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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Conservation of Mass

In continuum mechanics, the conservation of mass is assumed to be a postulate. In other words, for all material volumes $\mathscr{V}$, i.e., those containing the same particles for all times,
$$\frac{d}{d t} \int_{\mathscr{\Upsilon}} \rho d v=0,$$
where $\rho=\rho(x, y, z, t)$ is the density at the point $(x, y, z)$ at time $t$. By the transport theorem, the above equation reduces to
$$\int_{\mathscr{\gamma}}\left[\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})\right] d v=0 .$$
Since the size and shape of the material volume is arbitrary, a necessary and sufficient condition for the conservation of mass is the continuity equation:
$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})=\dot{\rho}+\rho \nabla \cdot \mathbf{v}=0,$$
where, as usual, the superposed dot denotes the material derivative.
In incompressible materials, only isochoric, i.e., volume preserving motions are possible and, since $\rho$ is a constant everywhere, conservation of mass implies and is guaranteed by
$$\nabla \cdot \mathbf{v}=v_{i j}^{j}=0,$$
where the Einstein summation convention has been employed. This convention is followed throughout in this monograph.
Since
$$\nabla \cdot \mathbf{v}=\operatorname{tr} \mathbf{L}=\frac{1}{2} \operatorname{tr} \mathbf{A}$$
where $\mathbf{L}$ is the velocity gradient and $\mathbf{A}$ is the first Rivlin-Ericksen tensor. Thus, an assertion equivalent to the conservation of mass in isochoric motions is that $\operatorname{tr} \mathbf{L}=\operatorname{tr} \mathbf{A}=0$

## 物理代写|流体力学代写Fluid Mechanics代考|Cauchy’s First Law of Motion

In Newtonian mechanics, external forces act on a body and the rate of change of the linear momentum of the body is due to these forces. In Lagrangian mechanics, the external forces are classified into forces of constraint and the remainder. In continuum mechanics, the outside world acts on a body through contact forces, i.e., by direct touch with the surface of a body, and non-contact forces, which act at a distance; the latter are grouped together as body forces.

Let $\mathscr{V}$ denote the volume occupied by the body $\mathscr{B}$ at time $t, \mathbf{t}$ the contact force per unit area on its surface $\mathscr{S}$ exerted by the outside world, and $\mathbf{b}$ the body force per unit mass. Then Newton’s second law of motion in an inertial frame of reference, as modified by Cauchy, states that the rate of change of the linear momentum is equal to the external forces on the body, i.e.,
$$\frac{d}{d t} \int_{\mathscr{r}} \rho \mathbf{v} d v=\int_{\mathscr{Y}} \mathbf{t} d S+\int_{\mathscr{r}} \rho \mathbf{b} d v$$
Using Reynolds’ transport theorem, one finds that the left side is
$$\int_{\mathscr{V}}\left[\left(\frac{d \rho}{d t}+\rho \nabla \cdot \mathbf{v}\right) \mathbf{v}+\rho \mathbf{a}\right] d v .$$
If one assumes that mass is conserved, one can see immediately from Eq. (3.1.3) that the equation of motion for a continuous medium now becomes
$$\int_{\mathscr{V}} \rho \mathbf{a} d v=\int_{\mathscr{S}} \mathbf{t} d S+\int_{\mathscr{Y}} \rho \mathbf{b} d v .$$
We wish to convert the surface integral in Eq. (3.3.3) to a volume integral through the divergence theorem in order to obtain a differential equation for the balance of linear momentum. To achieve this, note that on the boundary, the stress vector is given by
$$\mathbf{t}=\mathbf{t}(\mathbf{x}, t, \mathbf{n}) \quad \mathbf{x} \in \mathscr{S}$$

# 力学代考

## 物理代写|流体力学代写Fluid Mechanics代考|Conservation of Mass

$$\frac{d}{d t} \int_{\Upsilon} \rho d v=0$$

$$\int_{\gamma}\left[\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})\right] d v=0$$

$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})=\dot{\rho}+\rho \nabla \cdot \mathbf{v}=0$$

$$\nabla \cdot \mathbf{v}=v_{i j}^{j}=0,$$

$$\nabla \cdot \mathbf{v}=\operatorname{tr} \mathbf{L}=\frac{1}{2} \operatorname{tr} \mathbf{A}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Cauchy’s First Law of Motion

$$\frac{d}{d t} \int_{r} \rho \mathbf{v} d v=\int_{\mathscr{Y}} \mathbf{t} d S+\int_{r} \rho \mathbf{b} d v$$

$$\int_{\nu}\left[\left(\frac{d \rho}{d t}+\rho \nabla \cdot \mathbf{v}\right) \mathbf{v}+\rho \mathbf{a}\right] d v .$$

$$\int_{\mathscr{\nu}} \rho \mathbf{a} d v=\int_{\mathscr{S}} \mathbf{t} d S+\int_{\mathscr{Y}} \rho \mathbf{b} d v .$$

$$\mathbf{t}=\mathbf{t}(\mathbf{x}, t, \mathbf{n}) \quad \mathbf{x} \in \mathscr{S}$$

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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。

assignmentutor™您的专属作业导师
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