Variable viscosity in the Navier Stokes equations

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Discussion Overview

The discussion centers around the implications of variable viscosity in the Navier-Stokes equations, exploring the additional equations or considerations needed to solve these equations under varying viscosity conditions. The scope includes theoretical aspects, potential applications, and references to existing literature.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the additional equations required when viscosity is variable in the Navier-Stokes equations.
  • Another participant suggests that viscosity could depend on state variables such as pressure, temperature, and density, and mentions the need for energy balance equations for viscous heating.
  • A third participant agrees, stating that an equation relating viscosity to state variables is necessary and that the energy equation would be required for an exact solution.
  • Questions arise about the implications of an isothermal condition on viscosity, with some participants suggesting that incorporating isothermal constraints would be necessary.
  • One participant expresses skepticism about the occurrence of isothermal situations in nature due to viscous heating leading to temperature variations.
  • Another participant notes that viscosity in ferrofluids is dependent on the magnetic field, introducing a specific case of variable viscosity.
  • Several participants request references or papers related to variable viscosity, indicating a lack of readily available resources.

Areas of Agreement / Disagreement

Participants express differing views on the relevance and occurrence of isothermal conditions in practical scenarios, with some doubting its applicability due to the effects of viscous heating. There is no consensus on the necessity or implications of variable viscosity in isothermal situations.

Contextual Notes

Participants highlight the dependence of viscosity on temperature and other state variables, suggesting that assumptions about conditions (like isothermal) may limit the applicability of their discussions. The exploration of specific cases, such as ferrofluids, introduces additional complexity.

Who May Find This Useful

This discussion may be useful for researchers and students interested in fluid dynamics, particularly those exploring the complexities of variable viscosity in theoretical and applied contexts.

hoomanya
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Hi,

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

Thanks.
 
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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"
 
Last edited by a moderator:
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.
 
Thanks, and if the problem was isothermal, what then?
 
hoomanya said:
Thanks, and if the problem was isothermal, what then?

Then you incorporate that constraint.
 
Thanks,
Do you know any papers or any sources where this has been the case?
 
hoomanya said:
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.
 
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.
 
Viscosity in ferrofluids is generally dependent on the magnetic field.
 
  • #10
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.
 

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