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.
Origine | Fichiers produits par l'(les) auteur(s) |
---|---|
licence |