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Thermomechanically Excited Vortical Flow in a Hybrid-Oriented Nematic Channel

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Abstract

In this paper, we described numerically several scenarios of formation of vortex flows (VF) in microsized hybrid-oriented liquid crystal (HOLC) channels with orientation defects using a nonlinear generalization of the classical Ericksen–Leslie theory that allows taking into account termomechanical contribution, both in the expression for the shear stress and in the entropy balance equation. An analysis of the numerical results showed that there are two or one vortices in the HOLC channel although two vortices directed towards each other are generated at the initial stage of the VT formation Thermomechanically Excited Vortical Flow.

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FUNDING

This work was supported by the Ministry of Education and Science of Russian Federation, projects 3.11888.2018/11.12 and 3.9585.2017/8.9.

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Correspondence to A. V. Zakharov.

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Translated by A. Ivanov

APPENDIX

APPENDIX

The torque balance equation in the LC phase has the following form:

$${{{\mathbf{T}}}_{{{\text{el}}}}} + {{{\mathbf{T}}}_{{{\text{vis}}}}} + {{{\mathbf{T}}}_{{{\text{tm}}}}} = 0,$$

where Tel = \(\frac{{\delta {{\mathcal{W}}_{{{\text{el}}}}}}}{{\delta {\mathbf{\hat {n}}}}} \times {\mathbf{\hat {n}}}\) is elastic, Tvis = \(\frac{{\delta {{\mathcal{R}}^{{{\text{vis}}}}}}}{{\delta {{{{\mathbf{\hat {n}}}}}_{t}}}} \times {\mathbf{\hat {n}}}\) is viscous, and Ttm = \(\frac{{\delta {{\mathcal{R}}^{{{\text{tm}}}}}}}{{\delta {{{{\mathbf{\hat {n}}}}}_{t}}}} \times {\mathbf{\hat {n}}}\) is the thermomechanical contribution to the torque balance acting on the unit volume of the LC phase, respectively. Here, \({{{\mathbf{\hat {n}}}}_{t}}\)\(\frac{{d{\mathbf{\hat {n}}}}}{{dt}}\) is the material derivative of the vector \({\mathbf{\hat {n}}}\) = \({{n}_{x}}{\mathbf{\hat {i}}}\) + \({{n}_{z}}{\mathbf{\hat {k}}}\).

The linear balance momentum equation in the LC phase can be written in the form

$$\rho \frac{{d{\mathbf{v}}}}{{dt}} = \nabla \cdot \sigma ,$$

where \(\frac{{d{v}}}{{dt}} = \frac{{\partial {v}}}{{\partial t}} + u{{{v}}_{{,x}}} + w{{{v}}_{{,z}}}\), σ = σel + σvis + σtm\(\mathcal{P}\mathcal{E}\) is the full expression for the stress tensor (ST) consisting of the elastic σel = \( - \frac{{\partial {{\mathcal{W}}_{{{\text{el}}}}}}}{{\partial \nabla {\mathbf{\hat {n}}}}} \cdot {{(\nabla {\mathbf{\hat {n}}})}^{T}}\), viscous σvis = \(\frac{{\delta {{\mathcal{R}}^{{{\text{vis}}}}}}}{{\delta \nabla {\mathbf{v}}}}\) and thermomechanical σtm = \(\frac{{\delta {{R}^{{{\text{tm}}}}}}}{{\delta \nabla {\mathbf{v}}}}\) contributions to ST, respectively. Here, \(\mathcal{R}\) = \({{\mathcal{R}}^{{{\text{vis}}}}}\) + \({{\mathcal{R}}^{{{\text{tm}}}}}\) + \({{\mathcal{R}}^{{{\text{th}}}}}\) is the full Rayleigh dissipation function, \({{\mathcal{W}}_{{{\text{el}}}}}\) = \(\frac{1}{2}\)[K1 × (∇ · \({\mathbf{\hat {n}}}\))2 + K3(\({\mathbf{\hat {n}}}\) × ∇ × \({\mathbf{\hat {n}}}\))2] is the elastic energy density, K1 and K3 are the splay and bend elastic constants, \(\mathcal{P}\) is the hydrostatic pressure in the LC system, and \(\mathcal{E}\) is the unit tensor while

$${{\mathcal{R}}^{{{\text{vis}}}}} = \alpha {{({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}})}^{2}} + {{\gamma }_{1}}{{({{{\mathbf{\hat {n}}}}_{t}} - {{{\mathbf{D}}}_{a}} \cdot {\mathbf{\hat {n}}})}^{2}}$$
$$ + \;2{{\gamma }_{2}}({{{\mathbf{\hat {n}}}}_{t}} - {{{\mathbf{D}}}_{a}} \cdot {\mathbf{\hat {n}}}) \cdot ({{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}} - ({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}}){\mathbf{\hat {n}}})$$
$$ + \;{{\alpha }_{4}}{{{\mathbf{D}}}_{s}}{{{\mathbf{D}}}_{s}} + ({{\alpha }_{5}} + {{\alpha }_{6}})({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}})$$

is viscous, but

$$\frac{1}{\xi }{{\mathcal{R}}^{{{\text{tm}}}}} = ({\mathbf{\hat {n}}} \cdot \nabla T){{{\mathbf{D}}}_{s}}:{\mathbf{M}} + \nabla T \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{M}} \cdot {\mathbf{\hat {n}}} + ({\mathbf{\hat {n}}} \cdot \nabla T)$$
$$ \times \;({{{\mathbf{\hat {n}}}}_{t}} - {{{\mathbf{D}}}_{a}} \cdot {\mathbf{\hat {n}}} - 3{{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}} + 3({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}}){\mathbf{\hat {n}}}) \cdot {\mathbf{M}} \cdot {\mathbf{\hat {n}}}$$
$$ + \;{\mathbf{\hat {n}}}{{(\nabla {\mathbf{v}})}^{T}} \cdot {\mathbf{M}} \cdot \nabla T + \frac{1}{2}({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}})\nabla T \cdot {\mathbf{M}} \cdot {\mathbf{\hat {n}}}$$
$$ + \;{{{\mathbf{\hat {n}}}}_{t}} \cdot {\mathbf{M}} \cdot \nabla T + \frac{1}{2}{{\mathcal{M}}_{0}}\nabla T \cdot \nabla {\mathbf{v}} \cdot {\mathbf{\hat {n}}}$$
$$ + \;({\mathbf{\hat {n}}} \cdot \nabla T){{\mathcal{M}}_{0}}({\mathbf{\hat {n}}} \cdot {{{\mathbf{D}}}_{s}} \cdot {\mathbf{\hat {n}}}) + \frac{1}{2}{{\mathcal{M}}_{0}}{{{\mathbf{\hat {n}}}}_{t}} \cdot \nabla T$$

and

$${{\mathcal{R}}^{{{\text{th}}}}} = \frac{1}{T}{{({{\lambda }_{{||}}}({\mathbf{\hat {n}}} \cdot \nabla T))}^{2}} + {{\lambda }_{ \bot }}(\nabla T - {\mathbf{\hat {n}}}{{({\mathbf{\hat {n}}} \cdot \nabla T)}^{2}})$$

are thermomechanical and thermal contributions to the full expression for the Rayleigh function \(\mathcal{R}\), respectively.

Here, α1–α6 are the Leslie viscosity coefficients, γ1(T) and γ2(T) are the rotational viscosity coefficients of the LC system, ξ is the thermomechanical constant, and λ|| and λ are the thermal conductivity coefficients of the LC system along and across the direction of the director \({\mathbf{\hat {n}}}\), respectively. The tensors Ds = \(\frac{1}{2}\)[∇v + (∇v)T] and Da = \(\frac{1}{2}\)[∇v – (∇v)T] are symmetric and asymmetric contributions to the strain rate tensor, M = \(\frac{1}{2}\)[∇\({\mathbf{\hat {n}}}\) + (∇\({\mathbf{\hat {n}}}\))T], and \({{\mathcal{M}}_{0}}\) = ∇ · \({\mathbf{\hat {n}}}\) is the scalar invariant of the tensor M.

The heat conduction equation describing the change in the temperature field T(x, z, t) under the effect of the heat flux q through the lower boundary of the HOLC channel has the following form:

$$\rho {{C}_{P}}\frac{{dT}}{{dt}} = - \nabla \cdot {\mathbf{Q}},$$

where Q = \( - T\frac{{\delta \mathcal{R}}}{{\delta \nabla T}}\) is the heat flux in the LC system, and CP is the heat capacity coefficient, respectively.

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Zakharov, A.V. Thermomechanically Excited Vortical Flow in a Hybrid-Oriented Nematic Channel. Phys. Solid State 61, 1136–1143 (2019). https://doi.org/10.1134/S1063783419060295

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