Editing Navier–Stokes equations (section)

==Properties==
===Nonlinearity===

The Navier–Stokes equations are [[Nonlinearity|nonlinear]] [[partial differential equations]] in the general case and so remain in almost every real situation.<ref>{{cite book |title=Fluid Mechanics |first1=M. |last1=Potter |first2=D. C. |last2=Wiggert |series=Schaum's Outlines |publisher=McGraw-Hill |year=2008 |isbn=978-0-07-148781-8}}</ref><ref>{{cite book |title=Vectors, Tensors, and the basic Equations of Fluid Mechanics |first=R. |last=Aris |publisher=Dover Publications |year=1989 |isbn=0-486-66110-5 }}</ref> In some cases, such as one-dimensional flow and [[Stokes flow]] (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the [[turbulence]] that the equations model.

The nonlinearity is due to [[convective]] acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but [[laminar flow|laminar]] (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging [[nozzle]]. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.<ref>{{cite book |title=McGraw Hill Encyclopaedia of Physics |edition=2nd |first=C. B. |last=Parker |year=1994 |publisher=McGraw-Hill |isbn=0-07-051400-3 }}</ref>

===Turbulence===

[[Turbulence]] is the time-dependent [[Chaos theory|chaotic]] behaviour seen in many fluid flows. It is generally believed that it is due to the [[inertia]] of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the [[Reynolds number]] quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.<ref>Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, {{ISBN|3-527-26954-1}} (Verlagsgesellschaft), {{ISBN|0-89573-752-3}} (VHC Inc.)</ref>

The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or [[direct numerical simulation]]. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the [[Reynolds-averaged Navier–Stokes equations]] (RANS), supplemented with turbulence models, are used in practical [[computational fluid dynamics]] (CFD) applications when modeling turbulent flows. Some models include the [[Spalart–Allmaras turbulence model|Spalart–Allmaras]], [[k-omega turbulence model|{{mvar|k}}–{{mvar|ω}}]], [[turbulence kinetic energy|{{mvar|k}}–{{mvar|ε}}]], and [[SST (Menter’s Shear Stress Transport)|SST]] models, which add a variety of additional equations to bring closure to the RANS equations. [[Large eddy simulation]] (LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales.

===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>