Variable viscosity in the Navier Stokes equations

[hoomanya]

Hi,

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

Thanks.

 

[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.
(... quick google search...)
Here's an example pre-print which may enlighten:
http://arxiv.org/abs/physics/0302100"

 

[boneh3ad]

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.

 

[hoomanya]

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

 

[jambaugh]

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

Then you incorporate that constraint.

 

[hoomanya]

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

 

[jambaugh]

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

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.

 

[boneh3ad]

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.

 

[Bob S]

Viscosity in ferrofluids is generally dependent on the magnetic field.

 

[hoomanya]

Thanks jambaugh. I wouldn't be asking questions (or ask for references) here if had found something useful on Google.

Thanks very much everyone. I have a better idea now.

 

[boneh3ad]

Google Scholar:
http://scholar.google.com/scholar?q=Variable+viscosity&hl=en&btnG=Search&as_sdt=1,44&as_sdtp=on

AIAA (search variable viscosity):
http://aiaa.org/content.cfm?pageid=406