Moving Through Fluids | 流体中的运动

📚 Moving Through Fluids | 流体中的运动

When a solid body moves through a liquid or gas, its motion is resisted by the surrounding fluid. This resistance, called drag, arises from two main mechanisms: the viscosity of the fluid and the pressure difference created around the body. A-Level Physics focuses on modelling these forces to predict terminal velocity and to understand applications from falling raindrops to sedimenting particles.

当固体在液体或气体中运动时，其运动会受到周围流体的阻碍。这种阻碍称为阻力，主要来自两种机制：流体的黏度和物体周围产生的压力差。A-Level 物理重点在于建立这些力的模型，以预测终端速度并理解从下落雨滴到沉降颗粒等应用。

1. Fluids and drag forces | 流体与阻力

A fluid is any substance that can flow, so both liquids and gases are fluids. When a solid body moves relative to a fluid, the fluid exerts a drag force that always opposes the relative motion.

流体是任何可以流动的物质，因此液体和气体都是流体。当固体相对于流体运动时，流体会施加一个总是阻碍相对运动的阻力。

Drag force is not a single fixed quantity; it depends on the shape of the object, its speed, the density of the fluid, and the fluid’s viscosity. In many A-Level problems, drag increases with speed.

阻力不是固定不变的单一量；它取决于物体形状、速度、流体密度和流体黏度。在许多 A-Level 问题中，阻力随速度增大而增大。

At low speeds and for small objects, viscous drag dominates. At higher speeds, pressure drag due to turbulence becomes more significant. This distinction helps us decide whether to use a linear or quadratic drag model.

在低速和小物体情况下，黏性阻力占主导。在较高速度下，由湍流引起的压差阻力变得更加显著。这种区别有助于我们决定使用线性阻力模型还是平方阻力模型。

2. Laminar and turbulent flow | 层流与湍流

In laminar flow, fluid particles move in smooth, parallel layers with no mixing between layers. This occurs at low speeds and around streamlined shapes.

在层流中，流体粒子以平滑、平行的层运动，层与层之间没有混合。这发生在低速和流线型物体周围。

In turbulent flow, the fluid moves irregularly, forming eddies and vortices. Turbulence greatly increases the drag force because more kinetic energy is transferred to chaotic fluid motion.

在湍流中，流体不规则运动，形成涡流和漩涡。湍流会大大增加阻力，因为更多动能被转化为混乱的流体运动。

Feature	Laminar flow	Turbulent flow
Particle paths	Smooth and parallel	Irregular with eddies
Speed range	Low	High
Drag force	Smaller, often proportional to v	Larger, often proportional to v²

The type of flow around an object is indicated by the Reynolds number Re = ρ v L / η. Low Re corresponds to laminar flow, and high Re corresponds to turbulent flow.

物体周围的流动类型由雷诺数 Re = ρ v L / η 表示。低 Re 对应层流，高 Re 对应湍流。

In A-Level problems, a sphere moving slowly through oil is usually assumed to be in the laminar regime, so Stokes’ law can be applied.

在 A-Level 问题中，通常假设小球在油中缓慢运动处于层流状态，因此可以应用斯托克斯定律。

3. Viscosity | 黏度

Viscosity is a measure of a fluid’s internal friction or resistance to flow. A high-viscosity fluid, such as honey, resists motion more than a low-viscosity fluid, such as water.

黏度是流体内部摩擦或流动阻力的量度。高黏度流体（如蜂蜜）比低黏度流体（如水）更阻碍运动。

The SI unit of viscosity is the pascal second (Pa s). Viscosity usually decreases with increasing temperature for liquids, but increases with temperature for gases.

黏度的国际单位是帕斯卡·秒（Pa·s）。液体的黏度通常随温度升高而降低，而气体的黏度随温度升高而增大。

In a viscous fluid, adjacent layers move at different speeds and exert frictional forces on each other. This is why a sphere dragged through the fluid experiences a retarding force even in steady laminar flow.

在黏性流体中，相邻层以不同速度运动，并相互施加摩擦力。这就是为什么球体即使处于稳定的层流中，在流体中穿行时也会受到阻滞力。

4. Stokes’ law for a sphere | 球体的斯托克斯定律

For a small sphere moving slowly through a viscous fluid, the viscous drag force is given by Stokes’ law:

对于在黏性流体中缓慢运动的小球，黏性阻力由斯托克斯定律给出：

F = 6π η r v

Here F is the drag force, η is the viscosity of the fluid, r is the radius of the sphere, and v is its speed relative to the fluid. The law is valid only for laminar flow and low Reynolds number.

其中 F 是阻力，η 是流体黏度，r 是球体半径，v 是球体相对流体的速度。该定律仅适用于层流和低雷诺数条件。

Stokes’ law shows that drag is directly proportional to speed, not to speed squared. This linear relationship holds only at low speeds.

斯托克斯定律表明阻力与速度成正比，而不是与速度平方成正比。这种线性关系仅在低速时成立。

Example: A sphere of radius 2.0 × 10⁻³ m moves at 0.50 m s⁻¹ through a fluid of viscosity 0.80 Pa s. The drag force is F = 6π × 0.80 × 2.0 × 10⁻³ × 0.50 ≈ 0.015 N.

示例：一个半径为 2.0 × 10⁻³ m 的球体以 0.50 m s⁻¹ 的速度在黏度为 0.80 Pa s 的流体中运动。阻力为 F = 6π × 0.80 × 2.0 × 10⁻³ × 0.50 ≈ 0.015 N。

5. Upthrust and Archimedes’ principle | 浮力与阿基米德原理

An object immersed in a fluid also experiences an upward buoyancy force, called upthrust. Archimedes’ principle states that the upthrust is equal to the weight of the fluid displaced.

浸在流体中的物体还会受到向上的浮力，称为浮力。阿基米德原理指出，浮力等于物体排开流体的重量。

U = ρₗ V g

For a fully submerged object of volume V in a fluid of density ρₗ, the upthrust is U = ρₗ V g. This force reduces the object’s effective weight.

对于体积为 V、浸在密度为 ρₗ 的流体中的物体，浮力为 U = ρₗ V g。此力减小了物体的有效重量。

When a sphere falls through a fluid, the total downward force is its weight mg, while the upward forces are the upthrust U and the viscous drag F. If the object is less dense than the fluid, the upthrust can exceed the weight and the object rises.

当球体在流体中下落时，向下的力为重力 mg，而向上的力为浮力 U 和黏性阻力 F。如果物体密度小于流体密度，浮力可能大于重力，物体就会上升。

6. Forces on a body moving through a fluid | 物体在流体中运动的受力

Consider a sphere released from rest in a viscous fluid. Initially, its speed is zero, so the viscous drag is zero. The only upward force is the upthrust, so the sphere accelerates downward if its weight is greater than the upthrust.

考虑一个从静止释放到黏性流体中的球体。初始速度为零，因此黏性阻力为零。唯一的向上力是浮力，所以如果重力大于浮力，球体向下加速。

As the sphere gains speed, the drag force F = 6π η r v increases. The net force downward is:

随着球体获得速度，阻力 F = 6π η r v 增大。向下的合力为：

F_net = mg – U – F_drag

The acceleration becomes smaller as the drag grows, because a = F_net / m. This is why the object does not accelerate indefinitely under gravity.

随着阻力增大，加速度变小，因为 a = F_net / m。这就是物体不会在重力作用下无限加速的原因。

7. Terminal velocity | 终端速度

Terminal velocity is reached when the net force on the falling object becomes zero. At this point, the weight is exactly balanced by the upthrust and the viscous drag, so the object stops accelerating and moves at constant speed.

当下落物体的合力变为零时，就达到终端速度。此时重力恰好被浮力和黏性阻力平衡，物体停止加速并以恒定速度运动。

mg = U + F_drag

For most A-Level problems involving small spheres in viscous liquids, the upthrust is often small but must be included unless the question states otherwise.

对于大多数涉及小球在黏性液体中运动的 A-Level 问题，浮力通常较小，但除非题目另有说明，否则必须计入。

In air, the upthrust on a dense object such as a skydiver is usually negligible compared with the weight. In that case the terminal velocity condition becomes mg = F_drag, where F_drag is the air resistance.

在空气中，诸如跳伞者等致密物体所受的浮力

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Moving Through Fluids | 流体中的运动