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# 美国代写|流体力学代写Fluid Dynamics代写|Elementary Definitions

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## 美国代写|流体力学代写Fluid Dynamics代写|What is the Mechanical Definition of a Fluid?

The answer to this question may seem trivially obvious, but it turns out to be worth pondering. The term fluid encompasses both liquids and gases, the former being characterized by the existence of a free surface and the latter by the ease with which it may be compressed. Somewhat surprisingly, the macroscopic dynamics of both liquids and gases can be accounted for by more or less the same theory, with only modest differences in emphasis.

To construct such a theory we adopt the continuum approximation, which assumes that matter is smeared continuously across space. This approximation rests on the large difference between the molecular scale (the distance between molecules) and the characteristic distance over which the macroscopic properties of a fluid, such as density or pressure, vary. So, for example, the density of a fluid is defined as the mass per unit volume measured over a scale which is large enough for all molecular fluctuations to be smoothed out, yet small enough for the density, $\rho$, to be considered a smoothly varying function of position. Likewise, the stresses exerted by one part of a fluid on another are considered to be a smooth function of position, being defined as the force per unit area transmitted across a small plane surface within the fluid, the surface being infinitesimally small on the macroscopic scale, yet large on the molecular scale. These may be normal stresses, arising from forces perpendicular to the surface in question, or shear stresses, arising from tangential forces.

The distinction between solids and fluids is, at first sight, rather obvious; i.e. solids exhibit rigidity, while fluids readily deform when acted upon by a force. However, there are subtleties in this distinction that are worth noting. For example, we cannot distinguish between solids and liquids if only normal stresses are in play. Rather, it is the way in which these two states respond to an imposed shear stress that distinguishes between the two. Suppose, for example, that we have two cylinders, each sealed by a movable piston. One cylinder is filled with oil and the other with a cylindrical block of rubber. We now pressurize the contents of the two cylinders using the pistons. Evidently both systems behave in exactly the same way: when a compressive stress is imposed by the pistons, both the oil and the rubber compress a little, and then return to a state of static equilibrium.

## 美国代写|流体力学代写Fluid Dynamics代写|Fluid Statics and One Definition of Pressure

Before discussing the dynamics of fluids, perhaps it is worth making a few comments about hydrostatics, if only to reinforce the notions of pressure and of stresses acting within a fluid. Let us start with Pascal’s law for stationary fluids.

We have already seen that shear stresses are everywhere zero in a stationary fluid. Pascal’s law follows directly from this and states that the magnitude of the normal stress acting at any given point is independent of direction. The proof is trivial. Let us use $\sigma$ to denote a normal stress and reserve $\tau$ for shear stress. Consider a small wedge of fluid of mass $m$ surrounding the point of interest, as shown in Figure 1.2. Let the wedge have sides $\delta x, \delta y$, and $\delta z$, and let $\sigma_{x}, \sigma_{z}$, and $\sigma_{\alpha}$ be the normal stresses in the $x$ and $z$ directions and on the inclined surface. As there are no shear stresses, vertical equilibrium demands
$$\left[\sigma_{\alpha}(\delta L \delta y)\right] \cos \alpha=\sigma_{\alpha}(\delta x \delta y)=\sigma_{z}(\delta x \delta y)+\rho g\left(\frac{1}{2} \delta x \delta y \delta z\right)$$
and as the size of the wedge tends to zero the weight $m g$ drops out of (1.2) to give $\sigma_{\alpha}=\sigma_{z}$. Similarly, horizontal equilibrium requires $\sigma_{\alpha}=\sigma_{x}$. So, we conclude that because of the absence of shear stress, the normal stress, $\sigma$, is the same in all directions. In fluid dynamics the normal stresses are compressive, and so we define the fluid pressure to be $p=-\sigma$, and Pascal’s law is sometimes paraphrased as saying that the pressure at any given point is the same in all directions

## 美国代写|流体力学代写FLUID DYNAMICS代写|Different Categories of Fluid and of Fluid Flow

We now turn to dynamics. Before presenting a formal analysis of fluids in motion, it is, perhaps, useful to discuss the different approximations commonly employed in fluid dynamics, as well as the different regimes of behaviour normally encountered in practice. Perhaps the first important subdivision is between incompressible and compressible flows. It is tempting to associate the former with liquids and the latter with gases, but this is too simplistic. Compressibility cannot always be ignored in liquids (think of acoustic waves in the oceans), and, conversely, gasses in which the Mach number (the flow speed divided by the speed of sound) is less than $0.3$ usually exhibit negligible variation in density. In this text we consider only incompressible flows.

The second subdivision is between ideal and real fluids. As mentioned above, it is sometimes convenient to ignore the viscous stresses in a fluid on the grounds that the viscosity, which is always finite, is sufficiently small for those stresses to be negligible. A hypothetical fluid with zero viscosity is called an ideal fluid. Real fluids stick to solid surfaces (Figure 1.1), whereas an ideal fluid is free to slip over such a surface. Moreover, viscous stresses convert mechanical energy into heat, whereas ideal fluids conserve mechanical energy (in the absence of external forces). So, an inviscid fluid can be a useful idealization, but one that invariably breaks down near solid surfaces in motion.

## 美国代写|流体力学代写FLUID DYNAMICS代写|FLUID STATICS AND ONE DEFINITION OF PRESSURE

[σ一个(d大号d是)]因⁡一个=σ一个(dXd是)=σ和(dXd是)+ρG(12dXd是d和)

## 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。

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