Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity
Abstract
In the Navier--Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional consistency terms. In this context, the present work compares the classical symmetric-gradient diffusion term with more recent variable-viscosity generalizations of the Laplacian and rotational forms. We discuss, analyze and test their differences with respect to implementation, efficiency, numerical stability and outflow boundary conditions. With a focus on time-dependent flows, we consider second-order implicit-explicit (IMEX) temporal discretizations aimed at improving efficiency and numerical stability. Through a rigorous stability analysis, we show how selected explicit treatments can bypass algorithmic nonlinearities without inducing CFL conditions. Our numerical results highlight important differences between the three viscous formulations---especially in the presence of outflow boundaries, for which the generalized Laplacian form proves more suitable in diffusion-dominated regimes. %(as widely known for constant viscosity).
Related
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BibTeX
@misc{cisternas2026stressdivergence,
title = {{Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity}},
author = {Benjamin Cisternas and Douglas R. Q. Pacheco and Felipe Galarce and Ernesto Castillo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10919},
eprint = {2609.10919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10919},
}
RIS
TY - UNPB
TI - Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity
AU - Benjamin Cisternas
AU - Douglas R. Q. Pacheco
AU - Felipe Galarce
AU - Ernesto Castillo
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.10919
UR - https://arxiv.org/abs/2609.10919
AB - In the Navier--Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional consistency terms. In this context, the present work compares the classical symmetric-gradient diffusion term with more recent variable-viscosity generalizations of the Laplacian and rotational forms. We discuss, analyze and test their differences with respect to implementation, efficiency, numerical stability and outflow boundary conditions. With a focus on time-dependent flows, we consider second-order implicit-explicit (IMEX) temporal discretizations aimed at improving efficiency and numerical stability. Through a rigorous stability analysis, we show how selected explicit treatments can bypass algorithmic nonlinearities without inducing CFL conditions. Our numerical results highlight important differences between the three viscous formulations---especially in the presence of outflow boundaries, for which the generalized Laplacian form proves more suitable in diffusion-dominated regimes. %(as widely known for constant viscosity).
ER -