Generalized equations of hydrodynamics in fractional derivatives

TL;DR

This paper develops a unified framework for deriving generalized hydrodynamic equations with fractional derivatives, extending classical models to systems with fractal structures and non-Markovian memory effects.

Contribution

It introduces a method to derive non-Markovian hydrodynamics equations with fractional derivatives using the Liouville equation and NSO method, applicable to fractal media.

Findings

Derived non-Markovian hydrodynamic equations with fractional derivatives.

Obtained fractional Navier-Stokes equations for isothermal processes.

Explored models for frequency-dependent viscosity leading to fractional equations.

Abstract

We present a general approach for obtaining the generalized transport equations with fractional derivatives using the Liouville equation with fractional derivatives for a system of classical particles and the Zubarev non-equilibrium statistical operator (NSO) method within the Gibbs statistics. We obtain the non-Markov equations of hydrodynamics for the non-equilibrium average values of densities of particle number, momentum and energy of liquid in a spatially heterogeneous medium with a fractal structure. For isothermal processes ($\beta=1/k_{B}T =const$), the non-Markov Navier-Stokes equation in fractional derivatives is obtained. We consider models for the frequency dependence of memory function (viscosity), which lead to the Navier-Stokes equations in fractional derivatives in space and time.