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Null-divergence nature of the odd viscous stress for an incompressible liquid
Phys. Rev. Fluids 8, 014104 – Published 26 January, 2023
DOI: https://doi.org/10.1103/PhysRevFluids.8.014104
Abstract
What constitutes the “extra stress” exerted by a liquid endowed with odd viscosity [for instance, the tensile or compressive stress imparted on a cylinder rotating in a viscous liquid as was calculated by Avron, J. Stat. Phys. 92, 543 (1998)], requires clarification. The nature of this extra stress depends on the character of the boundary conditions. We show that when only velocities are prescribed on the boundaries, it is not the full anomalous (odd) stress tensor that generates this “extra” stress but only its null divergence part, that is, the part of the odd stress whose divergence vanishes identically. We demonstrate this fine point by calculating the extra stress for a viscous liquid between concentric rotating cylinders and the corrections to the viscosity coefficient in liquid suspensions. Similar conclusions can be reached for a liquid with periodic boundary conditions. On the other hand, when stresses are prescribed on some part of the boundary, the state of stress due to odd viscosity becomes ambiguous. In general, it is the whole Cauchy stress tensor that now depends on odd viscosity. There are exceptions, however, to this rule which we briefly discuss. Finally, the decomposition of the Cauchy stress tensor into a part that is operative in the Navier-Stokes equations, and into a null part (whose divergence vanishes identically) gives rise to new fluid flow behavior and recovers previous results from the literature as special cases.
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