Tumbling instability in a shearing nematic liquid crystal: Analysis of broadband dielectric results and theoretical treatment

The rotational diffusion coefficient $D_{\perp },$ the rotational viscosity coefficients ${\gamma }_i$ $\text{(i=1,2),}$ and the orientational relaxation time ${\tau }_{00}^1$ are investigated. ${\gamma }_i$ are calculated by a combination of an existing statistical–mechanical approach and of broadband experimental results of the complex permittivity for the polar liquid crystal 4-n-octyl-4^′-cyanobiphenyl (8CB). The flow dynamic behavior of the nematic liquid crystal in a shear flow, as well as the transition from a flow alignment regime to a tumbling instability are also investigated in 8CB near and far away of a charged surface. The relaxation of the alignment angle $\mathrm{\theta (\tau )}$ to its equilibrium ${\theta }_{\mathrm{eq}}$ in a layer up to $\text{∼1.5}$ μm is calculated using the Ericksen–Leslie theory for the case of planar alignment of 8CB at a charged indium tin oxide-coated glass plate and for the temperature range in which 8CB exhibits a nematic phase. In accordance with earlier Couette flow experiments, our calculations show a tumbling flow in the bulk nematic phase. The nature of the hydrodynamic instability under shear flow, in the vicinity, e.g., at a few tens of mK from a nematic–smectic A phase transition ${\mathrm{(T}}_{\mathrm{NA}})$ in the nematic phase, and far from $T_{\mathrm{NA}}$ is also discussed for 8CB.