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Abstract: In this work, we propose a new approach to modeling multiphase flow and solute transport through a stack of thin porous layers. Currently, numerical simulation of thin layers involves discretization across the layer thickness. In our new approach, thin porous layers are treated as a bunch of two-dimensional (2D) interacting continua. Macroscale balance laws are formulated in terms of thickness-averaged material properties. A number of exchange terms are employed to account for exchanges of mass, momentum, energy, and entropy between two neighboring layers. The entropy inequality is then exploited for obtaining constitutive equations to close the problem under study. As an example, simplified governing equations are derived for a system of air–water flow and heat transfer through two thin porous layers. In comparison to previous macroscale models, our model possesses the following distinctive advantages: (1) it is rigorous thermodynamics-based model; (2) it is formulated in terms of layer-thickness-averaged material properties which are usually easily measurable; and (3) it reduces 3D modeling to 2D leading to a very significant reduction of computational efforts. [Copyright &y& Elsevier]

Abstract: Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper''s methods. [Copyright &y& Elsevier]