Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

Numerical simulations of water–gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water–gas equations efficiently and accurately in nuclear underground waste disposal situations.     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.     

Global mode method for dynamic modeling of a flexible-link flexible-joint manipulator with tip mass

In this paper, the global mode method (GMM) is proposed to obtain a reduced-order analytical dynamic model for a signal flexible-link flexible-joint (SFF) manipulator. Firstly, the nonlinear partial differential equations (PDE) that govern the motion of the flexible link and flexible joint, respectively, are derived by applying the Hamilton principle. By combining the linearized governing equations of motion for a flexible link and the equation of motion for the flexible joint, the characteristic equation is obtained for the whole system. The natural frequencies and global mode shapes of the linearized model of the SFF manipulator are determined, and orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to truncate the nonlinear PDEs of the SFF manipulator to a nonlinear ordinary differential equation with a few degrees-of-freedom (DOF). For comparison, two other dynamic models of the SFF are derived by employing the assumed mode method (AMM) and finite element method (FEM). To verify the method proposed, the results from the GMM are compared with those obtained from the FEM. The effects of the link length and payload mass on the convergence of AMM model for the first two frequencies are investigated. Based on the dynamic models, obtained by GMM and AMM, dynamical responses for the system with different numbers of modes are worked out numerically, which are compared with those obtained from FEM. These comparisons show a good agreement between the results of the GMM and that of the FEM model, which indeed proved the accuracy and applicability of the GMM model.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Global well-posedness for a drug transport model in tumor multicell spheroids

In this paper we study global existence of solutions of a mathematical model for drug transport in tumor multicell spheroids. The model is a free boundary problem of a system of partial differential equations. It contains one nonlinear first-order equation describing the distribution of live tumor cells, and two nonlinear reaction diffusion equations describing the evolution of nutrient concentration and drug concentration, respectively. By using the method of characteristics for first-order equations, the $L^p$-theory for parabolic equations, the Banach fixed point theorem and the extension method, we prove that this problem has a unique global solution.     

Homogenization of compressible two-phase two-component flow in porous media

The paper is devoted to the homogenization of immiscible compressible two-phase two-component flow in heterogeneous porous media. We consider liquid and gas phases, two-component (water and hydrogen) flow in a porous reservoir with periodic microstructure, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. Phase exchange, capillary effects included by the Darcy–Muskat law and Fickian diffusion are taken into account. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water obeying the Henry law. The flow is then described by the conservation of the mass for each component. The microscopic model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion–convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using the two-scale convergence.     

精确计算Bingham流体圆管层流压降及速度分布规律的新方法

Bingham(宾汉)模型情况下,多采用通用公式进行圆管层流压降的解析计算,即将Bingham模型本构方程代入粘性流体圆管层流流动通用公式进行计算,仅能得到压降的解析解.新方法结合Bingham流体本构方程与运动方程,建立有关力学平衡方程,并运用代数方程的根式解理论对圆管层流流动时的非线性方程进行求解,可直接求得Bingham流体圆管层流压降及速度流核区半径的解析解,进一步可求得圆管层流速度解析解;Bingham流体圆管层流速度的直接影响因素为流量、塑性粘度和屈服值,研究发现速度流核宽度与屈服值成正比,与流量及塑性粘度成反比,且流核的宽度越大,流核区的速度越小.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Initial-boundary-value problem of hyperbolic equations for blood flow in a vessel

In this paper, we study an initial-boundary-value problem of a system of hyperbolic, partial-differential equations that models blood flow in a vessel. The one-spatial-dimensional model assumes that blood flow in the vessel is an incompressible, homogeneous, Newtonian fluid which has a small Womersley number. Boundary conditions with either the pressure or the flow rate at each end of the vessel are considered, and the existence of the global solution is obtained using a form of Glimm's finite-difference scheme.     

Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation

We consider a semilinear wave equation, defined on a two-dimensional bounded domain Ω, with a nonlinear dissipation. Our main result is that the flow generated by the model is attracted by a finite dimensional global attractor. In addition, this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation. To our knowledge this is a first result of this type in the context of higher dimensional wave equations.     

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Pointwise structure of a radiation hydrodynamic model in one-dimension

In this paper, we study the nonlinear stability and the pointwise structure around a constant equilibrium for a radiation hydrodynamic model in 1-dimension, in which the behavior of the fluid is described by a full Euler equation with certain radiation effect. It is well-known that the classical solutions of the Euler equation in 1-D may blow up in finite time for general initial data. The global existence of the solution in this paper means that the radiation effect stabilizes the system and prevents the formation of singularity when the initial data is small. To study the precise effect of the radiation in this model, we also treat the pointwise estimates of the solution for the original nonlinear problem by combining the Green's function for the linearized radiation hydrodynamic equations with the Duhamel's principle. The result in this paper shows that the pointwise structure of this model is similar to that of full Navier-Stokes equations in 1-D.     

Finite difference scheme of a model for nonlinear wave-wave interaction in ionic media

In the paper, the coupled 1D nonlinear Schr?dinger system (CNLS) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations. A new six-point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations.