## 物理代写|流体力学代写Fluid Mechanics代考|ENGR30002

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Effect of Solidity on Blade Profile Losses

Equation (5.181) exhibits a fundamental relationship between the lift coefficient, the solidity, the inlet and exit flow angle, and the loss coefficient $\zeta$. The question is, how the profile loss $\zeta$ will change if the solidity $\sigma$ changes. The solidity has the major influence on the flow behavior within the blading. If the spacing is too small, the number of blades is large and the friction losses dominate. Increasing the spacing, which is identical to reducing the number of blades, at first causes a reduction of friction losses. Further increasing the spacing decreases the friction losses and also reduces the guidance of the fluid that results in flow separation leading to additional losses. With definite spacing, there is an equilibrium between the separation and friction losses. At this point, the profile loss $\zeta=\zeta_{\text {tnctoon }}+\zeta_{\text {separaton }}$ is at a minimum. The corresponding spacing/chord ratio has an optimum, which is shown in Fig. 5.33. To find the optimum solidity for a variety of turbine and compressor cascades, a series of comprehensive experimental studies have been performed by several researchers. A detailed discussion of the results of these studies is presented in [23].

The relationship for the lift-solidity coefficient derived in the preceding sections is restricted to turbine and compressor stages with constant inner and outer diameters. This geometry is encountered in high pressure turbines or compressor components, where the streamlines are almost parallel to the machine axis. In this special case, the stream surfaces are cylindrical with almost constant diameter. In a general case such as the intermediate and low pressure turbine and compressor stages, however, the stream surfaces have different radii. The meridional velocity component may also change from station to station. In order to calculate the blade lift-solidity coefficient correctly, the radius and the meridional velocity changes must he taken into account. Detailed discussions on this and turbomachinery aero-thermodynamic topics are found in [23].

## 物理代写|流体力学代写Fluid Mechanics代考|Inviscid Potential Flows

As discussed in Chap. 4 , generally the motion of fluids encountered in engineering applications is described by the Navier-Stokes equations. Considering today’s computational fluid dynamics capabilities, it is possible to numerically solve the Navier-Stokes equations for laminar flows (no turbulent fluctuations), transitional flows (using appropriate intermittency models), and turbulent flow (utilizing appropriate turbulence models). Given today’s computational capabilities, one may argue at this juncture that there is no need to artificially subdivide the flow regime into different categories such as incompressible, compressible, viscid or inviscid ones. However, based on the degree of complexity of the flow under investigation, a computational simulation may take up to several days, weeks, and even months for direct Navier-stokes simulations (DNS). The difficulties associated with solving the Navier-Stokes equations are caused by the existence of the viscosity terms in the Navier-Stokes equations.

Measuring the velocity distributions encountered in engineering applications such as in a pipe flow, flow around a compressor or turbine blade, or along the wing of an aircraft, we find that the effect of viscosity is confined to a very thin layer called the boundary layer with a local thickness $\delta$. As we discuss in Chap. 11, comprehensive experimental investigations performed earlier by Prandtl $[26,27]$ show that the boundary layer thickness $\delta$ compared to the length $L$ of the subject under investigation is very small. In the vicinity of the wall, because of the no-slip condition, the velocity is $V_{\text {wall }}=0$. Moving away from the wall towards the edge of the boundary layer, the velocity continuously increases until it reaches the velocity at the edge of the boundary layer $V=V_\delta$. Within the boundary layer, the flow is characterized by non-zero vorticity $\nabla \times V \neq 0$. No major changes in velocity magnitude is expected outside the boundary layer, provided that the surface of the subject under investigation does not have a curvature. In case of surfaces with convex or concave curvatures, the velocity outside the boundary layer changes in lateral direction.

Outside the boundary layer, the effect of the viscosity can be neglected as long as the Reynolds number is high enough ( $\operatorname{Re}=100,000$ and above) indicating that the convective flow forces are much larger than the shear stress forces. Theoretically, the boundary layer thickness approaches zero as the Reynolds number tends to infinity. In this case, the flow can be assumed as irrotational, which is then characterized by zero vorticity $\nabla \times V=0$. Thus, as Prandtl suggested, the flow may be decomposed into two distinct regions, the vortical inner region, called the boundary layer, where the viscosity effect is predominant, and the non-vortical region outside the boundary layer.

The flow in the outer region can be calculated using the Euler equation of motion, while the boundary layer method can be applied for calculating the viscous flow within the inner region. Combining these two methods allows calculation of the flow field in a sufficiently accurate manner as long as the boundary layer is not separated. Figure $6.1$ exhibits the velocity distributions along the suction surface of an airfoil. While in case (a) the viscosity is accounted for, in case (b) it is neglected. Thus, the flow is assumed irrotational, which is characterized by $\nabla \times V=0$. As a consequence of this assumption, the velocity on the surface has a non-zero tangential component, which is in contrast to the reality. These type of flows are called potential flows which is the subject of the following sections.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|ZEIT2503

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Blade Force in an Inviscid Flow Field

Starting from a given turbine cascade with the inlet and exit flow angles shown in Fig. 5.27, the blade force can be obtained by applying the linear momentum principles to the control volume with the unit normal vectors and the coordinate system shown in Fig. 5.27. Applying Eq. (5.26), the blade inviscid force is obtained from:
$$\boldsymbol{F}i=\dot{m} \boldsymbol{V}_1-\dot{m} \boldsymbol{V}_2-\boldsymbol{n}_1 p_1 s h-\boldsymbol{n}_2 p_2 s h$$ with the subscript $i$ that refers to inviscid flow, $s$ as the spacing and $h$ as the blade height that can be assumed unity. The relationship between the control volume normal unit vectors and the unit vectors pertaining to the coordinate system is given by $\boldsymbol{n}_1=-\boldsymbol{e}_2$ and $\boldsymbol{n}_2=\boldsymbol{e}_2$. The velocities in Eq. (5.153) can be expressed in terms of circumferential as well as axial components: $$\boldsymbol{F}_i=-e_1 \dot{m}\left[\left(V{u 1}+V_{u 2}\right)\right]+e_2\left[\dot{m}\left(V_{a x 1}-V_{a x 2}\right)+\left(p_1-p_2\right) s h\right]$$
with $V_{a x 1}=V_{a x 2}$ as a result of incompressible flow assumption and $V_{u 1} \not \equiv V_{u 2}$ from Fig. 5.22. Equation (5.154) rearranged as:
$$\boldsymbol{F}i=-\boldsymbol{e}_1 \dot{m}\left(V{u 1}+V_{u 2}\right)+\boldsymbol{e}2\left(p_1-p_2\right) s h=e_1 F_u+e_2 F{a x}$$
with the circumferential and axial components
$$F_u=-\dot{m}\left(V_{u 1}+V_{u 2}\right) \text { and } F_{a x}=\left(p_1-p_2\right) s h .$$
The static pressure difference in Eq. (5.156) is obtained from the following Bernoulli equation:

\begin{aligned} p_{01} &=p_{02} \ p_1-p_2 &=\frac{1}{2} \rho\left(V_2^2-V_1^2\right)=\frac{1}{2} \rho\left(V_{u 2}^2-V_{u 1}^2\right) . \end{aligned}
Inserting the pressure difference along with the mass flow $\dot{m}=\rho V_{a x} s h$ into Eq. (5.156) and the blade height $h=1$, we obtain the axial as well as the circumferential components of the lift force:
$$\left.\begin{array}{l} F_{a x}=\frac{1}{2} \varrho\left(V_{u 2}+V_{u 1}\right)\left(V_{u 2}-V_{u 1}\right) s \ F_u=-\varrho V_{a x}\left(V_{u 2}+V_{u 1}\right) s \end{array}\right}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Blade Forces in a Viscous Flow Field

The working fluids in turbomachinery, whether air, combustion gas, steam or other substances, are always viscous. The blades are subjected to the viscous flow and undergo shear stresses with no-slip condition on blades, casing and hub surfaces, resulting in boundary layer developments. Furthermore, the blades have certain definite trailing edge thicknesses. These thicknesses together with the boundary layer thickness, generate a spatially periodic wake flow downstream of each cascade as shown in Fig. 5.30.
The presence of the shear stresses cause drag forces that reduce the total pressure. In order to calculate the blade forces, the momentum Eq. (5.153) can be applied to the viscous flows. As seen from Eq. (5.156), the circumferential component remains unchanged. The axial component, however, changes in accordance with the pressure difference as shown in the following relations:
\begin{aligned} F_u &=-\rho V_{a x}\left(V_{u 2}+V_{u 1}\right) s h \ F_{a x} &=\left(p_1-p_2\right) s h . \end{aligned}

The blade height $h$ in Eq. (5.169) may be assumed as unity. For a viscous flow, the static pressure difference cannot be calculated by the Bernoulli equation. In this case, the total pressure drop must be taken into consideration. We define the total pressure loss coefficient:
$$\zeta \equiv \frac{P_1-P_2}{\frac{1}{2} \varrho V_2^2}$$
with $P_1$ and $P_2$ as the averaged total pressure at stations 1 and 2 . Inserting for the total pressure the sum of static and dynamic pressures, we get the static pressure difference as:
$$p_1-p_2=\frac{\rho}{2}\left(V_2^2-V_1^2\right)+\zeta \frac{\rho}{2} V_2^2 .$$
Incorporating Eq. (5.171) into the axial component of the blade force in Eq. (5.169) yields:
$$F_{a x}=\frac{\rho}{2}\left(V_2^2-V_1^2\right) s+\zeta \frac{\rho}{2} V_2^2 s .$$
We introduce the velocity components into Eq. (5.172) and assume that for an incompressible flow the axial components of the inlet and exit flows are the same. As a result, Eq. (5.172) reduces to:
$$F_{a x}=\frac{\rho}{2}\left(V_{u 2}^2-V_{u 1}^2\right) s+\zeta \frac{\rho}{2} V_2^2 s .$$
The second term on the right-hand side exhibits the axial component of drag forces accounting for the viscous nature of a frictional flow shown in Fig. 5.31. Thus, the axial projection of the drag force is obtained from:
$$D_{a x}=\zeta \frac{\rho}{2} V_2^2 s$$

## 物理代写|流体力学代写流体力学代考|无粘流场中的叶片力

$$\boldsymbol{F}i=\dot{m} \boldsymbol{V}1-\dot{m} \boldsymbol{V}_2-\boldsymbol{n}_1 p_1 s h-\boldsymbol{n}_2 p_2 s h$$，下标$i$表示无粘流量，$s$为间距，$h$为可统一假设的叶片高度。控制体积法单位向量与坐标系中的单位向量之间的关系由$\boldsymbol{n}_1=-\boldsymbol{e}_2$和$\boldsymbol{n}_2=\boldsymbol{e}_2$给出。式(5.153)中的速度可以用周向分量和轴向分量表示:由于不可压缩流动假设，$$\boldsymbol{F}_i=-e_1 \dot{m}\left[\left(V{u 1}+V{u 2}\right)\right]+e_2\left[\dot{m}\left(V_{a x 1}-V_{a x 2}\right)+\left(p_1-p_2\right) s h\right]$$

$$\boldsymbol{F}i=-\boldsymbol{e}1 \dot{m}\left(V{u 1}+V{u 2}\right)+\boldsymbol{e}2\left(p_1-p_2\right) s h=e_1 F_u+e_2 F{a x}$$
，其中周向分量和轴向分量
$$F_u=-\dot{m}\left(V_{u 1}+V_{u 2}\right) \text { and } F_{a x}=\left(p_1-p_2\right) s h .$$

\begin{aligned} p_{01} &=p_{02} \ p_1-p_2 &=\frac{1}{2} \rho\left(V_2^2-V_1^2\right)=\frac{1}{2} \rho\left(V_{u 2}^2-V_{u 1}^2\right) . \end{aligned}

$$\left.\begin{array}{l} F_{a x}=\frac{1}{2} \varrho\left(V_{u 2}+V_{u 1}\right)\left(V_{u 2}-V_{u 1}\right) s \ F_u=-\varrho V_{a x}\left(V_{u 2}+V_{u 1}\right) s \end{array}\right}$$

## 物理代写|流体力学代写流体力学代考|叶片在粘性流场中的力

\begin{aligned} F_u &=-\rho V_{a x}\left(V_{u 2}+V_{u 1}\right) s h \ F_{a x} &=\left(p_1-p_2\right) s h . \end{aligned}

$$\zeta \equiv \frac{P_1-P_2}{\frac{1}{2} \varrho V_2^2}$$
，其中$P_1$和$P_2$为1站和2站的平均总压。将总压力插入静态压力和动态压力之和，我们得到静压差为:
$$p_1-p_2=\frac{\rho}{2}\left(V_2^2-V_1^2\right)+\zeta \frac{\rho}{2} V_2^2 .$$

$$F_{a x}=\frac{\rho}{2}\left(V_2^2-V_1^2\right) s+\zeta \frac{\rho}{2} V_2^2 s .$$

$$F_{a x}=\frac{\rho}{2}\left(V_{u 2}^2-V_{u 1}^2\right) s+\zeta \frac{\rho}{2} V_2^2 s .$$

$$D_{a x}=\zeta \frac{\rho}{2} V_2^2 s$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|ENGG2500

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Effect of Stage Load Coefficient on Stage Power

The stage load coefficient $\lambda$ defined in Eq. (5.139) is an important parameter which describes the stage capability to generate/consume shaft power. A turbine stage with low flow deflection, thus, low specific stage load coefficient $\lambda$, generates lower specific stage power $l_m$. To increase $l_m$, blades with higher flow deflection are used that produce higher stage load coefficient $\lambda$. The effect of an increased $\lambda$ is shown in Fig. $5.25$ where three different bladings are plotted. The top blading with the stage load coefficient $\lambda=1$ has lower deflection. The middle blading has a moderate flow deflection and moderate $\lambda=2$ which delivers the stage power twice as high as the top blading. Finally, the bottom blading with $\lambda=3$, delivers three times the stage power as the first one. In the practice of turbine design, among other things, two major parameters must be considered. These are the specific load coefficients and the stage polytropic efficiencies. Lower deflection generally yields higher stage polytropic efficiency, but many stages are needed to produce the required turbine power. However, the same turbine power may be established by a higher stage flow deflection and, thus, a higher $\lambda$ at the expense of the stage efficiency. Increasing the stage load coefficient has the advantage of significantly reducing the stage number, thus, lowering the engine weight and manufacturing cost. In aircraft engine design practice, one of the most critical issues besides the thermal efficiency of the engine, is the thrust/weight ratio. Reducing the stage numbers may lead to a desired thrust/weight ratio. While a high turbine stage efficiency has top priority in power eter for aircraft engine designers.

## 物理代写|流体力学代写Fluid Mechanics代考|Unified Description of Stage with Constant Mean Diameter

For a turbine or compressor stage with constant mean diameter (Fig. 5.27), we present a set of equations that describe the stage by means of the dimensionless parameters such as stage flow coefficient $\phi$, stage load coefficient $\lambda$, degree of reaction $r$, and the flow angles. From the velocity diagram with the angle definition in Fig. 5.27, we obtain the flow angles:
\begin{aligned} &\cot \alpha_2=\frac{U_2+W_{u 2}}{V_{a x}}=\frac{1}{\phi}\left(1+\frac{W_{u 2}}{U}\right)=\frac{1}{\phi}\left(1-r+\frac{\lambda}{2}\right) \ &\cot \alpha_3=-\frac{W_{u 2}-U_2}{V_{a x}}=-\frac{1}{\phi}\left(\frac{W_{u 3}-U}{U}\right)=\frac{1}{\phi}\left(1-r-\frac{\lambda}{2}\right) . \end{aligned}
Similarly, we find the other flow angles, thus, we summarize:
\begin{aligned} &\cot \alpha_2=\frac{1}{\phi}\left(1-r+\frac{\lambda}{2}\right), \cot \beta_2=\frac{1}{\phi}\left(\frac{\lambda}{2}-r\right) \ &\cot \alpha_3=\frac{1}{\phi}\left(1-r-\frac{\lambda}{2}\right), \cot \beta_3=-\frac{1}{\phi}\left(\frac{\lambda}{2}+r\right) . \end{aligned}
The stage load coefficient can be calculated from:
$$\lambda=\phi\left(\cot \alpha_2-\cot \beta_3\right)-1 .$$
As seen from Eq. (5.150), one is dealing with seven unknowns and only four equations. To obtain a solution, assumptions need to be made relative to the remaining three unknowns. These may include any of the following parameters: $\alpha_2, \beta_3, \phi, \lambda$, or $r$. The criteria for selecting these parameters are discussed in details in [23].
The preceding discussions that have led to Eqs. (5.150) and (5.151) deal with compressor and turbine stages with constant hub and tip diameters. These equations cannot be applied to cases where the diameter, circumferential, and meridional velocities are not constant. Examples are axial flow turbine and compressor types shown in Figs. $5.21$ and 5.22, radial inflow (centripetal) turbines, and centrifugal compressors. In these cases, the meridional velocity ratio and the diameter are no longer constant. The dimensionless parameters for these cases are summarized below:
$$\mu=\frac{V_{m 2}}{V_{m 3}}, \nu=\frac{R_2}{R_3}=\frac{U_2}{U_3}, \phi=\frac{V_{m 3}}{U_3}, \lambda=\frac{1_m}{U_3^2}, r=\frac{\Delta h^{\prime \prime}}{\Delta h^{\prime}+\Delta h^{\prime \prime}}$$

## 物理代写|流体力学代写流体力学代考|等平均直径级的统一描述

\begin{aligned} &\cot \alpha_2=\frac{U_2+W_{u 2}}{V_{a x}}=\frac{1}{\phi}\left(1+\frac{W_{u 2}}{U}\right)=\frac{1}{\phi}\left(1-r+\frac{\lambda}{2}\right) \ &\cot \alpha_3=-\frac{W_{u 2}-U_2}{V_{a x}}=-\frac{1}{\phi}\left(\frac{W_{u 3}-U}{U}\right)=\frac{1}{\phi}\left(1-r-\frac{\lambda}{2}\right) . \end{aligned}

\begin{aligned} &\cot \alpha_2=\frac{1}{\phi}\left(1-r+\frac{\lambda}{2}\right), \cot \beta_2=\frac{1}{\phi}\left(\frac{\lambda}{2}-r\right) \ &\cot \alpha_3=\frac{1}{\phi}\left(1-r-\frac{\lambda}{2}\right), \cot \beta_3=-\frac{1}{\phi}\left(\frac{\lambda}{2}+r\right) . \end{aligned}

$$\lambda=\phi\left(\cot \alpha_2-\cot \beta_3\right)-1 .$$

$$\mu=\frac{V_{m 2}}{V_{m 3}}, \nu=\frac{R_2}{R_3}=\frac{U_2}{U_3}, \phi=\frac{V_{m 3}}{U_3}, \lambda=\frac{1_m}{U_3^2}, r=\frac{\Delta h^{\prime \prime}}{\Delta h^{\prime}+\Delta h^{\prime \prime}}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Energy Equation in Rotating Frame of Reference

The energy equation for rotating frame of reference is simply obtained by multiplying the equation of motion with a differential displacement $d \mathbf{r}_{\mathbf{R}}^*=\mathbf{W d t}$ along the path of a particle that moves within a rotating frame of reference. It is given by,

$$\begin{gathered} \mathbf{W} d t \cdot\left[\frac{\partial \mathbf{W}}{\partial t}+\nabla\left(h+\frac{W^2}{2}-\frac{\omega^2 R^2}{2}+g 2\right)\right]= \ \mathbf{W} d t \cdot[2 \mathbf{W} \times \omega+\mathbf{W} \times(\nabla \times \mathbf{W})+\mathrm{T} \nabla \mathrm{s}-\mathbf{f}] \end{gathered}$$
Multiplying out and re-arranging the terms, we find:
$$\begin{gathered} \frac{\partial}{\partial t}\left(\frac{W^2}{2}\right)+d_R\left(h+\frac{W^2}{2}-\frac{\omega^2 R^2}{2}+g z\right)= \ \mathbf{W} d t \cdot[2 \mathbf{W} \times \omega+\mathbf{W} \times(\nabla \times \mathbf{W})+\mathrm{T} \nabla \mathrm{s}-\mathbf{f}] \end{gathered}$$
In Eq. (4.139), $d_R$ denotes the changes in a relative frame of reference. Since the vectors $\mathbf{W} \times \omega$ and $\mathbf{W} \times(\nabla \times \mathbf{W})$ are perpendicular to $\mathbf{W}$ their scalar products with $\mathbf{W}$ are zero. As a result, Eq. (4.139) reduces to:
$$\frac{\partial}{\partial t}\left(\frac{W^2}{2}\right)+d_R\left(h+\frac{W^2}{2}-\frac{\omega^2 R^2}{2}+g z\right)=d \mathbf{r}R^* \cdot(T \nabla s-\mathbf{f}) .$$ Multiplying out the right-hand side and considering the identity $d_R s=d \mathbf{r}{\mathbf{R}}^* \cdot(\nabla \mathbf{s})$, Eq. $(4.140)$ is modified as:
$$\frac{\partial}{\partial t}\left(\frac{W^2}{2}\right)+d_R\left(h+\frac{W^2}{2}-\frac{\omega^2 R^2}{2}+g z\right)=T d_R s-d t \mathbf{W} \cdot \mathbf{f}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Mass Flow Balance

We apply the Reynolds transport theorem by substituting the function $f(\boldsymbol{X}, t)$ in Chap. 2 by the density of the flow field:
$$m=\int_{v(t)} \rho(\boldsymbol{X}, t) d v$$
where the density generally changes with space and time. To obtain the integral formulation, the Reynolds transport theorem from Chap. 2 is applied. The requirement that the mass be constant leads to:
$$\frac{D m}{D t}=\int_{v(t)} \frac{\partial}{\partial t} \rho(\boldsymbol{X}, t) d v+\int_{S(t)} \rho(\boldsymbol{X}, t) \boldsymbol{V} \cdot \boldsymbol{n} d S=0$$
with $v(t)$ and $S(t)$ as the time dependent volume and surface of the integral boundaries. If the density does not undergo a time change (steady flow), the above equation is reduced to:
$$\int_{S(t)} \rho(\boldsymbol{X}, t) \boldsymbol{V} \cdot \boldsymbol{n} d S=0$$

For practical purposes, a fixed control volume is considered where the integration must be carried out over the entire control surface:
$$\int_{S_C} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S=\int_{S_{\text {Cin }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int_{S_{\text {Cout }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int_{S_{\text {C wall }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S=0 .$$
The control surface may consist of one or more inlets, one or more exits, and may include porous walls, as shown in Fig. 5.1. For such a case, Eq. (5.4) is expanded as:
\begin{aligned} &\int_{S_{\mathrm{Cin}1}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int{S_{\mathrm{Cin}2}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int{S_{\mathrm{Cout}1}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S \ &\quad+\int{S_{\mathrm{Cout}2}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int{S_{\mathrm{Cout}3}} \rho \boldsymbol{V} \cdot n d S+\int{S_{\mathrm{Cwall}}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S=0 \end{aligned}
As shown in Fig. $5.1$ and by convention, the normal unit vectors, $\boldsymbol{n}{\text {in }}, \boldsymbol{n}{\text {out }}, \boldsymbol{n}{\text {Wall }}$, point away from the region bounded by the control surface. Similarly, the tangential unit vectors, $t{\text {in }}, t_{\text {out }}, t_{\text {Wall, }}$, point in the direction of shear stresses. A representative example where the integral over the wall surface does not vanish is a film cooled turbine blade with discrete film cooling hole distribution along the blade suction and pressure surfaces, as shown in Fig. 5.2.

## 物理代写|流体力学代写Fluid Mechanics代考|Energy Equation in Rotating Frame of Reference

$$\frac{\partial}{\partial t}\left(\frac{W^2}{2}\right)+d_R\left(h+\frac{W^2}{2}-\frac{\omega^2 R^2}{2}+g z\right)=T d_R s-d t \mathbf{W} \cdot \mathbf{f}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Mass Flow Balance

$$m=\int_{v(t)} \rho(\boldsymbol{X}, t) d v$$

$$\frac{D m}{D t}=\int_{v(t)} \frac{\partial}{\partial t} \rho(\boldsymbol{X}, t) d v+\int_{S(t)} \rho(\boldsymbol{X}, t) \boldsymbol{V} \cdot \boldsymbol{n} d S=0$$

$$\int_{S(t)} \rho(\boldsymbol{X}, t) \boldsymbol{V} \cdot \boldsymbol{n} d S=0$$

$$\int_{S_C} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S=\int_{S_{\text {Cin }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int_{S_{\text {Cout }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int_{S_{\mathrm{C} \text { wall }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S=0$$

$$\int_{S_{\text {Cin1 }}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int S_{\text {Cin2 }} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+\int S_{\mathrm{Cout1}} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S \quad+\int S_{\mathrm{Cout} 2} \rho \boldsymbol{V} \cdot \boldsymbol{n} d S+$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|CIVL3612

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Continuity Equation in Rotating Frame of Reference

Inserting the velocity vector from Eq. (4.113) into the continuity equation for absolute frame of reference, Eq. (4.4), we obtain:
$$\frac{\partial \rho}{\partial t}+\nabla \cdot[\rho(\mathbf{W}+\omega \times \mathbf{r})]=0 .$$
When we expand the second term in Eq. (4.119), we find:
$$\frac{\partial \rho}{\partial t}+(\boldsymbol{\omega} \times \mathbf{r}) \cdot \nabla \rho+\mathbf{W} \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}+\rho \nabla \cdot(\boldsymbol{\omega} \times \mathbf{r})=0$$

After a simple rearrangement, Eq. (4.121) leads to:
$$\frac{\partial \rho}{\partial t}+(\mathbf{W}+\omega \times \mathbf{r}) \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}+\rho \nabla \cdot(\omega \times \mathbf{r})-0 .$$
It is necessary to discuss the individual terms in Eq. (4.120) before rearranging them. The first term indicates the time rate of change of density at a fixed station in an absolute (stationary) frame of reference. The second term involves the spatial change of density registered by a stationary observer. Combining the first and second terms expresses the time rate of change of the density within the rotating frame of reference:
$$\frac{\partial_R \rho}{\partial t} \equiv \frac{\partial \rho}{\partial t}+(\omega \times \mathbf{r}) \cdot \nabla \rho .$$
From Eq. (4.122), it becomes clear that in cases where the local change of the density in an absolute frame might be zero, $\partial \rho / \partial t=0$, in a rotating frame of reference, it will become a function of time $\partial \rho_R / \partial t \not \equiv 0$. Since the product $(\omega \times \mathbf{r}) \cdot \nabla \rho$ exhibits the circumferential change of the density in the rotating frame, it can vanish only if the flow within the rotating frame is considered axisymmetric. Since the last term in Eq. (4.120), $\nabla \cdot(\omega \times \mathbf{r})=0$, identically vanishes, the equation of continuity in a rotating frame reduces to:
$$\frac{\partial_R \rho}{\partial t}+\mathbf{W} \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}=\frac{\partial_{\mathrm{R}} \rho}{\partial \mathrm{t}}+\nabla \cdot(\rho \mathbf{W})=0 .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Equation of Motion in Rotating Frame of Reference

Replacing the acceleration in Eq. (4.22) by the expression obtained in Eq. (4.118):
$$\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \boldsymbol{\omega}}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\boldsymbol{\omega} \times \mathbf{r}) 2 \boldsymbol{\omega} \times \mathbf{W}=\frac{1}{\rho} \nabla \cdot \Pi+\mathbf{g}$$
and replacing stress tensor $\Pi$ by Eq. (4.35), $\Pi=-p \mathbf{I}+\lambda(\nabla \cdot \mathbf{V}) I+2 \mu \mathbf{D}$, Eq. (4.124) becomes:
\begin{aligned} &\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}= \ &\frac{1}{\rho} \nabla \cdot[-p \mathbf{I}+\lambda(\nabla \cdot \mathbf{V}) \mathbf{I}+2 \mu \mathbf{D}]+\mathbf{g} . \end{aligned}
Combining the last two terms in the bracket as $\nabla \cdot[\lambda(\nabla \cdot \mathbf{V}) \mathbf{I}+2 \mu \mathbf{D}] / \rho \equiv-\mathbf{f}$, and setting for $\mathbf{g}=-\nabla(\mathrm{gz})$, we re-arrange Equation (4.125) as:
$$\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \boldsymbol{\omega}}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}=-\frac{1}{\rho} \nabla p-\mathbf{f}-\nabla(\mathrm{gz}) .$$
The friction force $\mathbf{f}$ was given a negative sign since it opposes the flow motion and causes energy dissipation. Using the Clausius entropy relation, the pressure gradient can be expressed in terms of enthalpy and entropy gradients:
$$\delta q=T d s=d h-v d p .$$
The thermodynamic properties $s, h$, and $p$ are uniform continuous scalar point functions whose changes are expressed as:
$$d s=d \mathbf{X} \cdot \nabla \mathrm{s}, \mathrm{dh}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{h}, \mathrm{dp}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{p}, \mathrm{ds}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{s},$$
with $d \mathbf{X}$ as the differential displacement along the path of the fluid particle. We replace the quantities in Eq. (4.127) by those in Eq. (4.128) and arrive at:
$$d \mathbf{X} \cdot\left(\mathbf{T} \nabla \mathbf{s}-\nabla \mathbf{h}+\frac{\nabla \mathbf{p}}{\rho}\right)=0 .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Continuity Equation in Rotating Frame of Reference

$$\frac{\partial \rho}{\partial t}+\nabla \cdot[\rho(\mathbf{W}+\omega \times \mathbf{r})]=0$$

$$\frac{\partial \rho}{\partial t}+(\boldsymbol{\omega} \times \mathbf{r}) \cdot \nabla \rho+\mathbf{W} \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}+\rho \nabla \cdot(\boldsymbol{\omega} \times \mathbf{r})=0$$

$$\frac{\partial \rho}{\partial t}+(\mathbf{W}+\omega \times \mathbf{r}) \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}+\rho \nabla \cdot(\omega \times \mathbf{r})-0$$

$$\frac{\partial_R \rho}{\partial t} \equiv \frac{\partial \rho}{\partial t}+(\omega \times \mathbf{r}) \cdot \nabla \rho .$$

$$\frac{\partial_R \rho}{\partial t}+\mathbf{W} \cdot \nabla \rho+\rho \nabla \cdot \mathbf{W}=\frac{\partial_{\mathrm{R}} \rho}{\partial \mathrm{t}}+\nabla \cdot(\rho \mathbf{W})=0$$

## 物理代写|流体力学代写Fluid Mechanics代考|Equation of Motion in Rotating Frame of Reference

$$\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \boldsymbol{\omega}}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\boldsymbol{\omega} \times \mathbf{r}) 2 \boldsymbol{\omega} \times \mathbf{W}=\frac{1}{\rho} \nabla \cdot \Pi+\mathbf{g}$$

$$\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}=\quad \frac{1}{\rho} \nabla \cdot[-p \mathbf{I}+\lambda(\nabla \cdot \mathbf{V}) \mathbf{I}+$$

$$\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \boldsymbol{\omega}}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}=-\frac{1}{\rho} \nabla p-\mathbf{f}-\nabla(\mathrm{gz}) .$$

$$\delta q=T d s=d h-v d p$$

$$d s=d \mathbf{X} \cdot \nabla \mathrm{s}, \mathrm{dh}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{h}, \mathrm{dp}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{p}, \mathrm{ds}=\mathrm{d} \mathbf{X} \cdot \nabla \mathrm{s},$$

$$d \mathbf{X} \cdot\left(\mathbf{T} \nabla \mathbf{s}-\nabla \mathbf{h}+\frac{\nabla \mathbf{p}}{\rho}\right)=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|AMME2261

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Entropy Balance

The second law of thermodynamics expressed in terms of internal energy as
$$d s=\frac{\delta Q}{T}=\frac{d u+p d v}{T}$$
The infinitesimal heat $\delta Q$ added to or rejected from the system may include the heat generated by the irreversible dissipation process. Replacing the differential $d$ by the material differential operators, we arrive at:
$$T \frac{D s}{D t}=\frac{D u}{D t}+p \frac{D v}{D t} .$$
The right-hand side of Eq. (4.108) is expressed by Eq. (4.90) as:
$$\frac{D u}{D t}+p \frac{D v}{D t}=-\frac{1}{\rho} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{\rho} \mathbf{T}: \mathbf{D}$$
replacing the left-hand side of Eq. (4.109) by Eq. (4.108) results in
$$\rho \frac{D s}{D t}=-\frac{1}{T} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{T} \mathbf{T}: \mathbf{D} .$$
The second term on the right-hand side, which includes the second order friction tensor $\mathbf{T}$ is the dissipation function Eq. (4.74)
$$\rho \frac{D s}{D t}=-\frac{1}{T} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{T} \Phi .$$

This equation shows clearly that the total entropy change $D s / D t$ generally consists of two parts. The first part is the entropy change due to a reversible heat supply to the system (addition or rejection) and may assume positive, zero, or negative values. The second term exhibits the entropy production due to the irreversible dissipation and is always positive. Thus, Eq. (4.111) may be modified as:
$$\rho \frac{D s}{D t}=\rho\left(\frac{D s}{D t}\right){\mathrm{v}}+\rho\left(\frac{D s}{D t}\right){\mathrm{irr}}$$
with $\rho\left(\frac{D s}{D t}\right){\text {rev }}=-\frac{1}{T} \nabla \cdot \dot{\mathbf{q}}$ and $\rho\left(\frac{D s}{D t}\right){\text {irr }}=\frac{\Phi}{T}$. The reversible part exhibits the heat added/rejected reversibly to/from the system, thus the entropy change can assume positive or negative values, whereas, for the irreversible, the entropy change is always positive.

## 物理代写|流体力学代写Fluid Mechanics代考|Velocity and Acceleration in Rotating Frame

We consider now a rotating frame of reference that is attached to the rotor, thus, turns with an angular velocity $\omega$ about the machine axis. From a stationary observer point of view, a fluid particle that travels through a rotation frame has at an arbitrary time $t$, the position vector $\mathbf{r}$, and a relative velocity $\mathbf{W}$. In addition, it is subjected to the inherent rotation of the frame, causing the fluid particle to rotate with the velocity $\omega \times \mathbf{r}$. Thus, the observer located outside the rotating frame observes the velocity
$$\mathbf{V}=\mathbf{W}+\omega \times \mathbf{r} \text {. }$$
Inserting Eq. (4.113) into Eq. (4.16), the substantial acceleration is found
$$\frac{D \mathbf{V}}{D t}=\frac{\partial(\mathbf{W}+\omega \times \mathbf{r})}{\partial t}+(\mathbf{W}+\omega \times \mathbf{r}) \cdot \nabla(\mathbf{W}+\omega \times \mathbf{r}) .$$
We multiply Eq. (4.114) out and find
\begin{aligned} \frac{D \mathbf{V}}{D t} &=\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial(\omega \times \mathbf{r})}{\partial t}+\mathbf{W} \cdot \nabla \mathbf{W} \ &+\mathbf{W} \cdot \nabla(\omega \times \mathbf{r})+(\omega \times \mathbf{r}) \cdot \nabla \mathbf{W}+(\omega \times \mathbf{r}) \cdot \nabla(\omega \times \mathbf{r}) . \end{aligned}
Investigating the terms in Eq. (4.115), we begin with the second term on the righthand side
$$\frac{\partial(\omega \times \mathbf{r})}{\partial t}=\omega \times \frac{\partial \mathbf{r}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}=\frac{\partial \omega}{\partial t} \times \mathbf{r}$$

since in the first term on the right-hand side of Eq. (4.116) for a fixed radius vector $\partial \mathbf{r} / \partial \mathbf{t}=\mathbf{0}$. Furthermore, the last three terms of Eq. (4.115) are:
$$(\omega \times \mathbf{r}) \cdot \nabla \mathbf{W}=\omega \times \mathbf{W}, \mathbf{W} \cdot \nabla(\omega \times \mathbf{r})=\omega \times \mathbf{W},$$
$$\text { and }(\omega \times \mathbf{r}) \cdot \nabla(\boldsymbol{\omega} \times \mathbf{r})=\omega \times \omega \times \mathbf{r} \text {. }$$
Detailed derivations of Eq. (4.117) are given in Vavra [19]. Considering Eqs. (4.116) and (4.117), Eq. (4.115) becomes
$$\frac{D \mathbf{V}}{D t}=\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}$$

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

$$d s=\frac{\delta Q}{T}=\frac{d u+p d v}{T}$$

$$T \frac{D s}{D t}=\frac{D u}{D t}+p \frac{D v}{D t}$$

$$\frac{D u}{D t}+p \frac{D v}{D t}=-\frac{1}{\rho} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{\rho} \mathbf{T}: \mathbf{D}$$

$$\rho \frac{D s}{D t}=-\frac{1}{T} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{T} \mathbf{T}: \mathbf{D} .$$

$$\rho \frac{D s}{D t}=-\frac{1}{T} \nabla \cdot \dot{\mathbf{q}}+\frac{1}{T} \Phi .$$

$$\rho \frac{D s}{D t}=\rho\left(\frac{D s}{D t}\right) \mathrm{v}+\rho\left(\frac{D s}{D t}\right) \operatorname{irr}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Velocity and Acceleration in Rotating Frame

$$\mathbf{V}=\mathbf{W}+\omega \times \mathbf{r} .$$

$$\frac{D \mathbf{V}}{D t}=\frac{\partial(\mathbf{W}+\omega \times \mathbf{r})}{\partial t}+(\mathbf{W}+\omega \times \mathbf{r}) \cdot \nabla(\mathbf{W}+\omega \times \mathbf{r})$$

$$\frac{D \mathbf{V}}{D t}=\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial(\omega \times \mathbf{r})}{\partial t}+\mathbf{W} \cdot \nabla \mathbf{W} \quad+\mathbf{W} \cdot \nabla(\omega \times \mathbf{r})+(\omega \times \mathbf{r}) \cdot \nabla \mathbf{W}+(\omega \times \mathbf{r}) \cdot \nabla$$

$$\frac{\partial(\omega \times \mathbf{r})}{\partial t}=\omega \times \frac{\partial \mathbf{r}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}=\frac{\partial \omega}{\partial t} \times \mathbf{r}$$

$$(\omega \times \mathbf{r}) \cdot \nabla \mathbf{W}=\omega \times \mathbf{W}, \mathbf{W} \cdot \nabla(\omega \times \mathbf{r})=\omega \times \mathbf{W},$$
and $(\omega \times \mathbf{r}) \cdot \nabla(\boldsymbol{\omega} \times \mathbf{r})=\omega \times \omega \times \mathbf{r}$

$$\frac{D \mathbf{V}}{D t}=\frac{\partial \mathbf{W}}{\partial t}+\frac{\partial \omega}{\partial t} \times \mathbf{r}+\mathbf{W} \cdot \nabla \mathbf{W}+\omega \times(\omega \times \mathbf{r})+2 \omega \times \mathbf{W}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|ENGG2500

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Figure $1.9$ illustrates the nature of the statistically steady and unsteady flow types. As an example, Fig. 1.9a shows the velocity distribution of a statistically steady turbulent pipe flow with a constant mean. Figure $1.9 \mathrm{~b}$ represents the turbulent velocity of a statistically unsteady flow discharging from a container under pressure. As seen, the mean velocity is a function of time. A periodic unsteady turbulent flow through a reciprocating engine is represented by Fig. $1.9 \mathrm{c}$. In both unsteady cases, the unsteady mean is the result of an ensemble averaging process that we discuss in Chap. $10 .$
In Fig. 1.9, random fluctuations typical of a turbulent flow are superimposed on the mean flow. For steady or unsteady laminar flows where the Reynolds number is below the critical one, the velocity distributions do not have random component as shown in Fig. 1.10.

As briefly discussed in Sect. 1.2, there is a relationship between the shear stress $\tau_{21}$ and the deformation rate $d V_{1} / d x_{2}$. Fluids which exhibits a linear shear-deformation behavior are called Newtonian Fluids. There are, however, many fluids which exhibit a nonlinear shear- deformation behavior. Figure $1.11$ shows qualitatively the behavior of few of these fluids. More details are found among others in [6].

While the pseudoplastic fluids are characterized by a degressive slope, dilatant fluids exhibit progressive slops. For these type of fluids the shear stress tensor can be described as a polynomial function of deformation tensor, where the degree of polynomials and the coefficients are determined from experiments.

Those fluids with linear behavior which will not deform unless certain initial stress $\left(\tau_{21}\right)_{0}$ is exceeded are called Bingham fluids. It should be noted that most of the fluid used in engineering applications belong to the Newtonian Class.

## 物理代写|流体力学代写Fluid Mechanics代考|Tensors in Three-Dimensional Euclidean Space

In this section, we briefly introduce tensors, their significance to fluid dynamics and their applications. The tensor analysis is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics. Once the basics of tensor analysis are understood, the reader will be able to derive all conservation laws of fluid mechanics without memorizing any single equation. In this section, we focus on the tensor analytical application rather than mathematical details and proofs that are not primarily relevant to engineering students. To avoid unnecessary repetition, we present the definition of tensors from a unified point of view and use exclusively the three-dimensional Euclidean space, with $N=3$ as the number of dimensions. The material presented in this chapter has drawn from classical tensor and vector analysis texts, among others those mentioned in References. It is tailored to specific needs of fluid mechanics and is considered to be helpful for readers with limited knowledge of tensor analysis.

The quantities encountered in fluid dynamics are tensors. A physical quantity which has a definite magnitude but not a definite directionexhibits a zeroth-order tensor, which is a special category of tensors. In a $N$-dimensional Euclidean space, a zeroth-order tensor has $N^{0}=1$ component, which is basically its magnitude. In physical sciences, this category of tensors is well known as a scalar quantity, which has a definite magnitude but not a definite direction. Examples are: mass $m$, volume $v$, thermal energy $Q$ (heat), mechanical energy $W$ (work) and the entire thermo-fluid dynamic properties such as density $\rho$, temperature $T$, enthalpy $h$, entropy $s$, etc.
In contrast to the zeroth-order tensor, a first-order tensor encompasses physical quantities with a definite magnitude with $N^{1}\left(N^{1}-3^{1}-3\right)$ components and a definite direction that can be decomposed in $N^{1}=3$ directions. This special category of tensors is known as vector. Distance $\mathbf{X}$, velocity $\mathbf{V}$, acceleration $A$, force $F$ and moment of momentum $M$ are few examples. A vector quantity is invariant with respect to a given category of coordinate systems. Changing the coordinate system by applying certain transformation rules, the vector components undergo certain changes resulting in a new set of components that are related, in a definite way, to the old ones. As we will see later, the order of the above tensors can be reduced if they are multiplied with each other in a scalar manner. The mechanical energy $W=$ $\mathbf{F} \mathbf{X}$ is a representative example, that shows how a tensor order can be reduced. The reduction of order of tensors is called contraction.

## 物理代写|流体力学代写Fluid Mechanics代考|Tensors in Three-Dimensional Euclidean Space

。与零阶张量相比，一阶张量包含具有确定大小的物理量 $N^{1}\left(N^{1}-3^{1}-3\right)$ 成分和确定的分解方向 $N^{1}=3$ 方向。这种特殊类别的张量称为向量。距离 $\mathbf{X}$ ，速度 $\mathbf{V}$ ，加速度 $A$ ，力量 $F$ 和动量瞬间 $M$ 是几个例子。向量对于 给定类别的坐标系是不变的。通过应用某些变换规则来改变坐标系，矢量分量会发生某些变化，从而产生一组新 的分量，这些分量以一定的方式与旧的分量相关。正如我们稍后将看到的，如果将上述张量以标量方式相乘，则 可以减少它们的阶数。机械能 $W=\mathbf{F X}$ 是一个有代表性的例子，它展示了如何减少张量阶数。张量阶数的减少称为收缩。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|MECH3261

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Flow Classification

Laminar flow is characterized by the smooth motion of fluid particles with no random fluctuations present. This characteristic is illustrated in Fig. 1.3a by measuring the velocity distribution $\mathbf{V}=V(\mathbf{x})$ of a statistically steady flow at an arbitrary position vector $\mathbf{x}$. As Fig. $1.3$ reveals, the velocity distribution for laminar flow does not have any time-dependent random fluctuations. In contrast, random fluctuations are inherent characteristics of a turbulent flow. Figure $1.3 \mathrm{~b}$ shows the velocity distribution for a turbulent flow with random fluctuations. For a statistically steady flow, the velocity distribution is time dependent, given by $\mathbf{V}=V(\mathbf{x}, t)$.

It can be decomposed as a constant mean velocity $\bar{V}(\mathbf{x})$ and random fluctuations $\mathbf{V}^{\prime}(\mathbf{x}, \mathbf{t})$ :
$$V(\mathbf{x}, t)=\overline{\mathbf{V}}(x)+\mathbf{V}^{\prime}(\mathbf{x}, t) .$$
At this point, the question may arise under which condition the flow pattern may change from laminar to turbulent. To answer this question, consider the experiment by Reynolds [5] late nineteenth century, who injected dye streak into a pipe flow as shown in Fig. 1.4.

At a lower velocity, Fig. 1.4a, no fluctuation was observed and the dye filament followed the flow direction. At certain distances, the diffusion process that was gradually taking place caused a complete mixing of the dye with the main fluid. Increasing the velocity, Fig. 1.4b however, changed the flow picture completely. The orderly motion of the dye with a short laminar length, shown in Fig. 1.4b, changed into a transitional mode that started with a sinus-like wave, which we discuss in detail in Chap. 8. The transitional mode was followed by a strong fluctuating turbulent motion. This resulted in a rapid mixing of the dye with the main fluid. To explain this phenomenon, Reynolds introduced a dimensionless parameter, named after him later as the Reynolds number.

## 物理代写|流体力学代写Fluid Mechanics代考|Change of Density, Incompressible, Compressible Flow

Fluid density generally changes with pressure and temperature. As the Mollier diagram for steam shows, the density of water in the liquid state changes insignificantly with pressure. In contrast, significant changes are observed when water changes the state from liquid to vapor. A similar situation is observed for other gases.

Considering a statistically steady liquid flow with negligibly small changes in density, the flow is termed incompressible. For gas flows, however, the density change is a function of the flow Mach number.

Figure $1.8$ depicts relative changes of different flow properties as functions of the flow Mach number. Up to $M=0.3$, the relative changes of density may be considered negligibly small meaning that the flow may be considered incompressible. For Mach numbers $M>0.3$, density changes cannot be neglected. In case the flow velocity approaches the speed of sound, $M=1.0$, the flow pattern undergoes a drastic change associated with shock waves.

The density classification based on flow Mach number gives a practical idea about the density change. A more adequate definition whether the flow can be considered compressible or incompressible is given by the condition $D \rho / D t=0$, which in conjunction with the continuity equation results in $\nabla \cdot \mathbf{V}=0$. This is the condition for a flow to be considered incompressible. This issue is discussed in more detail in Chap. $4 .$

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

$$V(\mathbf{x}, t)=\overline{\mathbf{V}}(x)+\mathbf{V}^{\prime}(\mathbf{x}, t)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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 $\tau_{21}$ :
$$\tau_{21}=\mu \frac{d V_{1}}{d x_{2}} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|流体力学代写Fluid Mechanics代考|AMME2261

statistics-lab™ 为您的留学生涯保驾护航 在代写流体力学Fluid Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流体力学Fluid Mechanics代写方面经验极为丰富，各种代写流体力学Fluid Mechanics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Inherent Nonlinearity

Since the velocity field satisfies a linear differential equation, it would appear that linearity would prevail and the principle of superposition should apply. That is, if $u_{1}=u_{1}\left(y, G_{1}\right), u_{2}=u_{2}\left(y, G_{2}\right)$ are two velocity fields under the constant pressure drops per unit length $G_{1}, G_{2}$ respectively, superposition would mean that $u\left(y, G_{1}+\right.$ $\left.G_{2}\right)=u_{1}\left(y, G_{1}\right)+u_{2}\left(y, G_{2}\right)$. However, this is false because the location of the yield surface is not a linear function of the pressure drop, and the vanishing of the shear rate at the yield surface is crucial in determining the velocity field. To be precise, let the locations of the yield surfaces under the pressure drops $G_{1}$ and $G_{2}$ be $h_{1}$ and $h_{2}$ respectively. Thus,
$$h_{1}=\frac{\tau_{y}}{G_{1}}, \quad h_{2}=\frac{\tau_{y}}{G_{2}} .$$
However, the yield surface due to the pressure drop $\left(G_{1}+G_{2}\right)$ is located at $h$, given by
$$h=\frac{\tau_{y}}{G_{1}+G_{2}} \neq h_{1}+h_{2} .$$
A different way of understanding the nonlinearity is to look at Eq. (1.4.5). Without loss of generality, let $G_{1} \geq G_{2}$, and consider $y$ such that this point lies within the yielded zone whether the pressure drop per unit length is $G_{2}, G_{1}$, or $G_{1}+G_{2}$. That is
$$\frac{\tau_{y}}{G_{1}+G_{2}}<\frac{\tau_{y}}{G_{1}} \leq \frac{\tau_{y}}{G_{2}}<y<H .$$
Given this,
\begin{aligned} u_{1}\left(y, G_{1}\right)+u_{2}\left(y, G_{2}\right) &=\frac{G_{1}+G_{2}}{2 \eta}\left(H^{2}-y^{2}\right)-2 \frac{\tau_{y}}{\eta}(H-y) \ u\left(y, G_{1}+G_{2}\right) &=\frac{G_{1}+G_{2}}{2 \eta}\left(H^{2}-y^{2}\right)-\frac{\tau_{y}}{\eta}(H-y) \ & \neq u_{1}\left(y, G_{1}\right)+u_{2}\left(y, G_{2}\right) \end{aligned}
In fact, $u(y, 2 G) \neq 2 u(y, G)$. This loss of linearity rules out the application of Laplace transform methods to solve initial-boundary value problems in the flows of Bingham fluids; for additional reasons, see Sect. 6.3.

## 物理代写|流体力学代写Fluid Mechanics代考|Non-dimensionalisation

There are two distinct combinations of length, time and velocity scales associated with a Bingham fluid. One is the intrinsic set arising from the material properties, viz. the density $\rho$, the viscosity $\eta$ of the fluid, and the yield stress $\tau_{y}$. The second is induced by a given flow and we shall return to this later.

It is simple to note that an intrinsic mass $M$, length $L$ and time $T$ scales are given by
$$M \sim \eta^{3} / \sqrt{\rho \tau_{y}^{3}}, \quad L \sim \sqrt{\eta^{2} / \rho \tau_{y}}, \quad T \sim \eta / \tau_{y} .$$
The characteristic velocity $U$ derived from the above length and time scales is:
$$U \sim \sqrt{\tau_{y} / \rho} .$$
This can lead to a physically meaningless situation whereby a fluid at rest can have a non-zero characteristic velocity. Since $\tau_{y} / \eta \sim T^{-1}$, one can multiply it by a measure of length, such as the radius of a pipe or that of the edge of a square, and obtain a characteristic velocity [1]. Similarly, one can assume that a force acts on a fluid in the absence of any motion. Hence, only the flow induced non-dimensional entities are to be preferred.

That is, when solving initial-boundary value problems, it is preferable to replace the length, velocity, the pressure and stresses, and the time scale through nondimensional quantities, which are induced by the flow under consideration. As an example, consider the flow in a channel. Using the width $H$ of the channel and a characteristic velocity $U$ related to the flow rate, say, one can render the $(x, y)$ coordinates, the velocity $u$, and time $t$ into non-dimensional forms as follows:
$$\tilde{x}=x / H, \quad \tilde{y}=y / H, \quad \tilde{u}=u / U, \quad \tilde{t}=U t / H .$$
As far as the pressure $p$, and the wall shear stress $\sigma_{w}$ are concerned, we need a characteristic stress. This is provided by $\eta U / H$, for $U / H$ has the dimension of shear rate. Thus, one obtains
$$\tilde{p}=p H / \eta U, \quad \tilde{\sigma}{w}=\sigma{w} H / \eta U$$

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

$u\left(y, G_{1}+G_{2}\right)=u_{1}\left(y, G_{1}\right)+u_{2}\left(y, G_{2}\right)$. 然而，这是错误的，因为屈服面的位置不是压降的线性函数，并 且屈服面处剪切速率的消失对于确定速度场至关重要。准确地说，让屈服面在压力下降下的位置 $G_{1}$ 和 $G_{2}$ 是 $h_{1}$ 和 $h_{2}$ 分别。因此，
$$h_{1}=\frac{\tau_{y}}{G_{1}}, \quad h_{2}=\frac{\tau_{y}}{G_{2}} .$$

$$h=\frac{\tau_{y}}{G_{1}+G_{2}} \neq h_{1}+h_{2} .$$

$$\frac{\tau_{y}}{G_{1}+G_{2}}<\frac{\tau_{y}}{G_{1}} \leq \frac{\tau_{y}}{G_{2}}<y<H .$$

$$u_{1}\left(y, G_{1}\right)+u_{2}\left(y, G_{2}\right)=\frac{G_{1}+G_{2}}{2 \eta}\left(H^{2}-y^{2}\right)-2 \frac{\tau_{y}}{\eta}(H-y) u\left(y, G_{1}+G_{2}\right)=\frac{G_{1}+G_{2}}{2 \eta}\left(H^{2}\right.$$

## 物理代写|流体力学代写Fluid Mechanics代考|Non-dimensionalisation

$$M \sim \eta^{3} / \sqrt{\rho \tau_{y}^{3}}, \quad L \sim \sqrt{\eta^{2} / \rho \tau_{y}}, \quad T \sim \eta / \tau_{y} .$$

$$U \sim \sqrt{\tau_{y} / \rho} .$$

$$\tilde{x}=x / H, \quad \tilde{y}=y / H, \quad \tilde{u}=u / U, \quad \tilde{t}=U t / H .$$

$$\tilde{p}=p H / \eta U, \quad \tilde{\sigma} w=\sigma w H / \eta U$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。