A Navier-Stokes with spatially varying viscosity

[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.

 

[Chestermiller]

Any thoughts on how to derive this?

 

[taktoa]

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.

 

[taktoa]

There are a couple of papers on achieving this with the smoothed particle hydrodynamics method, but they don't talk about the actual PDE that these methods solve:

• https://www.dankoschier.de/resources/papers/WKBB18.pdf
• http://app.spheric-sph.org/sites/spheric/files/Spheric_Book.pdf
• https://journals.aps.org/pre/abstract/10.1103/PhysRevE.87.043301 (this one is about the Lattice Boltzmann method)

 

[Chestermiller]

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?

 

[Andy Resnick]

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.

 

[boneh3ad]

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.