# Should the non-relativistic Navier Stokes Equations be modified?

• Anashim
In summary, the conversation discusses the concept of choking mass flow, where the momentum density of a fluid reaches its maximum value in stationary conditions. This is closely related to the speed of sound and can be proven through thermodynamic considerations. The question is raised whether this also holds true in transient flows and if it implies a need to modify the Navier-Stokes equations to exhibit pseudo-Lorentzian symmetry instead of Galilean symmetry. This could lead to a more accurate description of transient flows and other phenomena such as shock waves.
Anashim
Choking mass flow seems to reflect the fact that fluid momentum density has a maximum value (in stationary conditions) equal to ##\rho_* c_*## where ##\rho_*## is the critical mass density and ##c_*## is the critical velocity, which is closely related to the speed of sound (see Landau-Lifchitz,"Fluid Mechanics", section 83).

If this result also held in transient flows, would it not imply that the Navier-Stokes Equations should be modified so that they explicitly exhibited pseudo-Lorentzian symmetry (in momentum density) instead of Galilean symmetry?

The momentum density field would then become explicitly causal.

The velocity field does not seem to make much sense if detached from the mass density field.

What I have in mind was the fact that, in a stationary fluid flow, from thermodynamic considerations, it can be proven that:

$$\frac{d(\rho v)}{dv}=\rho\big[1-\frac{v^2}{c^2}\big]$$

Landau and Lifhchitz, "Fluid Mechanics", section 83.

where ##\rho## is the mass density, ##v## the local velocity and ##c## the local speed of sound.

This indicates that the momentum density does have a maximum, at least in stationary flows (where the local velocity is equal to the local speed of sound).

Since the momentum Navier-Stokes equations are Galilean invariant, a sufficiently large pressure gradient should make possible to attain larger momentum densities and this contradicts the previous equation.

I was wondering if one should not enforce the maximum local momentum density as a pseudo-Lorentzian symmetry. (In incompressible flows, at least, I think that one does need to enforce this symmetry).

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In this case, the Navier-Stokes equations would be modified to include the effects of pseudo-Lorentzian symmetry. This might lead to a more accurate description of transient flows (and of other phenomena such as shock waves).

## 1. What are the non-relativistic Navier Stokes Equations?

The non-relativistic Navier Stokes Equations are a set of mathematical equations that describe the motion of fluids, such as liquids and gases, in non-relativistic (low-speed) conditions. They take into account the forces acting on a fluid, such as pressure, viscosity, and gravity, to determine the velocity and pressure of the fluid at any given point.

## 2. Why should the non-relativistic Navier Stokes Equations be modified?

The non-relativistic Navier Stokes Equations have been found to be inaccurate in certain scenarios, such as when dealing with high-speed or compressible fluids. In these cases, modifications to the equations are necessary to accurately predict the behavior of the fluid.

## 3. What are some proposed modifications to the non-relativistic Navier Stokes Equations?

There are several proposed modifications to the non-relativistic Navier Stokes Equations, including the introduction of terms for compressibility, turbulence, and non-Newtonian behavior. These modifications aim to improve the accuracy of the equations in a wider range of scenarios.

## 4. How do modifications to the non-relativistic Navier Stokes Equations affect the results?

The effects of modifications to the non-relativistic Navier Stokes Equations vary depending on the specific modification and the scenario being studied. In some cases, the modifications may have a minimal impact on the results, while in others they may significantly improve the accuracy of the predictions.

## 5. Are there any drawbacks to modifying the non-relativistic Navier Stokes Equations?

Yes, there are potential drawbacks to modifying the non-relativistic Navier Stokes Equations. One drawback is that the modified equations may become more complex and difficult to solve, which can increase the computational resources and time needed for simulations. Additionally, the modifications may not always accurately reflect the physical behavior of fluids, leading to incorrect predictions.

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