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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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Euler’s Equations

Euler’s Equations

(p.14) 2 Euler’s Equations
Fluid Mechanics

S. G. Rajeev

Oxford University Press

Euler derived the fundamental equations of an ideal fluid, that is, in the absence of friction (viscosity). They describe the conservation of momentum. We can derive from it the equation for the evolution of vorticity (Helmholtz equation). Euler’s equations have to be supplemented by the conservation of mass and by an equation of state (which relates density to pressure). Of special interest is the case of incompressible flow; when the fluid velocity is small compared to the speed of sound, the density may be treated as a constant. In this limit, Euler’s equations have scale invariance in addition to rotation and translation invariance. d’Alembert’s paradox points out the limitation of Euler’s equation: friction cannot be ignored near the boundary, nomatter how small the viscosity.

Keywords:   Euler’s equations, ideal fluid, Helmholtz equation, conservation laws, d’Alembert’s paradox, viscosity

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