A Note On Determining Projections for Non-Homogeneous Incompressible Fluids

TL;DR

This paper investigates the determination of degrees of freedom in non-homogeneous incompressible fluids with variable viscosity, extending existing frameworks to complex spatial and temporal viscosity variations.

Contribution

It applies the determining projection framework to fluids with spatially and temporally varying viscosity, providing bounds on the number of determining functionals for these complex cases.

Findings

Bounds established for time-varying viscosity similar to constant viscosity cases.

Exploration of preliminary ideas for analyzing space-varying viscosity.

Extension of determining projection methods to more complex fluid models.

Abstract

In this note, we consider a viscous incompressible fluid in a finite domain in both two and three dimensions, and examine the question of determining degrees of freedom (projections, functionals, and nodes). Our particular interest is the case of non-constant viscosity, representing either a fluid with viscosity that changes over time (such as an oil that loses viscosity as it degrades), or a fluid with viscosity varying spatially (as in the case of two-phase or multi-phase fluid models). Our goal is to apply the determining projection framework developed by the second author in previous work for weak solutions to the Navier-Stokes equations, in order to establish bounds on the number of determining functionals for this case, or equivalently, the dimension of a determining set, based on the approximation properties of an underlying determining projection. The results for the case of time-varying viscosity mirror those for weak solutions established in earlier work for constant viscosity. The case of space-varying viscosity, treated within a single-fluid Navier-Stokes model, is quite challenging to analyze, but we explore some preliminary ideas for understanding this case.