Editing Navier–Stokes equations (section)

===Applicability===
{{Further|Discretization of Navier–Stokes equations}}
Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.

The Navier–Stokes equations assume that the fluid being studied is a [[Continuum mechanics|continuum]] (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at [[Relativistic velocity|relativistic velocities]]. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, [[capillarity]] of internal layers in fluids appears for flow with high gradients.<ref>{{Citation
| last1    = Gorban
| first1   = A.N.
| last2   = Karlin
| first2  = I. V.
| year     = 2016
| title    = Beyond Navier–Stokes equations: capillarity of ideal gas
| type    = Review article
| journal = Contemporary Physics
| volume   = 58
| issue   = 1
| pages   = 70–90
| doi = 10.1080/00107514.2016.1256123 
| url = https://www.researchgate.net/publication/310825466 
| arxiv= 1702.00831
| bibcode= 2017ConPh..58...70G| s2cid = 55317543
}}</ref>  For large [[Knudsen number]] of the problem, the [[Boltzmann equation]] may be a suitable replacement.<ref>{{Citation
| last    = Cercignani
| first   = C.
| year     = 2002
| title    = Handbook of mathematical fluid dynamics 
| chapter  = The Boltzmann equation and fluid dynamics 
| volume  = 1
| publisher = North-Holland
| place    = Amsterdam
| pages = 1–70
| editor-last   = Friedlander
| editor-first  = S.
| editor2-last  = Serre
| editor2-first = D.
| isbn = 978-0444503305
}}</ref> 
Failing that, one may have to resort to [[molecular dynamics]] or various hybrid methods.<ref>{{Citation
| last1    = Nie
| first1   = X.B.
| last2   = Chen
| first2  = S.Y.
| last3   = Robbins
| first3  = M.O.
| year     = 2004
| title    =  A continuum and molecular dynamics hybrid method for micro-and nano-fluid flow
| type    = Research article
| journal = Journal of Fluid Mechanics
| volume  = 500
| pages = 55–64 
| doi = 10.1017/S0022112003007225
| doi-broken-date = 8 February 2025
| url = https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/a-continuum-and-molecular-dynamics-hybrid-method-for-micro-and-nano-fluid-flow/BE0D4513A0F90F844CD21D64F6D3F9EF
| bibcode= 2004JFM...500...55N
| s2cid = 122867563
}}</ref>

Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for [[Newtonian fluid]]s where the viscosity model is [[linear]]; truly general models for the flow of other kinds of fluids (such as blood) do not exist.<ref>{{Citation
| last    = Öttinger 
| first   = H.C.
| year     = 2012
| title    = Stochastic processes in polymeric fluids
| publisher = Springer Science & Business Media
| place    = Berlin, Heidelberg
| isbn = 9783540583530
| doi = 10.1007/978-3-642-58290-5
}}</ref>