IB Physics: Analysis of Fluid Motion Laws | IB物理：流体运动规律解析

📚 IB Physics: Analysis of Fluid Motion Laws | IB物理：流体运动规律解析

Fluid motion is a fascinating area of physics that connects microscopic molecular behaviour with macroscopic phenomena such as ocean currents, blood flow and aircraft lift. In the IB Physics syllabus, understanding the laws of fluid motion requires a firm grasp of pressure, buoyancy, continuity and Bernoulli’s principle.

流体运动是物理学中一个引人入胜的领域，它将微观分子行为与洋流、血流和飞机升力等宏观现象联系起来。在IB物理课程中，理解流体运动规律需要扎实掌握压强、浮力、连续性与伯努利原理。

1. Density and Pressure | 密度与压强

Density is defined as mass per unit volume, usually denoted by ρ (rho). For a fluid, density may vary with temperature and pressure, but in many IB problems it is treated as constant.

密度定义为单位体积的质量，通常用ρ表示。对于流体而言，密度可能随温度和压强变化，但在许多IB问题中被视为恒定。

Pressure is the normal force per unit area exerted by a fluid on a surface. The SI unit is the pascal (Pa), where 1 Pa = 1 N m⁻². In fluids at rest, pressure acts equally in all directions at a given depth.

压强是流体作用在单位面积上的法向力。国际单位是帕斯卡（Pa），1 Pa = 1 N m⁻²。在静止流体中，同一深度处压强向各个方向均相等。

Density formula: ρ = m / V
压强公式：ρ = m / V
Pressure at depth h: p = p₀ + ρgh
深度h处的压强：p = p₀ + ρgh

p = p₀ + ρgh, where p₀ is the atmospheric pressure at the surface.

p = p₀ + ρgh，其中p₀为液面处的大气压强。

2. Hydrostatic Pressure and Pascal’s Principle | 液体静压强与帕斯卡原理

Hydrostatic pressure arises from the weight of the fluid above a given point. Because liquids are nearly incompressible, the pressure depends only on depth and fluid density, not on the shape of the container.

液体静压强源于上方流体的重量。由于液体几乎是不可压缩的，压强仅取决于深度和流体密度，而与容器形状无关。

Pascal’s principle states that when an external pressure is applied to a confined fluid, the pressure change is transmitted undiminished throughout the fluid. This is the basis of hydraulic lifts and brakes.

帕斯卡原理指出：对密闭流体施加外部压强时，压强变化会毫无衰减地传递到流体各处。这是液压升降机和液压制动器的基础。

In a hydraulic system: F₁/A₁ = F₂/A₂
在液压系统中：F₁/A₁ = F₂/A₂
A small force applied over a small area can lift a large weight over a larger area.
在小面积上施加小力，可以在大面积上举起重物。

3. Archimedes’ Principle and Buoyancy | 阿基米德原理与浮力

Archimedes’ principle states that an object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.

阿基米德原理指出：完全或部分浸没在流体中的物体会受到向上的浮力，浮力大小等于物体排开流体的重量。

The buoyant force is independent of the object’s shape, density or depth. It depends only on the volume of fluid displaced and the gravitational field strength.

浮力与物体的形状、密度或浸没深度无关，只取决于排开流体的体积和重力场强度。

F_buoyancy = ρ_fluid × V_displaced × g

F_浮 = ρ_流体 × V_排 × g

If F_buoyancy > mg, the object floats.
若浮力大于mg，物体上浮。
If F_buoyancy = mg, the object remains in equilibrium.
若浮力等于mg，物体悬浮平衡。
If F_buoyancy < mg, the object sinks.
若浮力小于mg，物体下沉。

4. Ideal Fluid and Steady Flow | 理想流体与定常流动

An ideal fluid is assumed to be incompressible and non-viscous, and its flow is steady and irrotational. These simplifying assumptions allow us to apply conservations laws easily.

理想流体被假定为不可压缩、无粘性，且其流动是定常、无旋的。这些简化假设使我们能够方便地应用守恒定律。

Steady flow means that at any point, the velocity of the fluid particles is constant in time. Streamlines represent the paths of fluid particles in steady flow.

定常流动意味着在任意一点，流体质点的速度不随时间变化。流线代表定常流动中流体质点的运动路径。

Incompressible: density ρ remains constant throughout the fluid.
不可压缩：整个流体中密度ρ保持不变。
Non-viscous: no internal friction between fluid layers.
无粘性：流体层之间无内摩擦。

5. Equation of Continuity | 连续性方程

The equation of continuity is a statement of conservation of mass for an incompressible fluid. For a fluid flowing through a pipe of varying cross-sectional area A and velocity v, the volume flow rate must remain constant.

连续性方程是不可压缩流体质量守恒的表述。对于流经横截面积A和速度v变化的管道的流体，体积流量必须保持恒定。

A₁v₁ = A₂v₂

This implies that when the pipe narrows, the fluid must speed up, and when it widens, the fluid slows down.

这意味着当管道变窄时，流体必须加速；当管道变宽时，流体减速。

Volume flow rate Q = Av, measured in m³ s⁻¹.
体积流量Q = Av，单位为m³ s⁻¹。
Mass flow rate = ρAv, constant for ideal fluids.
质量流量 = ρAv，对理想流体恒定。

6. Bernoulli’s Equation | 伯努利方程

Bernoulli’s equation relates pressure, speed and height for an ideal fluid along a streamline. It is derived from the work-energy theorem and applies to incompressible, non-viscous fluids in steady flow.

伯努利方程将理想流体沿流线的压强、速度和高度联系起来。它由功-能定理推导而来，适用于不可压缩、无粘性、定常流动的流体。

p + ½ρv² + ρgh = constant

p + ½ρv² + ρgh = 常数

The three terms represent static pressure, dynamic pressure and gravitational potential energy per unit volume respectively.

这三项分别代表静压强、动压强和单位体积的重力势能。

If height h is constant: p + ½ρv² = constant.
若高度h不变：p + ½ρv² = 常数。
Higher speed means lower pressure in horizontal flow.
水平流动中，速度越大，压强越小。

7. Applications of Bernoulli’s Equation | 伯努利方程的应用

Bernoulli’s principle explains many real-life phenomena. In a venturi meter, a constriction causes the fluid speed to increase and pressure to drop, allowing flow speed to be measured.

伯努利原理解释了许多生活现象。在文丘里流量计中，收缩段使流体速度增大、压强降低，从而可以测量流速。

Aerofoil lift arises because air moves faster over the curved top surface than below, creating a pressure difference that produces an upward force.

机翼升力源于空气在弯曲的上表面流速比下表面快，从而产生压强差，形成向上的力。

Curved roof lifted off in high winds: fast air above creates low pressure.
大风掀翻弯曲屋顶：上方快速气流产生低压。
Atomizer / perfume spray uses moving air to reduce pressure and draw liquid up.
雾化器/香水喷雾利用流动空气降低压强从而吸上液体。
Sailing against the wind: sails act as aerofoils.
逆风航行：帆充当翼形。

8. Viscosity and Viscous Flow | 粘性与粘性流动

Real fluids have internal friction called viscosity. Viscosity measures a fluid’s resistance to deformation or flow. Honey has high viscosity, water has lower viscosity, and gases have very low viscosity.

真实流体具有称为粘性的内摩擦。粘性度量流体对形变或流动的抵抗程度。蜂蜜的粘性高，水的粘性较低，气体的粘性非常低。

For viscous flow through a pipe, the velocity is not uniform: it is maximum at the centre and zero at the walls. This parabolic profile is described by Poiseuille’s law for laminar flow.

对于管内粘性流动，速度不均匀：中心处最大，管壁处为零。这种抛物线分布可用层流时的泊肃叶定律描述。

Q = (πΔp r⁴) / (8ηL)

where Q is volume flow rate, Δp is pressure difference, r is pipe radius, η is viscosity, L is pipe length.
其中Q为体积流量，Δp为压强差，r为管道半径，η为粘性系数，L为管道长度。
Flow rate is very sensitive to pipe radius: doubling radius increases flow rate by a factor of 16.
流量对管道半径极其敏感：半径加倍使流量增大为原来的16倍。

9. Laminar and Turbulent Flow | 层流与湍流

Laminar flow is smooth and orderly, with fluid layers sliding past each other without mixing. Turbulent flow is chaotic, with eddies and vortices that cause energy losses.

层流是平滑有序的流动，流体层彼此滑动而不混合。湍流是混乱的流动，伴有涡流和旋涡，造成能量损失。

The Reynolds number, Re, is a dimensionless quantity that predicts the flow regime. It is given by:

雷诺数Re是无量纲量，用于预测流态。其表达式为：

Re = ρvd / η

Low Re (typically < 2000): laminar flow.
低雷诺数（通常小于2000）：层流。
High Re (typically > 4000): turbulent flow.
高雷诺数（通常大于4000）：湍流。
In between: transitional regime.
中间区域：过渡流态。

10. Terminal Speed and Stokes’ Law | 收尾速度与斯托克斯定律

When a sphere falls through a viscous fluid, it experiences three forces: weight, buoyancy and viscous drag. At first it accelerates, but the drag increases with speed until the net force is zero.

当小球在粘性流体中下落时，它受到三个力：重力、浮力和粘性阻力。起初它加速，但阻力随速度增大，直到合力为零。

Stokes’ law gives the viscous drag on a small sphere of radius r moving at speed v through a fluid of viscosity η:

斯托克斯定律给出半径为r的小球以速度v通过粘性系数为η的流体时所受的粘性阻力：

F_drag = 6πηrv

F_阻力 = 6πηrv

At terminal speed, the net force is zero, giving:

在收尾速度时合力为零，得到：

v_terminal = (2r²(ρ_sphere – ρ_fluid)g) / (9η)

v_收尾 = (2r²(ρ_球 – ρ_流体)g) / (9η)

11. Energy Considerations in Fluids | 流体中的能量分析

Bernoulli’s equation is essentially an energy conservation law per unit volume. In real fluids, viscous losses convert mechanical energy into internal energy, so the total head decreases along the flow.

伯努利方程本质上是单位体积的能量守恒定律。在真实流体中，粘性损耗将机械能转化为内能，因此总水头沿流动方向减小。

For IB problems, you should be able to compare points along a streamline, using the energy approach to determine unknown pressures, speeds or heights.

对于IB考题，你需要能比较沿流线的各点，利用能量方法求未知的压强、速度或高度。

Check units: each term in Bernoulli’s equation has units of pressure (Pa = J m⁻³).
检查单位：伯努利方程中每一项都有压强单位（Pa = J m⁻³）。
When viscous effects are significant, Bernoulli’s equation no longer holds exactly.
当粘性效应显著时，伯努利方程不再精确成立。

12. Common IB Exam Tips | IB考试常见要点

Fluid motion questions often combine continuity with Bernoulli’s equation. Always start by identifying the two points along one streamline, then list the known and unknown quantities.

流体运动题目常常将连续性方程与伯努利方程结合。务必先确定同一条流线上的两个点，然后列出已知量和未知量。

State assumptions clearly: ideal fluid, steady flow, incompressible.
清楚说明假设：理想流体、定常流动、不可压缩。
Remember that pressure can be absolute or gauge — stay consistent.
注意压强可以是绝对压强或表压——保持一致。
For floating problems, use Archimedes’ principle with the displaced volume only.
对于浮体问题，使用阿基米德原理时只考虑排开体积。
Do not confuse density of the object with density of the fluid in buoyancy calculations.
在浮力计算中不要把物体密度与流体密度混淆。

Q = Av = constant; p + ½ρv² + ρgh = constant

Q = Av = 常数；p + ½ρv² + ρgh = 常数

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IB Physics: Analysis of Fluid Motion Laws | IB物理：流体运动规律解析