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# 物理代写|热力学代写Thermodynamics代考|EGM-321 Maximum Economy: Yardangs, Rivers and the Human Blood

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## 物理代写|热力学代写Thermodynamics代考|Maximum Economy: Yardangs, Rivers and the Human Blood

We have discussed minimization of viscous heating power for Newtonian fluids. But even if human blood is non-Newtonian, when it comes to the circulation of the blood the general principle […] is that the form and the arrangement of the blood vessels are such that the circulation proceeds with a minimum of effort, and with a minimum of wall-surface, the latter condition leading to a minimum of friction and being therefore included in the first and, more generally we assume that the cost of operating a physiological system is a minimum, what we mean by the cost being measurable in calories and ergs, units whose dimensions are equivalent to those of work-see Chap. XV of [30].

This minimization, which is based on Murray’s tenet of ‘maximum economy’ $[31],{ }^{27}$ is rigorously proven nowhere, but all the experience and the very instinct of the physiologist tells him it is true; he comes to use it as a postulate […] and it does not lead him astray [30] and leads to correct predictions concerning the branching and the size of arteries. Since the temperature of the human body is constant and uniform, minimization of the amount of power dissipated through viscosity is equivalent to minimization of the amount of entropy produced per unit time by viscous dissipation. ${ }^{28}$

Even if $\operatorname{Re} \gg 1$, the idea that minimum friction rules the development of structures in a system with negligible $\nabla T$ is not limited to physiology. In hydrology, for example, MinEP has been postulated in Ref. [32] in order to describe the fractal pattern of river basins. In desertic areas, it has been empirically observed that yardangs (desertic landforms sculpted by wind erosion) evolve spontaneously into streamlined minimum-drag forms. When viewed from above, yardangs resemble the hull of a boat-a benchmark of reduced fluid dynamic resistance. Softer rocks are eroded and removed by the wind, and the harder material remains. The resulting pattern is the outcome of both geology (the original distribution of rocks) and fluid mechanics (the motion of the wind) [33]. ${ }^{29}$

## 物理代写|热力学代写Thermodynamics代考|Porous Media

Problems involving flows across porous media are of paramount practical relevance (think e.g. of oil wells, aquifer pollution, etc.), involve both viscosity and capillarity and have also been described with the help of a postulated MinEP-like principle [34].

A porous medium is defined as a solid permeated by an interconnected network of pores filled with a fluid (water, here and below). Darcy’s law states that
$$\mathbf{q}w=-K_w \nabla H_w \quad ; \quad \mathbf{q}_w \equiv \mathbf{v} \varphi_p \quad ; \quad H_w \equiv z+\frac{p}{\rho g}$$ in saturated, isotropic porous media. Here $\mathbf{v}, g, p, z, \rho, \varphi_p \equiv \frac{V_p}{V{\text {tot }}}, V_p$ and $V_{\text {tot }}$ are the water velocity $\mathbf{v}$, the absolute value of the acceleration of gravity, the pressure, the vertical coordinate, the water mass density, the porosity of the medium (which is supposed uniform below), the volume of voids and the total volume, respectively; $\mathbf{q}_w, H_w, K_w>0$ and $\frac{1}{K_w}$ are dubbed ‘volumetric water flux’, ‘hydraulic head’, ‘hydraulic conductivity’ and ‘hydraulic resistivity’, respectively. (If the medium is anisotropic $K$ is replaced by a tensor; nothing essential changes in the following). Here ‘saturated’ means that all the pores are filled and conducting, so that hydraulic conductivity is maximal. Two remarks are of interest: firstly, $\rho g H_w$ is a contribution to the enthalpy $h$ per unit mass in the energy balance of Sect. $4.2 .7$ in a uniform, constant, vertical gravitational field-the obvious example being Earth’s field. Secondly, Darcy’s law is formally analogous to Ohm’s law in a rigid conductor at rest (Sect. 5.3.1): $\mathbf{q}_w, \frac{1}{K_w}$ and $H_w$ play the role of electric current density, electric resistivity and electrostatic potential, respectively. This analogy suggests that some variational principle is lurking.

In Darcy’s original formulation $\frac{d K_w}{d t}=0, \frac{d K_w}{d\left|\mathbf{q}_w\right|}=0$ and Darcy’s law is linear. Just like its electrical counterpart, however, $K_w$ depends on a quantity (‘permeability’) which depends on the porous medium. ${ }^{30}$ If the permeability is too low, then Darcy’s law is no more linear, and $K_w$ is not negligible only if $\left|\mathbf{q}_w\right|$ exceeds a threshold. Another example of nonlinearity occurs at large $\left|\mathbf{q}_w\right|$ : too much water poured into the system, indeed, tends to destroy the porous structure of the solid and to form microfractures; the latter may coalesce (‘fingering’) into larger fractures which act as preferential channels for the water flow across the system. In this case, $\frac{d K_w}{d\left|\mathbf{q}_w\right|}>0$.

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## 物理代写|热力学代写THERMODYNAMICS代考|POROUS MEDIA

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$$\mathbf{q} w=-K_w \nabla H_w \quad ; \quad \mathbf{q}w \equiv \mathbf{v} \varphi_p \quad ; \quad H_w \equiv z+\frac{p}{\rho g}$$ 在饱和的各向同性多孔介质中。这里 $\mathbf{v}, g, p, z, \rho, \varphi_p \equiv \frac{V_p}{V \text { tot }}, V_p$ 和 $V{\text {tot }}$ 是水流速度 $\mathbf{v}$ ，重力加速度的绝对值，压力，垂直坐标，水的质量密度，介质的孔隙率 whichissupposeduniformbelow，分别为空隙体积和总体积； $\mathbf{q}_w, H_w, K_w>0$ 和 $\frac{1}{K_w}$ 分别被称为“体积水通量”、水头”、水力传导率”和“水力电阻率”。
Ifthemediumisanisotropic\$K\$isreplacedbyatensor; nothingessentialchangesinthe following. 这里的“饱和”是指所有孔隙都被填充和传导，因此水力 传导率最大。有两点值得注意：首先， $\rho g H_w$ 是对焓的贡南 $h$ Sect能量平衡中的每单位质量。4.2.7在一个均匀、恒定、垂直的引力场中一-一个明显的例子就是地球 的场。其次，达西定律在形式上类似于静止的刚性导体中的欧姆定律 Sect. 5.3.1: $\mathbf{q}_w, \frac{1}{K_w}$ 和 $H_w$ 分别起电流密度、电阻率和静电势的作用。这个类比表明潜伏着一 些变分原理。 太低，则达西定律不再是线性的，并且 $K_w$ 只有当 $\left|\mathbf{q}_w\right|$ 超过一个阈值。另一个非线性的例子发生在大范围内 $\left|\mathbf{q}_w\right|:$ 过多的水倒入系统中，确实会破坏固体的多孔结 构并形成微裂缝；后者可能合并’ fingering 进入较大的裂缝，这些裂缝作为水流穿过系统的优先通道。在这种情况下， $\frac{d K_a}{d\left|\mathbf{q}_w\right|}>0$.

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。