Rotational viscosity

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:

${\displaystyle {\frac {\partial J_{ij}}{\partial t}}+{\frac {\partial (v_{k}J_{ij})}{\partial x_{k}}}=\left(x_{j}{\frac {\partial P_{ki}}{\partial x_{k}}}-x_{i}{\frac {\partial P_{kj}}{\partial x_{k}}}\right)+(P_{ji}-P_{ij})}$

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 (${\displaystyle S_{ij}}$):

${\displaystyle J_{ij}=L_{ij}+S_{ij}}$

where the extrinsic angular momentum density is:

${\displaystyle L_{ij}=\rho (x_{i}v_{j}-x_{j}v_{i})}$

and ${\displaystyle \rho }$ is the mass density of the fluid element. The conservation of linear momentum equation is written:

${\displaystyle {\frac {\partial (\rho v_{i})}{\partial t}}+{\frac {\partial (\rho v_{i}v_{k})}{\partial x_{k}}}=-{\frac {\partial P_{ki}}{\partial x_{k}}}}$

and it can be shown that this implies that:

${\displaystyle {\frac {\partial L_{ij}}{\partial t}}+{\frac {\partial (v_{k}L_{ij})}{\partial x_{k}}}=\left(x_{j}{\frac {\partial P_{ki}}{\partial x_{k}}}-x_{i}{\frac {\partial P_{kj}}{\partial x_{k}}}\right)}$

Subtracting this from the equation for the conservation of angular momentum yields:

${\displaystyle {\frac {\partial S_{ij}}{\partial t}}+{\frac {\partial (v_{k}S_{ij})}{\partial x_{k}}}=P_{ji}-P_{ij}}$

Any situation in which this last term is zero will result in the total pressure tensor being symmetric, and the conservation of angular momentum equation will be redundant with the conservation of linear momentum. If, however, the internal rotational degrees of freedom of the particles are coupled to the flow (via the velocity term in the above equation), then the total pressure tensor will not be symmetric, with its antisymmetric component describing the rate of angular momentum exchange between the flow and the particle rotation.

In the linear approximation for this transport of angular momentum, the rate of flow is written:^[1]: p.308

${\displaystyle P_{ij}-P_{ji}=-\eta _{r}\left({\frac {\partial v_{i}}{\partial x_{j}}}-{\frac {\partial v_{j}}{\partial x_{i}}}-2\omega _{ij}\right)}$

where ${\displaystyle \omega _{ij}}$ is the average angular velocity of the rotating particles (as an antisymmetric tensor rather than a pseudovector) and ${\displaystyle \eta _{r}}$ is the rotational viscosity coefficient.