  In this work we study different Implicit-Explicit (IMEX) schemes for
incompressible flow problems with variable viscosity. Unlike most previous work
on IMEX schemes, which focuses on the convective part, we here focus on
treating parts of the diffusive term explicitly to reduce the coupling between
the velocity components. We present different, both monolithic and
fractional-step, IMEX alternatives for the variable-viscosity Navier--Stokes
system, analysing their theoretical and algorithmic properties. Stability
results are proven for all the methods presented, with all these results being
unconditional, except for one of the discretisations using a fractional-step
scheme, where a CFL condition (in terms of the problem data) is required for
showing stability. Our analysis is supported by a series of numerical
experiments.
