Roll convection of binary fluid mixtures in porous media

Abstract

We investigate theoretically the nonlinear state of ideal straight rolls in the RayleighBénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh-Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation of the streamlines is explained by means of the Darcy equation which is used to describe the transport of momentum. In addition to the properties of the rolls, their stability against arbitrary infinitesimal perturbations is investigated. We compute stability balloons for the pure fluid case as well as for a wide parameter range of Lewis numbers and separation ratios which are typical for binary gas and fluid mixtures. The stability regions of rolls are found to be restricted by a crossroll, a zigzag and a new type of oscillatory instability mechanism, which can be related to the crossroll mechanism.

Three-dimensional spatially localized binary-fluid convection in a porous medium

“…Finally, with the boundary conditions (1.5) the equations are also equivariant with respect to the midplane reflection z → 1 − z. The latter symmetry favours roll-like patterns (Golubitsky, Swift & Knobloch 1984;Umla et al 2010) and therefore plays an essential role in the observed patterns.…”

Three-dimensional convection of binary mixtures in porous media

“…For more details see Ref. [6]. We consider two infinite horizontal parallel plates perpendicular to a homogeneous gravitational field in z direction.…”

“…One can choose ≡ 0 and neglect Eq. (2.8a) as long as no transient behavior is investigated and there is no mean flow in the system [6]. The boundary condition on u implies = 0 at z = ± 1 2 .…”

“…[5]), it is also of interest to investigate Bénard rolls for binary fluids as has been done in Refs. [6,7]. In binary mixtures, the concentration field changes the spatiotemporal properties of convection because concentration differences lead to density gradients and thus contribute to the buoyancy.…”

“…Whereas advection and diffusion tend to equilibrate concentration gradients, the Soret effect sustains concentration differences being relevant for the structure and the dynamics of convection. The present authors have investigated mixtures with a positive Soret effect that saturate a porous medium [6]. Therein the heavier component is driven into the direction of lower temperatures.…”