Isotropic fluids only have two viscosities, intervening in shear and extensional deformations. For anisotropic media, such as nematic liquid crystals, more coefficients are needed, as shown by Mięsowicz in the late '30s: three shear viscosities, labeled \(\eta_1\) to \(\eta_3\), a fourth one \(\eta_{12}\) introduced later by Helfrich and a rotational viscosity, \(\gamma_1\). We do not worry here about extensional deformations.
The whole topic was put on a solid theoretical basis in the '60s by Leslie and Ericksen [brief and clear presentation here] who introduced six coefficients (\(\alpha_1\) to \(\alpha_6\)), only five of which are independent. As one can expect from dimensional analysis, the two sets of viscosities, \(\left \lbrace \eta_i, \gamma _1 \right \rbrace\) and \(\left \lbrace \alpha_j \right \rbrace\) are linearly related.
I only recently realized, while discussing with my former PhD advisor, that the difference between the two definitions is deeper than an arbitrary linear transformation. Mięsowicz had in mind clear experimental configurations, defined by the relative orientation of director, velocity and velocity gradient, while Leslie and Ericksen adopt a more formal approach, based on generalized hydrodynamics, as in the paper of Martin, Parodi and Pershan.
The twist (so to speak) is that the theoretical approach gives a clearer view of the various modes and the constraints on the coefficients, while the Mięsowicz configurations are very difficult to achieve in practice, precisely due to the coupling between flow and director orientation.
The twist (so to speak) is that the theoretical approach gives a clearer view of the various modes and the constraints on the coefficients, while the Mięsowicz configurations are very difficult to achieve in practice, precisely due to the coupling between flow and director orientation.
