Reynolds transport theorem for smooth deformations of currents on manifolds

Abstract

The Reynolds transport theorem for the rate of change of an integral over an evolving domain is generalized. For a manifold B, a differentiable motion m of B in the manifold S, an r-current T in B, and the sequence of images m(t) Y T of the current under the motion, we consider the rate of change of the action of the images on a smooth r-form in S. The essence of the resulting computations is that the derivative operator is represented by the dual of the Lie derivative operation on smooth forms.

Differential Geometry and Continuum Mechanics

“…Noting that material structure and the associated defects are viewed here as intrinsic to a body and unrelated to the kinematics of the body in space, in the following two subsections we consider the motion of material structure and defects resulting from a family of diffeomorphisms of the body. (See [FS13] for another application of the same mathematical notions.) In other words, the material structure, as represented by a smooth form and its exterior derivative or a de Rham current and its boundary, are carried with material diffeomorphisms.…”

New Conservation Laws Based on Generalised Reynolds Transport Theorems

“…By the analysis of sources, sinks and fluxes of the conserved quantity in a differential fluid element, the theorem can also be connected to corresponding differential equations for each phenomenon [2,3,4,5,6,7]. During the past half-century, the Reynolds transport theorem has been extended to domains with discontinuities [8,9], moving and smoothly-deforming control volumes [2,3,9], irregular and rough domains [10,11,12], two-dimensional domains [13,14,15,16,17], and differentiable manifolds or chains described by patchworks of local coordinate systems [18,19,20], the latter expressed in the formalism of exterior calculus. These formulations are restricted to one-parameter (temporal) mappings of the conserved quantity in volumetric space, induced by a velocity vector field.…”

A transport theorem for nonconvecting open sets on an embedded manifold

“…The transport theorems mentioned so far hold for evolving smooth domains, but results for irregular domains have been obtained. Falach and Segev established generalized transport theorems by modeling the domain of integration either as a de Rham current [9] or a flat chain [10] in the spirit of Federer's geometric measure theory [11]. In both cases, the domain was convecting according to a given flow map, though in [10] this map was only required to be Lipschitz.…”