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

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

Research on vortex flow has been initiated by the fundamental paper [36] of H.L.F. Helmholtz (1821-1894), a physicist and a professor of physiology and anatomy at the University of Königsberg, Bonn, Heidelberg and Berlin. In his paper, Helmholtz established his three theorems of vortex motion. Assuming incompressible frictionless fluids subjected to flow forces defined by a potential, Helmholtz [36] published a paper about the vortex motions in which he stated his vortex theorems. These theorems are translated from German and appear in an excellent textbook by Prandtl and Tietiens [31]. They reflect the quintessence of the vortex flow motion treated in Helmholtz original work. Before starting with the discussion of Helmholtz theorems, it is helpful to become familiar with the nomenclature sketched in Fig. 6.31.

A vortex line, Fig. 6.31a is a line tangent to the rotation vector $\nabla \times V$. The vortex lines may form a vortex tube, Fig. 6.3lb. Reducing the cross sections of a vortex tube to an infinitely small size, we obtain a vortex filament. Thus, a vortex filament is essentially a vortex tube with an infinitely small cross section but a finite value of circulation. This particular configuration allows one to apply the Stoke’s theorem Eq. (6.147) without integrating the rotation vector
$$\Gamma=(\nabla \times V) \cdot n d S .$$

Since the unit vector $\boldsymbol{n}$ is parallel to the rotation vector $\nabla \times \boldsymbol{V}$, we may re-arrange Eq. $(6.167)$
$$\nabla \times V=n \frac{\Gamma}{d S} .$$
Since $d s$ is, per definition, infinitely small and $\Gamma$ has a finite value, the rotation vector $\nabla \times V$ must be infinitely large, indicating that the vortex filaments represent a singularity. This and many other types of singularities are used for dealing with more complicated issues particularly in aerodynamics.

Ignoring the friction forces and assuming that there exists a potential acting on fluid particles, Helmholtz formulated in his original paper [36], three theorems: The first theorem states that no fluid particle can have a rotation if it did not originally rotate. This theorem reflects the physical content of the differential part of Eq. (6.166). The second theorem states that the fluid particles, which at any time are part of a vortex line, always belong to that same vortex line. This theorem is the consequence of the integral part of Eq. (6.166), stating that the circulation remains constant. The third theorem states that the product of the cross section area and angular velocity of an infinitely thin vortex filament remains constant over the whole length of the filament and keeps the same value even when the vortex moves. It further states that the vortex filament must, therefore, be either closed curves or end on the boundaries of the fluid.

## 物理代写|流体力学代写Fluid Mechanics代考|Induced Velocity Field, Law of Bio-Savart

We consider now an isolated vortex filament with the strength $\Gamma$ imbedded in an inviscid irrotational flow environment, as shown in Fig. 6.33.

In a distance $r$ from the point $A$, a differential element $d \xi$ of the vortex filament at the point $B$, induces a differential velocity vector field $d \boldsymbol{V}$. The velocity vector is perpendicular to the plane spanned by the normal unit vector $\boldsymbol{n}$ and the unit vector $\boldsymbol{e}$ in the $\boldsymbol{r}$-direction. The unit vector $\boldsymbol{n}$ is perpendicular to the infinitesimal cross section $d S$, whereas the unit vector $\boldsymbol{e}$ points from the center point of the element $A$ to the position $B$, where the velocity $d V$ is being induced. The relationship describing the velocity field is analogous to the one discovered by Bio and Savart through electrodynamic experiments. It describes the magnetic field induced by a current through a conducting wire. In an aerodynamic context, the conducting wire corresponds to the vortex filament, its current corresponds to the vortex strength $\Gamma$ and the induced magnetic field corresponds to the induced velocity field.

To present the derivation, first we provide the mathematical tool essential to arriving at the Bio-Savart law. Let us decompose an arbitrary vector point function $V$ into an irrotational part that can be expressed in terms of the gradient of a potential and a rotational or solenoidal part
$$\boldsymbol{V}=\nabla \Phi+\nabla \times \boldsymbol{U}$$
with $\Phi$ as a scalar potential and $\nabla \times \boldsymbol{U}$ as the solenoidal part of Eq. (6.186). Taking the curl of Eq. (6.186) gives
$$\nabla \times \boldsymbol{V}=\nabla \times(\nabla \Phi)+\nabla \times(\nabla \times \boldsymbol{U})=\nabla \times(\nabla \times \boldsymbol{U})$$
The first term on the right-hand side of Eq. (6.187) is the curl of the gradient of the scalar field $\Phi$ that identically vanishes. The divergence of Eq. (6.186) delivers
$$\nabla \cdot \boldsymbol{V}=\Delta \Phi+\nabla \cdot(\nabla \times \boldsymbol{U})=\nabla^2 \Phi=\Delta \Phi$$ with $\nabla \cdot \nabla \Phi=\Delta \Phi$ and $\nabla \cdot(\nabla \times \boldsymbol{U}) \equiv 0$. Equation $(6.188)$ is an inhomogeneous partial differential equation called Poisson’s equation. What makes the Poisson’s equation (6.188) a special case where $\Delta \Phi=0$, is the Laplace equation we treated in the preceding sections. Using the vector identity for $\nabla \times(\nabla \times \boldsymbol{U})=\nabla(\nabla \cdot \boldsymbol{U})-$ $\Delta \boldsymbol{U}, \mathrm{Eq} .(6.187)$ reads
$$\nabla \times \boldsymbol{V}=\nabla(\nabla \cdot \boldsymbol{U})-\Delta \boldsymbol{U}$$

## 物理代写|流体力学代写流体力学代考|Helmholtz定理

. . Helmholtz定理

$$\Gamma=(\nabla \times V) \cdot n d S .$$ 积分

$$\nabla \times V=n \frac{\Gamma}{d S} .$$

## 物理代写|流体力学代写流体力学代考|诱导速度场，生物-萨伐尔定律

$$\boldsymbol{V}=\nabla \Phi+\nabla \times \boldsymbol{U}$$

$$\nabla \times \boldsymbol{V}=\nabla \times(\nabla \Phi)+\nabla \times(\nabla \times \boldsymbol{U})=\nabla \times(\nabla \times \boldsymbol{U})$$

$$\nabla \cdot \boldsymbol{V}=\Delta \Phi+\nabla \cdot(\nabla \times \boldsymbol{U})=\nabla^2 \Phi=\Delta \Phi$$与$\nabla \cdot \nabla \Phi=\Delta \Phi$和$\nabla \cdot(\nabla \times \boldsymbol{U}) \equiv 0$。方程$(6.188)$是一个叫做泊松方程的非齐次偏微分方程。使泊松方程(6.188)成为$\Delta \Phi=0$的特例的是我们在前面几节中处理过的拉普拉斯方程。使用向量标识$\nabla \times(\nabla \times \boldsymbol{U})=\nabla(\nabla \cdot \boldsymbol{U})-$$\Delta \boldsymbol{U}, \mathrm{Eq} .(6.187)读取$$ \nabla \times \boldsymbol{V}=\nabla(\nabla \cdot \boldsymbol{U})-\Delta \boldsymbol{U}$\$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: