Non-linear stability in the B�nard problem for a double-diffusive mixture in a porous medium (original) (raw)
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The long-time behaviour of a triply convective-diffusive fluid mixture saturating a porous horizontal layer in the Darcy-Oberbeck-Boussinesq scheme, is investigated. It is shown that the L 2-solutions are bounded, uniquely determined (by the initial and boundary data) and asymptotically converging toward an absorbing set of the phase-space. The stability analysis of the conduction solution is performed. The linear stability is reduced to the stability of ternary systems of O.D.Es and hence to algebraic inequalities. The existence of an instability area between stability areas of the thermal Rayleigh number (''instability island"), is found analytically when the layer is heated and ''salted" (at least by one ''salt") from below. The validity of the ''linearization principle" and the global nonlinear asymptotic stability of the conduction solution when all three effects are either destabilizing or stabilizing, are obtained via a symmetrization.
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Carbon dioxide injected into saline aquifers dissolves in the resident brines increasing their density, which might lead to convective mixing. Understanding the factors that drive convection in aquifers is important for assessing geological CO 2 storage sites. A hydrodynamic stability analysis is performed for non-linear, transient concentration fields in a saturated, homogenous, porous medium under various boundary conditions. The onset of convection is predicted using linear stability analysis based on the amplification of the initial perturbations. The difficulty with such stability analysis is the choice of the initial conditions used to define the imposed perturbations. We use different noises to find the fastest growing noise as initial conditions for the stability analysis. The stability equations are solved using a Galerkin technique. The resulting coupled ordinary differential equations are integrated numerically using a fourth-order Runge-Kutta method. The upper and lower bounds of convection instabilities are obtained. We find that at high Rayleigh numbers, based on the fastest growing noise for all boundary conditions, both the instability time and the initial wavelength of the convective instabilities are independent of the porous layer thickness. The current analysis provides approximations that help in screening suitable candidates for homogenous geological CO 2 sequestration sites.
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The non-linear stability of plane parallel shear ows in an incompressible homogeneous uid heated from below and saturating a porous medium is studied by the Lyapunov direct method. In the Oberbeck-Boussinesq-Brinkman (OBB) scheme, if the inertial terms are negligible, as it is widely assumed in literature, we ÿnd global non-linear exponential stability (GNES) independent of the Reynolds number R. However, if these terms are retained, we ÿnd a restriction on R (depending on the inertial convective coe cient) both for a homogeneous uid and a mixture heated and salted from below. In the case of a mixture, when the normalized porosity is equal to one, the laminar ows are GNES for small R and for heat Rayleigh numbers less than the critical Rayleigh numbers obtained for the motionless state.
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The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initialvalue problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285-303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107) . For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.