Rotational viscosity

Viscosity is usually described as the property of a fluid which determines the rate at which local momentum differences are equilibrated. Rotational viscosity is a property of a fluid which determines the rate at which local angular momentum differences are equilibrated. It is only appreciable if there are rotational degrees of freedom for the fluid particles. In the classical case, by the equipartition theorem, at equilibrium, if particle collisions can transfer angular momentum as well as linear momentum, then these degrees of freedom will have the same average energy. If there is a lack of equilibrium between these degrees of freedom, then the rate of equilibration will be determined by the rotational viscosity coefficient.

The angular momentum density of a fluid element is written either as an antisymmetric tensor (${\displaystyle J_{ij}}$) or, equivalently, as a pseudovector. As a tensor, the equation for the conservation of angular momentum for a simple fluid with no external forces is written:

where ${\displaystyle v_{i}}$ is the fluid velocity and ${\displaystyle P_{ij}}$ is the total pressure tensor (or, equivalently, the negative of the total stress tensor). Note that the Einstein summation convention is used, where summation is assumed over pairs of matched indices. The angular momentum of a fluid element can be separated into extrinsic angular momentum density due to the flow (${\displaystyle L_{ij}}$) and intrinsic angular momentum density due to the rotation of the fluid particles about their center of mass (${\displaystyle S_{ij}}$):

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