Coefficient of Viscosity Formula: η = F/(A dv/dx), SI Unit

The coefficient of viscosity formula is $\eta = \frac{F}{A\,(dv/dx)}$ : the tangential force needed to keep one fluid layer sliding over another, divided by the contact area and by the velocity gradient between them. A large η means a thick, slow liquid like honey; a small η means a thin one like water.

For the common textbook set up, a flat plate dragged at speed v over a fluid film of thickness d, the gradient is simply $\frac{v}{d}$ , so $\eta = \frac{F d}{A v}$ . The calculator above uses this plate form. The broader idea of fluid friction is on the formula of viscosity page, and the whole chapter is summarised on the fluids formula sheet.

Quick facts

Formula
$\eta = \frac{F}{A\,(dv/dx)}$
Plate form
$\eta = \frac{F d}{A v}$
SI unit
pascal second (Pa s), also written N s/m² or kg/(m s)
CGS unit
poise (P): 1 P = 0.1 Pa s
Dimensional formula
[M¹ L⁻¹ T⁻¹]
Water at 20 °C
about 1.0 × 10⁻³ Pa s

What each letter in the coefficient of viscosity formula stands for

What η = F / (A · dv/dx) is built from
Symbol	Meaning	SI unit
$\eta$	Coefficient of viscosity, also called dynamic viscosity	Pa s
$F$	Force along the layers, parallel to the flow	N
$A$	Area of each layer that slides	m²
$\frac{dv}{dx}$	Change in speed per metre measured across the flow	s⁻¹

The formula comes from rearranging $F = \eta A \frac{dv}{dx}$ , Newton's law for viscous flow. The force is tangential: it acts along the surface, not into it. The velocity gradient formula page explains how to find dv/dx when the speed does not change evenly across the gap.

SI unit, poise and dimensions

Substitute units: newton divided by (square metre × per second) gives N s/m², and since 1 N/m² = 1 Pa, this is the pascal second. The CGS unit is the poise, one dyne second per square centimetre, and

$1\ \text{Pa s} = 10\ \text{poise} \qquad 1\ \text{centipoise} = 10^{-3}\ \text{Pa s}$

Water at room temperature is close to one centipoise, a handy figure to remember. For dimensions:

$[\eta] = \frac{[\text{M L T}^{-2}]}{[\text{L}^2][\text{T}^{-1}]} = [\text{M}^1\,\text{L}^{-1}\,\text{T}^{-1}]$

The step by step version, with other quantities checked against it, is on the dimensional formula of coefficient of viscosity page.

Typical coefficients of viscosity near room temperature
Fluid	η in Pa s
Air	1.8 × 10⁻⁵
Water	1.0 × 10⁻³
Blood	about 3 × 10⁻³ to 4 × 10⁻³
Castor oil	about 1
Glycerine	about 1.4

Solved problems

Example 1: a plate on an oil film

A plate of area 0.10 m² slides at a steady 0.05 m/s over a 2.0 mm layer of oil. A force of 0.50 N keeps it moving. Find the coefficient of viscosity of the oil.

Given: F = 0.50 N, A = 0.10 m², v = 0.05 m/s, d = 2.0 × 10⁻³ m.

$\eta = \frac{F d}{A v} = \frac{0.50 \times 2.0\times10^{-3}}{0.10 \times 0.05} = \frac{1.0\times10^{-3}}{5.0\times10^{-3}} = 0.20\ \text{Pa s}$

Example 2: force from a value in poise

An oil has viscosity 8.0 poise. A glass slide of area 250 cm² is pulled at 10 cm/s over a 1.0 mm film of this oil. What force is needed?

Convert: η = 8.0 P = 0.80 Pa s, A = 250 × 10⁻⁴ = 0.025 m², v = 0.10 m/s, d = 1.0 × 10⁻³ m.

$F = \frac{\eta A v}{d} = \frac{0.80 \times 0.025 \times 0.10}{1.0\times10^{-3}} = 2.0\ \text{N}$

Poise to pascal second: divide by 10. Many marks are lost by multiplying instead.

How to measure η in the lab

Parallel plate method. Measure F, A, v and the gap d, then use the plate form above.
Falling ball method. Time a small sphere at its steady speed and rearrange the terminal velocity formula: $\eta = \frac{2 (\rho - \sigma) g r^2}{9 v_t}$ .
Capillary flow method. Measure the volume per second through a narrow tube and use Poiseuille's equation, $Q = \frac{\pi P r^4}{8 \eta l}$ .

The falling ball method is the one most Class 11 practicals use, because only a jar, a stopwatch and a ruler are needed.

Effect of temperature and common slips

For liquids, η falls quickly as temperature rises: warm honey pours far faster than cold honey. For gases, η rises slightly with temperature because molecules move faster and carry more momentum between layers.

Using the plate speed as dv/dx. Divide the speed by the layer thickness first.
Leaving d in millimetres. Convert to metres or the answer is off by 1000.
Calling η the viscous force. η is a property of the fluid; the force also depends on area and gradient, as on the viscous force formula page.

Frequently asked questions

Which formula gives the coefficient of viscosity?

The formula of coefficient of viscosity is η = F / (A · dv/dx): tangential force per unit area for each unit of velocity gradient. For a plate moving at v over a layer of thickness d, η = Fd / (Av).

What is the SI unit of coefficient of viscosity?

The pascal second (Pa s), equal to N s/m² or kg m⁻¹ s⁻¹. The CGS unit is the poise, and 1 Pa s = 10 poise.

What is the dimensional formula of coefficient of viscosity?

[M¹ L⁻¹ T⁻¹]. Divide force [MLT⁻²] by area [L²] and then by velocity gradient [T⁻¹] to reach it.

Is coefficient of viscosity the same as viscosity?

In most school questions, yes. Coefficient of viscosity is the precise name for dynamic viscosity η. Kinematic viscosity, η divided by density, is a different quantity with unit m²/s.

Does the coefficient of viscosity change with temperature?

Yes. It decreases for liquids as they warm up and increases slightly for gases.