Longtime behavior of vertical throughflows for binary mixtures in porous layers (original) (raw)
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On the stability of vertical constant throughflows for binary mixtures in porous layers
International Journal of Non-Linear Mechanics, 2014
A system modeling fluid motions in horizontal porous layers, uniformly heated from below and salted from above by one salt, is analyzed. The definitely boundedness of solutions (existence of absorbing sets) is proved. Necessary and sufficient conditions ensuring the linear stability of a vertical constant throughflow have been obtained via a new approach. Moreover, conditions guaranteeing the global non-linear asymptotic stability are determined.
Instability of Vertical Constant Through Flows in Binary Mixtures in Porous Media with Large Pores
Mathematical Problems in Engineering
A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical fluid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability are found. When the layer is salted from above, a condition guaranteeing the occurrence of “cold” instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.
Transport in Porous Media, 2018
We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical highfrequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios ϕ, vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number Ra c is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain Ra c = 12/(ϕ(Le + 1) + 1) for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.
Unconditional nonlinear stability for convection in a porous medium with vertical throughflow
Acta Mechanica, 2007
Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.
Long-time behaviour of multi-component fluid mixtures in porous media
International Journal of Engineering Science, 2010
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.
International Journal of Non-Linear Mechanics, 2012
The long-time behavior of triply fluid mixtures saturating horizontal porous layers uniformly rotating around the vertical axis, according to the Brinkman law (holding for large pores), is investigated. The most destabilizing case of layers heated from below and salted from above by two salts, is considered. The ultimately boundedness of the solutions (existence of absorbing sets) is shown. A necessary and sufficient condition guaranteeing the global non-linear asymptotic L 2 -stability of the conduction solution is obtained.
On the stability of double diffusive convection in a porous layer with throughflow
Acta Mechanica, 2001
The effect of throughflow on the stability of double diffusive convection in a porous layer is investigated for different types of hydrodynamic boundary conditions. The lower and upper boundaries are assumed to be insulating to temperature and concentration perturbations. The resulting eigenvalue problem is solved by the Galerkin technique. The curvature of the basic temperature as well as solute concentration gradients significantly affects the stability of the system. It is observed that, for a suitable choice of parametric values, Hopf bifurcation occurs always prior to direct bifurcation, and the throughflow alters the nature of bifurcation. In contrast to the single component system, it is found that throughflow is (a) destabilizing even if the lower and upper boundaries are of the same type, and (b) stabilizing as well as destabilizing, irrespective of its direction, when the boundaries are of different types.
Non-linear stability in the B�nard problem for a double-diffusive mixture in a porous medium
Mathematical Methods in the Applied Sciences, 2001
The linear and non-linear stability of a horizontal layer of a binary uid mixture in a porous medium heated and salted from below is studied, in the Oberbeck-Boussinesq-Darcy scheme, through the Lyapunov direct method. This is an interesting geophysical case because the salt gradient is stabilizing while heating from below provides a destabilizing e ect. The competing e ects make an instability analysis di cult. Unconditional non-linear exponential stability is found in the case where the normalized porosity is equal to one. For other values of a conditional stability theorem is proved. In both cases we demonstrate the optimum result that the linear and non-linear critical stability parameters are the same whenever the Principle of Exchange of Stabilities holds.
Journal of Mathematical Fluid Mechanics, 2013
The problem is considered of thermal convection in a saturated porous medium contained in an infinite vertical channel with differentially heated sidewalls. The theory employed allows for different solid and fluid temperatures in the matrix. Nonlinear energy stability theory is used to derive a Rayleigh number threshold below which convection will not occur no matter how large the initial data. A generalized nonlinear analysis is also given which shows convection cannot occur for any Rayleigh number provided the initial data is suitably restricted.