  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.
