complete 2023/10/25 notes

src/20231025.md

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+## 2023/10/25
+
+### 1. Rigid body rotation with constant angular velocity $\vec \omega_0$ 以恒角速度$\vec \omega_0$转动的刚体
+
+$$\overrightarrow \nabla \times \left( \vec \omega_0 \times \vec r\right) = 2 \omega_0$$
+
+### 2. Field theory 场论
+
+$$\vec r = x \hat i + y \hat j + z \hat k$$
+
+$$r = \sqrt{x^2 + y^2 + z^2}$$
+
+Common conclusions in field theory:
+
+- $\overrightarrow \nabla \cdot \vec r = 3$
+- $\overrightarrow \nabla \times \vec r = \vec 0$
+- $\overrightarrow \nabla r = \hat{\vec r}$
+- $\nabla^2 r = \dfrac{2}{r}$
+- $\overrightarrow \nabla \vec r = \mathbf I$
+
+### 3. From Navier-Stokes equations to Bernoulli's principle (伯努利原理)
+
+The material derivative (物质导数) of $\vec v$ is shown as follows:
+
+$$\vec a = {\mathrm D \vec v \over \mathrm Dt} \equiv \underbrace{\partial \vec v \over \partial t}_\text{Local/Euler acceleration} + \underset{\text{平流加速度 (非线性)}}{\underbrace{\left(\vec v \cdot \overrightarrow \nabla \right) \vec v}_\text{advective acceleration}}$$
+
+Euler, 1750: infinitesimal 无穷小量
+
+Why does this form occur?
+
+Suppose we have a macroscopic tensor field (宏观张量场) $T$ with the sense that it depends only on position and time coordinates:
+
+$$T(t, \vec r(t))$$
+
+$${\mathrm dT \over \mathrm dt} = {\partial T \over \partial t} + {\partial T \over \partial \vec r} {\mathrm d \vec r \over \mathrm dt} = {\partial T \over \partial t} + \vec v {\partial T \over \partial \vec r} = \left({\partial \over \partial t} + \vec v \cdot {\partial \over \partial \vec r} \right) T = \left({\partial \over \partial t} + \vec v \cdot \overrightarrow \nabla \right) T.$$
+
+Using this we can get: If $\vec v = \vec v(\vec r(t), t)$, then $$\vec a = {\partial \vec v \over \partial t} + (\vec v \cdot \overrightarrow \nabla) \vec v.$$
+
+From the Navier-Stokes equations we can get:
+
+$$\vec a = - {1 \over \rho} \overrightarrow \nabla p + \nu \nabla^2 \vec v + \vec g,$$
+
+which become the Euler equations (fluid dynamics) when $\nu = 0.$
+
+By using lamb vector $$\left(\overrightarrow \nabla \times \vec v \right) \times \vec v = (\vec v \cdot \overrightarrow \nabla) \vec v - \overrightarrow \nabla \left( {1 \over 2} v^2 \right),$$
+
+we can get
+
+$${\partial \vec v \over \partial t} + \left(\overrightarrow \nabla \times \vec v \right) \times \vec v + \overrightarrow \nabla \left( {1 \over 2} v^2 \right) = -{1 \over \rho} \overrightarrow \nabla p + \mu \nabla^2 \vec v + \vec g.$$
+
+Under the following conditions:
+
+- The fluid flows in steady state (定常流动): $\dfrac{\partial \vec v}{\partial t} = 0$
+- Inviscid fluid (无黏液体): $\nu = 0$
+- Under conservative force field $\vec g = - \overrightarrow \nabla (gh)$
+- The fluid is incompressible (液体不可压缩)
+
+We can get:
+
+$$\left(\overrightarrow \nabla \times \vec v \right) \times \vec v + \overrightarrow \nabla \left( {1 \over 2} v^2 \right) = - \overrightarrow \nabla \left({p \over \rho} \right) - \overrightarrow \nabla (gh)$$
+
+$$\left(\overrightarrow \nabla \times \vec v \right) \times \vec v + \overrightarrow \nabla \left({p \over \rho} + {1 \over 2} v^2 + gh \right) = \vec 0$$
+
+$${\vec v \over |\vec v|} \cdot \left[ \left(\overrightarrow \nabla \times \vec v \right) \times \vec v + \overrightarrow \nabla \left({p \over \rho} + {1 \over 2} v^2 + gh \right) \right] = 0$$
+
+$${\vec v \over |\vec v|} \cdot \left[ \left(\overrightarrow \nabla \times \vec v \right) \times \vec v \right] + {\vec v \over |\vec v|} \cdot \overrightarrow \nabla \left({p \over \rho} + {1 \over 2} v^2 + gh \right) = 0$$
+
+Clearly,
+
+$${\vec v \over |\vec v|} \cdot \left[ \left(\overrightarrow \nabla \times \vec v \right) \times \vec v \right] = 0,$$
+
+because $\left(\overrightarrow \nabla \times \vec v \right) \perp \vec v.$
+
+$\displaystyle {\vec v \over |\vec v|} \cdot \overrightarrow \nabla$ is a directional derivative (方向导数).
+
+Integral along the streamline (沿流线积分), and we obtain 
+
+$${p \over \rho} + {1 \over 2} v^2 + gh = \text{const},$$
+
+which is called the **Bernoulli's principle (伯努利原理)**.
+
+Its other forms include:
+
+| Form | Name |
+| :-- | :-- |
+| $${p \over \rho} + {1 \over 2} v^2 + gh = \text{const}$$ | Energy form (per unit mass) |
+| $$p + {1 \over 2} \rho v^2 + \rho g h = \text{const}$$ | Pressure form |
+| $${p \over \rho g} + {v^2 \over 2g}  + h = \text{const}$$ | Head form (used in Hydraulic engineering 水利形式) |