Characterize the numerical and modeling properties of variable-viscosity viscous formulations

Characterize the numerical and modeling properties of the stress-divergence form and the variable-viscosity generalizations of the Laplacian and rotational forms of the incompressible Navier–Stokes equations, which remain incompletely understood and tested.

Background

The paper compares three consistent viscous formulations for incompressible Navier–Stokes flow with spatially variable viscosity: the classical stress-divergence form, the generalized Laplacian form, and the generalized rotational form. The authors emphasize that variable viscosity introduces additional viscosity-gradient terms and changes the associated algebraic structure, stability behavior, and natural outflow boundary conditions.

Although the paper provides a systematic comparison involving implementation, temporal stability, and numerical examples, it presents the broader numerical and modeling behavior of these formulations as not yet fully understood or tested, particularly across applications and flow regimes beyond those considered.

References

Although other forms have recently emerged that correctly generalize the Laplacian and rotational formulations, their numerical and modeling properties are still not fully understood or tested.

— Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity  (2609.10919 - Cisternas et al., 10 Sep 2026) in Section 1, Introduction

Temporal stability: according to our theoretical analysis, the GL and ROT methods might induce a non-physical energy growth for long-term simulations; however, we have not observed such phenomena in numerical experiments, which indicates that our analysis may not be sharp.

— Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity  (2609.10919 - Cisternas et al., 10 Sep 2026) in Section 6, Concluding remarks