Mechanical Properties of Fluids Class 11th Notes: Fluid Mechanics and Formulas

Pressure is the force applied perpendicular to a surface per unit area. 

Simply, Pressure is force acting per unit area on a surface.

P = F/A

Where:

P = Pressure (Pascal or N/m²)

F = force (Newton)

A = Area (m²)

• Pressure is a scalar quantity (has magnitude but no specific direction)
• SI unit: Pascal (Pa) = N/m²

Pascal’s Law

Pressure applied at one point in a fluid is transmitted equally in all directions.
Applied in, for example, Hydraulic lift, hydraulic brakes.

• Pressure Variation with Depth

When we go deeper into a fluid, pressure increases due to the weight of the fluid above.

P = P₀ + ρgh

Where:

• P = Pressure at depth h
• P₀ = Atmospheric pressure at the surface
• ρ = Density of fluid
• g = Acceleration due to gravity
• h = Depth below surface

Gauge vs Absolute Pressure

Gauge pressure excludes atmospheric pressure.
Absolute pressure = Gauge pressure + Atmospheric pressure.

Read more: Pressure in Fluids

Fluids in motion act differently; motion in fluids is called fluid dynamics. For example, when we open the tap slowly, we can see the smooth water flow, but as it goes faster, the smoothness of the water is gone.

Steady Flow: At any given point, the velocity of each passing fluid particle remains constant with time.

Streamline Flow: A curve whose tangent at any point gives the direction of fluid velocity at that point.

Properties of Streamlines:

• No two streamlines can cross each other
• Closer streamlines show higher velocity
• They form a permanent map of fluid flow

Equation of Continuity

Based on conservation of mass for incompressible fluids:

A₁v₁ = A₂v₂ = constant

Where,

• A = Cross-sectional area
• v = Fluid velocity

Equation of Continuity Applications:

• Water flows through pipes of varying diameter
• Blood flow through the arteries
• Air flow through wind tunnels

Types of Flow

Laminar Flow:

Smooth, parallel layers, Low velocities, and predictable patterns.

Turbulent Flow:

Chaotic, irregular motion, High velocities, Forms whirlpools and eddies, and occurs beyond critical velocity.

In a moving incompressible fluid, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant.

P + ½ρv² + ρgh = constant

This equation represents conservation of energy in fluid flow.

• P: Pressure energy per unit volume
• ½ρv²: Kinetic energy per unit volume
• ρgh: Potential energy per unit volume

Applications of Bernoulli’s Principle

1. Torricelli’s Law (Speed of Efflux)

For liquid flowing out of a tank through a small hole:

v = √(2gh)

It is the same as free-fall velocity, showing liquid emerges as if it fell freely through height h.

2. Venturi Meter

It is a device used to measure the rate of flow by measuring the pressure difference. Fluid moves faster in a constricted space, and pressure drops in a narrow section. Pressure difference is measured by a manometer, and flow rate is calculated using Bernoulli’s equation.

3. Dynamic Lift

It explains how airplane wings generate lift. Air moves faster over a curved upper surface. Higher velocity means lower pressure, so the pressure difference creates an upward force. 

4. Carburetor Operation

Air flows through a venturi (narrow section). Reduced pressure sucks fuel into the air stream and creates a proper air-fuel mixture for combustion.

Limitations of Bernoulli’s Equation

• Applies only to non-viscous fluids
• Valid for incompressible fluids
• Requires steady flow conditions
• Energy loss due to friction is not considered

Read more: Bernoulli’s Principle

Viscosity is the internal friction in fluids that stops relative motion between fluid layers.

The coefficient of viscosity is the ratio of shearing stress to strain rate in a fluid.

η = (F/A)/(dv/dy) = (F×d)/(A×v)

Where,

• η = Coefficient of viscosity
• F = Force required to move fluid layers
• A = Area of contact
• v = Relative velocity between layers
• d = Distance between layers

Units:

• SI unit: Poiseuille (Pl) = N⋅s/m² = Pa⋅s
• CGS unit: Poise (P)
• 1 Pl = 10 Poise

Viscous Flow Characteristics

Laminar Viscous Flow:

• Fluid moves in parallel layers
• Velocity varies linearly across layers
• Maximum velocity at center, zero at walls
• Like book pages sliding over each other

Factors Affecting Viscosity:

Factors that affect viscosity are Temperature, in liquids, viscosity decreases with temperature and in gases, viscosity increases with temperature. Pressure have a small effect in liquids, significant in gases and the Nature of fluid, Molecular structure and intermolecular forces.

Stokes’ Law

When a sphere moves through a viscous fluid, it experiences a drag force proportional to its velocity.

F = 6πηrv

Where,

• F = Viscous drag force
• η = Coefficient of viscosity
• r = Radius of sphere
• v = Velocity of sphere

Terminal Velocity

When a sphere falls through a viscous medium, initially accelerates due to gravity and drag force increases with velocity. Eventually reaches constant velocity or terminal velocity.

At terminal velocity: Weight = Buoyant force + Drag force

vₜ = (2r²(ρₛ - ρf)g)/(9η)

Where,

• vₜ = Terminal velocity
• ρₛ = Density of sphere
• ρf = Density of fluid

Surface tension arises because molecules at the liquid surface have higher potential energy than those in the interior.

Surface tension is the force per unit length acting in the plane of the liquid surface.

S = F/l = Energy/Area

Where,

• S = Surface tension (N/m or J/m²)
• F = Force acting on length l
• l = Length of contact line

Contact Angle

The angle between the tangent to the liquid surface and the solid surface (measured inside the liquid).

Acute Contact Angle (θ < 90°):

For example, Water on clean glass.

Obtuse Contact Angle (θ > 90°):

For example, Water on wax, mercury on glass.

Drops and Bubbles

Why are drops spherical?

A sphere has a minimum surface area for a given volume, Minimizes surface energy, and Gravity and other forces can distort this shape.

Pressure Inside Drops and Bubbles:

For a liquid drop: Pᵢ - P₀ = 2S/r

For a soap bubble: Pᵢ - P₀ = 4S/r

Where,

• Pᵢ = Pressure inside
• P₀ = Atmospheric pressure
• S = Surface tension
• r = Radius

Capillary Rise

Liquid rises in narrow tubes against gravity due to surface tension.

h = (2S cosθ)/(ρgr)

Where,

• h = Height of liquid column
• θ = Contact angle
• ρ = Density of liquid
• r = Radius of capillary tube

Real-life Applications:

• Water transport in plants
• Oil rising in lamp wicks
• Soil moisture movement

Applications of Surface Tension

1. Detergent Action:

• Detergent molecules have a dual nature. One end attracts water, the other attracts oil/grease.
• Reduces surface tension between water and oil
• Allows water to act and remove dirt

2. Floatation:

• Small objects can float on the water's surface because surface tension provides an upward force
• Examples: Water strider insects, a floating needle

3. Formation of Emulsions:

Surface-active agents help mix immiscible liquids, reduce interfacial tension, and stabilize droplet formation.

Practice class 11th physics, Mechanical Properties of Fluids, important formulas with application to quickly revise the chapter, and save time in exam preparation. The important formula helps you cover the Mechanical Properties of Fluids class 11th NCERT solutions and also helps in NEET and JEE exam preparation.

Concept	Formula	Applications
Pressure in a fluid	P = P₀ + hρg	Pressure increases with depth
Pascal’s Law	ΔP is transmitted equally in all directions	Hydraulic lift, brakes
Excess Pressure in a drop	ΔP = 2T / r	Due to surface tension
Excess Pressure in a Soap Bubble	ΔP = 4T / r	Two surfaces inside + outside
Capillary rise	h = (2T cosθ) / (ρ g r)	Explains capillarity in plants
Viscous force (Stokes’ law)	F = 6 π η r v	Viscous drag on a spherical body
Terminal velocity	vt = [2 r² (ρs – ρf) g] / (9 η)	Steady velocity of falling drop
Equation of Continuity	A₁ v₁ = A₂ v₂	Narrow tube → faster flow
Reynolds Number	Re = (ρ v D) / η	Laminar flow if Re < 2000
Bernoulli’s Equation	P + (½ ρ v²) + ρ g h = constant	Energy conservation in fluids
Pressure Energy per unit volume	P	Due to the applied pressure
Kinetic Energy per unit volume	½ ρ v²	Due to the velocity of flow
Potential Energy per unit volume	ρ g h	Due to the height in the gravity field

Check out class 11 physics notes of all chapters below:

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes