Table of Contents
Fluid Mechanics, Buoyancy and Archimedes’ Principle
1. Introduction
Fluid mechanics is the branch of physics that studies fluids at rest and in motion. A fluid is a substance that can flow and continuously deforms under an applied shear force.
Liquids and gases are both fluids. Fluid mechanics explains the behaviour of water in rivers, air around aircraft, blood in blood vessels, oil in pipelines and gases in industrial systems.
Two major branches of fluid mechanics are:
- Fluid statics: study of fluids at rest.
- Fluid dynamics: study of fluids in motion.
Buoyancy and Archimedes’ principle are fundamental concepts explaining why objects float, sink or remain suspended in fluids.
2. Basic Properties of Fluids
2.1 Density
Density is mass per unit volume.
[
\rho=\frac{m}{V}
]
It influences hydrostatic pressure, buoyancy and fluid behaviour.
2.2 Pressure
Pressure is force per unit area.
[
P=\frac{F}{A}
]
In a stationary fluid, pressure increases with depth.
2.3 Viscosity
Viscosity is a fluid’s resistance to flow due to internal friction. Honey is more viscous than water under comparable conditions.
2.4 Compressibility
Compressibility describes how much a fluid’s volume changes under pressure.
Gases are generally highly compressible, while liquids are comparatively incompressible.
2.5 Surface Tension
Surface tension is a property of liquid surfaces arising from cohesive intermolecular forces. It tends to minimise the surface area of a liquid.
3. Fluid Statics
Fluid statics deals with fluids in equilibrium.
3.1 Pressure in a Stationary Fluid
Pressure at a depth (h) in a liquid is:
[
P=\rho gh
]
This represents gauge pressure. Absolute pressure also includes atmospheric pressure.
3.2 Pressure Transmission
According to Pascal’s principle, pressure applied to an enclosed fluid is transmitted throughout the fluid.
Applications include hydraulic brakes, hydraulic lifts and hydraulic presses.
3.3 Pressure at the Same Level
In a connected stationary liquid of uniform density, pressure is equal at the same horizontal level.
This principle is used in communicating vessels and liquid-level systems.
4. Fluid Dynamics
Fluid dynamics studies fluids in motion.
4.1 Types of Fluid Flow
Steady Flow
Fluid properties at a particular point do not change with time.
Unsteady Flow
Fluid properties at a particular point change with time.
Laminar Flow
Fluid particles move in smooth, orderly layers with limited mixing between adjacent layers.
Turbulent Flow
Fluid particles move irregularly, producing eddies and mixing.
Compressible Flow
Fluid density changes significantly during flow.
Incompressible Flow
Fluid density remains approximately constant during flow. Many ordinary water-flow problems use this approximation.
4.2 Streamlines
A streamline is an imaginary curve whose tangent at any point indicates the direction of fluid velocity at that point.
In steady flow, streamlines do not intersect.
4.3 Equation of Continuity
The continuity equation expresses conservation of mass in fluid flow.
For an incompressible fluid:
[
A_1v_1=A_2v_2
]
Where:
- (A_1) and (A_2) = cross-sectional areas
- (v_1) and (v_2) = fluid velocities
The equation shows that fluid velocity increases when the cross-sectional area of a pipe decreases, assuming steady incompressible flow.
For a general compressible fluid:
[
\rho_1A_1v_1=\rho_2A_2v_2
]
4.4 Bernoulli’s Principle
Bernoulli’s principle states that for steady, incompressible, non-viscous flow along a streamline, the sum of pressure energy, kinetic energy and gravitational potential energy per unit volume remains constant.
[
P+\frac{1}{2}\rho v^2+\rho gh=constant
]
Where:
- (P) = pressure
- (\rho) = fluid density
- (v) = fluid velocity
- (g) = acceleration due to gravity
- (h) = height above a reference level
At the same height, an increase in fluid velocity is associated with a decrease in static pressure under the ideal assumptions of Bernoulli’s principle.
Applications:
- Venturi meters
- Atomisers and spray devices
- Carburettors
- Pitot tubes
- Certain aerodynamic explanations of lift
- Flow measurement systems
Real fluid systems may involve viscosity, turbulence and energy losses, so Bernoulli’s equation requires appropriate assumptions.
5. Buoyancy
5.1 Definition
Buoyancy is the upward force exerted by a fluid on an object partially or completely immersed in it.
This force is known as the buoyant force or upthrust.
Buoyancy occurs because fluid pressure increases with depth. The pressure acting on the lower part of an immersed object is therefore greater than the pressure acting on its upper part, producing a net upward force.
5.2 Factors Affecting Buoyant Force
Buoyant force depends on:
- Density of the fluid
- Volume of fluid displaced
- Acceleration due to gravity
It does not directly depend on the object’s mass, except through the effect of its immersed volume and equilibrium condition.
5.3 Archimedes’ Principle
Archimedes’ principle states:
An object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
[
F_B=\rho_f gV_d
]
Where:
- (F_B) = buoyant force
- (\rho_f) = density of the fluid
- (g) = acceleration due to gravity
- (V_d) = volume of displaced fluid
The principle applies to both liquids and gases.
5.4 Apparent Weight
An object immersed in a fluid appears to weigh less because the fluid exerts an upward buoyant force.
[
W_{apparent}=W_{actual}-F_B
]
Where:
- (W_{apparent}) = apparent weight
- (W_{actual}) = actual weight in air, neglecting air buoyancy
- (F_B) = buoyant force
The apparent weight is the tension or support force required to hold the object stationary in the fluid under these simplified conditions.
6. Floating, Sinking and Neutral Buoyancy
6.1 Floating Objects
An object floats when the buoyant force can balance its weight.
For a freely floating object at rest:
[
F_B=W
]
The weight of the displaced fluid therefore equals the object’s weight.
If an object floats in a liquid, its average density is generally lower than the density of that liquid.
6.2 Sinking Objects
An object sinks when its weight exceeds the maximum buoyant force available when it is fully submerged.
For an ordinary homogeneous object in a liquid, this occurs when its density is greater than the liquid’s density.
6.3 Neutral Buoyancy
An object is neutrally buoyant when its weight equals the buoyant force while fully immersed.
Its average density equals the density of the surrounding fluid.
Submarines regulate their buoyancy by adjusting the amount of water and air in their ballast tanks.
7. Floating of Ships and Submarines
7.1 Why Ships Float
Ships are made from materials such as steel, which are denser than water. However, a ship is hollow and contains a large volume of air.
The average density of the complete ship, including the enclosed air, can therefore be lower than the density of water.
A ship floats when it displaces enough water for the buoyant force to equal its weight.
7.2 Submarines
Submarines control their depth using ballast tanks.
- To dive, water enters the ballast tanks and increases the submarine’s average density.
- To rise, compressed air expels water from the tanks, reducing the submarine’s average density.
- To maintain a particular depth, the submarine adjusts its buoyancy to achieve approximate neutral buoyancy.
7.3 Plimsoll Line
A Plimsoll line is a marking on a ship’s hull indicating the maximum safe loading level under specified conditions.
It helps prevent dangerous overloading and takes account of variations in water density and environmental conditions.
8. Other Applications of Archimedes’ Principle
- Hydrometers measure the relative density of liquids.
- Lactometers are traditionally used to assess the relative density of milk.
- Ships and boats float by displacing water.
- Hot-air balloons rise because the buoyant force in surrounding air exceeds their weight.
- Submarines control depth through ballast systems.
- Density and volume of irregular objects can be determined using fluid displacement.
- Floating platforms and pontoons rely on buoyancy.
9. Worked Examples
Example 1: Buoyant Force
A body displaces 0.02 m³ of water. Calculate the buoyant force. Take water density as 1000 kg/m³ and (g=9.8,m/s^2).
[
F_B=\rho_f gV_d
]
[
F_B=1000\times9.8\times0.02
]
[
F_B=196,N
]
Answer: 196 N.
Example 2: Apparent Weight
An object weighs 100 N in air and experiences a buoyant force of 30 N in water. Calculate its apparent weight.
[
W_{apparent}=W_{actual}-F_B
]
[
W_{apparent}=100-30=70,N
]
Answer: 70 N.
Example 3: Fraction of an Object Submerged
An object has an average density of 600 kg/m³ and floats in water of density 1000 kg/m³. Determine the fraction of its volume submerged.
For a freely floating object:
[
\frac{V_{submerged}}{V_{total}}=
\frac{\rho_{object}}{\rho_{fluid}}
]
[
=\frac{600}{1000}=0.6
]
Answer: 60% of its volume is submerged.
Example 4: Continuity Equation
Water flows through a pipe with an area of 0.04 m² at a speed of 2 m/s. The pipe narrows to an area of 0.01 m². Calculate the new speed.
[
A_1v_1=A_2v_2
]
[
v_2=\frac{0.04\times2}{0.01}
]
[
v_2=8,m/s
]
Answer: 8 m/s.
10. Important Examination Distinctions
| Concept | Main idea |
|---|---|
| Fluid statics | Study of fluids at rest |
| Fluid dynamics | Study of fluids in motion |
| Buoyancy | Upward force exerted by a fluid |
| Archimedes’ principle | Buoyant force equals the weight of displaced fluid |
| Continuity equation | Conservation of mass in fluid flow |
| Bernoulli’s principle | Conservation of mechanical energy in ideal fluid flow |
| Laminar flow | Smooth, orderly fluid motion |
| Turbulent flow | Irregular fluid motion with mixing |
| Neutral buoyancy | Weight equals buoyant force while immersed |
11. 36 Key Takeaways for Exams
- Fluid mechanics is the study of fluids at rest and in motion.
- Liquids and gases are both fluids.
- Fluid statics studies fluids at rest.
- Fluid dynamics studies fluids in motion.
- Fluid pressure increases with depth.
- Pascal’s principle concerns pressure transmission in enclosed fluids.
- Steady flow has fluid properties that remain constant at a fixed point over time.
- Unsteady flow changes with time at a fixed point.
- Laminar flow is smooth and orderly.
- Turbulent flow involves irregular motion and mixing.
- Streamlines indicate the direction of fluid velocity.
- Streamlines do not intersect in steady flow.
- The continuity equation expresses conservation of mass.
- For incompressible flow, (A_1v_1=A_2v_2).
- Fluid velocity increases when the cross-sectional area decreases under steady incompressible flow.
- Bernoulli’s principle expresses conservation of mechanical energy in ideal fluid flow.
- Bernoulli’s equation includes pressure, kinetic and gravitational potential energy per unit volume.
- A Venturi meter uses pressure and velocity changes to measure fluid flow.
- Buoyancy is the upward force exerted by a fluid on an immersed object.
- Buoyancy results from the pressure difference between the lower and upper parts of an immersed body.
- Archimedes’ principle states that buoyant force equals the weight of displaced fluid.
- Buoyant force is calculated using (F_B=\rho_f gV_d).
- Buoyant force depends on fluid density and displaced volume.
- Apparent weight equals actual weight minus buoyant force.
- An object floats when buoyancy balances its weight.
- A freely floating object displaces a quantity of fluid whose weight equals its own weight.
- An object generally sinks when its average density exceeds the fluid density.
- Neutral buoyancy occurs when weight and buoyant force are equal while immersed.
- Ships float because their overall average density can be lower than that of water.
- Submarines control buoyancy through ballast tanks.
- A Plimsoll line indicates a ship’s safe loading limit under specified conditions.
- A hydrometer measures the relative density of liquids.
- A lactometer is traditionally used to assess the relative density of milk.
- Hot-air balloons rise because of buoyancy in the surrounding air.
- Fluid displacement can be used to determine the volume of irregular objects.
- Archimedes’ principle has important applications in ships, submarines, hydrometers and engineering.