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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|Translation, Deformation, Rotation

During a general three-dimensional motion, a fluid particle undergoes a translational and rotational motion which may be associated with deformation. The velocity of a particle at a given spatial, temporal position $(\boldsymbol{x}+d \boldsymbol{x}, t)$ can be related to the velocity at $(\boldsymbol{x}, t)$ by using the following Taylor expansion:

$$V(\mathbf{x}+\mathbf{d} x, \mathbf{t})=V(\mathbf{x}, \mathbf{t})+\mathbf{d} V$$
Inserting in Eq. (3.27) for the differential velocity change $d \boldsymbol{V}=d \boldsymbol{x} \cdot \nabla \boldsymbol{V}$, Eq. (3.27) is re-written as:
$$\boldsymbol{V}(\boldsymbol{x}+d \boldsymbol{x}, t)=\boldsymbol{V}(\boldsymbol{x}, t)+d \boldsymbol{x} \cdot \nabla \boldsymbol{V}$$
The first term on the right-hand side of Eq. (3.28) represents the translational motion of the fluid particle. The second expression is a scalar product of the differential displacement $d \boldsymbol{x}$ and the velocity gradient $\nabla \boldsymbol{V}$. We decompose the velocity gradient, which is a second order tensor, into two parts resulting in the following identity:
$$\nabla \boldsymbol{V}=\frac{1}{2}\left(\nabla \boldsymbol{V}+\nabla \boldsymbol{V}^{T}\right)+\frac{1}{2}\left(\nabla \boldsymbol{V}-\nabla \boldsymbol{V}^{T}\right) .$$
The superscript $T$ indicates that the matrix elements of the second order tensor $\nabla V^{T}$ are the transpositions of the matrix elements that pertain to the second order tensor $\nabla \boldsymbol{V}$. The first term in the right-hand side represents the deformation tensor, which is a symmetric second order tensor:
$$\boldsymbol{D}=\frac{1}{2}\left(\nabla \boldsymbol{V}+\nabla \boldsymbol{V}^{T}\right)=e_{i} e_{j} D_{i j}=\frac{1}{2} e_{i} e_{j}\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{\partial V_{j}}{\partial x_{i}}\right)$$
with components:
$$D_{i j}=\frac{1}{2}\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{\partial V_{j}}{\partial x_{i}}\right) .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Reynolds Transport Theorem

The conservation laws in integral form are, strictly speaking, valid for closed systems, where the mass does not cross the system boundary. In fluid mechanics, however, we are dealing with open systems, where the mass flow continuously crosses the system boundary. To apply the conservation laws to open systems, we briefly provide the necessary mathematical tools. In this section, we treat the volume integral of an arbitrary field quantity $f(\boldsymbol{X}, t)$ by deriving the Reynolds transport theorem. This is an important kinematic relation that we will use in Chap. $4 .$

The field quantity $f(\boldsymbol{X}, t)$ may be a zero ${ }^{\text {th }}$, first or second order tensor valued function, such as mass, velocity vector, and stress tensor. The time dependent volume under consideration with a given time dependent surface moves through the flow field and may experience dilatation, compression and deformation. It is assumed to contain the same fluid particles at any time and therefore, it is called the material volume. The volume integral of the quantity $f(\boldsymbol{X}, t)$ :
$$F(t)=\int_{v(t)} f(\boldsymbol{X}, t) d v$$ is a function of time only. The integration must be carried out over the varying volume $v(t)$. The material change of the quantity $F(t)$ is expressed as:
$$\frac{D F(t)}{D t}=\frac{D}{D t} \int_{v(t)} f(\boldsymbol{X}, t) d v .$$

# 力学代考

## 物理代写|流体力学代写Fluid Mechanics代考|Translation, Deformation, Rotation

$$V(\mathbf{x}+\mathbf{d} x, \mathbf{t})=V(\mathbf{x}, \mathbf{t})+\mathbf{d} V$$

$$\boldsymbol{V}(\boldsymbol{x}+d \boldsymbol{x}, t)=\boldsymbol{V}(\boldsymbol{x}, t)+d \boldsymbol{x} \cdot \nabla \boldsymbol{V}$$

$$\nabla \boldsymbol{V}=\frac{1}{2}\left(\nabla \boldsymbol{V}+\nabla \boldsymbol{V}^{T}\right)+\frac{1}{2}\left(\nabla \boldsymbol{V}-\nabla \boldsymbol{V}^{T}\right) .$$

$$\boldsymbol{D}=\frac{1}{2}\left(\nabla \boldsymbol{V}+\nabla \boldsymbol{V}^{T}\right)=e_{i} e_{j} D_{i j}=\frac{1}{2} e_{i} e_{j}\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{\partial V_{j}}{\partial x_{i}}\right)$$

$$D_{i j}=\frac{1}{2}\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{\partial V_{j}}{\partial x_{i}}\right) .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Reynolds Transport Theorem

$$F(t)=\int_{v(t)} f(\boldsymbol{X}, t) d v$$

$$\frac{D F(t)}{D t}=\frac{D}{D t} \int_{v(t)} f(\boldsymbol{X}, t) d v$$

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