I wanted to model a real life problem using the Navier-Stokes equations and was wondering what the assumptions made by the same are so that I could better relate my entities with a 'fluid' and make or set assumptions on them likewise. For example one of the assumptions of a Newtonian fluid is that the viscosity does not depend on the shear rate. Similarly what are the assumptions that are made on a fluid or how does the Navier-Stokes equations define a fluid for which the equation is applicable?
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3$\begingroup$ Are you asking what assumptions you can make to reduce the Navier-Stokes equations? Or are you asking, to what can the Navier-Stokes can be applied? $\endgroup$– Kyle KanosCommented Sep 24, 2013 at 15:58
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2$\begingroup$ It assumes the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term. See here en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations $\endgroup$– ShuchangCommented Sep 24, 2013 at 15:58
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4$\begingroup$ This might be extremely obvious, but the fluid must be reasonably approximated by a continuum. See Knudsen Number. $\endgroup$– OSECommented Sep 24, 2013 at 16:20
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$\begingroup$ @Kyle The latter $\endgroup$– Rohan PrabhuCommented Sep 25, 2013 at 5:57
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$\begingroup$ Maybe if you explained in a bit more detail what problem you are trying to solve you can get a better answer on what assumptions might or might not be good to model it. $\endgroup$– DerekCommented Jul 24, 2014 at 20:22
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4 Answers
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Most importantly, the Navier-Stokes equations are based on a continuum assumption. This means that you should be able to view the fluid as having properties like density and velocity at infinitely small points. If you look at e.g. liquid flows in nanochannels or gas flows in microchannels you could be in a regime where this assumption breaks down. As far as I know there is no hard limit for the continuum assumption, but the Knudsen number is a useful indicator.
Additionally there is, as @ShuchangZhang mentioned, an assumption on the nature of the stress in the fluid. Although I am not sure whether you could really call this an assumption or whether it should be considered a theory (like the NS equations itself).
The strongest assumptions are typically not in the Navier-Stokes equations themselves, but rather in the boundary conditions that should be applied in order to solve them. To give an example, whether the no-slip boundary condition (fluid velocity at the wall equals wall velocity) or the navier slip boundary condition (fluid velocity equals a scaled velocity gradient at the wall) has been a much debated subject, in particular for hydrophobic surfaces (see e.g. Phys. Rev. Lett. 94, 056102 (2005) and references therein and thereto)
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In non-relativistic physics, the Navier-Stokes equations emerge as an almost universal behaviour of rheological models when:
1a) Strain rates are small enough that the constitutive relation between strain rate and stresses is linear (Newtonian fluid limit).
2a) No fast external force is acting on the fluid, namely there is no elastic response in the fluid (e.g., water when perturbed at very high frequencies responds as an elastic body, but this is true also for simple liquids, see Frenkel's book, "Kinetic Theory of Liquids").
Alternatively, we may start from another perspective, that of kinetic theory approaches that try to justify hydrodynamics starting from more microscopic descriptions. In this case, Navier-Stokes appears when a double limit is taken (see this review):
1b) Each fluid element is close to thermal equilibrium.
2b) The evolution at the macroscopic level is slow with respect to the microscopic processes (collisions, reaction rates) that implement point (1b).
The Kundsen number argument - The Kundsen number is typically introduced to (partially) justify (1b). However, it is nothing more than an indicator, useful to have a feeling about the assumption that the fluid, at the scale of interest, is a continuum (this is, morally, point 1b). We may restate the Kundsen number argument in the following way: given that the substance is initially close to local thermodynamic equilibrium, we can expect Navier-Stokes to be valid if the spatial gradients in the flow are on typical lengths that are larger than an intrinsic microscopic scale (typically the mean free path).
The Kundsen number is not enough - The smallness of the Knudsen number alone is not sufficient. From the kinetic theory point of view, if the system is not driven slowly enough it will always display a huge number of degrees of freedom (because the distribution function of particles has no time to relax to a quasi-equilibrium one). In fact, hydrodynamics, like thermodynamics, deals with a small number of local macroscopic degrees of freedom: this only happens if the substance is almost relaxed from a kinetic theory perspective.
Moreover, even if the system is driven slowly and is relaxed so that hydrodynamics is possible, we may still need to include rheological corrections (visco-elastic and pseudo-plastic corrections, remember the example of the elastic response of water!).
Final message and relativistic extensions - Only when all conditions (1a),(2a), (1b), (2b) are met we can hope (rarely justify! only for gases there are rigorous theorems) that Navier-Stokes equations are valid at the macroscopic scale.
Unfortunately, extending Navier-Stokes hydrodynamics to special relativity is known to be problematic: this is discussed in the review Irreversible Thermodynamics and the stability of relativistic theories for dissipation, see also Is there a relativistic version of Navier-Stokes equations?
The workaround, in relativity, is to realize that the Knudsen number argument has to be extended to space-time gradients! This automatically takes into account both (1a) and (2a), while (1b) and(2b) have to be justified in terms of the microphysics of the particular substance. However, the final relativistic macroscopic theory is still not a Navier-Stokes description but rather a class of rheological models that encompass Israel-Stewart hydrodynamics (see this review).
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The Navier-Stokes equations assume (assuming we are looking at a vector conservative form):
- The continuum hypothesis, which is applicable for Knudsen numbers of much less than unity.
The Navier-Stokes equations must specify a form for the diffusive fluxes (e.g. otherwise you would have the Cauchy momentum equation not the Navier-Stokes momentum equation), e.g.
- Newtonian fluid for stress tensor or Cauchy's 2nd law, conservation of angular momentum
- Definition of the transport coefficients (e.g. viscosity)
The solution of the Navier-Stokes equations involves additional assumptions, (but this is separate from the equations themselves) e.g.
- An equation of state for closure (e.g. thermally perfect gas, calorically perfect gas)
- Stokes' assumption for zero bulk viscosity
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Navier Stokes equations can be exactly derived from kinetic equations. There are several assumptions regarding the reduction of the multiparticle distribution equations to a 1-particle distribution function $f(v, r)$. The Navier Stokes hydrodynamic limit is reached for systems that do not vary widely both in time and space, in order for the kinetic steady state to be locally reached. All the hydrodynamic coefficients like viscosity, diffusion, etc. can be calculated from the kinetic theory.
References:
From Boltzmann Kinetics to the Navier-Stokes Equations without a Chapman-Enskog Expansion, Benoit Cushman-Roisin and Brenden P. Epps (2018)
From Newton to Navier-Stokes: HOW TO CONNECT FLUID MECHANICS EQUATIONS FROM MICROSCOPIC TO MACROSCOPIC SCALES, Isabelle Gallagher (2019)
From Kinetic Theory to Navier–Stokes Hydrodynamics, Sauro Succi, Oxford University Press (2018)
