Unconditional Nonlinear Stability in Temperature-Dependent Viscosity Flow in a Porous Medium

The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L ³ or L ⁴ norms. It is also shown that L ² theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.     

饱和纳米流体多孔介质中竖直嵌入板上边界层的非Newton流

在层流条件下,对饱和多孔介质中的竖直板,研究幂指数型非Newton流的自由对流热交换.非Newton纳米流体服从幂指数型的数学模型,模型综合考虑了Brown运动和热泳的影响.通过相似变换,将问题的偏微分控制方程组,转化为常微分方程组,得到了常微分方程组的数值解.数值解依赖于幂指数n,Lewis数Le,浮力比Nr,Brown运动参数Nb,以及热泳参数Nt.在n和Le的不同取值下,研究并讨论了对相关流体性质参数的影响和简化的Nusselt数.     

Infinite Interval Problems Modeling the Flow of a Gas Through a Semi-Infinite Porous Medium

Various existence results are presented for boundary-value problems on the infinite interval. In particular, our theory includes a discussion of a problem arising in the unsteady flow of a gas through a semi–infinite porous medium.     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Local Aronson-Benilan Estimates for Porous Medium Equations under Ricci Flow

In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type inequality.     

非Darcy多孔介质中的非线性对流

利用温度-浓度-密度关系,研究非Darcy多孔介质中的自由对流问题.对于不同的惯性参数、传递参数、Rayleigh数、Lewis数、Soret数和Dufour数,分析了非线性温度参数和浓度参数对非线性对流的影响.浮力对对流起着辅助的附加作用,当惯性作用不计时,切向速度随着非线性温度和浓度的增加而急剧地增加.然而,当惯性效应不为0时,非线性温度和浓度对切向速度的影响是有限的.对两个传递参数、惯性影响参数以及控制非线性温度和浓度的其他参数,取不同的数值时,浓度分布有点儿变化,并在不同的范围内传播.随着非线性温度和浓度的增加,传热/传质在很大的范围内变化,这取决于是Dacry多孔介质,还是非Darcy多孔介质.当所有的影响(惯性的影响、两个传递系数的影响、Soret和Dufour的影响)同时为0/不为0,在非线性温度/浓度参数以及浮力的共同作用下,分析了传热/传质的变化.发现在Darcy多孔介质中,温度和浓度以及它们的交叉扩散,对传热/传质的影响,要比非Darcy多孔介质要大.发现了浮力的负面作用,随着非线性温度系数的增加,传热/传质率是提高的,而随着非线性浓度系数的增加,传热/传质率是下降的.     

Computation of Travelling Combustion Waves in a Porous Medium

Numerical solutions for travelling combustion waves in a porous medium are sought. The algorithm of computation is based on a shooting method used in an existence proof. The numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.     

Traveling Wave Solutions for Combustion in a Porous Medium

Traveling wave solutions are sought for a model of combustion in a porous medium. The problem is formulated as a nonlinear eigenvalue problem for a system of ordinary differential equations of order four, defined over an infinite interval. A shooting method is used to prove existence, and a priori bounds for the solution and parameters are obtained.     

多孔介质中的粘性流体在径向伸展薄片间作磁流体动力学流动时的Dufour和Soret效应

在一个充满不可压缩、粘性、导电流体的多孔介质空间中,以两个无限伸展的薄片为边界,研究Dufour和Sorer数对其间二维磁流体动力学稳定流动的影响,数学分析是在有粘性耗散、Joule热和一级化学反应下进行.通过适当的变换,将动量、能量和浓度定律所表示的偏微分控制方程组,变换为常微分方程组.利用同伦分析法(HAM)求解该方程组,保证了级数解的收敛性.分析了显现参数对无量纲速度、温度和浓度场的影响,同时对表面摩擦因数、Nusselt数和Sherwood数的影响进行了分析.     

分形多孔介质中的奇异扩散

分形多孔介质和均质多孔介质相比具有许多特殊的性质,它在各个不同的尺度上有相互钳套的自相似结构.孔隙分形中的粒子扩散和经典的Fick扩散不同,其均方位移服从分形幂律关系.据此对孔隙分形中的粒子扩散利用随机过程的统计方法建立了奇异扩散的理论模型,讨论了奇异扩散的非马尔可夫性质和分形性质.     

Unsteady Porous Flow with a Free Surface

Three different numerical methods for solving unsteady two-dimensionalporous flow problems with a free surface are presented. Thevelocity potential is expressed as the solution to a variationalproblem which is solved by a Rayleigh-Ritz expansion, a Kantorovichexpansion and a co-ordinate transformation method. The freesurface equation is solved by a Crank-Nicolson procedure. Thethree methods were tested on the same set of problems and theobtained results are virtually identical.     

填充烧结铜球的T型管中流体混合的大涡模拟

在FLUENT软件平台上,运用大涡模拟湍流模型及Smagorinsky-Lilly亚格子尺度模型,对填充有烧结铜球多孔介质的T型管道内冷热流体混合过程的流动与传热情况进行了数值计算,与未填充多孔介质时混合区域内的平均温度和温度波动、平均速度和速度波动等数据进行了对比,并对温度波动进行了功率谱密度分析．数值结果表明,多孔介质可有效削弱T型通道流体混合区域内的温度和速度波动,有效降低1 Hz至10 Hz频域中的温度波动的功率谱密度．     

多孔介质中的B rinkman-Forchheimer模型的收敛性

本文研究Brinkman-Forchheimer方程组的结构稳定性.首先我们得到一些有用的先验界,然后在此基础上推出解所满足的微分不等式,最后建立了解对Brinkman系数v的收敛性结果.     

A General Fractional Porous Medium Equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: \input amssym $$\left\{ {\matrix{ {{{\partial u} \over {\partial t}} + \left( { ‐ \Delta } \right)^{\sigma /2} \left( {\left| u \right|^{m ‐ 1} u} \right) = 0,} \hfill & {x \in {\Bbb R} ^N ,\,\,t > 0,} \hfill \cr {u\left( {x,0} \right) = f\left( x \right),} \hfill & {x \in {\Bbb R} ^N .} \hfill \cr } } \right.$$ We consider data \input amssym $f\in L^1(\Bbb{R}^N)$ and all exponents $0<\sigma<2\;and\;m>0$ . Existence and uniqueness of a strong solution is established for $ m > {m_\ast}={(N-\sigma)_+}/N$ , giving rise to an L¹‐contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range ${0 < m} \le {m_\ast}$ existence and uniqueness happen under some restrictions, and the properties of the solutions are different from the ones for the case above m_*. We also study the dependence of solutions on f, m, and σ. Moreover, we consider the above questions for the problem posed in a bounded domain. © 2012 Wiley Periodicals, Inc.     

Evolution of a Condensation Surface in a Porous Medium near the Instability Threshold

We consider the dynamics of a narrow band of weakly unstable and weakly nonlinear perturbations of a plane phase transition surface separating regions of soil saturated with water and with humid air; during transition to instability, the existing stable position of the phase transition surface is assumed to be sufficiently close to another phase transition surface that arises as a result of a turning point bifurcation. We show that such perturbations are described by a Kolmogorov–Petrovskii–Piskunov type equation.     

A Comparison Theorem for a Nonlinear system Arising in Porous Medium Combustion

Comparison theorems for the initial value finite domain one dimensional heat equation with a discontinuous forcing term are extended to a coupled system of a heat equation and an ordinary differential equation in space, rather than the usual ordinary differential equation in time, that arises in combustion theory.     

Numerical Analysis of the Interphase Interaction in a Porous Elastic Fluid-Saturated Medium

We consider dynamical processes in a two-phase porous fluid-saturated medium. The equation of the Biot–Frenkel model, which accounts for the influence of the elastic, inertial, and viscous interaction between the liquid and solid phases, is used for modeling the dynamics of the soil layer (flat deformation) with the finite-element method in cases of steady and nonstationary effects. For a layer under a uniaxial stress, we give numerical examples of the displacements of the soil skeleton and interstitial fluid.