基于分数阶本构模型的梳状反常扩散与热传导研究

发布时间：2018-01-02 00:14

本文关键词：基于分数阶本构模型的梳状反常扩散与热传导研究　出处：《北京科技大学》2017年博士论文　论文类型：学位论文

【摘要】：反常扩散与热传导是重要的研究领域,在科学和工程上有非常广泛的应用.本论文研究的反常扩散是基于梳状模型,它是一种特殊形式的随机行走,其特殊性在于沿着x方向的运输只可能发生在支柱上,沿着y方向的运输垂直于支柱.它有广泛的应用前景,涉及模拟癌细胞的扩散、渗流集群的扩散、沿着尖刺状树突的扩散,量子力学的研究等等,吸引了很多学者对其进行研究.Fick定律和Fourier定律是研究传热传质问题的最基本的定律,其本构模型呈现线性关系.但是,它们对应着无穷的传播速度,这违背了因果关系准则.本论文从两个方面修正经典的本构模型,一是引进松弛参数,克服了经典本构模型的不足,使得新形成的控制方程同时具有抛物和双曲特性,二是引进分数阶算子,所形成的方程由局部的微分形式转变成非局域的积分形式,使得传递过程同时具有记忆特性和非局域特性.本论文将修正的分数阶本构模型应用到梳状反常扩散与热传导研究中,主要分为两部分,一部分是将时间和空间分数阶Fick模型,时间分数阶Cattaneo模型,一维和二维分数阶Cattaneo-Christov模型分别应用到梳状反常扩散的研究中;另一部分是将一维分数阶Cattaneo-Christov模型应用到热传导模型的研究中.应用解析方法和数值方法对控制方程求解,解析方法采用积分变换方法,用到了 Lapace变换和Fourier变换,数值方法采用数值差分方法,其中,时间分数阶导数的离散用L1定义和L2定义近似,空间分数阶导数用移位的Grunwald公式近似.通过所求的解画图,重点用图形分析不同参数对粒子或者温度分布以及x轴上粒子总数与均方位移的影响,并对其物理特性进行详细的分析与讨论。
[Abstract]:Anomalous diffusion and heat conduction are important research fields and widely used in science and engineering. The anomalous diffusion in this paper is based on comb model and is a special form of random walk. Its particularity lies in that the transportation along x direction can only occur on the pillar, and the transportation along the y direction is perpendicular to the pillar. It has a wide application prospect, which involves simulating the diffusion of cancer cells and the diffusion of percolation clusters. Along the spiny dendrites diffusion, the study of quantum mechanics and so on, attracted many scholars to study it. Fick's law and Fourier's law are the most basic laws to study heat and mass transfer. Their constitutive models are linear. However, they correspond to infinite propagation velocity, which violates the causality criterion. In this paper, the classical constitutive model is modified from two aspects, one is the introduction of relaxation parameters. It overcomes the shortcomings of the classical constitutive model and makes the newly formed governing equations have parabolic and hyperbolic properties at the same time. The second is the introduction of fractional order operators. The resulting equation is transformed from a local differential form to a nonlocal integral form. In this paper, the modified fractional constitutive model is applied to the study of comb anomalous diffusion and heat conduction, which is divided into two parts. Part of the time and space fractional Fick model, time fractional order Cattaneo model. The one-dimensional and two-dimensional fractional Cattaneo-Christov models are applied to the study of comb anomalous diffusion, respectively. In the other part, the one-dimensional fractional Cattaneo-Christov model is applied to the study of heat conduction model, and the analytical method and numerical method are used to solve the governing equation. The analytical method uses integral transformation, Lapace transform and Fourier transform, and numerical method uses numerical difference method. The discretization of time fractional derivative is approximated by L1 definition and L2 definition, and the spatial fractional derivative is approximated by shifted Grunwald formula. The effects of different parameters on particle or temperature distribution and the total number of particles and mean square displacement on x axis are analyzed and discussed in detail.
【学位授予单位】：北京科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：TK124

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