On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion

TL;DR

This paper introduces a new classification of incompressible fluids based on their stress-velocity gradient relations and develops a mathematical theory for activated Euler fluids under various boundary conditions.

Contribution

It provides a novel classification scheme for incompressible fluids and establishes a mathematical framework for activated Euler fluids with diverse boundary conditions.

Findings

Classified various incompressible fluids including Euler, Navier-Stokes, and power-law fluids.

Developed a mathematical theory for activated Euler fluids with different boundary conditions.

Analyzed steady and unsteady flows of these fluids in three-dimensional domains.

Abstract

In the first part of the paper we provide a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. The considered class includes Euler fluids, Navier-Stokes fluids, classical power-law fluids as well as stress power-law fluids, and their various generalizations including the fluids that we refer to as activated fluids, namely fluids that behave as an Euler fluid prior activation and behave as a viscous fluid once activation takes place. We also present a classification concerning boundary conditions that are viewed as the constitutive relations on the boundary. In the second part of the paper, we develop a robust mathematical theory for activated Euler fluids associated with different types of the boundary conditions ranging from no-slip to freeslip and include Navier's slip as well as stick-slip. Both steady and unsteady flows of such fluids in three-dimensional domains are analyzed.