Navier-Stokes with spatially varying viscosity

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• taktoa
In summary, the author suggests that an equation similar to Navier-Stokes could be derived that incorporates spatially varying viscosity. However, this would require additional equations relating viscosity to other variables, and the author is still waiting for a response from the original poster.
taktoa
TL;DR Summary
Is there an equation, similar to Navier-Stokes, that encompasses incompressible fluid flow with spatially varying viscosity?
Does anyone know of a differential equation, similar to Navier-Stokes, that encompasses incompressible fluid flow with spatially varying viscosity? Viscosity is treated as a global constant in NS; I've found some papers online that address NS with viscosity as a function of velocity, but I can't seem to find any on NS with viscosity as a function of position.

Any thoughts on how to derive this?

Well, I was thinking you could consider separate instances of NS with different viscosities in a grid, and then take the limit as the grid size goes to zero. You'd obviously also need boundary conditions that enforce continuity between the NS instances. I have no idea how to get a sensible PDE out of that though.

In the x-direction, the terms involving viscous stresses in the equation of motion (in Cartesian coordinates) are:

$$\frac{\partial \tau_{xx}}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}+\frac{\partial \tau_{xz}}{\partial z}$$Do you know how each component of the stress tensor is expressed in terms of viscosity and the velocity gradients?

taktoa said:
Summary: Is there an equation, similar to Navier-Stokes, that encompasses incompressible fluid flow with spatially varying viscosity?

I think if you want to be serious about this, you need to (more) carefully consider the functional form of viscosity- for example, the viscosity can be temperature dependent, so a viscosity gradient would exist when there is a thermal gradient (https://www.nonlin-processes-geophys.net/10/545/2003/npg-10-545-2003.pdf). Similarly, the viscosity could vary with relative concentrations of (miscible?) components of a multiphase fluid: http://www.engr.mun.ca/muzychka/ETFS2008.pdf

Just stating "the viscosity varies with position" is problematic when discussing fluid flow.

There is no reason the Navier-Stokes equations can't handle flows with spatial and/or temporal variations in viscosity. The only difference is that viscosity becomes a variable instead of a constant. Of course that means one more equation is required relating viscosity to other variables like temperature.

Chestermiller
We haven't heard a peep from the OP since Sunday. I'm still waiting for a response from him on my questions in post #5.

1. What is Navier-Stokes with spatially varying viscosity?

Navier-Stokes with spatially varying viscosity is a mathematical model used to describe the motion of fluids, taking into account the variations in viscosity throughout the fluid's volume.

2. How does Navier-Stokes with spatially varying viscosity differ from the standard Navier-Stokes equations?

The standard Navier-Stokes equations assume a constant viscosity throughout the fluid, while Navier-Stokes with spatially varying viscosity accounts for the fact that viscosity can vary in different parts of the fluid.

3. What are the applications of Navier-Stokes with spatially varying viscosity?

Navier-Stokes with spatially varying viscosity is used in various fields such as fluid dynamics, meteorology, and oceanography to model the behavior of fluids with non-uniform viscosity.

4. What are the challenges in solving Navier-Stokes with spatially varying viscosity?

One of the main challenges in solving Navier-Stokes with spatially varying viscosity is the complexity of the equations, which require advanced mathematical techniques and computational methods to obtain solutions.

5. Are there any simplified versions of Navier-Stokes with spatially varying viscosity?

Yes, there are simplified versions of Navier-Stokes with spatially varying viscosity that make certain assumptions or approximations to make the equations more manageable. However, these simplified versions may not accurately capture the full behavior of the fluid.

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