FLUID STATICS AND SURFACE TENSION

Introduction to Fluids

A fluid is any substance that can flow and take the shape of its container. Both liquids and gases are fluids. The study of fluids at rest is called fluid statics or hydrostatics, while the study of fluids in motion is called fluid dynamics.

Fluids are everywhere around us—water in oceans and rivers, air in the atmosphere, blood flowing through our veins, oil in engines, and even the magma beneath Earth's crust. Understanding fluid behavior is essential in:

Engineering: Design of dams, ships, submarines, hydraulic systems, pumps
Medicine: Blood flow, IV drips, respiratory systems, drug delivery
Meteorology: Weather patterns, atmospheric pressure, wind flow
Aviation: Aircraft design, lift generation, fuel systems
Daily Life: Drinking straws, syringes, water supply systems, drainage

In this chapter, we will explore fundamental principles governing fluids:

Archimedes' Principle - Why objects float or sink
Pascal's Law - How hydraulic systems work
Surface Tension - Why water forms droplets
Viscosity - Why honey flows slower than water
Bernoulli's Equation - How airplanes fly

4.1 Archimedes' Principle and Pascal's Law

Archimedes' Principle

Archimedes' Principle Statement:

"When a body is wholly or partially immersed in a fluid, it experiences an upward force (upthrust or buoyant force) equal to the weight of the fluid displaced by it."

Mathematical Form:

F_b = ρ_fluid × V_displaced × g

or simply: F_b = W_{fluid displaced}

Where:

F_b = Buoyant force (upthrust) in Newtons
ρ_fluid = Density of fluid (kg/m³)
V_displaced = Volume of fluid displaced (m³)
g = Acceleration due to gravity (9.8 m/s²)

Archimedes' Principle Illustration

Explanation of Archimedes' Principle

When an object is immersed in a fluid:

Pressure increases with depth: The bottom surface experiences greater pressure than the top surface
Net upward force: This pressure difference creates an upward force on the object
Buoyant force magnitude: This upward force equals the weight of fluid displaced
Three possibilities:
- If W > F_b: Object sinks (ρ_object > ρ_fluid)
- If W = F_b: Object floats in equilibrium (ρ_object = ρ_fluid)
- If W < F_b: Object floats partially submerged (ρ_object < ρ_fluid)

Applications of Archimedes' Principle

1. Ships and Boats: Steel is denser than water, but ships float because they displace a large volume of water (hollow structure). The weight of water displaced equals the ship's weight.

2. Submarines: Control depth by adjusting water in ballast tanks. Taking in water increases weight (sinks), expelling water decreases weight (rises).

3. Hot Air Balloons: Heated air inside is less dense than surrounding air. Buoyant force from displaced air lifts the balloon.

4. Hydrometers: Instruments to measure liquid density. Floats deeper in less dense liquids, shallower in denser liquids.

5. Icebergs: About 90% of ice is submerged because ice density (917 kg/m³) is about 0.92 times water density (1000 kg/m³).

Pascal's Law

Pascal's Law Statement:

"Pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container."

Mathematical Form:

P₁ = P₂

F₁/A₁ = F₂/A₂

Therefore: F₂ = F₁ × (A₂/A₁)

This is the principle behind hydraulic systems: a small force applied over a small area can produce a large force over a large area.

Pascal's Law - Hydraulic System

Applications of Pascal's Law

1. Hydraulic Lift: Used in car repair shops. Small force on small piston lifts heavy vehicles on large piston.

2. Hydraulic Brakes: Pressing brake pedal creates pressure transmitted equally to all wheel cylinders, applying brakes uniformly.

3. Hydraulic Press: Used to compress materials, make metal sheets, extract oils. Can generate enormous forces.

4. Hydraulic Jack: Used to lift heavy loads like cars, trucks with minimal effort.

5. Dental Chairs: Height adjustment uses hydraulic mechanism for smooth, controlled movement.

Key Differences: Archimedes' Principle vs Pascal's Law

Archimedes: Relates to buoyant force on immersed objects (vertical forces)
Pascal: Relates to pressure transmission in enclosed fluids (all directions)
Archimedes: Explains floating and sinking
Pascal: Explains hydraulic force multiplication

4.2 Important Definitions in Fluid Statics

1. Upthrust (Buoyant Force)

Definition: The upward force exerted by a fluid on an object immersed in it is called upthrust or buoyant force.

Upthrust = Weight of fluid displaced

F_b = ρ_fluid V_displaced g

Characteristics:

Always acts vertically upward
Acts through the center of buoyancy
Independent of depth of immersion (for fully submerged objects)
Depends on volume displaced and fluid density
Independent of object's material or shape

2. Pressure in Fluid

Definition: Pressure at a point in a fluid is defined as the normal force acting per unit area at that point.

P = F/A

Unit: Pascal (Pa) = N/m² = kg/(m·s²)

Pressure at Depth h:

P = P₀ + ρgh

Where:

P = Total pressure at depth h
P₀ = Atmospheric pressure at surface (≈ 101,325 Pa = 1 atm)
ρ = Density of fluid
g = Acceleration due to gravity
h = Depth below surface

Important Points:

Pressure increases linearly with depth
Pressure acts equally in all directions at a given depth
Pressure at the same horizontal level in a connected fluid is same
Pressure is independent of shape of container

3. Buoyancy

Definition: The tendency of a fluid to exert an upward force on an object placed in it is called buoyancy.

Buoyancy is the phenomenon responsible for making objects float or feel lighter in fluids. It arises due to pressure difference between top and bottom surfaces of immersed objects.

Factors Affecting Buoyancy:

Volume of object: Larger volume → greater buoyant force
Density of fluid: Denser fluid → greater buoyant force
Gravitational field: Stronger gravity → greater buoyant force

Does NOT depend on:

Depth of immersion (for fully submerged objects)
Mass or density of the object
Shape of the object

4. Center of Buoyancy

Definition: The point at which the buoyant force can be considered to act is called the center of buoyancy. It is the centroid (geometric center) of the displaced fluid volume.

Center of Gravity and Center of Buoyancy

Key Points:

For a homogeneous object, center of buoyancy coincides with geometric center
For floating objects, only the submerged portion determines center of buoyancy
Center of buoyancy may change as object tilts or sinks deeper

5. Metacenter

Definition: When a floating body is tilted slightly from its equilibrium position, the center of buoyancy shifts. The point of intersection of the original vertical line through the center of buoyancy and the new vertical line through the shifted center of buoyancy is called the metacenter (M).

Metacenter and Stability

Conditions for Stability:

Stable Equilibrium: M is above G → Restoring couple brings ship back to upright position
Unstable Equilibrium: M is below G → Capsizing couple overturns the ship
Neutral Equilibrium: M coincides with G → Ship remains in tilted position

Metacentric Height (GM):

GM = Distance between G and M

Larger GM → More stable ship
For stability: GM should be positive (M above G)
Ships are designed to have low center of gravity and high metacenter

Practical Importance:

Ship Design: Naval architects carefully calculate metacentric height to ensure ships remain stable even in rough seas. Cargo ships have ballast tanks that can be filled to lower center of gravity and increase stability.

Why tall ships are less stable: Higher mast and superstructure raises center of gravity (G), reducing GM and making ship less stable.

4.3 Law of Floatation

Law of Floatation Statement:

"A body floats in a fluid if the weight of the body equals the weight of the fluid displaced by its submerged part."

Mathematical Form:

Weight of floating body = Weight of fluid displaced

W = F_b

ρ_body V_body g = ρ_fluid V_submerged g

Therefore: ρ_body V_body = ρ_fluid V_submerged

Condition for Floating

For an object to float in equilibrium:

V_submerged / V_body = ρ_body / ρ_fluid

Implications:

If ρ_body < ρ_fluid: Object floats partially submerged
If ρ_body = ρ_fluid: Object floats completely submerged (just below surface)
If ρ_body > ρ_fluid: Object sinks

Practical Examples

Example 1: Iceberg in Seawater

Density of ice = 917 kg/m³
Density of seawater = 1025 kg/m³

Fraction submerged = ρ_ice / ρ_seawater = 917/1025 ≈ 0.895 = 89.5%

Result: About 90% of iceberg is underwater, only 10% visible above surface. This is why icebergs are dangerous to ships—most of their mass is hidden!

Example 2: Human Body in Water

Average human body density ≈ 985 kg/m³
Density of fresh water = 1000 kg/m³

Fraction submerged = 985/1000 = 0.985 = 98.5%

Result: A person floats with about 98.5% of body submerged. With lungs full of air, density decreases further, making floating easier.

Applications of Law of Floatation

1. Ship Design (Plimsoll Line):

Ships have load line markings showing safe loading limits
In freshwater, ship sinks deeper than in seawater (lower density)
Overloading lowers freeboard (height above water), reducing stability

2. Hot Air Balloons:

Heated air inside has lower density than surrounding air
Buoyant force exceeds weight, causing balloon to rise
Temperature control allows altitude adjustment

3. Submarines:

Ballast tanks control overall density
Fill with water: density increases → submarine dives
Blow out water with compressed air: density decreases → submarine surfaces

4. Fish Swim Bladder:

Gas-filled organ that controls buoyancy
Inflate bladder: density decreases → fish rises
Deflate bladder: density increases → fish sinks

5. Hydrometer:

Instrument to measure liquid density
Floats deeper in less dense liquids
Scale on stem directly reads density or specific gravity
Used to test battery acid, alcohol content, milk purity

Important Note for NEB Exams:

The law of floatation is a direct consequence of Archimedes' principle applied to the special case of floating objects in equilibrium. Remember:

Only the submerged volume contributes to buoyancy
The weight equals buoyant force (equilibrium condition)
The ratio of densities gives fraction submerged

4.4 Surface Tension

Definition: Surface tension is the property of a liquid by virtue of which its free surface behaves like an elastic stretched membrane tending to contract and occupy minimum surface area.

Surface Tension (T or γ) = Force per unit length

T = F / L

Unit: N/m or J/m²

Molecular Explanation

Molecular Theory of Surface Tension

Explanation of Surface Tension

Molecules in the bulk of liquid (A):

Surrounded equally by other molecules in all directions
Cohesive forces (attractive forces between like molecules) balance out
Net force on molecule = 0

Molecules at the surface (B):

Have molecules only below and beside them, not above
Experience net inward (downward) cohesive force
Surface molecules are pulled inward, creating tension in surface
Surface behaves like a stretched elastic membrane

Why surface contracts: To minimize surface area (and thus surface energy), the liquid surface tends to contract and assume the shape with minimum area for given volume (sphere for free droplets).

Examples of Surface Tension

1. Water Droplets are Spherical:

Sphere has minimum surface area for given volume
Surface tension makes water form spherical drops
Small drops are nearly perfect spheres

2. Insects Walking on Water:

Water striders, pond skaters can walk on water surface
Surface tension creates elastic "skin" that supports weight
Insect's legs dent but don't break the surface

3. Needle Floating on Water:

Though denser than water, a carefully placed needle can float
Surface tension supports it if weight doesn't break surface
Surface deforms to create upward component of tension

4. Soap Bubbles:

Thin film of soap solution forms spherical bubbles
Two surfaces (inner and outer) → double surface tension
Internal pressure slightly higher than external

5. Mercury Droplets:

Mercury has very high surface tension (485 mN/m)
Forms nearly perfect spherical droplets
Doesn't wet glass (high cohesion, low adhesion to glass)

6. Raindrops:

Small raindrops are spherical due to surface tension
Large drops become flattened due to air resistance
Very large drops break apart into smaller droplets

Factors Affecting Surface Tension

Factor	Effect on Surface Tension	Explanation
Temperature	Increases → Decreases	Higher temperature increases molecular kinetic energy, weakening intermolecular forces
Impurities (soap, detergent)	Decreases significantly	Surfactants reduce cohesive forces between water molecules
Dissolved salts	Slightly increases	Salts strengthen hydrogen bonding in water
Nature of liquid	Varies	Mercury > Water > Alcohol > Ether (decreasing order)

Surface Tension Values

Liquid	Surface Tension (mN/m) at 20°C
Mercury	485
Water	72.8
Glycerine	63
Olive oil	32
Ethanol	22.3
Ether	17

4.5 Relation Between Surface Energy and Surface Tension

Surface tension can also be defined in terms of energy. The molecules at the surface have higher potential energy than those in the bulk because they experience a net inward force.

Surface Energy: The extra energy that surface molecules possess compared to interior molecules is called surface energy. It is the work done in increasing the surface area of a liquid by one unit area.

Derivation of Relation

Consider: A rectangular wire frame with a movable wire of length L sliding on it, forming a liquid film.

Step 1: Force due to surface tension on the wire:

Since film has two surfaces (upper and lower):

F = 2TL

(where T = surface tension)

Step 2: Work done in moving the wire through distance dx:

dW = F × dx = 2TL × dx

Step 3: Increase in surface area:

Both surfaces increase, so:

dA = 2L × dx

Step 4: Work done per unit area increase:

dW/dA = (2TL × dx)/(2L × dx) = T

Step 5: This work is stored as surface energy:

Surface Energy per unit area = T

Relation Between Surface Energy and Surface Tension:

Surface Tension = Surface Energy per unit area

T = E/A

Units:

As force per unit length: N/m
As energy per unit area: J/m²

Both are dimensionally equivalent: 1 N/m = 1 J/m²

Work Done Against Surface Tension

Physical Interpretation

Two Equivalent Definitions of Surface Tension:

1. Force Definition (Mechanical):
Surface tension is the force per unit length acting perpendicular to any line on the surface.

T = F/L (units: N/m)

2. Energy Definition (Thermodynamic):
Surface tension is the work done (energy expended) in increasing the surface area by one unit area.

T = E/A (units: J/m²)

Why they're equivalent:

Creating new surface requires moving molecules from bulk to surface
This requires work against cohesive forces
This work is stored as potential energy in surface molecules
The surface naturally tends to minimize its area (minimize energy)

Important for NEB Exams:

Remember to multiply by 2 for soap films (two surfaces)
Unit conversion: 1 N/m = 1 J/m² (dimensionally equivalent)
The derivation showing T = E/A is frequently asked (5-6 marks)
Surface energy makes liquids behave as if surface is under tension

4.6 Angle of Contact and Capillarity

Angle of Contact

Definition: The angle of contact (θ) between a liquid and a solid surface is the angle between the tangent to the liquid surface at the point of contact and the solid surface, measured inside the liquid.

Angle of Contact for Different Liquid-Solid Pairs

Understanding Wetting and Non-wetting

Two types of intermolecular forces:

Cohesive force: Attractive force between molecules of the same substance (like to like)
Adhesive force: Attractive force between molecules of different substances (unlike to unlike)

Condition	Angle of Contact	Behavior	Example
Adhesion > Cohesion	θ < 90° (Acute)	Liquid wets the surface Concave meniscus	Water on glass (θ ≈ 0°) Water on clean metal
Cohesion > Adhesion	θ > 90° (Obtuse)	Liquid doesn't wet surface Convex meniscus	Mercury on glass (θ ≈ 138°) Water on waxed surface
Adhesion = Cohesion	θ = 90°	Neutral wetting Flat meniscus	Pure water on silver

Capillarity

Definition: The phenomenon of rise or depression of a liquid in a narrow tube (capillary tube) is called capillarity or capillary action.

Capillary Rise and Depression

Formula for Capillary Rise/Depression

Derivation: Consider a capillary tube of radius r placed vertically in a liquid.

Step 1: At equilibrium, the upward force due to surface tension equals the weight of liquid column.

Step 2: Upward force due to surface tension:

F = T × perimeter × cos θ

F = T × (2πr) × cos θ

Step 3: Weight of liquid column:

W = mass × g = volume × density × g

W = πr²h × ρ × g

Step 4: At equilibrium: F = W

T × 2πr × cos θ = πr²hρg

Step 5: Solving for h:

h = (2T cos θ)/(rρg)

Ascent Formula (Capillary Rise/Depression):

h = (2T cos θ) / (rρg)

Where:
h = height of rise (positive) or depression (negative)
T = surface tension of liquid
θ = angle of contact
r = radius of capillary tube
ρ = density of liquid
g = acceleration due to gravity

Special Cases

Case	Condition	Result
Water in glass	θ ≈ 0°, cos θ ≈ 1	h = 2T/(rρg) (Maximum rise)
Mercury in glass	θ ≈ 138°, cos θ < 0	h is negative (Depression)
θ = 90°	cos 90° = 0	h = 0 (No rise or depression)
Smaller tube	r decreases	h increases (h ∝ 1/r)

Examples of Capillary Action

1. Water Rising in Plants:

Plants draw water from soil through capillary action in xylem vessels
Combined with transpiration pull, water reaches leaves even in tall trees
Narrow xylem tubes (10-400 μm) provide significant capillary rise

2. Oil Lamp Wicks:

Porous wick draws oil upward through capillary action
Oil continuously rises to flame, sustaining combustion
Cotton threads have numerous narrow spaces for capillary rise

3. Paper Towel Absorption:

Paper fibers have narrow spaces between them
Water rises through these spaces via capillary action
This makes paper towels effective for soaking up spills

4. Soil Moisture Movement:

Water moves through soil pores by capillary action
Important for plant root water absorption
Sandy soil (large pores) has less capillary action than clay

5. Ink in Fountain Pens:

Capillary action draws ink from reservoir to nib
Continuous flow maintained through narrow channels
Design ensures consistent ink delivery

6. Sponges:

Numerous small pores throughout sponge structure
Capillary action pulls water into pores
Can absorb many times their own weight in water

Important Points for NEB Exams:

Capillary rise/depression is inversely proportional to tube radius: h ∝ 1/r
For θ < 90°: liquid rises (positive h)
For θ > 90°: liquid depresses (negative h)
Remember: cos θ term determines whether rise or depression occurs