On the Helmholtz Theorem and Its Generalization for Multi-Layers

Abstract

The decomposition of a vector field to its curl-free and divergencefree components in terms of a scalar and a vector potential function, which is also considered as the fundamental theorem of vector analysis, is known as the Helmholtz theorem or decomposition. In the literature, it is mentioned that the theorem was previously presented by Stokes, but it is also mentioned that Stokes did not introduce any scalar and vector potentials in his expressions, which causes a contradiction. Therefore, in this article, Stokes's and Helmholtz's representations are examined in detail to reveal and emphasize their differences, similarities and important points. The Helmholtz theorem is obtained for all kinds of spaces by using the theory of distributions in a comprehensive and rigorous manner with detailed explanations, which also leads to a new surface version of the Helmholtz theorem or a new surface decomposition, resulting in the canonical form; hence, it is different than the one suggested previously in terms of two scalar functions. The generalized form of the Helmholtz theorem is also presented by employing the same approach when there is a multi-layer on the surface of discontinuity, which also corresponds to the extension of the theorem to fields with singularities of higher order. ARTICLE HISTORY

The generation and diffusion of vorticity in three-dimensional flows: Lyman's flux

“…The interface circulation also generalises the Biot-Savart integral to interfaces with slip. The Biot-Savart law is an application of Helmholtz's theorem, which has been extended to general interfaces by Kustepeli (2016). For an incompressible velocity field in an infinite domain, their expression reduces to…”

“…The kinematic properties presented here demonstrate that γ generalises the vorticity field across a tangential discontinuity. Using distribution theory, the generalised curl operator is (Kustepeli 2016)…”

“…The interface circulation also generalises the Biot–Savart integral to interfaces with slip. The Biot–Savart law is an application of Helmholtz's theorem, which has been extended to general interfaces by Kustepeli (2016). For an incompressible velocity field in an infinite domain, their expression reduces to In order to compute the induced velocity field, contributions from the interface vortex sheet must be included, to capture a velocity discontinuity on the interface.…”

“…The kinematic properties presented here demonstrate that generalises the vorticity field across a tangential discontinuity. Using distribution theory, the generalised curl operator is (Kustepeli 2016) where denotes the regular value (i.e. the vorticity in each fluid), and is the Dirac delta distribution on .…”

Josephson-Anderson Relation and the Classical D’Alembert Paradox

“…See [48] for an engaging discussion of the history of the Helmholtz decomposition and a useful survey of the mathematical literature, and see also [49]…”