Coefficient of Viscosity Dimensional Formula [ML⁻¹T⁻¹]

The coefficient of viscosity dimensional formula is [M¹ L⁻¹ T⁻¹], or [ML⁻¹T⁻¹] in short form. It tells you that the stickiness of a fluid scales with mass to the first power and falls off with one power of length and one power of time.

The coefficient η (eta) is defined by Newton's law of viscosity: the drag force between fluid layers is proportional to the layer area and to how fast the speed changes from one layer to the next (the velocity gradient). Honey has a large η, water a small one, but both carry the same bracket. Below are the derivation, the unit Pa s and the poise, a Stokes' law check and a list of slips. For numbers, see the coefficient of viscosity formula page.

Quick facts

Dimensional formula
[M¹ L⁻¹ T⁻¹], short form [ML⁻¹T⁻¹]
Defining formula
$F = \eta A \frac{dv}{dx}$
SI unit
Pa s = N s m⁻² = kg m⁻¹ s⁻¹
CGS unit
poise; 1 Pa s = 10 poise
Common symbol
η (eta)
Chapter
Mechanical properties of fluids, Class 11

Deriving the coefficient of viscosity dimensional formula

Start from the law that defines η, then isolate it.

Viscous force: $F = \eta A \frac{dv}{dx}$ , so $\eta = \frac{F}{A \, (dv/dx)}$ .
Force: $[F] = [M L T^{-2}]$ . Area: $[A] = [L^{2}]$ .
Velocity gradient is speed per distance: $[dv/dx] = [L T^{-1}] \div [L] = [T^{-1}]$ .
Area times gradient is $[L^{2} T^{-1}]$ . Flip it and multiply by the force: $[\eta] = [L^{-2} T] \cdot [M L T^{-2}] = [M L^{-1} T^{-1}]$ .

The velocity gradient step is where most errors start. It is not a velocity; it has units of s⁻¹, the same as a frequency. A minus sign sometimes appears in the force law to show the drag opposes motion, but a sign has no effect on dimensions. The velocity gradient page explains that ratio further.

Pa s, the poise and how they relate

Turn [ML⁻¹T⁻¹] into base units: kg m⁻¹ s⁻¹. Group them another way and you get N s m⁻², which is a pascal second (Pa s). Both are the SI unit of the coefficient of viscosity. Pa s is the form most answer keys print.

Convert 1 Pa s into the CGS unit, the poise

The poise is g cm⁻¹ s⁻¹. Raise each conversion factor to its own power from the bracket.

$1\ \text{Pa s} = (1000\ \text{g})(100\ \text{cm})^{-1}(1\ \text{s})^{-1} = 10\ \text{g cm}^{-1}\text{s}^{-1}$ .

So 1 Pa s = 10 poise, and 1 centipoise = 10⁻³ Pa s. Water near room temperature is close to 1 centipoise.

Matching brackets and a look alike

Quantities with the same bracket as viscosity, [M¹ L⁻¹ T⁻¹]
Quantity	How it is built	Why it matches
Dynamic viscosity of any fluid	shear stress ÷ velocity gradient	The definition itself
Pressure × time	$P \, t$	[ML⁻¹T⁻²] × [T]
Stress × time	$\sigma \, t$	Stress has the pressure bracket

The pressure times time row gives a neat way to remember the result: viscosity is shear stress divided by a rate, s⁻¹, so it is a stress multiplied by a time. Compare that with the dimensional formula of pressure.

Kinematic viscosity is a different quantity. It is η divided by density, ν = η/ρ, with dimensions [ML⁻¹T⁻¹] ÷ [ML⁻³] = [L²T⁻¹] and unit m²/s. Do not mix the two in an answer.

Two fluid laws put to the test

Check Stokes' law, F = 6πηrv, for a sphere falling through a fluid

Right side: $[\eta][r][v] = [M L^{-1} T^{-1}][L][L T^{-1}]$ .

Collect powers: M¹, L⁻¹⁺¹⁺¹ = L¹, T⁻¹⁻¹ = T⁻². Result $[M L T^{-2}]$ .

That is a force, so the law is dimensionally consistent. The 6π is a number and cannot be tested.

Check the flow rate relation Q = πPr⁴/(8ηl)

Top: $[M L^{-1} T^{-2}][L^{4}] = [M L^{3} T^{-2}]$ . Bottom: $[M L^{-1} T^{-1}][L] = [M T^{-1}]$ .

Ratio: $[L^{3} T^{-1}]$ , a volume per second. Here P is the pressure difference along the tube, and the result is the expected flow rate.

Errors that cost marks

Writing [MLT⁻¹]. That is momentum. Viscosity has length to the power −1.
Treating dv/dx as a velocity. It is [T⁻¹]; skipping the division by distance gives the wrong answer.
Mixing up dynamic and kinematic viscosity. η is [ML⁻¹T⁻¹]; ν is [L²T⁻¹].
Converting Pa s to poise by 100. The factor is 10, from 10³ g divided by 10² cm.

The viscosity bracket is collected with the other fluid quantities on the dimensional formulas page.

Frequently asked questions

In SI, what unit does η take?

Pa s. Written with base units that is kg m⁻¹ s⁻¹, and it can also appear as N s m⁻².

Is the dimensional formula of viscosity different from that of the coefficient?

No. When a question says viscosity it normally means the coefficient η, so the answer is [M¹ L⁻¹ T⁻¹]. Only kinematic viscosity differs, at [L²T⁻¹].

How many poise make one pascal second?

Ten. One poise is 0.1 Pa s, and one centipoise is 0.001 Pa s.

Is the coefficient of viscosity a scalar?

Yes. For ordinary fluids η is a single number that depends on the fluid and its temperature, with no direction.

Why is the velocity gradient [T⁻¹]?

It is a change in speed, [LT⁻¹], divided by the distance between layers, [L]. The lengths cancel and only one over time remains.