物理代写|流体力学代写Fluid Mechanics代考|Kinematics

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• Foundations of Data Science 数据科学基础

物理代写|流体力学代写Fluid Mechanics代考|Material and Spatial Descriptions

Kinematics is the study of the motion of a fluid, without considering the forces which cause this motion, that is without considering the equations of motion. It is natural to try to carry over the kinematics of a mass-point directly to the kinematics of a fluid particle. Its motion is given by the time dependent position vector $\vec{x}(t)$ relative to a chosen origin.

In general we are interested in the motion of a finitely large part of the fluid (or the whole fluid) and this is made up of infinitely many fluid particles. Thus the single particles must remain identifiable. The shape of the particle is no use as an identification, since, because of its ability to deform without limit, it continually changes during the course of the motion. Naturally the linear measure must remain small in spite of the deformation during the motion, something that we guarantee by idealizing the fluid particle as a material point.

For identification, we associate with each material point a characteristic vector $\vec{\xi}$. The position vector $\vec{x}$ at a certain time $t_{0}$ could be chosen, giving $\vec{x}\left(t_{0}\right)=\vec{\xi}$. The motion of the whole fluid can then be described by
$$\vec{x}=\vec{x}(\vec{\xi}, t) \quad \text { or } \quad x_{i}=x_{i}\left(\xi_{j}, t\right)$$

物理代写|流体力学代写Fluid Mechanics代考|Pathlines, Streamlines, Streaklines

The differential Eq. (1.10) shows that the path of a point in the material is always tangential to its velocity. In this interpretation the pathline is the tangent curve to the velocities of the same material point at different times. Time is the curve parameter, and the material coordinate $\vec{\xi}$ is the family parameter.

Just as the pathline is natural to the material description, so the streamline is natural to the Eulerian description. The velocity field assigns a velocity vector to every place $\vec{x}$ at time $t$ and the streamlines are the curves whose tangent directions are the same as the directions of the velocity vectors. The streamlines provide a vivid description of the flow at time t.

If we interpret the streamlines as the tangent curves to the velocity vectors of different particles in the material at the same instant in time we see that there is no connection between pathlines and streamlines, apart from the fact that they may sometimes lie on the same curve.

By the definition of streamlines, the unit vector $\vec{u} /|\vec{u}|$ is equal to the unit tangent vector of the streamline $\vec{\tau}=\mathrm{d} \vec{x} /|\mathrm{d} \vec{x}|=\mathrm{d} \vec{x} / \mathrm{d} s$ where $\mathrm{d} \vec{x}$ is a vector element of the streamline in the direction of the velocity. The differential equation of the streamline then reads
$$\frac{\mathrm{d} \vec{x}}{\mathrm{~d} s}=\frac{\vec{u}(\vec{x}, t)}{|\vec{u}|}, \quad(t=\text { const })$$
or in index notation
$$\frac{\mathrm{d} x_{i}}{\mathrm{~d} s}=\frac{u_{i}\left(x_{j}, t\right)}{\sqrt{u_{k} u_{k}}}, \quad(t=\mathrm{const})$$
Integration of these equations with the “initial condition” that the streamline emanates from a point in space $\vec{x}{0}\left(\vec{x}(s=0)=\vec{x}{0}\right)$ leads to the parametric representation of the streamline $\vec{x}=\vec{x}\left(s, \vec{x}{0}\right)$. The curve parameter here is the arc length $s$ measured from $x{0}$, and the family parameter is $\dot{x}_{0}$.

物理代写|流体力学代写Fluid Mechanics代考|Differentiation with Respect to Time

In the Eulerian description our attention is directed towards events at the place $\vec{x}$ at time $t$. However the rate of change of the velocity $\vec{u}$ at $\vec{x}$ is not generally the acceleration which the point in the material passing through $\vec{x}$ at time $t$ experiences. This is obvious in the case of steady flows where the rate of change at a given place is zero. Yet a material point experiences a change in velocity (an acceleration) when it moves from $\vec{x}$ to $\vec{x}+\mathrm{d} \vec{x}$. Here $\mathrm{d} \vec{x}$ is the vector element of the pathline. The changes felt by a point of the material or by some larger part of the fluid and not the time changes at a given place or region of space are of fundamental importance in the dynamics. If the velocity (or some other quantity) is given in material coordinates, then the material or substantial derivative is provided by (1.6). But if the velocity is given in field coordinates, the place $\vec{x}$ in $\vec{u}(\vec{x}, t)$ is replaced by the path coordinates of the particle that occupies $\vec{x}$ at time $t$, and the derivative with respect to time at fixed $\vec{\xi}$ can be formed from
$$\frac{\mathrm{d} \vec{u}}{\mathrm{~d} t}=\left{\frac{\partial \vec{u}{\vec{x}(\vec{\xi}, t), t}}{\partial t}\right}_{\vec{\xi}}$$

or
$$\frac{\mathrm{d} u_{i}}{\mathrm{~d} t}=\left{\frac{\partial u_{i}\left{x_{j}\left(\xi_{k}, t\right), t\right}}{\partial t}\right}_{\xi_{k}}$$
The material derivative in field coordinates can also be found without direct reference to the material coordinates. Take the temperature field $T(\vec{x}, t)$ as an example: we take the total differential to be the expression
$$\mathrm{d} T=\frac{\partial T}{\partial t} \mathrm{~d} t+\frac{\partial T}{\partial x_{1}} \mathrm{~d} x_{1}+\frac{\partial T}{\partial x_{2}} \mathrm{~d} x_{2}+\frac{\partial T}{\partial x_{3}} \mathrm{~d} x_{3}$$
The first term on the right-hand side is the rate of change of the temperature at a fixed place: the local change. The other three terms give the change in temperature by advancing from $\vec{x}$ to $\vec{x}+\mathrm{d} \vec{x}$. This is the convective change. The last three terms can be combined to give $\mathrm{d} \vec{x} \cdot \nabla T$ or equivalently $\mathrm{d} x_{i} \partial T / \partial x_{i}$. If $\mathrm{d} \vec{x}$ is the vector element of the fluid particle’s path at $\vec{x}$, then (1.10) holds and the rate of change of the temperature of the particle passing $\vec{x}$ (the material change of the temperature) is
$$\frac{\mathrm{d} T}{\mathrm{~d} t}=\frac{\partial T}{\partial t}+\vec{u} \cdot \nabla T$$
or
$$\frac{\mathrm{d} T}{\mathrm{~d} t}=\frac{\partial T}{\partial t}+u_{i} \frac{\partial T}{\partial x_{i}}=\frac{\partial T}{\partial t}+u_{1} \frac{\partial T}{\partial x_{1}}+u_{2} \frac{\partial T}{\partial x_{2}}+u_{3} \frac{\partial T}{\partial x_{3}}$$

物理代写|流体力学代写Fluid Mechanics代考|Material and Spatial Descriptions

X→=X→(X→,吨) 或者 X一世=X一世(Xj,吨)

物理代写|流体力学代写Fluid Mechanics代考|Pathlines, Streamlines, Streaklines

dX→ ds=在→(X→,吨)|在→|,(吨= 常量 )

dX一世 ds=在一世(Xj,吨)在ķ在ķ,(吨=C○ns吨)

物理代写|流体力学代写Fluid Mechanics代考|Differentiation with Respect to Time

\frac{\mathrm{d} \vec{u}}{\mathrm{~d} t}=\left{\frac{\partial \vec{u}{\vec{x}(\vec{\xi} , t), t}}{\partial t}\right}_{\vec{\xi}}\frac{\mathrm{d} \vec{u}}{\mathrm{~d} t}=\left{\frac{\partial \vec{u}{\vec{x}(\vec{\xi} , t), t}}{\partial t}\right}_{\vec{\xi}}

\frac{\mathrm{d} u_{i}}{\mathrm{~d} t}=\left{\frac{\partial u_{i}\left{x_{j}\left(\xi_{k} , t\right), t\right}}{\partial t}\right}_{\xi_{k}}\frac{\mathrm{d} u_{i}}{\mathrm{~d} t}=\left{\frac{\partial u_{i}\left{x_{j}\left(\xi_{k} , t\right), t\right}}{\partial t}\right}_{\xi_{k}}

d吨=∂吨∂吨 d吨+∂吨∂X1 dX1+∂吨∂X2 dX2+∂吨∂X3 dX3

d吨 d吨=∂吨∂吨+在→⋅∇吨

d吨 d吨=∂吨∂吨+在一世∂吨∂X一世=∂吨∂吨+在1∂吨∂X1+在2∂吨∂X2+在3∂吨∂X3

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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。