Non-Fourier heat transfer at small scales of time and space

Abstract

Heat transfer at small spatial and temporal scales presents differences compared to larger scales, where the Fourier law applies with very good representativeness. The best-known deviation concerns the behavior of materials with a very small time constant, where the Fourier law leads to a paradox. But there is another difficulty linked to the writing of the flux as derived from a single scalar potential. Like any vector, the heat flux is the sum of two components, one with curl-free and another with divergence-free, formally as a Hodge-Helmholtz decomposition.Discrete mechanics derives an equation of motion based on the conservation of the acceleration on a straight line, where the proper acceleration of the material medium is equal to the sum of the accelerations applied to it. This equation is presented as an alternative to the Navier-Stokes equation for fluid motions, but also reflects the conservation of the heat flux. The formulation proposed on this basis makes it possible to calculate the heat flux and to upgrade the scalar and vector potentials explicitly.