Fluid Dynamics: theory | Physics Circle

CONCEPT

Types of Flow, Reynolds Number & Continuity

In one line
Slow flow is orderly and fast flow breaks up. In steady flow, the same volume passes every section, so av = constant.

STREAMLINE

2.1.1 Streamline flow

Each element follows the one before it

Each element follows the same path with the same velocity as the element before it at that point.
A streamline is a path whose tangent at any point gives the direction of flow.
Two streamlines cannot cross.
Crowded streamlines mean larger velocity.

LAMINAR

2.1.2 Laminar and turbulent flow

Laminar: liquid moves in layers that do not mix. Velocity stays below the critical velocity.
Turbulent: above the critical velocity the motion becomes irregular.
Most of the energy then goes into eddies, as beside a bridge pillar.

Eddies behind a pillar

CRITICAL VELOCITY

2.1.3 Critical velocity and Reynolds number

Critical velocity: the speed up to which flow stays streamline.
Reynolds number R_n is a pure number: inertial force per unit area ÷ viscous force per unit area.

R_n = ρvdη

v_c = R_n ηρd

d is the diameter of the tube.
v_c is directly proportional to η, and inversely proportional to ρ and d.

R_n	Flow
below 1000	streamline (steady)
1000 to 2000	unsteady, changing
above 2000	turbulent

Some books give 2000 and 3000 as the limits instead. MHT CET questions follow this table.

CONTINUITY

2.1.4 Equation of continuity

Tube of varying cross-section

Mass in per second = mass out per second.

a₁ρ₁v₁ = a₂ρ₂v₂

For an incompressible liquid, ρ₁ = ρ₂.

a₁v₁ = a₂v₂ ⇒ av = constant ⇒ a ∝ 1v

Area against speed: a hyperbola

Circular pipe: v ∝ 1r² ∝ 1d².
Several outlets: n a₂v₂ = a₁v₁ (spray gun, sprinkler).
Rate of flow Q = av.

EXAMPLES

2.1.5 Everyday examples

Hills: narrow, shallow river, faster water. Plains: wide, deep river, slower water.
Water from a tap speeds up as it falls, so the stream becomes narrower.

JEE · NEET

Corner

Force of a jet on a wall

Mass striking per second = ρAv.
Force = rate of change of momentum.

F = ρAv² (water stopped)

Water bouncing back at the same speed: F = 2ρAv².

Falling stream from a tap

The stream narrows

v₂² = v₁² + 2gh · A₁v₁ = A₂v₂

Practise PYQs on this topicTypes of Flow, Reynolds Number & Continuity · 22 questions with solutions

CONCEPT

Bernoulli’s Equation

In one line
For an ideal fluid in steady flow, pressure + potential + kinetic energy per unit volume stays constant.

ENERGY

2.2.1 Energy of a flowing fluid

Per	Pressure	Potential	Kinetic
total	PV	mgh	12mv²
unit mass	Pρ	gh	12v²
unit volume	P	ρgh	12ρv²

THEOREM

2.2.2 Bernoulli’s theorem

Liquid pushed through a tube

Conditions: incompressible, non-viscous, steady flow, no source or sink.
Work done by pressure: W = (P₁ − P₂)V.
This work raises the PE and KE: W = ΔU + ΔK.

P₁ + ρgh₁ + 12ρv₁² = P₂ + ρgh₂ + 12ρv₂²

P + ρgh + 12ρv² = constant

It is conservation of mechanical energy for a moving fluid.

HEAD

2.2.3 Other forms

Pρ + gh + 12v² = constant

Pρg + h + v²2g = constant

Pρg is pressure head, h is gravitational head, v²2g is velocity head.

HORIZONTAL

2.2.4 Horizontal pipe

h is the same on both sides, so ρgh cancels.

P₁ − P₂ = 12ρ(v₂² − v₁²)

Join this with continuity, a₁v₁ = a₂v₂.
Narrow part: largest v, smallest P.
Wide part: smallest v, largest P.

FLOW RATE

2.2.5 Pressure difference and flow rate

P₁ − P₂ = ρQ²2(1a₂² − 1a₁²)

VALVE

2.2.6 Closed pipe and valve

Valve shut: v = 0, gauge reads P₁.
Valve open: gauge falls to P₂.

v = √(2(P₁ − P₂)ρ)

Practise PYQs on this topicBernoulli's Equation · 25 questions with solutions

CONCEPT

Applications of Bernoulli’s Theorem

In one line
Everywhere it is the same story: fast fluid, low pressure, and the pressure difference pushes things.

BOATS

2.3.1 Two boats moving side by side

Fast water between them

Water between them moves faster than on the outer sides.
Pressure between them falls, so they are pushed towards each other.
Two hanging balls with an air stream between them also come closer.

AEROPLANE

2.3.2 Aeroplane wing

Air travels faster over the top

Air above the wing moves faster than air below.
The pressure difference gives an upward dynamic lift.

Lift = A × 12ρ(v_up² − v_down²)

Level flight: lift = weight.
Use the total wing area.
For a small difference: ΔP ≈ ρ v Δv.

ATOMISER

2.3.3 Atomiser

Scent spray, paint gun, carburettor

Piston blows high-speed air over the top of a tube dipped in the liquid.
Low pressure there makes the liquid rise, and it leaves as fine drops.

ROOFS

2.3.4 Roofs in a wind storm

Pressure inside is atmospheric

Fast wind over the roof gives low pressure above it.
Pressure inside stays P₀.

F_up = 12ρv² × A

MAGNUS

2.3.5 Magnus effect

Spin changes the air speed above and below the ball

A spinning ball curves from its usual path.
Speeds of air above and below differ by the spin: v + rω and v − rω.
The pressure difference gives the sideways force.

VENTURIMETER

2.3.6 Venturimeter

Pressure difference read as a height h

Measures the rate of flow of a liquid in a pipe.
Works on Bernoulli’s theorem.
Pressure is lower at the narrow part.

P₁ − P₂ = hρg

V = a₁a₂ √(2hga₁² − a₂²)

JEE · NEET

Corner

Magnus effect, curving sideways

Back-spin: the ball stays up longer and the pitch curves less.
Top-spin: the force acts downwards and shortens the flight.
Spin about a vertical axis: out-swing or in-swing.

Practise PYQs on this topicApplications of Bernoulli's Theorem · 10 questions with solutions

CONCEPT

Velocity of Efflux (Torricelli)

In one line
Liquid leaves a hole at depth h with the speed of a body falling freely through h: v = √(2gh).

TORRICELLI

2.4.1 Torricelli’s theorem

Hole at depth h below the free surface

Bernoulli just inside and outside the hole, with both surfaces at P₀.

P₀ + hρg = P₀ + 12ρv² ⇒ v = √(2gh)

Independent of the nature of the liquid, the amount of liquid and the area of the hole.
Rate of flow: Q = a√(2gh).
h is the depth below the free surface.

Deeper hole, larger speed: v₁ < v₂ < v₃

If extra pressure P acts on the surface: ½ρv² = P + hρg.

RANGE

2.4.2 Range on the ground

Time to fall from height (H − h): t = √(2(H − h)g).

x = vt = 2√(h(H − h))

x_max = H at h = H2

Largest range at h = H2

Holes equally far from the top and the bottom

Range is the same for depth h and depth (H − h).
If the vessel stands on a block, use the total height of the hole above the ground.

TWO HOLES

2.4.3 Equal flow from two holes

a₁√h₁ = a₂√h₂

Square hole of side a: area a². Circular hole: area πr².

STEADY LEVEL

2.4.4 Level held steady

Inflow = outflow: Q = a√(2gh), so h = Q²2ga².

FALL OF LEVEL

2.4.5 Rate of fall of level

Continuity between tank and hole.

Adydt = A₀v

EMPTYING

2.4.6 Time to empty a tank

Tank area A, hole area A₀

t = AA₀ √(2g) [√H − √H′]

t_empty = AA₀ √(2Hg)

t ∝ √H: four times the height takes twice the time.

JEE · NEET

Corner

Thrust on the tank

Two holes on opposite sides

Each jet pushes the tank back with F = ρAv² = 2ρAg × depth.

net F = 2ρAg(h₂ − h₁)

Practise PYQs on this topicVelocity of Efflux (Torricelli) · 29 questions with solutions

EXAM-DAY CARD

Just see this before the exam

R_n = ρvdη

Reynolds

a₁v₁ = a₂v₂

continuity

P + ρgh + 12ρv²

Bernoulli

F = ½ρv²A

roof, lift

v = √(2gh)

efflux

x_max = H

range

FORMULA SHEET

Whole chapter on one page

Flow

R_n = ρvdη

v_c = R_nηρd

< 1000 steady, > 2000 turbulent

Continuity

a₁v₁ = a₂v₂, Q = av

v ∝ 1r²

n holes: n a₂v₂ = a₁v₁

Bernoulli

P + ρgh + ½ρv² = constant

P₁ − P₂ = ½ρ(v₂² − v₁²)

v = √(2(P₁ − P₂)ρ)

Uses

Lift = A ½ρ(v_u² − v_d²)

Roof: F = ½ρv²A

Venturi: P₁ − P₂ = hρg

Efflux

v = √(2gh)

x = 2√(h(H − h))

x_max = H at h = H2

Tank

a₁√h₁ = a₂√h₂

t = AA₀√(2Hg)

t ∝ √H

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

v ∝ r⁻¹ for continuity

Narrow part has more pressure

Efflux depth from the bottom

R_n with radius instead of diameter

CORRECT ✓

v ∝ {F("1","r²")}

Narrow part: largest v, smallest P

h is measured from the free surface

R_n = ρvdη, d = diameter