical simulations. The changes in the observed patterns with increasing ellipticity are emphasized. In some cases, the deformation of the cylinder results in the presence of oscillations

Spatially localized states in Marangoni convection in binary mixtures

Two-dimensional Marangoni convection in binary mixtures is studied in periodic domains with large spatial period in the horizontal. For negative Soret coefficients convection may set in via growing oscillations which evolve into standing waves. With increasing amplitude these waves undergo a transition to traveling waves, and then to more complex waveforms. Out of this state emerge stable stationary spatially localized structures embedded in a background of small amplitude standing waves. The relation of these states to the time-independent spatially localized states that characterize the so-called pinning region is investigated by exploring the stability properties of the latter, and the associated instabilities are studied using direct numerical simulation in time