The Clebsch transformation and its capabilities towards fluid and solid mechanics

Abstract

AbstractIn fluid dynamics, Clebsch made use of the representation for the velocity field $\vec u$ in terms of three potentials Φ, α, β in order to construct a first integral of the equations of motion in case of an inviscid flow with vortices. Apart from this, he received a self‐adjoint form of the equations allowing for deriving them from a variational formulation. In latter times the Clebsch transformation has been applied to different physical problems, for instance to baroclinic flow, Maxwell equations in classical electrodynamics [1], in Magnetohydrodynamics and even quantum theory within the context of a quantization of vortex tubes. Viscous flow, however, has not yet been formulated in terms of Clebsch variables to our best knowledge. It is the aim of this paper to demonstrate how Clebsch variables can be applied to viscous flow on the one hand, leading to a first integral of Navier‐Stokes equations as a first example. As a second example, solid mechanics is considered: by making use of an analogy between vortices in fluid flow on the one hand and dislocations in crystals on the other hand, a dynamic theory of dislocations can be established by using a certain modification of the Clebsch transformation. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)