# Heat Driven Flows in Microsized Nematic Volumes: Computational Studies and Analysis

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Heat Driven Nematic Flow in Rectangular Microfluidic Channel

#### 2.1. Formulation of the Balance of the Momentum and Torque Equations and Conductivity Equation for Nematic Fluids Confined in Rectangular Channel

#### 2.2. Orientational Relaxation of the Director, Velocity, and Temperature Fields in Rectangular Microsized Nematic Channels

- A. Slow heating mode

^{2}), caused by the laser irradiation focused on the upper boundary of the HAN channel, at different times, from ${\tau}_{1}=0.00042\phantom{\rule{3.33333pt}{0ex}}(\sim $$54\phantom{\rule{3.33333pt}{0ex}}$μs) [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], respectively, is shown in Figure 2a. Figure 2b shows the same as in Figure 2a, but at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], corresponding to the cooling mode. Here, the following times were used ${\tau}_{i+1}={2}^{i}{\tau}_{1}$, $i=1,\cdots ,5$, whose values increase from curve (1) to curve (6), and ${\tau}_{1}=0.00042$, and times ${\tau}_{i+1}={2}^{i-6}{\tau}_{7}$, $i=7,\cdots ,10$, whose values increase from curve (7) to curve (11), and ${\tau}_{7}=0.014$, respectively, whereas the time ${\tau}_{R}$ denotes the relaxation time of the LC system, and ${\tau}_{in}\sim {\tau}_{6}\sim 0.0134$ (∼$1.9$ ms) is the duration of the energy injection into the HAN channel across the upper boundary by the infrared laser with the power ${P}_{0}=14.3$ mW. The distance z dependence of the polar angle $\theta (\tau ,x=0.5,z)$ is characterized by the monotonic increase $\theta $ from $\theta (x=0.5,z=0)=0$, on the lower cooler boundary of the HAN channel, to $\theta (x=0.5,z=1)=\frac{\pi}{2}$, on the upper warmer boundary, respectively. The evolution of the polar angle $\theta (\tau ,x,z=0.97)$ to its equilibrium distribution ${\theta}_{eq}(x,z=0.97)=\theta (\tau ={\tau}_{6},x,z=0.97)$ along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary of the channel $(z=0.97)$, during the slow heating mode at different times, from ${\tau}_{1}=0.00042\phantom{\rule{3.33333pt}{0ex}}(\sim $$54\phantom{\rule{3.33333pt}{0ex}}$μs) [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], is shown in Figure 3a, whereas the cooling mode, at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], is shown in Figure 3b, respectively. Both the slow heating (see Figure 3a) and cooling (see Figure 3b) modes are characterized by nonsymmetric profile of $\theta (\tau ,x,z=0.97)$ with respect to the middle part $(x=0.5)$ of the HAN channel, which is caused by nonsymmetric effect of the velocity field components $u(\tau ,x,z)$ (see Figure 4a,b and Figure 5a,b) and $w(\tau ,x,z)$ (see Figure 6a,b and Figure 7a,b), respectively.

^{2}), caused by the laser beam, at different times, from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], respectively, is shown in Figure 8a.

- B. Fast heating mode

^{2}) (which is 8 times greater than Q in the first case A), caused by the laser beam focused on the upper boundary, are shown in Figure 10a,b, respectively.

^{2}), caused by the laser beam at different times, from ${\tau}_{1}=0.00004\phantom{\rule{3.33333pt}{0ex}}(\sim $$6\phantom{\rule{3.33333pt}{0ex}}$μs) [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively, corresponding to the fast heating mode, is shown in Figure 11a.

^{2}) at different times, from ${\tau}_{1}=0.00004\phantom{\rule{3.33333pt}{0ex}}(\sim $$6\phantom{\rule{3.33333pt}{0ex}}$μs) [curve (1)] to ${\tau}_{5}=0.0006\phantom{\rule{3.33333pt}{0ex}}(\sim $$90\phantom{\rule{3.33333pt}{0ex}}$μs) [curve (5)], respectively, corresponding to the fast heating mode is shown in Figure 13a.

- (i)
- A fast time scale, with the duration of the dimensional heat flux $q\sim 2.95\times {10}^{-2}$ mW/μm
^{2}over time ${t}_{in}\sim 90\phantom{\rule{3.33333pt}{0ex}}$μs, and the laser power in ${P}_{0}=115$ mW, and - (ii)
- A slow time scale, with the duration of the dimensional heat flux $q\sim 3.7\times {10}^{-3}$ mW/μm
^{2}over time ${t}_{in}\sim 1.9$ ms, and the laser power in ${P}_{0}=14.3$ mW, respectively [21].

#### 2.3. Laser Excited Vortical Flow in Microsized Nematic Channels

^{2}). In general, under the above conditions, the picture of warming is such that only a small part of the nematic volume is involved in the heating process, while a large part of the volume of the fluid were not heated. It should be noted that in both cases the area of the greatest heating was shifted in the direction in which the heat flux was directed due to laser radiation.

#### 2.4. Laser Excited Motion of Nematics Confined in Microsized Channel with a Free Surface

- A. Slow heating mode when a laser beam is focused in the interior of the HAN channel

- B. Fast heating mode when a laser beam is focused inside of an HAN channel

#### 2.5. How the Depth of Laser Radiation Focus Affects the Nature of Hydrodynamic Flows in Nematic Channel

## 3. Heat Driven Nematic Flow in Cylindrical Microfluidic Channel

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The coordinate system used for theoretical analysis. The x-axis is taken as being parallel to the director directions on the upper surface, $\theta (z,t)$ is the angle between the director $\widehat{\mathbf{n}}$ and the unit vector $\widehat{\mathbf{k}}$, respectively. Both the heat flux $\mathbf{q}$ and the unit vector $\widehat{\mathbf{k}}$ are directed normal to the horizontal boundaries of the liquid crystal (LC) channel.

**Figure 2.**(

**a**) The distance z dependence of the polar angle $\theta (\tau ,x=0.5,z)$ (in rad) during its evolution to the equilibrium distribution ${\theta}_{eq}(x=0.5,z)=\theta (\tau ={\tau}_{6},x=0.5,z)$ in the middle part of the hybrid aligned nematic (HAN) channel [21], corresponding to the slow heating mode is given starting from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], respectively. (

**b**) The same as in (

**a**), but at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], corresponding to the cooling mode.

**Figure 3.**(

**a**) The distance x dependence of the polar angle $\theta (\tau ,x,z=0.97)$ (in rad) during its evolution to the equilibrium distribution ${\theta}_{eq}(x,z=0.97)=\theta (\tau ={\tau}_{6},x,z=0.97)$ [21], along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary $(z=0.97)$, corresponding to the slow heating mode. The different times are given starting from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], respectively. (

**b**) The same as in (

**a**), but for the cooling mode at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], respectively.

**Figure 4.**(

**a**) The distance z dependence of the horizontal component $u(\tau ,x=0.5,z)$ of the velocity $\mathbf{v}$ during its evolution to the equilibrium distribution ${u}_{eq}(x=0.5,z)=u(\tau ={\tau}_{6},x=0.5,z)$ across the HAN channel [21]. The different times during the slow heating mode are given starting from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}=0.0134$ [curve (6)], respectively. (

**b**) The same as in (

**a**), but for the cooling mode at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], respectively.

**Figure 5.**The same as in Figure 4a,b, but the evolution of the horizontal component $u(\tau ,x,z=0.97)$ of the velocity $\mathbf{v}$ to its equilibrium distribution ${u}_{eq}(x,z=0.97)=u(\tau ={\tau}_{6},x,z=0.97)$ [21], along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary of the HAN channel $(z=0.97)$, during both the slow heating (

**a**) and cooling (

**b**) modes.

**Figure 6.**(

**a**) The distance z dependence of the vertical component $w(\tau ,x=0.5,z)$ of the velocity $\mathbf{v}$ during its evolution to the equilibrium distribution ${w}_{eq}(x=0.5,z)=u(\tau ={\tau}_{6},x=0.5,z)$ across the HAN channel [21]. The different times during the slow heating mode are given starting from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}=0.0134$ [curve (6)], respectively. (

**b**) The same as in (

**a**), but for the cooling mode at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], respectively.

**Figure 7.**The same as in Figure 6a,b, but the evolution of the vertical component $w(\tau ,x,z=0.97)$ of the velocity $\mathbf{v}$ to its equilibrium distribution ${w}_{eq}(x,z=0.97)=w(\tau ={\tau}_{6},x,z=0.97)$, along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary $(z=0.97)$, during both the slow heating (

**a**) and cooling (

**b**) modes [21].

**Figure 8.**(

**a**) The distance z dependence of the temperature field $\chi (\tau ,x=0.5,z)$ during its evolution to the equilibrium distribution ${\chi}_{eq}(x=0.5,z)=\chi (\tau ={\tau}_{6},x=0.5,z)$ in the middle part of the HAN channel [21], during the slow heating mode at different times, from ${\tau}_{1}=0.00042$ [curve (1)] to ${\tau}_{6}={\tau}_{in}=0.0134$ [curve (6)], respectively. (

**b**) The same as in (

**a**), but for the cooling mode at different times, from ${\tau}_{7}=0.014$ [curve (7)] to ${\tau}_{11}={\tau}_{R}=0.22$ [curve (11)], respectively.

**Figure 9.**The same as in Figure 8a,b, but the distance x dependence of the temperature field $\chi (\tau ,x,z=0.97)$ during its evolution to the equilibrium distribution ${\chi}_{eq}(x,z=0.97)=\chi (\tau ={\tau}_{6},x,z=0.97)$ along the width of the HAN channel $(0\le x\le 1)$, in the vicinity of the upper warmer restricted surface $(z=0.97)$, both during the slow heating (

**a**) and cooling (

**b**) modes [21].

**Figure 10.**(

**a**) The equilibrium distribution of the polar angle ${\theta}_{eq}(x=0.5,z)$ in the middle part of the HAN channel, under the effect of the dimensionless heat flow $Q=3.54\phantom{\rule{3.33333pt}{0ex}}(\sim $$2.95\times {10}^{-2}$ mW/μm

^{2}) [21]. (

**b**) The equilibrium distribution of the polar angle ${\theta}_{eq}(x,z=0.97)$ along the width of the channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary of the HAN channel $(z=0.97)$, under the same conditions as in (

**a**).

**Figure 11.**(

**a**) The evolution of the temperature field $\chi (\tau ,x=0.5,z)$ to its equilibrium distribution ${\chi}_{eq}(x=0.5,z)=\chi (\tau ={\tau}_{5},x=0.5,z)$ in the middle part of the HAN channel, under the effect of the dimensionless heat flux $\mathbf{Q}=3.54\phantom{\rule{3.33333pt}{0ex}}(\sim $$2.95\times {10}^{-2}$ mW/μm

^{2}) [21]. The fast heating mode is characterized by the sequence of times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively, whereas the cooling mode (

**b**) is characterized by the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], respectively.

**Figure 12.**(

**a**) The same as in Figure 11a, but the evolution of the temperature field $\chi (\tau ,x,z=0.97)$ to its equilibrium distribution ${\chi}_{eq}(x,z=0.97)=\chi (\tau ={\tau}_{5},x,z=0.97)$ along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary $(z=0.97)$ [21] at different times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively, is shown for the fast heating mode. (

**b**) The same as in (

**a**), but the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], corresponds to the cooling mode.

**Figure 13.**(

**a**) The evolution of the horizontal component $u(\tau ,x=0.5,z)$ of the velocity $\mathbf{v}$ in the middle part of the HAN channel ($x=0.5$) to the equilibrium distribution ${u}_{eq}(x=0.5,z)=u(\tau ={\tau}_{5},x=0.5,z)$ across the HAN channel [21] at different times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively, corresponds to the fast heating mode. (

**b**) The same as in (

**a**), but the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], respectively, is corresponding to the cooling mode.

**Figure 14.**(

**a**) The same as in Figure 13a, but the distance x dependence of the horizontal component $u(\tau ,x,z=0.97)$ of the velocity $\mathbf{v}$ during its evolution to the equilibrium distribution ${u}_{eq}(x,z=0.97)=u(\tau ={\tau}_{5},x,z=0.97)$, along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary $(z=0.97)$ [21], is shown for the fast heating mode at different times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively. (

**b**) The same as in (

**a**), but the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], respectively, corresponds to the cooling mode.

**Figure 15.**(

**a**) The distance z dependence of the vertical component $w(\tau ,x=0.5,z)$ of the velocity $\mathbf{v}$ during its evolution to the equilibrium distribution ${w}_{eq}(x=0.5,z)=w(\tau ={\tau}_{5},x=0.5,z)$ across the HAN channel, corresponding to the fast heating mode [21], is given at different times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively. (

**b**) The same as in (

**a**), but the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], corresponds to the cooling mode.

**Figure 16.**(

**a**) The same as in Figure 15a, but the distance x dependence of the vertical component $w(\tau ,x,z=0.97)$ of the velocity $\mathbf{v}$ during its evolution to the equilibrium distribution ${w}_{eq}(x,z=0.97)=w(\tau ={\tau}_{5},x,z=0.97)$, along the width of the HAN channel $(0\le x\le 1)$ in the vicinity of the upper warmer boundary $(z=0.97)$, during the fast heating mode [21] is given at different times, from ${\tau}_{1}=0.00004$ [curve (1)] to ${\tau}_{5}=0.0006$ [curve (5)], respectively. (

**b**) The same as in (

**a**), but the sequence of times, from ${\tau}_{6}=0.001$ [curve (6)] to ${\tau}_{12}={\tau}_{R}=0.064$ [curve (12)], corresponds to the cooling mode.

**Figure 18.**Distribution of the velocity field $\mathbf{v}=u\widehat{\mathbf{i}}+w\widehat{\mathbf{k}}=-\nabla \times \widehat{\mathbf{j}}\psi $ in the microscopic HAN channel with the orientational defect located at $-L\le x\le L,z=-1$, when the heat flux $\mathbf{q}$ is directed at two values of the angle $\alpha $: (

**a**) 20${}^{\circ}$ and (

**b**) 40° [12,26,27], respectively. The heating occurs during time ${\tau}_{in}=1.6\times {10}^{-4}\phantom{\rule{3.33333pt}{0ex}}(\sim $0.29 ms), whereas the value of dimensionless heat flux coefficient ${\mathcal{Q}}_{0}$ is equal to 0.05. Here 1 mm of the arrow length is equal to $1.8\phantom{\rule{3.33333pt}{0ex}}$μm/s [13,26,27].

**Figure 23.**Temperature field distribution $\chi (\tau ={\tau}_{in},x,z)$ over the HAN channel near the orientational defect located at $-L\le x\le L,z=-1$, when the heat flux $\mathbf{q}$ is directed at two values of the angle $\alpha $ [13]: (

**a**) 20° and (

**b**) 160°, respectively. The heating occurs during time ${\tau}_{in}=1.6\times {10}^{-4}\phantom{\rule{3.33333pt}{0ex}}(\sim $0.29 ms), whereas the value of dimensionless heat flux coefficient ${Q}_{0}$ is equal to 0.05.

**Figure 24.**(

**a**) The distance x dependence of both the dimensionless height $h\left(\tau ,x\right)$ of the LC/air interface and the dimensionless temperature $\chi \left(\tau ,x\right)$ (

**b**) on the free LC/air interface $\mathsf{\Gamma}$, during the slow heating mode with ${\delta}_{5}=7$ and ${\tau}_{in}=0.01$, at different times ${\tau}_{i}={2}^{i}\times {10}^{-5}\phantom{\rule{3.33333pt}{0ex}}(i=1,\cdots ,10)$ [11,12], respectively. The numbering of the curves increases from $i=1$ to $i=10$.

**Figure 25.**The same as in Figure 24, but the distance x dependence of both the horizontal $u\left(\tau ,x\right)$ (

**a**) and vertical $w\left(\tau ,x\right)$ (

**b**) components of the velocity vector $\mathbf{v}=u\widehat{\mathbf{i}}+w\widehat{\mathbf{k}}=-\nabla \times \widehat{\mathbf{j}}\psi $ on the LC/air interface during the slow heating mode [11,12].

**Figure 26.**(Slow heating mode). Distribution of the velocity field $\mathbf{v}=u\widehat{\mathbf{i}}+w\widehat{\mathbf{k}}=-\nabla \times \widehat{\mathbf{j}}\psi $ in the HAN channel after the slow heating mode during $\tau ={\tau}_{in}$ [11,12]. Here, 1 mm of the arrow length is equal to $0.04\phantom{\rule{3.33333pt}{0ex}}$μm/s.

**Figure 27.**(Fast heating mode). Distribution of the velocity field $\mathbf{v}=u\widehat{\mathbf{i}}+w\widehat{\mathbf{k}}=-\nabla \times \widehat{\mathbf{j}}\psi $ in the HAN channel after heating during $\tau ={\tau}_{in}$ [12]. Here, 1 mm of the arrow length is equal to $0.4\phantom{\rule{3.33333pt}{0ex}}$μm/s.

**Figure 28.**Distribution of the dimensionless temperature $\chi \left(\tau ,x=0.0,z\right)$ along the z-axis ($-1.0\le z\le 1.0$), when the laser beam is focused in the center ($x=0.0$) of the HAN channel, at different depths [11,12]: (

**a**) ${z}_{0}=0.80$, and (

**b**) ${z}_{0}=0.90$, respectively. The numbering of the curves increases from $i=6$ to $i=10$.

**Figure 30.**Distribution of the horizontal $u\left(\tau ,x=0.0,z\right)$ component of the velocity $\mathbf{v}$ along the z-axis ($-1.0\le z\le 1.0$), when the laser beam is focused in the center ($x=0.0$) of the HAN channel, at different depths [11,12]: (

**a**) ${z}_{0}=0.80$ and (

**b**) ${z}_{0}=0.90$, respectively. The numbering of the curves increases from $i=6$ to $i=10$.

**Figure 32.**Distribution of the vertical $w\left(\tau ,x=0.0,z\right)$ component of the velocity $\mathbf{v}$ along the z-axis ($-1.0\le z\le 1.0$), when the laser beam is focused in the center ($x=0.0$) of the HAN channel, at different depths [11,12]: (

**a**) ${z}_{0}=0.80$ and (

**b**) ${z}_{0}=0.90$, respectively. The numbering of the curves increases from $i=6$ to $i=10$.

**Figure 34.**(

**a**) The distance r dependence of the angle $\theta (\tau ,r)$ across the HAN cavity between two $a\le r\le a+1$ infinitely long coaxial cylinders, under the influence of the temperature gradient $\nabla \chi $, directed from the cooler inner (${T}_{in}=0.97$) to warmer outer (${T}_{out}=0.9862$) cylinders, at different times ${\tau}_{k}=\frac{k}{10}{\tau}_{R}$ ($k=1,\cdots ,10$), whose values increase from curve (1) to curve (10) [28]. Here ${\tau}_{R}=0.2$. (

**b**) The same as in (

**a**), but the distance r dependence of the velocity $u(\tau ,r)$. All calculations were carried out for $a=1.0$.

**Figure 36.**The distance r dependence of the equilibrium velocity ${u}_{eq}\left(r\right)$ across the nematic cavity between two $a\le r\le a+1$ infinitely long coaxial cylinders, with the anchoring hybrid condition in the form of Equation (42), under influence of the $\nabla \chi $, directed from the cooler (warmer) inward ${\chi}_{1}$ (${\chi}_{2}$) to warmer (cooler) outward ${\chi}_{2}$ (${\chi}_{1}$) cylinders (see curves (2) and (1), respectively), calculated for a number of values of a [28]: (

**a**) 1.0 and (

**b**) 0.2, respectively.

**Figure 40.**Dependence of ${u}_{eq}^{max}\left(a\right)$ on the dimensionless size of the nematic cavity $a=\frac{{R}_{1}}{{R}_{2}-{R}_{1}}$, with the anchoring hybrid condition in the form of Equation (43), for two cases: first, (

**a**) when the heating mode is directed from inward to outward bounding cylinders (${\chi}_{r=a+1}>{\chi}_{r=a}$), second, (

**b**) when the heating mode is directed from outward to inward bounding cylinders (${\chi}_{r=a}>{\chi}_{r=a+1}$) [28], respectively.

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**MDPI and ACS Style**

Śliwa, I.; Zakharov, A.V.
Heat Driven Flows in Microsized Nematic Volumes: Computational Studies and Analysis. *Symmetry* **2021**, *13*, 459.
https://doi.org/10.3390/sym13030459

**AMA Style**

Śliwa I, Zakharov AV.
Heat Driven Flows in Microsized Nematic Volumes: Computational Studies and Analysis. *Symmetry*. 2021; 13(3):459.
https://doi.org/10.3390/sym13030459

**Chicago/Turabian Style**

Śliwa, Izabela, and Alex V. Zakharov.
2021. "Heat Driven Flows in Microsized Nematic Volumes: Computational Studies and Analysis" *Symmetry* 13, no. 3: 459.
https://doi.org/10.3390/sym13030459