MATHEON-ICM WORKSHOP on Free Boundaries and Materials Modeling - Abstract
Danielewski, Marek
The volume continuity equation can be used to define the material (volume) frame of reference in the multicomponent, compressible systems [1]. The volume velocity is a unique frame of reference for all internal forces and processes, e.g., the mass diffusion, heat flow, etc. It allows the translation of the Newton's discrete mass-point molecular mechanics into continuum mechanics and the use of the Cauchy linear momentum equation of fluid mechanics and Navier-Lamé equation of mechanics of solids. No basic changes are required in the foundations of linear irreversible thermodynamics except recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in every continuum. Proposed modifications of Navier-Lamé and energy conservation equations are self-consistent with the literature for solid-phase continua dating back to Onsager, the classical interdiffusion experiments of Kirkendall, their subsequent interpretation by Darken in terms of diffuse volume transport and generalized formulation [2]. The method allows for self-consistent description of the wide class of processes, from mechano-chemistry to the transport phenomena at Planck scale.
The Planck--Kleinert Crystal hypothesis will be presented for an cubic fcc crystal [3]. The energy, momentum, and mass transport are described using the derived balance equations. The transverse wave is the electromagnetic wave, the collective movement of mass in the crystal is equivalent to the particle (body) and such an approach enables derivation of the Schrödinger equation. The diffusing interstitial Planck particles create a gravity field. The presented method allows for a self-consistent phenomenological description of multiscale phenomena and opens vast number of entirely new possibilities.
[1] M. Danielewski and B. Wierzba, Mechano-chemistry; diffusion in Multi-component compressible mixtures, Physica A, 387, 745--756 (2008).
[2] M. Danielewski, K. Holly and W. Krzyżański, Initial Boundary-Value Problem of Interdiffusion in r - Component ( $rgeq 2$ ) Mixture, in press.
[3] M. Danielewski, The Planck-Kleinert Crystal, Z. Naturforsch. 62a, 564-568 (2007).