# Flow and Nematic Director Profiles in a Microfluidic Channel: The Interplay of Nematic Material Constants and Backflow

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## Abstract

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## 1. Introduction

- (a)
- We compute a phase plane in terms of ${L}^{\ast}$ and ${L}_{2}$, for a fixed ${p}_{x}$, which demarcates regions of fluid flow in the direction of decreasing pressure from regions of fluid flow in the direction of increasing pressure and this flow reversal is a clear manifestation of backflow.
- (b)
- We compute the total flow rates in different parameter regimes. In particular, we show that backflow can be attained for a window of values of ${L}^{\ast}$ i.e., ${L}_{crit,1}^{\ast}<{L}^{\ast}<{L}_{crit,2}^{\ast}$ and these critical values depend on ${p}_{x},{L}_{2}$, the boundary conditions and other material parameters.
- (c)
- We study two different kinds of boundary conditions for $\theta $—Dirichlet and mixed boundary conditions. The mixed boundary conditions are phrased in terms of an anchoring coefficient B on the bottom surface and accompanied by a Dirichlet condition on the top surface. The mixed boundary conditions offer greater scope for tuning the solution landscape.
- (d)
- We perform some investigations on how we can choose a suitable initial condition to attain the discontinuous solution for $\theta $ at long times, and this may be useful for studying multistability in such model settings.

## 2. Theory

## 3. Results

#### 3.1. Comparison of the Flow and No–Flow Situation

#### 3.2. Effect of the Winding Number $\omega $

#### 3.3. Effect of the Parameter ${L}^{\ast}$

#### 3.4. Dynamic Evolution of the Spatial Profiles

#### 3.5. Effect of the Initial Condition

## 4. Steady-State Analysis

#### 4.1. Continuous Solutions in $\theta $

#### Small-${L}^{\ast}$ Limit

#### 4.2. Discontinuous Solutions in $\theta $

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- De Gennes, P.-G.; Prost, J. The Physics of Liquid Crystals (International Series of Monographs On Physics); Oxford University Press: Cary, NC, USA, 1995. [Google Scholar]
- Brochard, F. Backflow Effects in Nematic Liquid Crystals. Mol. Cryst. Liquid Cryst.
**1973**, 23, 51–58. [Google Scholar] [CrossRef] - Mieda, Y.; Furutani, K. Micromanipulation method using backflow effect of liquid crystals. In Proceedings of the 2006 International Symposium on Micro-NanoMechatronics and Human Science, Nagoya, Japan, 5–8 November 2006; pp. 1–6. [Google Scholar]
- Vanbrabant, P.J.M.; Beeckman, J.; Neyts, K.; James, R.; Fernandez, F.A. Effect of material properties on reverse flow in nematic liquid crystal devices with homeotropic alignment. Appl. Phys. Lett.
**2009**, 95, 151108. [Google Scholar] [CrossRef] [Green Version] - Leslie, F.M. Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal.
**1968**, 28, 265–283. [Google Scholar] [CrossRef] - Ericksen, J.L. Equilibrium theory of liquid crystals. In Advances in Liquid Crystals; Elsevier: New York, NY, USA, 1976; Volume 2, pp. 233–298. [Google Scholar]
- Beris, A.N.; Edwards, B.J. Thermodynamics of Flowing Systems: With Internal Microstructure; Number 36; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Crespo, M.; Majumdar, A.; Ramos, A.M.; Griffiths, I.M. Solution landscapes in nematic microfluidics. Phys. D Nonlinear Phenom.
**2017**, 351, 1–13. [Google Scholar] [CrossRef] - Majumdar, A.; Zarnescu, A. Landau-de Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond. Arch. Ration. Mech. Anal.
**2010**, 196, 227–280. [Google Scholar] [CrossRef] - Tóth, G.; Denniston, C.; Yeomans, J.M. Hydrodynamics of topological defects in nematic liquid crystals. Phys. Rev. Lett.
**2002**, 88, 105504. [Google Scholar] [CrossRef] [PubMed] - Chen, G.-Q.; Majumdar, A.; Wang, D.; Zhang, R. Global existence and regularity of solutions for active liquid crystals. J. Differ. Equat.
**2017**, 263, 202–239. [Google Scholar] [CrossRef] [Green Version] - Denniston, C.; Orlandini, E.; Yeomans, J.M. Lattice Boltzmann simulations of liquid crystal hydrodynamics. Phys. Rev. E
**2001**, 63, 056702. [Google Scholar] [CrossRef] [PubMed] - Sonnet, A.M.; Maffettone, P.L.; Virga, E.G. Continuum theory for nematic liquid crystals with tensorial order. J. Non-Newtonian Fluid Mech.
**2004**, 119, 51–59. [Google Scholar] [CrossRef] - Xiao, Y. Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model. J. Differ. Equat.
**2017**, 262, 1291–1316. [Google Scholar] [CrossRef] [Green Version] - Luo, C.; Majumdar, A.; Erban, R. Multistability in planar liquid crystal wells. Phys. Rev. E
**2012**, 85, 061702. [Google Scholar] [CrossRef] [PubMed] - Kusumaatmaja, H.; Majumdar, A. Free energy pathways of a multistable liquid crystal device. Soft Matter
**2015**, 11, 4809–4817. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Canevari, G.; Majumdar, A.; Spicer, A. Order reconstruction for nematics on squares and hexagons: A Landau–de Gennes study. SIAM J. Appl. Math.
**2017**, 77, 267–293. [Google Scholar] [CrossRef] - Bisi, F.; Gartland, E.C., Jr.; Rosso, R.; Virga, E.G. Order reconstruction in frustrated nematic twist cells. Phys. Rev. E
**2003**, 68, 021707. [Google Scholar] [CrossRef] [PubMed] - Lin, F.H.; Liu, C. Static and dynamic theories of liquid crystals. J. Partial Differ. Equat.
**2001**, 14, 289–330. [Google Scholar] - Biscari, P.; Sluckin, T.J. A perturbative approach to the backflow dynamics of nematic defects. Eur. J. Appl. Math.
**2012**, 23, 181–200. [Google Scholar] [CrossRef] - Marenduzzo, D.; Orlandini, E.; Yeomans, J.M. Hydrodynamics and rheology of active liquid crystals: A numerical investigation. Phys. Rev. Lett.
**2007**, 98, 118102. [Google Scholar] [CrossRef] [PubMed] - Blanc, C.; Svenšek, D.; Žumer, S.; Nobili, M. Dynamics of nematic liquid crystal disclinations: The role of the backflow. Phys. Rev. Lett.
**2005**, 95, 1–4. [Google Scholar] [CrossRef] [PubMed] - Paicu, M.; Zarnescu, A. Energy dissipation and regularity for a coupled Navier–Stokes and q-tensor system. Arch. Ration. Mech. Anal.
**2012**, 203, 45–67. [Google Scholar] [CrossRef] - Sengupta, A.; Tkalec, U.; Ravnik, M.; Yeomans, J.M.; Bahr, C.; Herminghaus, S. Liquid crystal microfluidics for tunable flow shaping. Phys. Rev. Lett.
**2013**, 110, 048303. [Google Scholar] [CrossRef] [PubMed] - Mondal, S.; Majumdar, A.; Griffiths, I.M. Nematohydrodynamics for Colloidal Self-Assembly and Transport Phenomena. arXiv, 2017; arXiv:1707.09015. [Google Scholar]
- Emmrich, E.; Klapp, S.H.; Lasarzik, R. Nonstationary models for liquid crystals: A fresh mathematical perspective. arXiv, 2017; arXiv:1708.06937. [Google Scholar]
- Wang, W.; Zhang, P.; Zhang, Z. Rigorous derivation from Landau-de Gennes theory to Ericksen–Leslie theory. SIAM J. Math. Anal.
**2013**, 47, 127–158. [Google Scholar] [CrossRef] - Batista, V.M.; Blow, M.L.; Gama, M.M.T. The effect of anchoring on the nematic flow in channels. Soft Matter
**2015**, 11, 4674–4685. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Giomi, L.; Mahadevan, L.; Chakraborty, B.; Hagan, M.F. Banding, excitability and chaos in active nematic suspensions. Nonlinearity
**2012**, 25, 2245. [Google Scholar] [CrossRef] - Pryor, R.W. Multiphysics Modeling Using COMSOL: A First Principles Approach; Jones & Bartlett Publishers: Burlington, MA, USA, 2009. [Google Scholar]
- Mondal, S.; De, S. Effects of non-Newtonian power law rheology on mass transport of a neutral solute for electro-osmotic flow in a porous microtube. Biomicrofluidics
**2013**, 7, 044113. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**The effect of the fluid flow on (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$, and (

**c**) the velocity ($\tilde{u}$), at equilibrium, for the case of symmetric boundary condition (Equation (15)). The values of the parameters used are $\omega =1/2$, ${L}^{\ast}={10}^{-3}$ and ${L}_{1}={10}^{-6}$. (Here, and elsewhere, we plot the profiles at $\tilde{t}=10$, after which time we find the solutions have relaxed to a steady state from the initial configuration, Equations (18)–(20).) Analytic solutions are given in Section 4 and Section 4.1.

**Figure 3.**The effect of the fluid flow on (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$, and (

**c**) the velocity $\tilde{u}$, at equilibrium $(\tilde{t}=10)$, for the case of asymmetric boundary condition (Equation (16)). The values of the parameters used are $B=1/3$, $\omega =1/2$, ${L}^{\ast}={10}^{-3}$ and ${L}_{1}={10}^{-6}$.

**Figure 4.**Plot of (

**a**) the total volumetric flow rate and (

**b**) the wall shear stress (which relates to the skin friction coefficient through Equation (25)) as a function of ${L}_{2}$ for different values of the constant B in the symmetric (Dirichlet) case, Equation (15), and the asymmetric case, Equation (16), for $\theta $. The total flow rate is scaled with the equivalent Poiseuille flow rate for a Newtonian fluid, ${\int}_{-1}^{1}\tilde{u}{|}_{{L}_{2}=0}\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}\tilde{y}=-2{\tilde{p}}_{x}/3$. The solid and the dotted lines correspond to the negative and positive values of ${\tilde{p}}_{x}$ respectively. The values of the parameters used are $|{\tilde{p}}_{x}|=1$, $\omega =1/2$, ${L}^{\ast}={10}^{-3}$ and ${L}_{1}={10}^{-6}\ll 1$.

**Figure 5.**Phase space plot of the parameters (${L}^{\ast}$ and ${L}_{2}$) for no overall mass flow rate. Here ${\tilde{p}}_{x}\phantom{\rule{3.33333pt}{0ex}}>\phantom{\rule{3.33333pt}{0ex}}0$. The curve (solid) corresponding to $B=1/3$ is for the asymmetric boundary condition (Equation (16)). The dotted curve is for the case of the symmetric (Dirichlet) boundary condition (Equation (15)). The values of the parameters used are ${\tilde{p}}_{x}=1$, $\omega =1/2$ and ${L}_{1}={10}^{-6}$.

**Figure 6.**The effect of the winding number $\omega $ on (

**a**) the director orientation ($\theta $), (

**b**) the order parameter ($\tilde{s}$) and (

**c**) the velocity ($\tilde{u}$), at equilibrium $(\tilde{t}=10)$. In this case, we have considered the symmetric boundary condition in $\theta $ (Equation (15)). The values of the parameters used are ${L}^{\ast}={10}^{-3}$, ${\tilde{p}}_{x}=-10$, ${L}_{2}=1$, and ${L}_{1}={10}^{-6}$. The legends of all the sub-figures are the same as in (

**a**).

**Figure 7.**The effect of the winding number $\omega $ on (

**a**) the director orientation ($\theta $), (

**b**) the order parameter ($\tilde{s}$) and (

**c**) the velocity ($\tilde{u}$), at equilibrium $(\tilde{t}=10)$. In this case, we have considered the asymmetric boundary condition in $\theta $ (Equation (16)). The values of the parameters used are $B=1/3$, ${L}^{\ast}={10}^{-3}$, ${\tilde{p}}_{x}=-10$, ${L}_{2}=1$, and ${L}_{1}={10}^{-6}$. The legends of all the sub-figures are the same as in (

**a**).

**Figure 8.**The effect of the parameter ${L}^{\ast}$ on (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$ and (

**c**) the velocity $\left(\tilde{u}\right)$, at equilibrium $(\tilde{t}=10)$, in the case of the symmetric boundary conditions for $\theta $ (Equation (15)). The values of the parameters used are $\omega =1/2$, ${L}_{2}=1$, ${\tilde{p}}_{x}=-10$ and ${L}_{1}={10}^{-6}$. The legends of all the sub-figures are the same as in (

**a**).

**Figure 9.**The effect of the parameter ${L}^{\ast}$ on (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$ and (

**c**) the velocity $\left(\tilde{u}\right)$, at equilibrium $(\tilde{t}=10)$, in the case of the asymmetric boundary conditions for $\theta $ (Equation (16)). The values of the parameters used are $\omega =1/2$, $B=1/3$, ${L}_{2}=1$, ${\tilde{p}}_{x}=-10$ and ${L}_{1}={10}^{-6}$. The legends of all the sub-figures are the same as in (

**a**).

**Figure 10.**The effect of the parameter ${L}^{\ast}$ on the net fluid flow rate at equilibrium $(\tilde{t}=10)$, for the asymmetric (Equation (16)) and symmetric case (Equation (15)). The values of the parameters used are $B=1/3$, $\omega =1/2$, ${L}_{2}=1$, ${\tilde{p}}_{x}=-10$ and ${L}_{1}={10}^{-6}$. The total flow rate is scaled with the equivalent Poiseuille flow rate, ${\int}_{-1}^{1}\tilde{u}{|}_{{L}_{2},{L}_{1}=0}\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}\tilde{y}=-2{\tilde{p}}_{x}/3$.

**Figure 11.**The dynamic evolution of (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$ and (

**c**) the velocity profile $\left(\tilde{u}\right)$ for the symmetric case (Equation 15). The values of the parameters used are $\omega =1/2$, ${L}^{\ast}={10}^{-3}$, ${L}_{2}=10$, ${\tilde{p}}_{x}=-10$ and ${L}_{1}={10}^{-6}$. The legends of all the sub-figures are the same as in (

**a**).

**Figure 12.**The dynamic evolution of (

**a**) the director orientation $\left(\theta \right)$, (

**b**) the order parameter $\left(\tilde{s}\right)$ and (

**c**) the velocity profile $\left(\tilde{u}\right)$ for the asymmetric case (Equation (16)). The values of the parameters used are $B=1/3$, $\omega =1/2$, ${L}^{\ast}={10}^{-3}$, ${L}_{2}=10$, ${\tilde{p}}_{x}=-10$ and ${L}_{1}={10}^{-6}$. The profiles of $\theta $ are asymmetric (around $\tilde{y}=0$) because of the inhomogeneity in the $\theta $ boundary conditions (Equation (16)). The legends of all the sub-figures are the same as in (

**a**).

**Figure 13.**Equilibrium profiles of (

**a**) the director orientation, $\theta $ and (

**b**) the order parameter $\tilde{s}$ for two different initial conditions for $\tilde{s}$ and a linear initial profile for $\theta $, without any fluid flow. The blue curves are the equilibrium profiles for $\theta $ and $\tilde{s}$ for $\tilde{s}(\tilde{y},0)={\tilde{y}}^{2}$ and the green curves are the equilibrium profiles for $\tilde{s}(\tilde{y},0)=1$. We have considered the symmetric condition for $\theta $ given by Equation (15). The values of the parameters used are $\omega =1/2$ and ${L}^{\ast}=0.03$.

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**MDPI and ACS Style**

Mondal, S.; Griffiths, I.M.; Charlet, F.; Majumdar, A.
Flow and Nematic Director Profiles in a Microfluidic Channel: The Interplay of Nematic Material Constants and Backflow. *Fluids* **2018**, *3*, 39.
https://doi.org/10.3390/fluids3020039

**AMA Style**

Mondal S, Griffiths IM, Charlet F, Majumdar A.
Flow and Nematic Director Profiles in a Microfluidic Channel: The Interplay of Nematic Material Constants and Backflow. *Fluids*. 2018; 3(2):39.
https://doi.org/10.3390/fluids3020039

**Chicago/Turabian Style**

Mondal, Sourav, Ian M. Griffiths, Florian Charlet, and Apala Majumdar.
2018. "Flow and Nematic Director Profiles in a Microfluidic Channel: The Interplay of Nematic Material Constants and Backflow" *Fluids* 3, no. 2: 39.
https://doi.org/10.3390/fluids3020039