Hydrodynamic and hydromagnetic stability

Three‐dimensional flows of an inviscid incompressible fluid and an inviscid subsonic compressible gas are considered and it is demonstrated how the WKB method can be used for investigating their stability. The evolution of rapidly oscillating initial data is considered and it is shown that in both cases the corresponding flows are unstable if the transport equations associated with the wave which is advected by the flow have unbounded solutions. Analyzing the corresponding transport equations, a number of classical stability conditions are rederived and some new ones are obtained. In particular, it is demonstrated that steady flows of an incompressible fluid and an inviscid subsonic compressible gas are unstable if they have points of stagnation.

Local stability conditions in fluid dynamics

Previous studies of fluid convection in porous media have considered the onset of convection in isotropic systems and the steady convection in anisotropic systems. This paper bridges between these and develops new results for the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. These results are relevant to sedimentary formations where the average vertical permeability is some fraction γ of the average horizontal permeability. Linear and global stability analyses are used to define the critical time tc at which the instability occurs as a function of γ and the dimensionless Rayleigh-Darcy number Ra* for both thermal and solute-driven convection in an infinite horizontal slab. Numerical results and approximate analytical solutions are obtained for both a slab of finite thickness and the limit of large slab thickness. For a thick slab, the increase in tc as γ decreases is approximately given by (1+γ)4∕(16γ2). One important application is to the geological storage of carbon dioxide where it is shown that the use of an effective vertical permeability in estimating the critical time is only valid for low permeabilities. The time scale for the onset of convection in geological storage can range from less than a year (for high-permeability formations) to decades or centuries (for low-permeability ones).

Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions

Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers

Numerical study of natural convection in a vertical porous annulus with discrete heating

This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. The authors derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.

An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

Summary and conclusions The present work is the first study of Rayleigh-Be'nard convection in an anisotropic porous medium with oblique principal axes. The analysis is restricted to transversely isotropic media with isotropic thermal conductivity. Qualitatively new flow patterns occur at the onset of convection. If the transverse permeability is larger than the longitudinal permeability, the planes of motion are tilted, but the cell walls are vertical as usual. On the other hand, if the longitudinal permeability is the larger one, the flow occurs in vertical planes, but the cell walls are tilted. 'The preference for these different patterns is explained as a preference for flow directions with as small a tangential permeability as possible. This preference also gives rise to a tendency of concentration of the flow along the cell boundaries when the anisotropy increases, in the case when the cell walls are tilted. The tilt is primarily of mechanical origin, because a vertical forcing due

Onset of convection in an anisotropic porous medium with oblique principal axes

Effects of anisotropy on convective flow through porous media

A necessary condition for linear stability of steady inviscid helical gas flows is found by the generalized progressing-wave expansion method The criterion obtained is compared with the known Richardson number criteria giving sufficient conditions for stability

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A Note on the Stability of Steady Inviscid Helical Gas Flows

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.

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L. Storesletten

Papers

An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

Onset of convection in an anisotropic porous medium with oblique principal axes

Effects of anisotropy on convective flow through porous media

A Note on the Stability of Steady Inviscid Helical Gas Flows

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer