Fluids and Conservation Laws · 流体与守恒定律
Thumb over the hose — the water shoots out
- Put your thumb over the end of a garden hose and the water suddenly sprays far and fast.
- You narrowed the opening, and the same flow had to speed up to get through.
- Moving fluids obey conservation laws: mass and energy are both preserved.
- These give two powerful rules — continuity and Bernoulli's principle.
用拇指按住水管口——水喷射而出
- 用拇指按住花园水管的末端,水突然又远又快地喷出来。
- 你缩小了开口,同样的流量必须加速才能通过。
- 流动的流体遵守守恒定律:质量和能量都被保持。
- 这给出两条有力的规则——连续性和伯努利原理。
Continuity: what goes in must come out
- For a steady flow, the fluid passing each point per second is the same everywhere.
- Continuity: $A_1 v_1 = A_2 v_2$ — area times speed is constant along the pipe.
- So a narrow section forces a faster flow, and a wide one a slower flow.
- It is just conservation of mass: no fluid piles up or vanishes.
连续性:进去多少必须出来多少
- 对稳定流动,每秒经过每一点的流体处处相同。
- 连续性:$A_1 v_1 = A_2 v_2$——沿管道,面积乘以速率恒定。
- 所以窄的一段迫使流动更快,宽的一段则更慢。
- 这只是质量守恒:没有流体堆积或消失。
Follow the flow through a pipe · 追踪管道中的流动
Step through a narrowing pipe and see how the flow speeds up where the pipe is narrow. · 逐步展示变窄的管道，观察流体在狭窄处流速加快。
Water at $2\ \tfrac{\text{m}}{\text{s}}$ flows from a pipe of area $6\ \text{cm}^2$ into one of area $2\ \text{cm}^2$. What is the new speed, in $\tfrac{\text{m}}{\text{s}}$? · 流速为$2\ \tfrac{\text{m}}{\text{s}}$的水从横截面积为$6\ \text{cm}^2$的管道流入横截面积为$2\ \text{cm}^2$的管道。新流速是多少$\tfrac{\text{m}}{\text{s}}$？
$A_1 v_1 = A_2 v_2 \Rightarrow 6 \times 2 = 2 v_2 \Rightarrow v_2 = 6\ \tfrac{\text{m}}{\text{s}}$.
The continuity equation $A_1 v_1 = A_2 v_2$ is a statement of the conservation of: · 连续性方程$A_1 v_1 = A_2 v_2$是关于____守恒的表述。
No fluid piles up or disappears — it is conservation of mass. · 没有流体堆积或消失——这是质量守恒。
In a steady flow, fluid moves faster through a narrower section of pipe. · 在稳态流动中，流体通过较窄的管道截面时流速更快。
Continuity ($A_1 v_1 = A_2 v_2$) forces a higher speed where the area is smaller. · 连续性方程（$A_1 v_1 = A_2 v_2$）迫使面积较小处流速较高。
Bernoulli: faster flow, lower pressure
- Bernoulli's principle comes from conserving energy in a flowing fluid.
- Where a fluid flows faster, its pressure is lower; where it slows, pressure rises.
- It is the energy trade-off: speeding up the flow costs pressure energy.
- This is why a fast stream of air has a low pressure that things get pushed into.
伯努利:流动越快,压强越低
- 伯努利原理来自在流动流体中守恒能量。
- 流体流动更快处,它的压强更低;变慢处,压强升高。
- 这是能量的权衡:加快流动要付出压强能量。
- 这就是为什么快速气流有低压,东西会被推向那里。
By Bernoulli's principle, where a fluid flows faster, its pressure is: · 根据伯努利原理，流体流速快的地方，其压强：
Faster flow means lower pressure — the energy trade-off of Bernoulli's principle. · 流速越快，压强越低——这是伯努利原理的能量权衡。
Bernoulli's principle comes from conserving ____ in a flowing fluid. · 伯努利原理源于流动流体中的____守恒。
It is the energy conservation law for a flowing fluid. · 它是流动流体的能量守恒定律。
Select all · 所有 correct statements about flowing fluids. · 选择所有关于流动流体的正确陈述。
Continuity (mass) speeds flow in narrow pipes; Bernoulli (energy) links faster flow to lower pressure. · 连续性方程（质量）使窄管中的流速加快；伯努利原理（能量）将流速加快与压强降低联系起来。
Conservation laws explain everyday flow
- An aeroplane wing speeds air over the top, lowering the pressure there, producing lift.
- A spinning ball curves because it makes the air faster on one side than the other.
- A chimney draws better in wind, because fast air over the top lowers the pressure.
- Both continuity and Bernoulli are conservation laws wearing fluid clothing.
守恒定律解释日常流动
- 飞机机翼让空气在上方流得更快,降低那里的压强,产生升力。
- 旋转的球会拐弯,因为它使一侧的空气比另一侧快。
- 烟囱在有风时抽得更好,因为顶部的快气流降低了压强。
- 连续性和伯努利都是穿着流体外衣的守恒定律。
Bernoulli's principle is often stated backwards. It is faster flow ⇒ lower pressure, not higher. And continuity ($A_1 v_1 = A_2 v_2$) means fluid speeds up in a narrow pipe — the opposite of the intuition that "less room means slower".
伯努利原理常被说反。它是流动越快 ⇒ 压强越低,而非更高。而连续性($A_1 v_1 = A_2 v_2$)意味着流体在窄管中加速——与"空间越小越慢"的直觉相反。
Water flows at $2\ \tfrac{\text{m}}{\text{s}}$ through a pipe of area $6\ \text{cm}^2$, then into a narrow section of area $2\ \text{cm}^2$.
- Continuity: $A_1 v_1 = A_2 v_2 \Rightarrow 6 \times 2 = 2 \times v_2$.
- $v_2 = \dfrac{12}{2} = 6\ \tfrac{\text{m}}{\text{s}}$ — three times faster in the narrow part.
水以 $2\ \tfrac{\text{m}}{\text{s}}$ 流过面积 $6\ \text{cm}^2$ 的管道,然后进入面积 $2\ \text{cm}^2$ 的窄段。
- 连续性:$A_1 v_1 = A_2 v_2 \Rightarrow 6 \times 2 = 2 \times v_2$。
- $v_2 = \dfrac{12}{2} = 6\ \tfrac{\text{m}}{\text{s}}$——在窄段快了三倍。
Flowing fluids conserve mass and energy. Continuity ($A_1 v_1 = A_2 v_2$) means a narrower pipe gives a faster flow. Bernoulli's principle says faster flow means lower pressure. Together they explain lift, curveballs and much of everyday fluid flow.
流动的流体守恒质量和能量。连续性($A_1 v_1 = A_2 v_2$)意味着更窄的管道给出更快的流动。伯努利原理说流动越快,压强越低。两者一起解释了升力、曲线球以及许多日常流体流动。