The speed of light is not really relevant in low Mach number flow. From the assumption of incompressible flow, you get infinite speed of sound. That might be more of an issue.

My research is about ventilation of tunnels. Typical air flow velocities range from 0 to 25m/s. We consider this incompressible flow. I only had an issue with this assumption, when I analysed very long tunnels (12km and more). For long tunnels (or let's say for large flow domains), the speed of sound becomes relevant, even for very small flow velocities. The information of flow change at one portal needs time to reach the other portal.

BTW, the same question can be asked for mechanics: Would you take relativistic effects into account when you analyse the sandwich falling from the table?

Any model is an approximation. We just have to be aware of the approximation and of the limits of the model.

The speed of light is not really relevant in low Mach number flow. From the assumption of incompressible flow, you get infinite speed of sound. That might be more of an issue.

My research is about ventilation of tunnels. Typical air flow velocities range from 0 to 25m/s. We consider this incompressible flow. I only had an issue with this assumption, when I analysed very long tunnels (12km and more). For long tunnels (or let's say for large flow domains), the speed of sound becomes relevant, even for very small flow velocities. The information of flow change at one portal needs time to reach the other portal.

BTW, the same question can be asked for mechanics: Would you take relativistic effects into account when you analyse the sandwich falling from the table?

Any model is an approximation. We just have to be aware of the approximation and of the limits of the model.

Yes, indeed I wrote about the speed of sound as the issue. The model assumes that the pressure waves travels instantaneously (a=Inf) in the fluid whereas we have the finite light speed as limit in any medium. Your example of the propagation of pressure waves in the tunnel is well suited to understand that the low-Mach model and the zero-Mach model are really different.

In terms of a mechanic-based problem, we could wonder what about a solid body that translate rigidly: if we act with a force on an edge to translate, does this force is transmitted instantaneously along the whole body?

I would fully agree with Ingo Riess but - instead of the notion "approximation" - I prefer the notion "idealization". E.g. an empirical relation is piecewise linear, which is a most simple idea of an actually cubic relation etc. etc.

Filippo Maria Denaro : my compliment for posing a fine question which NEEDS to be raised when our scientific culture shall not decay completely ...

I crossed exactly this issue when I thought about high-Re turbulence (see att.) where the number of empirical (but in no way trivial) models is very high, with so many parameters: the solution of the dilemma was the idealization Re = \infty, that exists in superfluids only, where quantum mechnics is a topic.

In my opinion, the principles of fluid dynamics should not conflict with the laws of physics.

a2 =dp/drho|s=const. ->+Inf.

The above equation physically does not mean that M = 0, but is close to zero. Therefore, in my opinion, what we need to ask is not the contradiction between both the theory of fluid dynamics and the laws of physics, but the contradiction between the theory of fluid dynamics and the laws of physics on mathematical theory, at least, at least, how do we interpret a mathematical condition or equation....

In the book of Landau there is a section explaining the framework of the assumption but still has the concept of a = +Inf without any comment about the violation of the limiting light velocity.

Now I wonder what about an energetic consequence of that, for example considering the inviscid assumption for which Bernoulli integral can be invoked.

This integral assumes rho=constant (or M=0) to be exact. That means the fact that there is no sense in taking derivative of the Bernouilli equation with respect to the density. Otherwise, if we assume a small but finite value of the Mach number we should come back to the Crocco relation. And now dp/drho can be high but finite.

What if we rethink to the assessed theories if we re-introduce a finite propagation of the pressure waves also when the convective velocity is small?

Please be careful with the meaning of zero and infinity. We never know the value of infinity physically exactly. Especially if they comes from the limit of 1/linf. or the limit of 1/zero. Please also pay attention to the utilization of Taylor series. In most problems, we just only its low order term (the linear term). That's my simple opinion....

Nazaruddin Sinaga

M=0 can be only due to the assumption of infinite sound velocity. But in any case, even assuming the limit concept, why it is accepted the sound velocity greater than the light one?

"why it is accepted the sound velocity greater than the light one? "

Maybe Helmut Ziegfeld Baumert or Tapan K. Sengupta

Thank you very much Tapan K. Sengupta

I always like your answers and comments, you are so smart, at least for me.

Have a nice night.......

After a look at some literature, I cannot add more to this issue, I still wonder if we should reconsider the Mach limit to a small but finite value and taking into account the real pressure waves aslo at low speed flow...

This issue is not related to the present discussion, however what do you mean exactly for:

" How can we say that there is a linear change along the edge-length of CV" ?

I do not understand why you use a first order expansion, is it a numerical issue in the discretization?.... however, open a discussion if you have doubts about this problem

Dear Filippo Maria Denaro ,

This is a very good question and the answers will have many far reaching consequences for our understanding of fluids.

In the most general sense, only the Lorentz invariant, relativistic fluid model is truly consistent with the currently known laws of physics. The equations of this model are often simplified by requiring only the Galilean invariance leading to a well-known compressible fluid model, valid for a large range of the Mach numbers. Stepping down the ladder of realism, it is often assumed that the fluid is incompressible. This assumption not only creates problems with the infinite speed of sound but is also incompatible with thermodynamics. More generally, we can say that the incompressible fluid model describes the topological transformation of the continuum; this concept is very well connected to the topological fluid dynamics introduced by K. Moffatt.

Many aspects of fluid flow are described by the incompressible fluid idealization, including even fluid systems having a relatively large spatial extent. In the atmospheric sciences, incompressible models have often been applied to shallow convection studies. The use of such a model was justified using the scale analysis. In our time, however, there is an evident trend to study the atmospheric flows using the complete system of fluid equations without the simplifying assumptions.

Looking at the different mathematical models of the continuum, I find that simplifying assumptions can often be justified by scale analysis and the rules of symmetry. It is probably better to understand a simpler model than to use a complicated model without understanding.

Thanks Janusz Pudykiewicz for your contribution. My question is exactly in terms of a possible warning in using a too much idealized model. As you wrote, in the incompressible model there is no thermodynamics pressure (it is only a lagrangian multiplier) and that is coherent to the assumption dp/drho->Inf. On the other hand, the model has also no energy equation. But the scaling analysis, as I generally know, is constrained by the assumption that convective velocity

Dear Tapan K. Sengupta

actually my position is not exactly the opposite... Let me clarify better. The hystory showed that the inviscid model was able to do some good work but only when Prandtl introduced the concept of BL we were able to reconcile the fact that the flow velocitu at the wall can be equal to the velocity of the wall and the tangential stress is not infinite. This is an example of an acceptable model that, however, required to be revised in terms of a different model, that is the BL concept.

Some months ago an interesting discussion on RG already highlighted a topic about the finite limite in a velocity propagation in case of heat transfer. See

https://www.researchgate.net/post/Special_Relativity_regarding_to_Heat_propagation_What_is_the_speed_of_Heat

You see here the same point is raised, a model where a finite velocity propagation should be considered.

Now my question is: are we sure that we can disregard the finite velocity propagation of acoustic waves from the solution of the velocity field just assuming M=0 (as I wrote before, I can see a scaling analysis in terms of v

Yes, I was thinking about a possible model with a relaxation parameter in the the waves velocity propagation in such a way that the sound velocity is finite but we can still invoke the divergence-free constraint as continuity equation.

This is not a classical low-Mach model that allows for density to change.

Tapan K. Sengupta

let me use again the example of the BL. We solve for the viscous region by assuming that the external inviscid solution is already determined. But that is a zero-th order approximation (in terms of matched asymptotic expansion) since the external solution should be recomputed from the viscous field and repeat again until convergence. I think about the assumption of infinite sound velocity as a zero-th order solution that should be then corrected until a finite sound velocity is obtained.

On the other hand, if we want to rigorously introduce the incompressible flow model from the thermodynamics, when we set homogeneous density and temperature then there is nothing else that to set Grad p =0 in the momentum equation. The fact we retain a term like grad phi (nothing to do with the pressure) is more a mathematical artifice that can be seen in the framework of the Hodge decomposition.

I cannot say if the present near and far field decompositoon are a good model in terms of approximation to the reality...

Dear Filippo Maria Denaro

Thanks for the comment. Although approximations of fluid equations are commonly used, very often with great success, we must absolutely avoid approximations that violate the basic principles of physics. The main reason for using simplified equations is the desire to perform analytical manipulations or to obtain equations that are easier to integrate numerically. For example, in meteorology, the incompressible and anelastic approximations are often described as "soundproof" (in the anelastic approximation, we replace div (v) = 0 by div (\rho v) = 0 ). With the advent of better time-stepping schemes, we can avoid filtered equations.

Despite the problems noted in your question, the incompressible approximation is commonly used in a wide range of fluid models, from basic studies of turbulence to the atmospheric boundary layer. Even astrophysical applications are full of examples of "soundproof models".

The history of further improvements to the inelastic approximation is consistent with your idea of ​​modifying the fluid equations.

The issue of the definition of sound speed is also relevant in our discussion. The attached paper introduces some useful concepts in the context of geophysics.