Effects of flow on the dynamics of a ferromagnetic nematic liquid crystal

We investigate the effects of flow on the dynamics of ferromagnetic nematic liquid crystals. As a model, we study the coupled dynamics of the magnetization, $\mathbf{M}$, the director field, $\mathbf{n}$, associated with the liquid crystalline orientational order, and the velocity field, $\mathbf{v}$. We evaluate how simple shear flow in a ferromagnetic nematic is modified in the presence of small external magnetic fields, and we make experimentally testable predictions for the resulting effective shear viscosity: an increase by a factor of 2 in a magnetic field of about 20 mT. Flow alignment, a characteristic feature of classical uniaxial nematic liquid crystals, is analyzed for ferromagnetic nematics for the two cases of magnetization in or perpendicular to the shear plane. In the former case, we find that small in-plane magnetic fields are sufficient to suppress tumbling and thus that the boundary between flow alignment and tumbling can be controlled easily. In the latter case, we furthermore find a possibility of flow alignment in a regime for which one obtains tumbling for the pure nematic component. We derive the analogs of the three Miesowicz viscosities well-known from usual nematic liquid crystals, corresponding to nine different configurations. Combinations of these can be used to determine several dynamic coefficients experimentally.

Figure 1

A schematic representation of the staggered grid used to solve Eqs. (3, 4, 5). The director field, the magnetization field, and the (quasi)currents are defined on the points (black circles), while the velocity field is defined on the crosses located between the points. There are two crosses outside of the physical space, which are present in order to satisfy the boundary condition for the velocity field.

Figure 2

Profiles of $v_x/v_{0x}$, $M_z/M_0$, and $n_z$ at a shear rate ${\mathrm{\Gamma }}_x=1{\mathrm{s}}^{-1}$ with (a) ${\mu }_{0}H=10\mathrm{mT}$ and (b) ${\mu }_{0}H=-10\mathrm{mT}$.

Figure 3

The normalized $x$ component of the velocity field $v_x/v_{0x}$ for different values of the applied magnetic field at a shear rate ${\mathrm{\Gamma }}_x=1{\mathrm{s}}^{-1}$.

Figure 4

The behavior of the effective viscosity for (a) small and (b) large values of the applied magnetic field at oppositely equal shear rates. The dashed lines represent two of the Miesowicz viscosities (${\eta }_{xx}$ and ${\eta }_{zz}$), defined in Sec. 4.

Figure 5

Profiles of $v_x/v_{0x}$, $M_z/M_0$, and $n_z$ at a shear rate ${\mathrm{\Gamma }}_x=1{\mathrm{s}}^{-1}$ with (a) ${\mu }_{0}H=50$ mT and (b) ${\mu }_{0}H=1$ T. The boundary layers for (a) the lower magnetic field, ${\xi }^l$, and (b) the higher magnetic field, ${\xi }^q$, can be estimated from Eqs. (31) and (32), respectively.

Figure 6

Miesowicz viscosities in a usual nematic liquid crystal. The director field is indicated in orange.

Figure 7

Analogs of the Miesowicz viscosities in a ferromagnetic nematic liquid crystal, when the director (orange) is along the (a) $x$, (b) $y$, and (c) $z$ axis. The magnetization is shown in red.

Figure 8

The theoretical dependence of the magnetization ${\psi }^{+}$ (blue curve) and director ${\theta }^{+}$ (red dashed curve) angles in degrees as functions of the dimensionless ratio of the characteristic magnetic shear rate [Eq. (62)] and the applied shear rate, ${\mathrm{\Gamma }}_x^H/{\mathrm{\Gamma }}_x$.

Figure 9

The theoretical time dependence of $n_z$ in the middle of the cell, for the cases with and without inclusion of flow, ${\mu }_{0}H=5\mathrm{mT}$. In the former case, only the flow alignment tensor ${\lambda }_{ijk}$ and the viscous tensor ${\nu }_{ijkl}^D$ are taken into account in the dynamics.

Figure 10

The theoretical time dependence of the (a) $z$ and (b) $y$ component of the director field at ${\mu }_{0}H=5$ mT for two different values of ${\lambda }_3^D$.

Figure 11

Backflow profiles in terms of the Ericksen numbers ${\mathrm{Er}}_x$ and ${\mathrm{Er}}_y$, Eq. (74), at four moments of time; ${\mu }_{0}H=5\mathrm{mT}$, ${\lambda }_3^D{M}_0=0.7$.

Figure 12

The $x$ and $y$ components of the stress tensor divergence (a) ${\sigma }_{xz}^{\prime }=\partial {\sigma }_{xz}/\partial z$ and (b) ${\sigma }_{yz}^{\prime }=\partial {\sigma }_{yz}/\partial z$, at $z=d/2$, in units of characteristic divergence of the elastic stress $K/d^3$ for each of the three different backflow-driving stress tensor contributions explained in the text.