Newton's Law of Viscosity

Equivalent forms

\tau = \eta \dot{\gamma}

\nu = \eta / \rho

Stress and strain rate are linearly proportional — the defining trait of a Newtonian fluid.

Symbol	Name	SI unit	Dimension	Range
$\eta$	Dynamic Viscosity Fluid viscosity	Pa·s	M·L⁻¹·T⁻¹	0.0001 – 10
$dv/dy$	Velocity Gradient Rate of change of velocity perpendicular to flow	1/s	T⁻¹	0 – 1000
$\tau$	Shear Stressoutput Tangential force per unit area	Pa	M·L⁻¹·T⁻²	0 – 1000

Unit systems

Symbol	SI	CGS	Imperial
$\eta$	Pa·s	P	lb/(ft·s)
$dv/dy$	1/s	1/s	1/s
$\tau$	Pa	dyn/cm²	psi

Where it holds

Newtonian fluids (water, air, light oils) under simple shear at moderate strain rates. Excludes shear-thinning (blood, ketchup), shear-thickening (cornstarch), viscoelastic (polymers), and Bingham plastics (toothpaste).

Dimensional analysis

$[\eta ]\cdot [dv/dy] = (M\cdot L^{-1}\cdot T^{-1})(T^{-1}) = M\cdot L^{-1}\cdot T^{-2} \checkmark$ (pressure/stress)

Discovery

Isaac Newton · 1687

Newton proposed in Principia that the resistance arising from lack of slipperiness in a fluid is proportional to the velocity gradient. The concept of dynamic viscosity emerged from this insight.

Try this

Why does honey resist your spoon while water doesn't?

Two plates separated by 0.001 m of oil (viscosity 0.1 Pa·s) move at relative velocity 0.5 m/s. What shear stress does the oil produce on each plate?

Research status: stable

Real-world applications

Lubrication of engines and bearings
Designing pipelines and pumping systems
Polymer extrusion and molding
Atmospheric and oceanic boundary layers

Common misconceptions

Viscosity is NOT density — water and oil have similar density but very different viscosity
All real fluids have viscosity; an 'inviscid fluid' is an idealization
Many everyday fluids (blood, ketchup, paint) are NOT Newtonian — their viscosity varies with strain rate

Experimental verification

Couette viscometers (concentric cylinders) and capillary viscometers measure viscosity directly via this law. Water at

20 ^{\circ}C

:

\eta = 0.001 Pa\cdot s

. Glycerin:

1.5 Pa\cdot s

. Air:

1.8 \times 10^{-5} Pa\cdot s

. These values are tabulated to 4-5 significant figures.

Derivation

Consider two parallel plates with a fluid between them.

The top plate moves at velocity v, the bottom is stationary.

Molecular momentum transfer between adjacent fluid layers creates a tangential force.

Newton postulated that this force per unit area

(\tau )

is proportional to the velocity gradient (dv/dy).

The proportionality constant

\eta is

the dynamic viscosity, a material property.

Kinematic viscosity

\nu = \eta /\rho

measures momentum diffusivity.

Limiting cases

dv/dy \to 0

⟶

\tau \to 0

No relative motion means no shear stress.

\eta \to 0

⟶

\tau \to 0

An inviscid fluid (idealization) transmits no shear.

Non-Newtonian⟶

\tau \neq \eta \cdot dv/dy

Ketchup, blood, and polymer melts have viscosity that depends on strain rate.

What if…

What if the gap halves?

Velocity gradient doubles, so shear stress doubles. Thinner films are stickier.

What if the fluid is non-Newtonian?

$Use \tau = K\cdot (dv/dy)^{n}$ (power-law model). Shear-thinning (n<1) reduces resistance at high strain; shear-thickening (n>1) increases it.

What if temperature rises?

Liquid viscosity falls rapidly $(\eta$ drops 50% per $30 ^{\circ}C$ for water), gas viscosity rises slowly. Engine oils have viscosity-index additives to combat this.

1

Oil between parallel plates

Given ·

\eta:: 0.1
dv/dy:: 500

Find · \tau

Steps

Velocity gradient $= 0.5\,\mathrm{m/s} / 0.001\,\mathrm{m} = 500\,\mathrm{s}^{-1}$
$\tau = \eta \cdot dv/dy = 0.1 \times 500$
$\tau = 50\,\mathrm{Pa}$

Result ·

\tau = 0.1 \times 500 = 50\,\mathrm{Pa}

2

Force on a moving plate

Given ·

\eta:: 0.1
dv/dy:: 500

Find · F per

m^{2}

Steps

$\tau$ from above is 50 Pa
Force per unit area $= 50\,\mathrm{N/m}^{2}$
For a $0.1\,\mathrm{m}^{2}$ plate, $F = 5\,\mathrm{N}$

Result ·

F/A = \tau = 50\,\mathrm{Pa}

, so 50 N per

m^{2} of

plate area

Stokes' Law→Poiseuille's Law→Reynolds Number→