# What are the assumptions of the Navier-Stokes equations?

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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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? –  Kyle Kanos Sep 24 '13 at 15:58
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 –  Shuchang Sep 24 '13 at 15:58
This might be extremely obvious, but the fluid must be reasonably approximated by a continuum. See Knudsen Number. –  OSE Sep 24 '13 at 16:20
@Kyle The latter –  Rohan Prabhu Sep 25 '13 at 5:57
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. –  Derek Jul 24 '14 at 20:22

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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Thank you so much for the detailed reply. –  Rohan Prabhu Sep 25 '13 at 6:18

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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