Formula of Viscosity: Newton's Law, Units and Examples

The formula of viscosity comes from Newton's law of viscous flow, $F = \eta A \frac{dv}{dx}$ , which gives $\eta = \frac{F}{A\,(dv/dx)}$ . Viscosity η is the internal friction of a fluid: the resistance one layer feels when it slides past the next.

Pour water and then syrup from the same cup. The syrup creeps because its layers drag hard on each other, so it has the bigger η. Every liquid and gas has some viscosity, and it decides how fast oil drains from an engine, how blood moves through arteries and how quickly a marble sinks in shampoo. This page covers the law, kinematic viscosity and the flow equations built on it; the coefficient of viscosity formula page focuses on the plate method and unit conversions.

Quick facts

Newton's law
$F = \eta A \frac{dv}{dx}$
Viscosity
$\eta = \frac{F}{A\,(dv/dx)}$
SI unit
Pa s (N s m⁻²); CGS unit poise, 1 P = 0.1 Pa s
Dimensional formula
[M¹ L⁻¹ T⁻¹]
Kinematic viscosity
$\nu = \frac{\eta}{\rho}$ , unit m²/s
Type
Scalar property of the fluid

What is the formula of viscosity, explained

Imagine a liquid as a deck of thin cards. The bottom card sticks to the container wall and stays still. The top card moves with the flow. Each card in between moves a bit faster than the one under it. The change in speed per unit distance across the cards is the velocity gradient, dv/dx. Each layer pulls back on the faster one above, and that pull is the viscous force.

Parts of Newton's law of viscosity
Symbol	Name	SI unit
$F$	Viscous force, acting along the layers	N
$\eta$	Viscosity, a property of the fluid	Pa s
$A$	Contact area of the layers	m²
$\frac{dv}{dx}$	Velocity gradient across the flow	s⁻¹

A minus sign is sometimes written, $F = -\eta A \frac{dv}{dx}$ , to show the force opposes the relative motion. For magnitudes, drop it. The velocity gradient formula page shows how to work out dv/dx from speeds and distances.

Dynamic and kinematic viscosity

η is called dynamic viscosity. Engineers also use kinematic viscosity, which divides by density:

$\nu = \frac{\eta}{\rho}$

It is measured in m²/s and has dimensions [M⁰ L² T⁻¹]. Kinematic viscosity tells you how quickly motion spreads through a fluid under its own weight, which is why it appears on motor oil grades and in pipe flow charts.

Example: kinematic viscosity of an engine oil

An oil has η = 0.25 Pa s and density 880 kg/m³. Find its kinematic viscosity.

$\nu = \frac{0.25}{880} = 2.8 \times 10^{-4}\ \text{m}^2/\text{s}$

For comparison, water is about 1.0 × 10⁻⁶ m²/s, so the oil is nearly 300 times more sluggish.

Worked examples with Newton's law

Example 1: finding η from a force

Two layers of a lubricant, each of area 0.20 m², have a velocity gradient of 50 s⁻¹ between them. The viscous force is 0.40 N. Find the viscosity.

$\eta = \frac{F}{A\,(dv/dx)} = \frac{0.40}{0.20 \times 50} = 0.040\ \text{Pa s}$

Example 2: flow through a thin tube

Water (η = 1.0 × 10⁻³ Pa s) flows through a tube of radius 0.50 mm and length 20 cm with a pressure difference of 1600 Pa across its ends. Find the volume flow rate using Poiseuille's equation.

Convert: r = 5.0 × 10⁻⁴ m, l = 0.20 m. Then r⁴ = 6.25 × 10⁻¹⁴ m⁴.

$Q = \frac{\pi P r^4}{8 \eta l} = \frac{\pi \times 1600 \times 6.25\times10^{-14}}{8 \times 1.0\times10^{-3} \times 0.20} = 2.0 \times 10^{-7}\ \text{m}^3/\text{s}$

That is about 0.20 mL every second. Halving the radius would cut the flow by a factor of 16.

Radius enters Poiseuille's equation to the fourth power, so convert it to metres with care before raising it.

Stokes' law and falling objects

A small sphere of radius r moving at speed v through a fluid feels a drag of $F = 6 \pi \eta r v$ . This is Stokes' law, and it holds when the flow stays smooth. When the drag plus the buoyant force equal the weight, the sphere stops accelerating; the speed at that point is given by the terminal velocity formula. Measuring that speed is the simplest way to find η in a school lab. Whether the flow stays smooth at all is judged by $R_e = \frac{\rho v D}{\eta}$ , the Reynolds number, which leads to the critical velocity formula. All of these sit together on the fluids revision sheet.

Temperature, fluids that break the rule, and traps

Heating a liquid weakens the attraction between its molecules, so its viscosity drops. Gases do the opposite: their viscosity rises with temperature. Fluids that follow F = ηA(dv/dx) with a fixed η, such as water and air, are called Newtonian. Ketchup, paint and cornflour paste are non Newtonian: their effective viscosity changes with how hard you stir them.

Confusing η with ν. Check the unit: Pa s or m²/s.
Forgetting the gradient. Viscous force depends on how quickly speed changes across the gap, not on the speed alone.
Using diameter in Stokes' law. The r there is the radius.

Frequently asked questions

What is the formula of viscosity?

From Newton's law F = ηA(dv/dx), viscosity is η = F / (A · dv/dx). It measures the force per area needed for each unit of velocity gradient.

What is the SI unit of viscosity?

Viscosity is measured in pascal seconds (Pa s). One poise, the CGS unit, equals 0.1 Pa s. Kinematic viscosity uses m²/s.

What is Newton's law of viscosity?

The viscous force between two layers of fluid is proportional to their area of contact and to the velocity gradient between them: F = ηA(dv/dx).

What is the difference between viscosity and density?

Density is mass per volume and says how heavy a fluid is. Viscosity says how strongly it resists flowing. Oil is less dense than water yet more viscous.

Why does honey flow faster when warm?

Warming the honey weakens the forces between its molecules, so the layers slide past one another more easily and its viscosity falls.