Euler equations (fluid dynamics)
Partial differential equations governing adiabatic and inviscid fluid flow.
A set of partial differential equations describing fluid motion without heat transfer or viscosity is what the Euler equations are.[1] The Navier–Stokes equations, when both viscosity and thermal conductivity are absent, reduce to the Euler equations.[1]
Quick Facts
- Field
- Fluid dynamics
- Type
- Partial differential equations
Facts from the source article.
Key facts and history
Euler first presented his work to the Berlin Academy in 1752, and the equations appeared in published form in his 1757 article "Principes généraux du mouvement des fluides".[2] During the second half of the 19th century, it was found that the energy balance equation must be kept for compressible flows.[3] Many fundamental questions about the mathematical properties of these equations remain open.[4] The regularity and possible finite-time singularity formation of three-dimensional solutions are important open questions.[4]
Leonhard Euler is the namesake of these equations.[1] Before Euler's work, the Bernoulli family and Jean le Rond d'Alembert had already made relevant contributions.[2] Only the wave equation predates the Euler equations as an example of a partial differential equation.[5] In fluid dynamics texts, the term "the compressible Euler equations" is commonly used for the complete system that includes an energy equation.[6]
What the cited sources add
Nguyen's lecture notes describe the Eulerian description of fluids as introduced by Euler in 1755.[7] The notes define an ideal fluid by setting the Cauchy stress tensor to a form involving only pressure.[7] The momentum equations for ideal fluids together with the continuity equations are referred to as the Euler equations.[7] The American Meteorological Society defines the Froude number as the nondimensional ratio of inertial force to gravity force.[8]
Nguyen's notes state that the mathematical justification of continuum fluid dynamics from discrete molecular dynamics remains an outstanding unsolved problem.[7] When the Froude number is high, gravitational effects are comparatively minor.[9]
References
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: Toro 1999, p. 24 (original not read by us) en.wikipedia.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: Euler 1757; Christodoulou 2007, p. ; Darrigol, "From Newton's mechanics to Euler's equations", Physica D: Nonlinear Phenomena, 2008, p. 1855–1869 (original not read by us) en.wikipedia.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: Christodoulou 2007, p. (original not read by us) en.wikipedia.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: Elgindi, "Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$", Annals of Mathematics, 2021-11-01 (original not read by us) en.wikipedia.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: "Bulletin of the American Mathematical Society", pubs.ams.org (original not read by us) en.wikipedia.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: Anderson 1995, p. (original not read by us) en.wikipedia.org
- Nguyen, "Math 505, Mathematical Fluid Mechanics: Notes 1", Pennsylvania State University. facts only, no text reused. sites.psu.edu
- "Froude number", American Meteorological Society. facts only, no text reused. glossary.ametsoc.org
- Wikipedia, "Euler equations (fluid dynamics)" (CC BY-SA 4.0), which cites: "Froude number", American Meteorological Society (we also read that source) en.wikipedia.org
Each numbered sentence above was checked against the source it cites before publication. Sentences citing Wikipedia adapt text from the English Wikipedia article Euler equations (fluid dynamics) (contributors), used under CC BY-SA 4.0; this entry is available under the same licence. Other sources were used for facts only; no text or images are copied from them.
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