# Why does viscosity have units of pressure times time?

Pascal-seconds is a unit of viscosity. Why?

I know that viscosity represents the thickness of a fluid, or its ease of flow, but I don't understand why that is represented by these parameters.

Gold Member
Viscosity is essentially a proportionality parameter relating shear stress to the velocity gradient in a fluid. Viscosity, therefore, must have units that make the equation work. Shear stress is defined as
$$\tau = \mu \nabla \vec{v}$$
where the units are, therefore
$$[\text{Pa}] = [\text{Pa}\cdot\text{s}]\dfrac{[\text{m}/\text{s}]}{[\text{m}]}$$

haruspex
Homework Helper
Gold Member
2020 Award
I take the question to be seeking some underlying reason why viscosity should be pressure * time, as though there is some set up in which the relationship emerges naturally without any other dimensions being involved along the way.
I don't think such can be found. The two distance measurements that dimensionally cancel are both in the velocity gradient, so a more fundamental question is why a velocity gradient should have dimension of 1/time, or frequency. The two distances are perpendicular to each other, which suggests to me the cancellation has no deep meaning.
Similar coincidences arise, e.g. angular momentum and action; note that one is a vector while the other is a scalar. OTOH, angular momentum is quantised, so maybe there is a deep meaning in that case.

Yes. For one of my classes we have to memorize lots of constants and conversions and I'm looking for a way to really understand the units of viscosity. I know the relationships but I was wondering if there's more to it. Sounds like not.

I also asked a professor today and he said I should think about it as the relationship between shear stress and shear rate and get the units from that instead of trying to derive a relationship from the units.

Having known the result, I guess we can reason out to an extent why the unit involves the product of pressure (which is basically force on a unit area) and time. We can get a measure of the thickness of the fluid by studying its reaction to an applied force. If I apply a greater force, I will get a greater resistance to its flow. As far as the time is concerned, you can study how much time it takes to flow a particular distance (or area, since we took force per unit area). More viscous fluids will take a longer time to flow. Since both factors mean greater viscosity, their being multiplied is acceptable.

Then again, I'm saying this because I already know the units.

Andy Resnick
Pascal-seconds is a unit of viscosity. Why?

Be careful- there are two 'flavors' of viscosity, 'kinematic' (typically units of Poise) and 'dynamic' (typically units of Stokes). Viscosity refers to the diffusion of momentum.

Poise has units of M/LT (say, kg/m*s) and is thus equivalent to both pressure*time and momentum/area. Stokes has units of L^2/T, say m^2/s.

Chestermiller
Mentor
In the simplest geometry, the so-called shear rate is equal to the rate of change of tangential velocity with respect to distance from the wall: dv/dy. According to Newton's law of viscosity, the shear stress at the wall is equal to the viscosity times the shear rate at the wall.

τ = η (dv/dy)

Shear stress has units of m/(l s2)

Stress has the same units as pressure.

Shear rate has units of (1/s)

For Newtonian fluids, Newton's law of viscosity in 3D is much more complicated than this (and involves the stress tensor and the rate of deformation tensor), but the general idea and the units are the same.

Gold Member
You flipped kinematic and dynamic there, Andy. I figured if just clear that up.

Andy Resnick