A mapping from Schrodinger equation to Navier–Stokes equations through the product-like fractal geometry, fractal time derivative operator and variable thermal conductivity

Abstract

In this study, the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media is combined with the concept of the fractal time derivative operator. This combination is used to construct a map between the Schrödinger equation which governs the wave function of a quantum–mechanical system and the Navier–Stokes equations, which are the fundamental partial differential equations that describe the flow of incompressible fluids. Several interesting features are found. In particular, for the case of a variable thermal conductivity and special numerical values of the fractal parameters in the theory, it is observed that the entropy density in the semiclassical approximation of any stationary state may be not be constant in time. The decrease in the entropy over time leads in our approach to a decrease in the thermal conductivity with distance, a scenario which takes place in material sciences.