Viscosity: theory | Physics Circle

CONCEPT

Viscosity & Stokes’ Law

In one line
Viscosity is internal friction between layers of a fluid. On a slow sphere it gives a force F = 6πηrv.

VISCOSITY

4.1.1 What is viscosity?

A layer of liquid drags on the next layer

The property of a fluid that opposes relative motion between its layers.
The viscous force is tangential to the layers.
Layer in contact with a fixed surface is at rest; speed grows with height.
Unlike solid friction, it depends on area and on the velocity gradient, not on normal reaction.

NEWTON

4.1.2 Newton’s law of viscous flow

Velocity grows from the bottom layer upward

F = −ηA dvdy

dvdy is the velocity gradient, A the area of the layer.
Linear profile: gradient = vd, so v ∝ height.

Shear stress = η dvdy

COEFFICIENT

4.1.3 Coefficient of viscosity

η = Shear stressVelocity gradient

System	Unit
SI	N s m⁻² = Pa s = decapoise
CGS	poise = dyne s cm⁻²

1 Pa s = 10 poise

Dimensions: [ML⁻¹T⁻¹].
η depends on the nature of the fluid and its temperature.

TEMPERATURE

4.1.4 Effect of temperature

Fluid	Heated	Why
liquid	η falls	cohesion weakens
gas	η rises	more molecular collisions

Hot water flows faster than cold water.
For a gas, η ∝ √T.
Gases have far smaller viscosity than liquids.
Ideal fluid: η = 0.

STOKES

4.1.5 Stokes’ law

Forces on a sphere falling in a fluid

F = 6πηrv

Viscous force on a small sphere of radius r moving at speed v.
F ∝ r and F ∝ v. It always opposes the motion.
At constant velocity the sphere has no net force.

FORCE

4.1.6 Viscous force at terminal velocity

F = W − U = mg(1 − σρ)

Ball rising steadily in a denser liquid: F = U − W.
Liquid n times as dense as the ball: F = (n − 1)W.
Heat produced per second = F v ∝ r⁵.

JEE · NEET

Corner

Poiseuille’s formula

Flow is fastest along the axis

V = πPr⁴8ηl

V: volume of liquid flowing per second.
Valid for steady flow in a narrow horizontal tube.

Liquid resistance

R = 8ηlπr⁴ ⇒ V = PR

Series: same flow

P = P₁ + P₂ · R = R₁ + R₂

Parallel: same pressure

V = V₁ + V₂ · 1R = 1R₁ + 1R₂

Practise PYQs on this topicViscosity & Stokes' Law · 33 questions with solutions

CONCEPT

Terminal Velocity

In one line
A falling sphere speeds up until weight balances upthrust and viscous force. Then it moves at a steady terminal velocity.

FORCES

4.2.1 Forces on the sphere

Weight W = 43πr³ρg, down.
Upthrust U = 43πr³σg, up.
Viscous force F = 6πηrv, up. It grows with v.

FORMULA

4.2.2 Terminal velocity

6πηrv + 43πr³σg = 43πr³ρg

v = 2r²(ρ − σ)g9η

ρ: density of the sphere. σ: density of the fluid.
v ∝ r², v ∝ (ρ − σ), v ∝ g, v ∝ 1η.
ρ > σ: falls. ρ < σ: rises (air bubble, v is upward).

GRAPH

4.2.3 Velocity against time

Rises from zero and levels off

Velocity starts at zero and approaches v asymptotically.
Acceleration falls from g′ to zero.
A steady pull twice the effective weight gives the same speed upward.

RATIOS

4.2.4 Quick ratios

Change	Result
same liquid, same material	v₁ : v₂ = r₁² : r₂²
same size, same liquid	v₁ : v₂ = (ρ₁ − σ) : (ρ₂ − σ)
same speed	r²(ρ − σ) is the same
mass m to 8m	r doubles, v is 4 times

Rain drops: v ∝ r² and surface area ∝ r², so areas go as speeds.
Volumes go as v^3/2.

FALL

4.2.5 Falling freely first

Ball falls through h, then keeps a constant speed in the liquid.

√(2gh) = 2r²(ρ − σ)g9η

BUBBLE

4.2.6 Air bubble rising

Density of air is negligible

η = 2r²σg9v

MERGING

4.2.7 Drops merging

n drops merge: R = n^1/3 r.

v′ = n^2/3 v

n	2	8	27	64	125
v′ ÷ v	2^2/3	4	9	16	25

SPLITTING

4.2.8 Drop splitting

Drop broken into n droplets: v′ = v ÷ n^2/3.

EXPERIMENT

4.2.9 Measuring viscosity

Measure v for balls of different r; v against r is a parabola.
The result depends on temperature, as η does.
Can be used to assess the density of a liquid.
Limits: small sphere, slow speed, wide vessel.

Practise PYQs on this topicTerminal Velocity · 54 questions with solutions

EXAM-DAY CARD

Just see this before the exam

F = ηAdvdy

Newton

1 Pa s = 10 poise

units

F = 6πηrv

Stokes

v = 2r²(ρ − σ)g9η

terminal

v′ = n^2/3v

merging

V = πPr⁴8ηl

Poiseuille

FORMULA SHEET

Whole chapter on one page

Viscosity

F = ηA dvdy

[η] = ML⁻¹T⁻¹

1 Pa s = 10 poise

Stokes

F = 6πηrv

F = mg(1 − σρ)

power ∝ r⁵

Terminal

v = 2r²(ρ − σ)g9η

v ∝ r², v ∝ (ρ − σ)

bubble: η = 2r²σg9v

Merging

R = n^1/3 r

v′ = n^2/3 v

8 drops: 4v

Temperature

liquid: η falls on heating

gas: η rises, η ∝ √T

ideal fluid: η = 0

Tubes

V = πPr⁴8ηl

series: same V

parallel: same P

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

F at terminal velocity = mg

v ∝ ρ, ignoring the liquid

n drops merged: v′ = n^1/3v

Heating raises η for all fluids

CORRECT ✓

F = mg(1 − σρ)

v ∝ (ρ − σ)

v′ = n^2/3v

Liquid: falls. Gas: rises