Editing Fluid dynamics (section)

===Inviscid versus viscous versus Stokes flow===

The dynamic of fluid parcels is described with the help of [[Newton's second law]]. An accelerating parcel of fluid is subject to inertial effects.

The [[Reynolds number]] is a [[dimensionless quantity]] which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number ({{math|''Re'' ≪ 1}}) indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called [[Stokes flow|Stokes or creeping flow]].

In contrast, high Reynolds numbers ({{math|''Re'' ≫ 1}}) indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an [[inviscid flow]], an approximation in which viscosity is completely neglected. Eliminating viscosity allows the [[Navier–Stokes equations]] to be simplified into the [[Euler equations (fluid dynamics)|Euler equations]]. The integration of the Euler equations along a streamline in an inviscid flow yields [[Bernoulli's equation]]. When, in addition to being inviscid, the flow is [[Lamellar field|irrotational]] everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called [[potential flow]]s, because the velocity field may be expressed as the [[gradient]] of a potential energy expression.

This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the [[no-slip condition]] generates a thin region of large strain rate, the [[boundary layer]], in which [[viscosity]] effects dominate and which thus generates [[vorticity]]. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict [[Drag (physics)|drag forces]], a limitation known as the [[d'Alembert's paradox]].

A commonly used<ref>{{Cite journal|last=Platzer|first=B.|date=2006-12-01|title=Book Review: Cebeci, T. and Cousteix, J., Modeling and Computation of Boundary-Layer Flows|url=http://dx.doi.org/10.1002/zamm.200690053|journal=ZAMM|volume=86|issue=12|pages=981–982|doi=10.1002/zamm.200690053|bibcode=2006ZaMM...86..981P |issn=0044-2267}}</ref> model, especially in [[computational fluid dynamics]], is to use two flow models: the Euler equations away from the body, and [[boundary layer]] equations in a region close to the body. The two solutions can then be matched with each other, using the [[method of matched asymptotic expansions]].