For the incompressible Navier--Stokes system with variable viscosity and density, we propose and analyse an IMEX framework treating the convective and diffusive terms semi-implicitly. This extends to variable density and second order in time some methods previously analysed for variable viscosity and constant density. We present three new schemes, one of which is a fully decoupled fractional-step method. All of them share the novelty that the viscous term is treated in an implicit-explicit (IMEX) fashion which, if combined with an explicit treatment of the pressure, decouples the velocity components. Unconditional temporal stability is proved for all three variants. Furthermore, the system to solve at each time step is linear, thus avoiding the costly solution of nonlinear problems even if the viscosity follows a non-Newtonian rheological law. Our presentation is restricted to the semi-discrete case, only considering the time discretisation. In this way, the results herein can be applied to any spatial discretisation. We validate our theory through numerical experiments considering finite element methods in space. The tests range from simple manufactured solutions to complex two-phase viscoplastic flows.
Espinoza-Contreras et al. (Wed,) studied this question.