Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model? We know that the Standard Model is a theory about almost everything (except gravity).  So it should be the basis of fluid mechanics, which is a macroscopic theory from experiences.  So is it possible that we can derive equations of fluid mechanics from the Standard Model?
If the answer is yes, please give a simple example.
If the answer is no, what is the reason that prevent the derivations to be reality.
 A: From your comment :

So is it possible to prove the consistence of fluid mechanics with the Standard Model?

The standard model is consistent with special relativity and quantum theory.  We know those explain everything our normal fluid equations deal with because it just atoms, ions and electrons, so it's a very safe bet that it's consistent with normal fluid mechanics.
A direct proof, however, would be rather insane and provide a different result from standard fluid mechanics because it would include terms that model conditions and particles at extreme energies that are irrelevant for normal fluid mechanics.  We'd end up with some insane equations that modelled everything, e.g. neutrino fluids or Higgs particle fluids and mixtures of all of these things.  You'd end up discarding most of what you found (assuming someone could do such hideous math) to reduce it to a form related to normal fluids.
We have separate physics for macroscopic objects precisely because that's the most sane way to work.
A: 
If the answer is yes, please give a simple example.


If the answer is no, what is the reason that prevent the derivations to be reality.

The short answer is "no." The long answer is "yes."
Here's a sketch of the long answer.

*

*The standard model includes the electro-weak Lagrange density which, after symmetry breaking, includes terms than make up the QED Lagrange density for electrons and photons:
$$
\bar \psi(i \gamma\cdot D -m)\psi -\frac{1}{4}F^2
$$


*We ignore everything but the above term because we will assume that we are working at length scales much larger than the the inverse weak masses (so those forces fall off exponentially and we don't care).


*We will also ignore the strong part of the standard model (for the same reason). So, similarly those forces are confined to nuclei. And at these large length scales we now assume nuclei can be treated effectively as non-dynamical (quantum mechanically) point charges. I.e., they are just there to create a potential for our dynamic electrons.


*Let's further take the non-relativistic limit of the QED Lagrangian and also restrict ourselves to a single Hilbert space (say of N particles), which is fine since we are non-relativistic now and can't create any particles.


*Let's also change from the Lagrange formalism to the Hamiltonian formalism and write down the resulting Hamiltonian. This leads us to the usual electromagnetic/solid-state Hamiltonian:
$$
\hat H = \sum_i^N\left(\frac{(\mathbf p_i+e\mathbf A/c)^2}{2m} + V(\mathbf r_i)\right) + \sum_{i\neq j}\frac{e^2}{2|\mathbf r_i - \mathbf r_j|}
$$
where $p_i$ and $r_i$ are operators. And where
$$
V(\mathbf r) = -\sum_{k}\frac{Z_ke^2}{|\mathbf r - \mathbf R_k|}
$$
where $Z_k$ and $\mathbf R_k$ describe the charges and positions (and so the external potential) of whatever nuclei condensed out of the strong part of the interaction.


*Now, we need to get out of the quantum regime and into the classical. So, take the classical limit and find out that the Hamiltonian operator is associated with a similar-looking classical Hamiltonian (not operator), which can be written as:
$$
H(p,r) = \sum_i^N\left(\frac{(\mathbf p_i+e\mathbf A/c)^2}{2m} + V(\mathbf r_i)\right) + \sum_{i\neq j}\frac{e^2}{2|\mathbf r_i - \mathbf r_j|}\;,
$$
where $p_i$ and $r_i$ are not operators, but are the momenta and positions of the classical particles.


*Now we need to get to some kind of continuum limit. We can start to do this by re-writing the Hamiltonian in terms of the density:
$$
H(p,r) = T(p) + \int \rho(\mathbf r)v(\mathbf r) + \int\int \rho(\mathbf r)\rho(\mathbf r')\frac{e^2}{|\mathbf r-\mathbf r'|}\;,
$$
where
$$
\rho(\mathbf r,t) = \sum_i \delta(\mathbf r - \mathbf r_i(t))\;,
$$
is the classical density, which obeys:
$$
\frac{\partial \rho}{\partial t} = \nabla \cdot J\;,
$$
where
$$
J = \sum_i v_i \delta(\mathbf r - \mathbf r_i(t))
$$


*Do the rest of the continuum limit and make other approximations to get to fluid dynamics. ;)


*Profit!
A: The answer is no, here is why.
The Standard Model lets us predict (among other things) experimental outcomes of tests run in particle accelerators, at the scale length of ~much smaller than a proton and truly gigantic energy scales (billions of electron volts), where the number of particles in the system is of order ~a few. It was not invented to tell us anything at all about the behavior of macroscopic objects like a bucketful of water/glycerine mixture or honey flowing through pipes or air flowing over a wing at supersonic speeds, where the typical length scale is of order ~one baseball diameter and the energy scale is of order ~a couple of electron volts, and the number of particles in the system is of order ~10exp23.
That said, if you had a superduper megacomputer that could model those 10^23 particles individually and track their movements in 3-D space with one picosecond time resolution and one angstrom spatial resolution, you might be able to observe the emergence of macroscopic behavior patterns like viscosity, surface tension, heat capacity, shear stresses and so on, but then again you might not.
That would be akin to painting the Golden Gate Bridge with a toothpick tip dipped in paint: not definitively ruled out by mathematics, but a fool's errand nonetheless.
A: One way to derive fluid dynamics is to start from the equations of motion for $N$ particles, and use these to compute the evolution of average quantities (like the density) of the distribution of particles. Then, one makes the approximation that the evolution of the average quantities will not depend on higher order statistics of the distribution, which describe complicated interactions between particles (so-called collision terms). In this sense, a lot of the details of the interactions between particles is actually irrelevant in fluid mechanics, by design. You would still get an identical long-distance fluid mechanical set of equations, even if the Standard Model were replaced by another local theory of physics obeying the same symmetries. In an important sense, the details of what lies underneath do not matter. What matters are the symmetries of the underlying processes, and the fact that the interactions are local.
The effect of the layers of physics below the fluid description, which nest like Russian dolls at least until you reach the Standard Model, is wrapped up in parameters like the viscosity of water, which are measured. In principle, you could try to compute these quantities from a deeper theory. There are, indeed, papers that try to do calculations like this, such as https://pubs.acs.org/doi/abs/10.1021/acs.jpclett.9b02913, although not starting from the Standard Model, but from "one level down" (interactions between molecules).
If you could simulate a large number of Standard Model particles, with sufficient temporal and spatial resolution to capture all the interactions, and enough temporal and spatial range to be able to measure fluid effects, there's no reason to expect you could not, say, calculate the viscosity of water from the Standard Model. However, there are not enough computational resources on Earth to do this calculation. To give you an idea of the magnitude of this challenge, calculating the mass of the proton numerically with QCD (using so-called "lattice QCD" simulations) is extremely difficult and only accomplished in the past few years. Hopefully it goes without saying that there are many, many orders of magnitude between the size of a proton and the size of a bathtub.
There are some cases where Standard Model particles are "one level down" from a fluid-level description of a system, and therefore Standard Model calculations can be usefully done to estimate some of the properties of the fluid. For example, the quark-gluon plasma, a soup of quarks and gluons, which behaves approximately as a Fermi liquid, and has properties like phase transitions and viscosity (actually it has zero viscosity, which is interesting in itself), and an equation of state. Note that the last reference is a set of slides describing a way to estimate the equation of state of the quark-gluon plasma using Lattice QCD, which is a calculation of fluid properties starting from the Standard Model.
A: The Navier-Stokes equations are not a theory based only on experience/measurement, rather it are the laws of Newtonian mechanics applied to fluid continuous media. Here is one random article Google gave me on how Euler derived his equations (Euler's equations are a precursor of the modern Navier-Stokes equations).
I'm sure you can find derivations of Newtonian mechanics from the standard model if you look deep enough, though I couldn't find anything rigorous in 5 minutes of googling. (I did find this video that tries to explain the derivation of classical mechanics from quantum mechanics.) Just put that together with the derivation of Euler's / Navier-Stokes equations and you have your answer.
A: You must really think in continuous steps.

*

*We know how to go from the Standard model (or specifically QED) to non-relativistic Quantum mechanics (the Schrödinger equation) with some intermediate steps like Dirac equation taken as a wave equation for an electron and the Pauli equation. We can compute hyperfine corrections to various electron energy levels in a hydrogen atom, but they will be irrelevant later.


*We know how to approximately compute many body quantum systems (e.g. atoms and molecules) using methods like Hartree-Fock or DFT that are derived from the Schrödinger equation for many particles..


*We can approximate the interactions of gas molecules (or atoms of a monoatomic gas) with the Hartree-Fock or DFT methods and derive intermolecular potentials including the Van der Waals forces.


*The quantum mechanics tells us about the statistics of many body systems. For gases at normal temperatures the Boltzmann statistics (Maxwell-Boltzmann distribution) works very well. Therefore we can model many-particle systems using the Boltzmann equation. The Boltzmann equation already describes fluid dynamics and can be simulated numerically in methods such as the Lattice Boltzmann method. The collision term depends on the statistics, which will be Maxwell-Boltzmann for normal gases.


*We can consider the gas to be continuum and derive the macroscopic quantities in the procedure such as the Chapman–Enskog theory. The starting point is the Boltzmann equation and approximate interaction terms between the molecules, the results are the momentum flux (stress) tensor and the heat flux vector. The expressions contain terms that we call the viscosity coefficient and the thermal conductivity coefficient and the predictions for their values. Once you have the fluxes, you basically have the Navier-Stokes equations that speak purely about continuum and do not know anything about molecules.
A: I think one way to think about this would be to recover the Schrodinger equation from a non-relativistic limit of the Dirac equation, and to then recover Newton's second law from the $\hbar\rightarrow0$ limit for the dynamics a coherent state in the Schrodinger equation. I believe the Navier-Stokes equations are then derived from N2L.
