A network model for two-fluid,two-phase flow

A systematic derivation is given of two-fluid conservation laws on an arbitrary network. These laws are shown to conserve the mass and total energy of the mixture in the network. Numerical simulations are presented to illustrate their use.     

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

A new model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media is considered. The main feature of this model is the introduction of a new global pressure and the full equivalence to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation (the global pressure equation) and a nonlinear diffusion-convection one (the saturation equation). Under some realistic assumptions on the data, we show an existence result with the help of appropriate regularizations and a time discretization. We use suitable test functions to get a priori estimates. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.     

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ(t) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ(t) and the heat flux vector q (t). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated ?ojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.     

A New Finite Element Space for Expanded Mixed Finite Element Method

In this article, we propose a new finite element space Λ$_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space Λ$_h$ is designed in such a way that the strong requirement V$_h\subset$Λ$_h$ in [9] is weakened to {v$_h\in$V$_h$; divv$_h$=0}$\subset$Λ$_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new Λ$_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d, d=2,3$ and for triangular partitions in $\mathbb{R}^2$. Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.     

Quenching rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.     

A blow-up criterion for a 2D viscous liquid-gas two-phase flow model

In this paper, we prove a blow-up criterion in terms of the upper bound of the liquid mass for the strong solution to the two-dimensional (2D) viscous liquid-gas two-phase flow model in a smooth bounded domain. The result also applies to three-dimensional (3D) case.     

Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.     

Finite time blow-up for the harmonic map heat flow

Summary We consider the harmonic map heat flow from the three-dimensional ball to the two-sphere. We establish the existence of regular initial data leading to blow-up in finite time.     

A hyperbolic model of two-phase flow: global solutions for large initial data

The paper deals with a simple nonlinear hyperbolic system of conservation laws modeling the flow of an inviscid fluid. The model is given by a standard isothermal p-system of the gasdynamics, for which phase transitions of the fluid are taken into consideration via a third homogeneous equation. We focus on the case of initial data consisting of two different phases separated by an interface. By means of an adapted version of the front tracking algorithm, we prove the global-in time existence of weak entropic solutions under suitable assumptions on the (possibly large) initial data.     

Linear and nonlinear Finite Element modeling of coupled electro-magneto-mechanical multifield problems

In this paper we present the theoretical background and application of Finite Element algorithms for linear and nonlinear problems of multiple field coupling. They enable the prediction of the electromagnetomechanical behavior of materials and structures and supply useful tools for the optimization of multifunctional composites. First, linear three-field coupling is presented within the context of a Finite Element implementation. Then, a homogenization procedure is discussed. Finally, a micromechanical model for nonlinear ferroelectric constitutive behavior and its numerical realization are outlined. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fully coupled method for simulation of wave-current-seabed systems

Coastal flow involves surface wave propagation, current circulation, and seabed evolution, and its prediction remains challenging when they strongly interact with each other, especially during extreme events such as tsunami and storm surge. We propose a fully coupled method to simulate motion of wave-current-seabed systems and associated multiphysics phenomena. The wave action equation, the shallow water equations, and the Exner equation are respectively used for wave, current, and seabed morphology, and the discretization is based on a second-order, flux-limiter, finite difference scheme previously developed for current-seabed systems. The proposed method is tested with analytical solutions, laboratory measurements, and numerical solutions obtained with other schemes. Its advantages are demonstrated in capturing interplay among wave, current, and seabed; it has the capability of first-order upwind schemes to suppress artificial oscillations as well as the accuracy of second-order schemes in resolving flow structures.     

Finite time blow-up for the Yang-Mills heat flow in higher dimensions

We consider the -gradient flow associated with the Yang-Mills functional, the so-called Yang-Mills heat flow. In the setting of a trivial principal SO(n)-bundle over in dimension n greater than 4, we show blow-up in finite time for a class of SO(n)-equivariant initial connections. Received November 18, 1999; in final form December 23, 1999 / Published online February 5, 2001     

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.     

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.     

A three-phase FE-model for the simulation of partially saturated soils

While for many numerical simulations in geotechnics use of a two-phase model is sufficient, separate consideration of all three phases is mandatory for numerical simulations of partially saturated soils subjected to compressed air. This is a common technique frequently applied for temporary ground support in tunnelling. For the numerical simulation of tunnelling using compressed air, a multiphase model for soft soils is developed, in which the individual constituents of the soil – the soil skeleton, the fluid and the gaseous phase – and their interactions are considered. The three phase model is formulated within the framework of the Theory of Porous Media (TPM), based upon balance equations and constitutive relations for the soil constituents and their mixture. Water is modelled as an incompressible and air as a compressible phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media

When air or oxygen is injected into a petroleum reservoir, and oxidation or combustion is induced, a combustion front forms if heat loss to the surrounding rock formation is negligible. Here, we employ a simple model for combustion, which takes into account oil viscosity reduction, but neglects gas density dependence on temperature and uses a simplified oxidation reaction. We show that for small heat loss, this combustion front is actually the lead part of a pulse, while the trailing part of the pulse is a slow cooling process. If the heat loss is too large, we show that such a pulse does not exist. The proofs use geometric singular perturbation theory and center manifold reduction.Dedicated to Constantine Dafermos on his 60^th birthdayThis work was supported in part by: CNPq under Grant 300204/83-3; CNPq/NSF under Grant 91.0011/99-0; MCT under Grant PCI 650009/97-5; FINEP under Grant 77.97.0315.00; FAPERJ under Grants E-26/150.936/99 and E-26/151.893/2000; NSF under Grant DMS-9973105.