📚 Motion of Objects in Fluids | 物体在流体中的运动规律
When an object moves through a fluid (a liquid or a gas), it experiences a resistive force that opposes its motion. This force is known as fluid drag or viscous drag, and it plays a crucial role in determining the motion of everything from falling raindrops to submarines.
当物体在流体（液体或气体）中运动时，会受到一个阻碍其运动的阻力，称为流体阻力或粘性阻力。这个力在决定从下落雨滴到潜艇等各种物体运动状态时起着至关重要的作用。
1. What is Fluid Resistance? | 什么是流体阻力？
Fluid resistance is the force that opposes the relative motion between a solid object and the fluid through which it moves. It arises because the fluid layers in contact with the object are dragged along, creating shear stresses and energy losses.
流体阻力是阻碍固体物体与其所穿过流体之间相对运动的力。它的产生是因为与物体接触的流体层被物体拖曳，从而产生切应力和能量损失。
In A-Level physics, we usually distinguish between two main types of fluid resistance: viscous drag (dominant at low speeds and small sizes) and inertial drag or form drag (dominant at high speeds and larger objects).
在A-Level物理中，我们通常区分两种主要类型的流体阻力：粘性阻力（在低速和小尺寸时占主导）和惯性阻力或形状阻力（在高速和较大物体时占主导）。
2. Viscosity and Viscous Drag | 粘度与粘性阻力
Viscosity is a measure of a fluid’s internal friction. A highly viscous fluid, such as honey, resists relative motion between its layers more than a low-viscosity fluid like water.
粘度是衡量流体内部摩擦程度的一个物理量。像蜂蜜这样的高粘度流体，其层与层之间相对运动所受的阻力比水这样的低粘度流体更大。
When a small sphere moves slowly through a viscous fluid, the fluid adheres to the sphere’s surface. This creates a velocity gradient across the fluid layers, producing a tangential shear force on the sphere known as viscous drag.
当一个小球在粘性流体中缓慢运动时，流体附着在球体表面，在流体各层之间形成速度梯度，从而对球体产生切向剪切力，即粘性阻力。
F = 6πηrv
For a sphere of radius r moving with speed v through a fluid of viscosity η, the viscous drag is given by Stokes’ law:
对于半径为 r、以速度 v 在粘度为 η 的流体中运动的球体，其粘性阻力由斯托克斯定律给出：
F = 6πηrv
This equation shows that viscous drag is directly proportional to the radius, the speed, and the viscosity. It is valid only for laminar (streamline) flow and for small Reynolds numbers.
该方程表明粘性阻力与半径、速度和粘度均成正比。它仅适用于层流（流线型流动）和较小雷诺数的情况。
3. Stokes’ Law Applications and Conditions | 斯托克斯定律的应用与条件
Stokes’ law is widely used to determine the viscosity of fluids, to model the settling of particles in suspensions, and to estimate the terminal velocity of small droplets in a gas.
斯托克斯定律广泛用于测定流体粘度、模拟悬浮液中颗粒的沉降，以及估算小液滴在气体中的终端速度。
The law assumes that the particle is a rigid sphere, that the fluid is incompressible and homogeneous, and that the flow is laminar with no turbulence. In practice, this means the Reynolds number must be less than about 0.1 for Stokes’ law to be accurate.
该定律假设颗粒是刚性球体，流体不可压缩且均匀，并且流动为层流而无湍流。实际上，这意味着雷诺数需小于约0.1，斯托克斯定律才准确。
For larger objects or higher speeds, the inertial resistance becomes significant and the drag force is better described by a quadratic dependence on speed.
对于较大的物体或较高的速度，惯性阻力变得显著，阻力更好地用速度的二次方关系来描述。
4. Drag Force at High Speeds | 高速时的阻力
When an object moves quickly through a fluid, the fluid in front is pushed aside and a turbulent wake forms behind the object. The drag force in this regime is mainly due to pressure differences between the front and back of the object.
当物体在流体中快速运动时，前方的流体被推开，物体后方形成湍流尾迹。此时阻力主要来源于物体前后之间的压力差。
The magnitude of this inertial drag is given by:
这种惯性阻力的大小由下式给出：
F = ½ CD ρ A v²
where CD is the drag coefficient, ρ is the fluid density, A is the cross-sectional area perpendicular to motion, and v is the speed.
其中 CD 是阻力系数，ρ 是流体密度，A 是垂直于运动方向的横截面积，v 是速度。
This quadratic drag dominates for large objects like cars, aeroplanes, and skydivers. Streamlining reduces CD by allowing the fluid to flow smoothly around the object, minimising the wake.
这种二次方阻力对汽车、飞机、跳伞运动员等大型物体占主导。流线型设计通过让流体顺畅地流过物体周围并最小化尾迹来降低 CD。
5. Terminal Velocity – Falling Objects | 终端速度——下落物体
Consider a small sphere falling from rest through a viscous fluid. The forces acting are its weight W, the upthrust U (buoyancy), and the viscous drag F.
考虑一个小球从静止开始在粘性流体中下落。作用在其上的力有重力 W、浮力 U（上推力）和粘性阻力 F。
Initially, the sphere accelerates because the downward resultant force is W − U. As the speed increases, the drag force increases until the resultant force becomes zero. The sphere then falls with a constant maximum speed called the terminal velocity.
最初，小球因向下的合力 W − U 而加速。随着速度增大，阻力增大，直到合力变为零。此时小球以恒定的最大速度下落，该速度称为终端速度。
At terminal velocity:
在终端速度时：
W − U = 6πηrvt
Using the expressions for weight and upthrust, we can derive:
利用重力和浮力的表达式，我们可以推导出：
vt = [2r²(ρs − ρf)g] / (9η)
Here, ρs is the density of the sphere and ρf is the density of the fluid.
这里，ρs 是球体的密度，ρf 是流体的密度。
6. Velocity-Time Graph for a Falling Object | 下落物体的速度-时间图像
For an object falling through a fluid, the velocity-time graph has a characteristic shape. The gradient (acceleration) starts at g and gradually decreases to zero as the drag force builds up.
对于在流体中下落的物体，其速度-时间图像具有典型形状。斜率（加速度）从 g 开始，随着阻力逐渐增大而减小到零。
The graph approaches a horizontal asymptote at the terminal velocity vt. The time taken to reach terminal velocity depends on the mass, radius, and viscosity of the surrounding fluid.
图像趋近于在终端速度 vt 处的水平渐近线。达到终端速度所需的时间取决于物体的质量、半径以及周围流体的粘度。
For a high-viscosity fluid, terminal velocity is reached quickly, while in a low-viscosity fluid it may take longer. This can be investigated experimentally using a falling-ball viscometer.
在高粘度流体中，终端速度很快达到；而在低粘度流体中，所需时间可能更长。这可以通过落球粘度计进行实验研究。
7. Laminar and Turbulent Flow | 层流与湍流
Fluid flow can be laminar (streamline) or turbulent. In laminar flow, fluid particles move in smooth parallel layers with no mixing across layers. In turbulent flow, the motion is erratic, with vortices and eddies that dissipate large amounts of energy.
流体流动可以是层流（流线型）或湍流。层流中，流体粒子沿平滑的平行层运动，层间没有混合。湍流中，运动不规则，存在漩涡和涡流，会耗散大量能量。
Whether flow is laminar or turbulent is determined by the dimensionless Reynolds number Re:
流动是层流还是湍流由无量纲的雷诺数 Re 决定：
Re = ρvd / η
where d is a characteristic length (e.g., diameter of a pipe or sphere). Low Reynolds numbers indicate laminar flow; high Reynolds numbers indicate turbulent flow.
其中 d 是特征长度（例如管道直径或球的直径）。低雷诺数表示层流，高雷诺数表示湍流。
8. Reynolds Number and Flow Regime | 雷诺数与流动状态
For flow in a smooth pipe, the transition from laminar to turbulent flow typically occurs at a Reynolds number around 2000–4000. For a sphere moving through a fluid, turbulence begins around Re ≈ 1.
对于光滑管道内的流动，从层流向湍流的转变通常发生在雷诺数约为2000–4000时。对于球体在流体中运动，湍流在 Re ≈ 1 左右开始出现。
The Stokes’ law formula F = 6πηrv is only valid for Re < 0.1. For higher Reynolds numbers, the drag coefficient changes and the quadratic drag formula becomes more appropriate.
斯托克斯定律公式 F = 6πηrv 仅在 Re < 0.1 时有效。对于更高的雷诺数，阻力系数发生变化，二次方阻力公式更为适用。
In practice, the drag coefficient CD varies with Re. For Stokes flow, CD = 24/Re. At very high Reynolds numbers, CD becomes roughly constant.
实际上，阻力系数 CD 随 Re 变化。在斯托克斯流动中，CD = 24/Re。在极高雷诺数下，CD 近似为常数。
9. Upthrust and Its Role in Fluid Motion | 浮力及其在流体运动中的作用
An object submerged in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid, according to Archimedes’ principle.
根据阿基米德原理，浸没在流体中的物体受到一个竖直向上的浮力，其大小等于物体排开的流体所受的重力。
For a falling sphere, the upthrust is:
对于下落的小球，浮力为：
U = (4/3)πr³ρfg
This upthrust reduces the effective weight of the object, lowering its terminal velocity. In a dense fluid, an object may even float if its density is less than that of the fluid.
这个浮力减小了物体的有效重力，从而降低其终端速度。在密度较大的流体中，如果物体密度小于流体密度，它甚至会漂浮。
When studying the motion of objects in fluids, it is important to always include upthrust in the force balance, especially for gases where it is sometimes neglected, and for liquids where it is significant.
在研究物体在流体中的运动时，务必在力平衡中始终包含浮力，尤其是在气体中有时被忽略，而在液体中则很重要。
10. Practical Applications | 实际应用
-
Falling-ball viscometry: Measuring the terminal velocity of a ball falling through a liquid allows the viscosity of the liquid to be calculated using Stokes’ law.
落球粘度测量：测量小球在液体中下落的终端速度，即可利用斯托克斯定律计算液体的粘度。
-
Raindrop formation: Small raindrops fall with low terminal velocities because of viscous drag, while larger drops fall faster. This affects the formation and distribution of rainfall.
雨滴形成：小雨滴因粘性阻力而以较低的终端速度下落，而较大的雨滴下落更快。这影响降水的形成和分布。
-
Sedimentation: In water purification, particles settle due to gravity, and their settling rates are governed by Stokes’ law and the balance of forces.
沉降：在水净化中，颗粒因重力而沉降，其沉降速率由斯托克斯定律和力的平衡决定。
-
Air resistance and parachute design: Parachutes are designed with large drag coefficients and large surface areas to ensure a safe, low terminal velocity for skydivers.
空气阻力与降落伞设计：降落伞被设计成具有大阻力系数和大表面积，以确保跳伞者以安全、较低的终端速度着陆。
-
Submarine and vehicle design: Streamlined shapes reduce form drag, saving fuel and increasing speed. This is a direct application of the principles of inertial drag.
潜艇与车辆设计：流线型形状减少了形状阻力，节省燃料并提高速度。这是惯性阻力原理的直接应用。
11. Exam Tips and Common Mistakes | 考试提示与常见错误
In CIE A-Level Physics, questions on this topic often require you to sketch the velocity-time graph for a falling sphere, explain the forces involved, and calculate terminal velocity. Be sure to clearly label the forces and state the conditions for Stokes’ law.
在CIE A-Level物理中，此类题目通常要求你画出下落小球的速度-时间图像，解释涉及的力，并计算终端速度。务必清晰标出各力，并说明斯托克斯定律的适用条件。
Common mistakes include forgetting the upthrust, using Stokes’ law for high-speed motion, ignoring the density of the fluid in the terminal velocity formula, and confusing laminar with turbulent drag.
常见错误包括忘记浮力、对高速运动使用斯托克斯定律、在终端速度公式中忽略流体密度，以及混淆层流与湍流的阻力。
Always check the units: viscosity is measured in Pa·s (or N s m⁻²) in the SI system. Remember that 1 Pa·s = 1 kg m⁻¹ s⁻¹.
始终检查单位：粘度在国际单位制中单位为Pa·s（或 N s m⁻²）。记住 1 Pa·s = 1 kg m⁻¹ s⁻¹。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导