本文關(guān)鍵詞：基于分?jǐn)?shù)階本構(gòu)模型的梳狀反常擴(kuò)散與熱傳導(dǎo)研究　出處：《北京科技大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 分?jǐn)?shù)階導(dǎo)數(shù) 反常擴(kuò)散 熱傳導(dǎo) 梳狀模型 本構(gòu)模型

【摘要】：反常擴(kuò)散與熱傳導(dǎo)是重要的研究領(lǐng)域,在科學(xué)和工程上有非常廣泛的應(yīng)用.本論文研究的反常擴(kuò)散是基于梳狀模型,它是一種特殊形式的隨機(jī)行走,其特殊性在于沿著x方向的運(yùn)輸只可能發(fā)生在支柱上,沿著y方向的運(yùn)輸垂直于支柱.它有廣泛的應(yīng)用前景,涉及模擬癌細(xì)胞的擴(kuò)散、滲流集群的擴(kuò)散、沿著尖刺狀樹突的擴(kuò)散,量子力學(xué)的研究等等,吸引了很多學(xué)者對(duì)其進(jìn)行研究.Fick定律和Fourier定律是研究傳熱傳質(zhì)問(wèn)題的最基本的定律,其本構(gòu)模型呈現(xiàn)線性關(guān)系.但是,它們對(duì)應(yīng)著無(wú)窮的傳播速度,這違背了因果關(guān)系準(zhǔn)則.本論文從兩個(gè)方面修正經(jīng)典的本構(gòu)模型,一是引進(jìn)松弛參數(shù),克服了經(jīng)典本構(gòu)模型的不足,使得新形成的控制方程同時(shí)具有拋物和雙曲特性,二是引進(jìn)分?jǐn)?shù)階算子,所形成的方程由局部的微分形式轉(zhuǎn)變成非局域的積分形式,使得傳遞過(guò)程同時(shí)具有記憶特性和非局域特性.本論文將修正的分?jǐn)?shù)階本構(gòu)模型應(yīng)用到梳狀反常擴(kuò)散與熱傳導(dǎo)研究中,主要分為兩部分,一部分是將時(shí)間和空間分?jǐn)?shù)階Fick模型,時(shí)間分?jǐn)?shù)階Cattaneo模型,一維和二維分?jǐn)?shù)階Cattaneo-Christov模型分別應(yīng)用到梳狀反常擴(kuò)散的研究中;另一部分是將一維分?jǐn)?shù)階Cattaneo-Christov模型應(yīng)用到熱傳導(dǎo)模型的研究中.應(yīng)用解析方法和數(shù)值方法對(duì)控制方程求解,解析方法采用積分變換方法,用到了 Lapace變換和Fourier變換,數(shù)值方法采用數(shù)值差分方法,其中,時(shí)間分?jǐn)?shù)階導(dǎo)數(shù)的離散用L1定義和L2定義近似,空間分?jǐn)?shù)階導(dǎo)數(shù)用移位的Grunwald公式近似.通過(guò)所求的解畫圖,重點(diǎn)用圖形分析不同參數(shù)對(duì)粒子或者溫度分布以及x軸上粒子總數(shù)與均方位移的影響,并對(duì)其物理特性進(jìn)行詳細(xì)的分析與討論。
[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.
【學(xué)位授予單位】：北京科技大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：TK124

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