# Variable viscosity in the Navier Stokes equations

1. Jul 21, 2011

### hoomanya

Hi,

If the viscosity in the NS equations was a variable, what extra equation is used to solve the NS equations?

Thanks.

2. Jul 21, 2011

### jambaugh

I would think the functional dependency of the viscosity on state variables, e.g. pressure, temperature, density, and their equation of state (e.g. ideal gas law or liquid thermal expansion).

You would probably also have to figure in energy balance equations for viscous heating.
Here's an example pre-print which may enlighten:
http://arxiv.org/abs/physics/0302100" [Broken]

Last edited by a moderator: May 5, 2017
3. Jul 21, 2011

jambaugh hit the nail on the head pretty much. You would need an equation relating viscosity to state variables. The energy balance portion of his answer would be taken care of by the energy equation, which would also be necessary to solve for an exact answer.

4. Jul 22, 2011

### hoomanya

Thanks, and if the problem was isothermal, what then?

5. Jul 22, 2011

### jambaugh

Then you incorporate that constraint.

6. Jul 25, 2011

### hoomanya

Thanks,
Do you know any papers or any sources where this has been the case?

7. Jul 25, 2011

### jambaugh

I seriously doubt an isothermal situation will come up in nature. Given viscous heating there will be variations in temperature. As far as papers go... you can google search as well as I can so I leave it to you.

8. Jul 25, 2011

Not to mention that if the system is isothermal, I can't think of a really good reason why viscosity would vary. Viscosity depends almost entirely on temperature.

9. Jul 25, 2011

### Bob S

Viscosity in ferrofluids is generally dependent on the magnetic field.

10. Jul 25, 2011