Equations of transfer in non-local media

Abstract

The transfer equations and Fourier law analogues have been obtained for three types of non-local media: media with heat memory, spatially non-local media and media with a discrete structure. The conditions are specified under which these equations reduce into each other or to familiar transfer equations, such as the classical parabolic-type transport equation and ‘telegraph’ equation. It is shown that the type of partial differential equations derived from discrete transfer equations is governed by the limiting transition law, i.e. by the relationship between the time, τ, and space, h, scales of the medium internal structure. In the case of the ‘diffusional’ law of limiting transition, when the thermal diffusivity coefficient a = h 2 4τ = const for τ, h → 0, the discrete equations yield parabolic-type partial differential equations, whereas with the ‘wave’ law of limiting transition, when the heat wave speed c = h 2τ = const < ∞ for τ, h → 0, they yield partial differential equations of hyperbolic type.