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Pré-Publication, Document De Travail Année : 2024

The property v⊥∇ × v is true, in general; application to Navier-Stokes equations

Résumé

The orthogonality of a vector and its curl, not established within the framework of the continuous medium hypothesis in three dimensions of space, is discussed from the Helmholtz-Hodge theorem where any real vector can be decomposed into a component divergence-free and another curl-free. The cross product of a real vector and its curl shows that the result can be considered as a quaternion whose set forms an algebra over the field of reals. For it to be a vector in three-dimensional space, it is necessary to introduce a compatibility condition where the vector and its curl are orthogonal. The application to the Navier-Stokes equations is straightforward, the non-linear terms expressing inertia in fluid flows can be regarded as rotations in real three-dimensional space. A simple example illustrates the close link between the Helmholtz-Hodge theorem and the profound meaning of inertia.
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hal-04617953 , version 1 (19-06-2024)

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  • HAL Id : hal-04617953 , version 1

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Jean-Paul Caltagirone. The property v⊥∇ × v is true, in general; application to Navier-Stokes equations. 2024. ⟨hal-04617953v1⟩
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