Mathematical Modelling of Strongly Non-Equilibrium Transfer Processes at Nanoscopic Scale

Abstract

Abstract-Based on the concepts of local non-equilibrium thermodynamics, mathematical models for heat, mass and momentum transfer have been developed with space-time nonlocality taken into consideration. Derivation of transfer differential equations is based on taking into account both specific flows (of heat, mass and momentum) and gradients of corresponding values in diffusion laws by Fourier, Fick and Newton. The research of accurate analytical solutions of the derived models enabled us to find new change patterns of the required parameters for small and ultra small values of time and space variables, as well as for high-speed processes, the change time of which is comparable to relaxation time. Particularly, from the investigation of an accurate analytical solution, it has been found that there is a time delay for Derichlet's boundary condition acceptance, which demonstrates that due to body's resistance to heat penetration, its instant heat-up is impossible, irrespective of any heat exchange with the environment. Therefore, wall heat exchange factor depends not only on heatexchange conditions (medium speed, viscosity, etc.), but also on the body's physical properties. So, firstly, it is a time-variant and, secondly, it can not exceed a certain limit value, established for each particular case. The conducted research of the rod oscillations with relaxation phenomena, taken into consideration, discovered bifurcation oscillations (beat) appearing under the external harmonic load in cases when the difference between the frequency of eigen-oscillations of the rod and constrained load oscillations is insufficient.