Bifurcation phenomena in turbulent flows: Functional analysis and non-Newtonian fluids

Abstract

This study investigates the behavior of turbulence in non-Newtonian fluids through a rigorous mathematical framework, focusing on the generalized Navier-Stokes equations. We present a weak formulation of these equations, considering the non-Newtonian characteristics of the fluid, and explore their implications in theoretical and numerical contexts. The study employs the Galerkin approximation method to solve the equations in irregular domains, highlighting the challenges posed by non-Newtonian fluids and the complexity of turbulence. A key result of this work is the formulation of a new theorem on the existence and uniqueness of weak solutions for a specific class of non-Newtonian fluids under given conditions. The theorem is derived using functional analysis techniques, including Sobolev spaces, and provides a solid foundation for the numerical methods used in the analysis. Through this theoretical work, we demonstrate the onset of turbulence in non-Newtonian fluids and the critical parameters governing the transition. The study also discusses bifurcation phenomena and energy balance equations, offering new insights into the mechanisms of turbulence in these complex fluids. This research contributes to the understanding of fluid dynamics in non-Newtonian contexts, providing a theoretical framework that can be extended to various practical applications, such as in industrial processes and environmental modeling.