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Euler equations (fluid dynamics)
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==Exact solutions== All [[potential flow]] solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic.{{sfn|Marchioro|Pulvirenti |1994|p=33}} [[File:OS schematic.svg|thumb|right|300px|A two-dimensional parallel shear-flow.]] Solutions to the Euler equations with [[vorticity]] are: * parallel [[shear flow]]s – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a [[Cartesian coordinate system]] <math>(x,y,z)</math> the flow is for instance in the <math>x</math>-direction – with the only non-zero velocity component being <math>u_x(y,z)</math> only dependent on <math>y</math> and <math>z</math> and not on <math>x.</math>{{sfn|Friedlander |Serre|2003|p=298}} * [[Arnold–Beltrami–Childress flow]] – an exact solution of the incompressible Euler equations. * Two solutions of the three-dimensional Euler equations with [[cylindrical symmetry]] have been presented by Gibbon, Moore and Stuart in 2003.{{sfn|Gibbon |Moore |Stuart|2003|p=}} These two solutions have infinite energy; they blow up everywhere in space in finite time.
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