Approaches for Modeling of the Complex Heat and Fluid Flows and Another Physical Processes of Fractal Nature

TL;DR

This paper explores modeling complex heat and fluid flows in fractal media using fractional differential equations, addressing dynamic changes in fractal properties during physical processes like cooling and solidification.

Contribution

It introduces models based on fractional differential equations with variable order to accurately describe evolving fractal systems in thermal hydraulic processes.

Findings

Fractional models effectively describe heat and fluid flows in fractal media.

Dynamic fractal properties require variable-order fractional derivatives.

The approach addresses complex, evolving physical processes in natural and technological systems.

Abstract

The paper deals with the fundamental problem of a modeling of the physical, in particular, thermal hydraulic processes, in various media of fractal structure of the natural, technological and technical systems and devices. The examples of a few complex systems (mainly, the physical processes) are shown as the different fractal structures, e.g. the ones obtained by cooling and solidification of the multiphase mixtures. Some materials reveal the structural features having a significant influence on the behaviors of these materials. The thermal hydraulic processes which are going during a melts cooling, with the further water and steam flow through a porous or a channeled media, are presenting the characteristic examples. The best models of the thermal hydraulic and other physical processes in such fractal media should be the ones based on the fractional differential equations. An order of the fractional derivatives in time and space may change in the process. For example, by a cooling of a melt, with a change of the fractal properties of the system, it is going as follows: first, the vapor, water, melt mixture is interacting, then a formation of the solid structure is performing due to solidification of a melt, and a vapor flows through the permeable structure developed in a process. The dynamically changing fractal systems (dynamic fractals) must be modeled by the corresponding equations changing according to the evolving process. Therefore, it is not surprising that so far some of such tasks have not been properly resolved even in a simplified formulation. There are many similar problems in the modern physics, technology, etc., which are discussed in the paper.