Variable viscosity in the Navier Stokes equations

In summary: The viscosity of a fluid is a measure of its resistance to flow. Viscosity is affected by a variety of factors, including temperature, pressure, and shear stress.
  • #1
hoomanya
90
0
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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  • #2
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"
 
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  • #3
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
Thanks, and if the problem was isothermal, what then?
 
  • #5
hoomanya said:
Thanks, and if the problem was isothermal, what then?

Then you incorporate that constraint.
 
  • #6
Thanks,
Do you know any papers or any sources where this has been the case?
 
  • #7
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.
 
  • #8
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
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.
 

1. What is variable viscosity in the Navier Stokes equations?

Variable viscosity in the Navier Stokes equations refers to the concept that the viscosity of a fluid can change or vary depending on factors such as temperature, pressure, and velocity. It is a crucial parameter in fluid dynamics and plays a significant role in determining the behavior of fluids.

2. How is variable viscosity incorporated into the Navier Stokes equations?

Variable viscosity is incorporated into the Navier Stokes equations through the addition of the term "dynamic viscosity" (μ) in the equations. This term takes into account the variations in viscosity and allows for a more accurate representation of the fluid flow.

3. What are the effects of variable viscosity on fluid flow?

The effects of variable viscosity on fluid flow can be significant, as it can affect the speed, direction, and turbulence of the flow. In some cases, variable viscosity can also lead to the formation of vortices and other complex flow patterns.

4. How do scientists measure variable viscosity in fluids?

Scientists measure variable viscosity in fluids using various techniques such as rheology, which involves measuring the response of a fluid to an applied force, and viscometry, which measures the resistance of a fluid to flow. These techniques help determine the relationship between viscosity and the factors that cause it to vary.

5. What are some real-world applications of understanding variable viscosity in the Navier Stokes equations?

The understanding of variable viscosity in the Navier Stokes equations has numerous real-world applications, including fluid dynamics in engineering and design, weather prediction and modeling, and the study of ocean currents and turbulence. It is also crucial in industries such as aerospace, automotive, and oil and gas.

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