Challenges in Classical and Non-Newtonian Fluid Mechanics A Mathematical Perspective

Abstract

Challenges in classical and non-Newtonian fluid mechanics are abundant, particularly when viewed from a mathematical perspective. Classical fluid mechanics, based on Newtonian principles, deals with the behavior of simple fluids like water and air. However, challenges arise when dealing with complex fluids that do not follow Newton's linear relationship between stress and strain rate.One significant challenge is the mathematical description of non-Newtonian fluids, which often require sophisticated constitutive equations to capture their behavior accurately. These equations must account for shear-thinning, shear-thickening, viscoelasticity, and other complex rheological properties. Finding appropriate mathematical models that can predict these behaviors under various conditions is a formidable task.Additionally, solving the governing equations of fluid mechanics, the Navier-Stokes equations, poses significant mathematical challenges, especially in the context of non-Newtonian fluids. These equations are notoriously difficult to solve analytically and often require numerical methods, which can be computationally intensive. Understanding the stability and turbulence of non-Newtonian flows is another mathematical challenge. Turbulent flows of complex fluids exhibit unique characteristics that differ from those of Newtonian fluids, requiring advanced mathematical tools to analyze and predict their behaviour.