### 物理代写|流体力学代写Fluid Mechanics代考|CHNG2801

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• (Generalized) Linear Models 广义线性模型
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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|Continuum Hypothesis

The random motion mentioned above, however, does not allow to define a molecular velocity at a fixed spatial position. To circumvent this dilemma, particularly for gases, we consider the mass contained in a volume element $\delta V_{G}$ which has the same order of magnitude as the volume spanned by the mean free path of the gas molecules. The volume $\delta V_{G}$ has a comparable order of magnitude for a molecule of a liquid $\delta V_{L}$. Thus, a fluid can be treated as a continuum if the volume $\delta V_{G}$ occupied by the mass $\delta m$ does not experience excessive changes. This implies that the ratio $$\rho=\lim {\delta V{G} \rightarrow 0}\left(\frac{\delta m}{\delta V_{G}}\right)$$
does not depend upon the volume $\delta V_{G}$. This is known as the continuum hypothesis that holds for systems, whose dimensions are much larger than the mean free path of the molecules. Accepting this hypothesis, one may think of a fluid particle as a collection of molecules that moves with a velocity that is equal to the average velocity of all molecules that are contained in the fluid particle. With this assumption, the density defined in Eq. (1.1) is considered as a point function that can be dealt with as a thermodynamic property of the system. If the $\mathrm{p}-\mathrm{v}-\mathrm{T}$ behavior of a fluid is given, the density at any position vector $\mathbf{x}$ and time $t$ can immediately be determined by providing an information about two other thermodynamic properties. For fluids that are frequently used in technical applications, the $\mathrm{p}-\mathrm{v}-\mathrm{T}$ behavior is available from experiments in the form of $\mathrm{p}-\mathrm{v}, \mathrm{h}$-s, or T-s tables or diagrams. For computational purposes, the experimental points are fitted with a series of algebraic equations that allow a quick determination of density by using two arbitrary thermodynamic properties.

## 物理代写|流体力学代写Fluid Mechanics代考|Molecular Viscosity

Molecular viscosity is the fluid property that causes friction. Figure $1.1$ gives a clear physical picture of the friction in a viscous fluid. A flat plate placed at the top of a particular viscous fluid is moving with a uniform velocity $V_{1}=U$ relative to the stationary bottom wall.
The following observations were made during experimentation:

1. In order to move the plate, a certain force $F_{1}$ must be exerted in $x_{1}$-direction.
2. The fluid sticks to the plate surface that moves with the velocity $\mathbf{U}$.
3. The velocity difference between the stationary bottom wall and the moving top wall causes a velocity change which is, in this particular case, linear.
4. The force $F_{1}$ is directly proportional to the velocity change and the area of the plate.

These observations lead to the conclusion that one may set:
$$F_{1} \propto A \frac{d V_{1}}{d x_{2}}$$
Multiplying the proportionality (1.2) by a factor $\mu$ which is the substance property viscosity, results in an equation for the friction force in $\mathrm{x}{1}$-direction: $$F{1}=\mu A \frac{d V_{1}}{d x_{2}} .$$
The subsequent division of Eq. (1.3) by the plate area $A$ gives the shear stress component $\tau_{21}$ :
$$\tau_{21}=\mu \frac{d V_{1}}{d x_{2}} .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Continuum Hypothesis

$$\rho=\lim \delta V G \rightarrow 0\left(\frac{\delta m}{\delta V_{G}}\right)$$

## 物理代写|流体力学代写Fluid Mechanics代考|Molecular Viscosity

1. 为了移动盘子，一定的力 $F_{1}$ 必须发挥 $x_{1}$-方向。
2. 流体粘附在随速度运动的板表面上U.
3. 静止底骍和移动顶壁之间的速度差导致速度变化，在这种特殊情况下，速度变化是线性的。
4. 力量 $F_{1}$ 与速度变化和板面积成正比。
这些观察得出的结论是，人们可以设定:
$$F_{1} \propto A \frac{d V_{1}}{d x_{2}}$$
将比例 (1.2) 乘以一个因子 $\mu$ 这是物质的特性粘度，导致摩擦力方程x1-方向：
$$F 1=\mu A \frac{d V_{1}}{d x_{2}} .$$
等式的后续除法。(1.3) 按板块面积 $A$ 给出剪应力分量 $\tau_{21}$ :
$$\tau_{21}=\mu \frac{d V_{1}}{d x_{2}} .$$

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## MATLAB代写

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