# When are the Navier Stokes equations invalid?

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1. May 9, 2017

### ramzerimar

I'm studying the Navier Stokes equations right now, and I've heard that those set of equations are invalid in some situations (like almost any mathematical formulation for a physics problem). I would like to know in which situations I cannot apply the NS equations, and what is the common procedure when something like that happens (like, there are any other set of PDEs that describe fluid flow in situations where the NS equations are invalid?).

2. May 9, 2017

Presumably you made some assumptions when studying the derivation of those equations. Can you list what those assumptions are? That will give you a pretty big hint about what the equations' limitations are.

3. May 9, 2017

### muscaria

One example it can't be applied in is when the system is not Galilean invariant. This occurs for a fluid which is subject to a nonlinear vector potential which depends on the density $\rho$ of the fluid. In this situation, the kinetic energy density of the fluid is nonlinear in $\rho$ and the pressure of the fluid depends explicitly on the flow.
The way forward in these kind of situations is to start from first principles without assuming anything in particular..In your case Cauchy's equation would do.

4. May 9, 2017

### Staff: Mentor

The Navier Stokes equations apply to fluids described by Newton's law of viscosity (i.e., Newtonian Fluids). If the fluid exhibits more complicated behavior than that of a Newton fluid (a Newtonian fluid is one for which the stress tensor in linearly proportional to the rate of deformation tensor), the Navier Stokes equations will not apply. Such fluids are called non-Newtonian fluids, and include viscoelastic fluids and purely viscous non-Newtonian fluids. Examples of such fluids are polymer melts and solutions, and suspensions. And, of course, the Navier Stokes equations don't apply to solids.

5. May 11, 2017

### ramzerimar

I've read that one of the assumptions is that the NS equations are for a incompressible flow. So, for dealing with compressible flow problems, the NS equations cannot be applied?

6. May 11, 2017

### Staff: Mentor

The NS equations can be applied to incompressible flows.

7. May 11, 2017

### ramzerimar

To analyze compressible flow I would need to rewrite the NS equations or just add the energy equation to the mix? The difference that I see is that, in compressible flow, the density also changes and so becomes a variable, and I would need another equation to describe the flow.

8. May 11, 2017

### Staff: Mentor

Yes, and you would also have to use the compressible form of the continuity equation. And you might have to allow for temperature-dependent viscosity.

9. May 12, 2017

There are the basic $u$, $v$, $w$, $p$, $\rho$, and $T$ variables. Additionally, you could have $\mu$ (dynamic viscosity), $\lambda$ (second coefficient of viscosity), $c_p$, $c_v$, $\kappa$ (thermal conductivity), or even $X_i$ (mass fractions of various chemical species). The Navier-Stokes equations can technically apply to problems involving all of those variables, both compressible and incompressible.