# Units of viscosity

1. Mar 2, 2015

### richard9678

I don't understand units associated with kinematic viscosity.

Let's recap dynamic viscosity (η):

η = Shear stress (τ) / rate of shear strain (m/s)/m)

η = F/A / (velocity (m/s) / gap distance (m))

η = Force( N) / Area (m2) / (velocity (m/s) / gap distance (m))

Stress (F/A) is pressure, Newtons per metre squared, so unit is Pascals. When you cancel the two m's out in the divisor, you are left with:

η = N/m2 or Pascals / s (Pascal Seconds).

The equation to convert dynamic viscosity to kinematic velocity is:

ν = η / ρ

ν = η N m-2 s / ρ Kg m-3

I get this. The numerator is saying Newtons per square meter (pressure in Pascals) per second (Pascal seconds) and the divisor is saying Kg per cubic metre)

Since 1 Kg = N m-1 s2, this simplifies to dimesions of m2 s-1

I don't get how things simplify.

I know that the Stoke is the unit, and it's a unit of position and time.

Thanks.

Last edited: Mar 2, 2015
2. Mar 2, 2015

### richard9678

1 stoke is 1 centimeter squared per second. I'm having difficulty in conceptualising what that means.

Velocity is metres per second.

Acceleration is meters per second per second.

ν = η / ρ is relating shear characteristics (dynamic viscosity) to fluid acceleration (mass density).

Still trying to conceptualise the stoke as a dimension.

The SI unit of kinematic viscosity is m2/s.

The cgs physical unit for kinematic viscosity is the stoke.

Last edited: Mar 2, 2015
3. Mar 2, 2015

### richard9678

Actually, right now I could do to concentrate on using the formula correctly.

ν = η / ρ

The answer needs to be in centistokes.

How the heck to get centistokes.

An oil might have a dynamic viscosity of 0.25 Pascal Seconds.

The oil might have a mas of 900Kg per cubic meter.

Okay, ν = η / ρ = 0.250 / 900 = 0.00027.

The answer is wrong for centistokes. 270 or 27 might be about right. I must figure out why I'm not getting the right answer.

I think the numerator might be 106 larger than it should be.

Stokes are in CGS units. Might be a clue.

Last edited: Mar 2, 2015
4. Mar 2, 2015

### Andy Resnick

It is a little tricky, to be sure- viscosity can be thought of as 'diffusion of momentum':

http://www.quora.com/How-is-viscosity-the-diffusion-of-momentum [Broken]

As a practical matter, I find it best to consistently simplify the units into MLT (rather than Pa, N, kg, etc...) to make sure everything works out.

Last edited by a moderator: May 7, 2017
5. Mar 2, 2015

### richard9678

ν = η / ρ

Before I go to bed:

For centistokes.

I think η needs to be in centipoise (1 cP = 0.001 Pa second) and ρ in g /cm-3.

Where η is 0.25 Pacal seconds, and ρ 900 Kg / m-3, I think the answer should be 27 centistokes.

Last edited: Mar 2, 2015
6. Mar 2, 2015

### Staff: Mentor

OK. I'm going to do this in cgs units, which are the units that I like to work with. I leave it up to you to convert to metric.

Force: $dynes=\frac{gm-cm}{sec^2}$

Stress and Pressure: $\frac{dynes}{cm^2}=\frac{gm}{cm-sec^2}$

Velocity: $\frac{cm}{sec}$

Velocity gradient: $\frac{cm}{sec-cm}=\frac{1}{sec}$

Dynamic Viscosity: $Poise=\frac{dynes}{cm^2-sec}=\frac{gm}{cm-sec}$

Dynamic Viscosity: $centipoise = 0.01 Poise$

Density: $\frac{gm}{cm^3}$

Kinematic Viscosity: $Stokes = \frac{Poise-cm^3}{gm}=\frac{dynes-cm}{gm-sec}=\frac{cm^2}{sec}$

Kinematic Viscosity: $centistokes=0.01Stokes$

Hope this helps.

Chet

7. Mar 3, 2015

### richard9678

Let me see if I can get this.

Up till now, all my physics calculations are worked out using SI units. That means I'm entering meters, Kg and seconds. I do this for working out dynamic viscosity η.

But, now I've come across a formula to work out centistokes (a unit of kinematic viscosity) from dynamic viscosity and mass, which is a unit worked out using cgs units.

Therefore I must make changes when using the equation ν = η / ρ.

Lets take ρ, mass first. 900Kg per m-3 in SI units is 900. So, what's that in cgs? 1 x cm-3 is 106 times smaller than m-3. So, in cgs 900Kg becomes 9 x 10-4Kg. Were still in Kg, so, to show in grams we must multiply by 103. So, our figure should be 0.9g cm-3

Correct I think so far.

8. Mar 3, 2015

### richard9678

The issue now is what units should be being used for η.

I believe it is true, that the dynamic viscosity for engine oil will be in the tenths of a Pascal-second range.

To work out kinetic viscosity in stokes we would enter the POISE for η. A poise is 0.1 Pascal-second.

But, we need to work with centistokes. So, we need to be working in centipoise that is 1 x 10-2 of a Poise (cP).

1 cP is therefore 10-3 times a Pascal second.

Going back to our original equation: ν = η / ρ = 0.250 / 900 = 0.00027.

η should be in cP therefore 250. ρ should be in g / cm-3 therefore 0.9.

ν = η / ρ = 250 / 0.9 = 277.7 centistokes (kinetic viscosity).

I think that is correct.

So, if you get η in Pascal-seconds, from earlier calculations, you have to multiply η by 1000, when using the formula to obtain centistiokes.

Last edited: Mar 3, 2015
9. Mar 3, 2015