# Investigation of Shear-Driven and Pressure-Driven Liquid Crystal Flow at Microscale: A Quantitative Approach for the Flow Measurement

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{3}/α

_{2}> 0. The flow behavior of nematic liquid crystals can be generally categorized into two major types through the signs of the two Leslie viscosities α

_{2}and α

_{3}. If α

_{3}/α

_{2}> 0, then the liquid crystal is flow aligning, otherwise it is flow tumbling [25]. Next, we addressed two typical steady 2D cases: the shear-driven (Couette) and the pressure-driven (Poiseuille) flow of nematic liquid crystal in a parallel microchannel respectively. Then we applied justified assumptions for simplification purpose before solving them numerically. The director profile is calculated for various pressure gradients and shear stresses. For pressure-driven flow cases, we find that under the limitation of strong anchoring and weak flow effects, flow alignment is not presented. In fact, the director field is majorly determined by the boundary conditions. For other cases, the results clearly show the influence of flow condition on reorientation of director field, which provide guideline for flow measurements at microscale [26,27,28].

## 2. Mathematical Models

**n**called the director.

#### 2.1. Couette Flow

_{11}= K

_{22}= K

_{33}≡ K, where K

_{11}is the splay coefficient, K

_{22}is the twist coefficient, and K

_{33}is the bend coefficient. (Currie, P. K., and G. P. Macsithigh, 1979).

#### 2.2. Poiseuille Flow

^{3}/K is a dimensionless pressure gradient.

## 3. Results

#### 3.1. Couette Flow

_{3}/α

_{2})

^{0.5}] (${\varnothing}_{c}$= 0.22, in this article) with the increasing velocity, which agrees with the continuum theory proposed by Leslie and Ericksen [30]. Leslie defined the Leslie angle as: “for which there is no hydrodynamic torque on the director in simple shear flow of an infinitely thick sample”. It is found in this article that the Leslie angle still exists in microchannel.

#### 3.2. Poiseuille Flow

## 4. Discussion

## 5. Conclusions

_{3}/α

_{2})

^{0.5}] with the increasing velocity. (ii) When the velocity exceeds the threshold, the profiles will lose its stability. Vthreshold is about 10

^{−3}–10

^{−4}m/s (depend on different parameters). If the velocity exceeds the threshold, the director deviates from the plane of shear. The deformation takes a form of director rotation about the axis perpendicular to the layer plane. As a result, transverse flow arises. The method of nondimensionalizing and the numerical approach proposed in this article will be a potential technique in liquid crystal research. The coupling influence what we analyze is might be able to give clues to the flow dynamics.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{F}is the usual nematic free energy density. The non-zero contribution to the rate of strain tensor A and vorticity tensor W are

_{0}, and hence, by (12), the pressure takes the form

_{2}, (21) by n

_{1}and then subtracting the resulting equations to produce, after much tedious algebra, the single equation

_{i}and n

_{i}using (8) and (1) finally reduce this equation to

_{1}and K

_{3}are necessarily non-negative.

## Appendix B

_{2}and Equation (11)*n

_{1}, after the simplification process we can have

- (i)
- one constant approximation applied to the Oseen–Zocher–Frank elastic free energyK
_{11}= K_{22}= K_{33}≡ K, where K_{11}is the splay coefficient, K_{22}is the twist coefficient, and K_{33}is the bend coefficient - (ii)
- Parodi’s relationship${\mathsf{\alpha}}_{6}={\mathsf{\alpha}}_{5}+{\mathsf{\alpha}}_{3}+{\mathsf{\alpha}}_{2}$
- (iii)
- The viscous coefficient${\text{}\mathsf{\alpha}}_{1}$ = 0

## Abbreviations

n | Director |

${F}_{i}$ | External body force per unit mass |

${G}_{i}$ | Generalized body force |

ρ | Density |

${W}_{F}$ | Elastic energy density for nematic |

λ | Lagrange multiplier |

p | Pressure |

${t}_{ij}$ | Viscous stress |

${g}_{i}$ | Vector |

K | Free energy |

${\mathsf{\alpha}}_{\mathrm{i}}$ | Viscosity coefficient |

$\overline{v}$ | Velocity of director |

h | Channel half-width |

Φ | Angle with z axis |

c | Constan |

g | Dimensionless pressure gradient |

G | Pressure gradient |

V | Velocity of the upper plate in Couette flow |

g* | Critical dimensional pressure gradient |

Vthreshold | Threshold velocity of the upper plane |

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**Figure 1.**Schematics and description of the mathematical model of a nematic liquid crystal in the Couette flow. The upper plate is moved to the right at a constant velocity.

**Figure 2.**Schematics and the mathematical description of a nematic liquid crystal in the Poiseuille flow.

**Figure 3.**(

**a**) Weak flow director profile across the dimensionless channel width −1≤ z ≤ 1 under strong anchoring and g = 25 (Anderson, T. G. et al., 2015). (

**b**) The calculated director profile of strong anchoring at g = 25.

**Figure 4.**(

**a**) Director profiles at different velocities of the upper plate. (

**b**) The maximum directional angle at different velocities of the upper plate.

**Figure 5.**Strong and weak flow solutions at different dimensional pressure gradients. (

**a**–

**d**) correspond to the condition of g = 5, 10, 20, and 25 respectively.

**Figure 6.**Velocity profiles and director fields at strong anchoring condition. (

**a**–

**c**): The solution of strong anchoring at g = 10. (

**a**) Velocity distributions over the channel width; (

**b**) angle of the director; (

**c**) director distributions over the microchannel. (

**d**–

**f**): Director and flow distributions within the microchannel under the pressure of g = 25.

**Figure 7.**The solution of strong solution at g = 50. (

**a**) The velocity profile; (

**b**) angle of the director; (

**c**) the distribution of the director at g = 50.

**Figure 8.**Positional sensitivity of the directional profile of the LC flow. (

**a**) and (

**b**) are sensitivities of the LC in Couette flow at various velocities of the upper plate; (

**c**) and (

**d**) represent the sensitivities at various dimensional pressure gradient in Poiseuille flow.

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**MDPI and ACS Style**

Zhu, J.; Tang, R.; Chen, Y.; Yin, S.; Huang, Y.; Wong, T.
Investigation of Shear-Driven and Pressure-Driven Liquid Crystal Flow at Microscale: A Quantitative Approach for the Flow Measurement. *Micromachines* **2021**, *12*, 28.
https://doi.org/10.3390/mi12010028

**AMA Style**

Zhu J, Tang R, Chen Y, Yin S, Huang Y, Wong T.
Investigation of Shear-Driven and Pressure-Driven Liquid Crystal Flow at Microscale: A Quantitative Approach for the Flow Measurement. *Micromachines*. 2021; 12(1):28.
https://doi.org/10.3390/mi12010028

**Chicago/Turabian Style**

Zhu, Jianqin, Runze Tang, Yu Chen, Shuai Yin, Yi Huang, and Teckneng Wong.
2021. "Investigation of Shear-Driven and Pressure-Driven Liquid Crystal Flow at Microscale: A Quantitative Approach for the Flow Measurement" *Micromachines* 12, no. 1: 28.
https://doi.org/10.3390/mi12010028