  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
multi-component 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.
