Angular-momentum conservation in discretization of the Navier-Stokes equation for viscous fluids

Although the Navier-Stokes equation (NSE) is derived under angular-momentum conservation (AMC), numerical simulation methods often lack it. Here we reveal that AMC violations result from implementation of the degenerated viscous terms of NSE. To maintain AMC, these degenerated terms must be separately integrated in accordance with their stress origins. As observed in particle-based hydrodynamics methods, the violation causes artificial rotations in multicomponent fluids with different viscosities. At the interface between two fluids or with a mobile solid object, AMC must be satisfied, whereas AMC can be neglected in bulk fluids. We also clarify that the condition for constant fluid rotation as a rigid body in a container rotating at a constant speed is not the AMC of the stresses, but the invariance of the viscous forces under a global rotation. To confirm our theory, we simulated the circular laminar flows of single- and binary-component fluids using two-dimensional Lagrangian finite-volume methods. The results show excellent agreement with the analytical predictions for fluids with and without AMC.