# When are the Navier Stokes equations invalid?

• ramzerimar
The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.In summary, the Navier-Stokes equations can be applied to incompressible flows, but not to compressible flows. They also have limitations in terms of the type of fluid they can describe, such as non-Newtonian fluids. Some examples of when the equations may not apply include when dealing with compressible flow or when the fluid behavior is not adequately modeled by a continuum. Additionally, the Prantdl number, which relates to heat transfer behavior, is not relevant when considering the mechanical behavior of viscoelastic fluids.

#### 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?).

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.

ramzerimar said:
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?).
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.

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.

jack action
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.

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?

ramzerimar said:
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?
The NS equations can be applied to incompressible flows.

Chestermiller said:
The NS equations can be applied to incompressible flows.
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.

ramzerimar said:
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.
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.

The Navier-Stokes equations are absolutely valid for compressible flows. They are a little more complicated since you can't make so many simplifying assumptions and they require a greater number of supplemental equations since there many more variables in the overall problem (there are a minimum of 6 and can be a dozen or more depending on the conditions).

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.

The two most important limitations on the Navier-Stokes equations is that they only apply to (a) fluids that can adequately be modeled by a continuum and (b) Newtonian fluids. Some examples of when the continuum breaks down are in the upper atmosphere where density is so low that the mean free path is comparable to the flow scales or in microfluidics when the flow scales are so small that they are comparable to the mean free path.

Ravi Singh choudhary and Chestermiller
Chestermiller said:
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.
Respected professor
i just want to know the examples of viscoelastic fluids with the values of Prandtl number . I saw a lot of articles , where viscoelastic fluids are used with Prandtl number 0.7 , 1 and 3 etc . which are shocking for me because these Prandtl numbers are for air and water and air ,water are Newtonian fluids .

Idrees Afridi said:
Respected professor
i just want to know the examples of viscoelastic fluids with the values of Prandtl number . I saw a lot of articles , where viscoelastic fluids are used with Prandtl number 0.7 , 1 and 3 etc . which are shocking for me because these Prandtl numbers are for air and water and air ,water are Newtonian fluids .
The Prantdl number relates to the heat transfer behavior of the fluid, not the mechanical behavior. The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.

Chestermiller said:
The Prantdl number relates to the heat transfer behavior of the fluid, not the mechanical behavior. The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.
Professor thanks... but by taking Pr =0.7 means v r talking about air ... is ve take a air as viscoelastic fluid...

If i want to discuss the thermal behaviour of viscoelastic fluid then kindly suggest me a suitable Prandtl number . For which i have a justification that why i am taking this Prandtl number ?

Idrees Afridi said:
If i want to discuss the thermal behaviour of viscoelastic fluid then kindly suggest me a suitable Prandtl number . For which i have a justification that why i am taking this Prandtl number ?
This is too far removed from the original theme of the present thread to continue along these lines. Please start a new thread that addresses the topic of how to analyze heat transfer to viscoelastic fluids.

ramzerimar said:
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?).
I have recently solved the Navier-Stokes d.e. for air and water and other Madelung's fluids.
Those fluids are characterized by having a (well-behaved and eventually time-dependent) density. Please do see my publication cited as

R. Meulens , "A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation", AIP Advances 12, 015308 (2022) https://doi.org/10.1063/5.0074083