The evolution of the molecular relaxation parameters described by the Cole–Cole model at the isotropic-nematic phase transition*

This paper presents the dielectric properties of the isotropic liquid and nematic phase at the phase transition. One strong molecular relaxation is observed in both phases. It is interpreted as related to the relaxation around a short molecular axis, due to the fact that molecules possess a strong longitudinal dipole moment. In the isotropic liquid the relaxation is described by the Debye model, while after entering the nematic phase (at cooling) relaxation becomes described by the Cole–Cole model. The distribution parameter of the Cole–Cole model changes from 0.05 (10 degrees above the temperature of the Iso-N transition) to 0.09 (exactly at the phase transition Iso-N), and finally, it reaches 0.35 (10 degrees below the Iso-N transition). Additionally, we observe that ion contribution to the dielectric response is not influenced by the phase transition. All relaxation parameters are discussed within the context of the phase transition phenomena.     

Applications of a Cole–Hopf transform to the 3D Navier–Stokes equations

The Navier–Stokes equations written in the vector potential can be recast as non-linear Schrödinger equations at imaginary times, i.e. heat equations with a potential term, using the Cole–Hopf transform. On this basis, we study two kinds of Navier–Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection takes place, whereas the signature of the non-linear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magneto–hydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions.