Why the Euler equations are wrong
Abstract
Euler's equations for incompressible fluids are incomplete; the material derivative formulated by Leonhard Euler does not account for advection in certain flows, which causes the solution to remain fixed in its initial state. The non-orthogonality of the inertia terms leads, under certain circumstances, to the elimination of the solenoidal inertia terms and imparts a certain directional rigidity to flows subject to the constraint of incompressibility. This same form of inertia appears in other laws of mechanics, such as the Navier-Stokes equations, raising questions regarding the completeness of these laws. An alternative approach based on discrete mechanics and the principle of conservation of acceleration, resolves this issue by expressing the inertial terms through an orthogonal Helmholtz-Hodge decomposition into two components. The case of a rotating fluid filament in a cylindrical cavity clearly illustrates how the stretching of vortices, which is not accounted for in Euler's equations, initially develops according to the discrete law of motion. This flow culminates in an explosion within in finite time.
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BibTeX
@misc{caltagirone2026euler,
title = {{Why the Euler equations are wrong}},
author = {Jean-Paul Caltagirone},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05772563v3},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Why the Euler equations are wrong
AU - Jean-Paul Caltagirone
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05772563v3
AB - Euler's equations for incompressible fluids are incomplete; the material derivative formulated by Leonhard Euler does not account for advection in certain flows, which causes the solution to remain fixed in its initial state. The non-orthogonality of the inertia terms leads, under certain circumstances, to the elimination of the solenoidal inertia terms and imparts a certain directional rigidity to flows subject to the constraint of incompressibility. This same form of inertia appears in other laws of mechanics, such as the Navier-Stokes equations, raising questions regarding the completeness of these laws. An alternative approach based on discrete mechanics and the principle of conservation of acceleration, resolves this issue by expressing the inertial terms through an orthogonal Helmholtz-Hodge decomposition into two components. The case of a rotating fluid filament in a cylindrical cavity clearly illustrates how the stretching of vortices, which is not accounted for in Euler's equations, initially develops according to the discrete law of motion. This flow culminates in an explosion within in finite time.
ER -