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August  2018, 23(6): 2177-2192. doi: 10.3934/dcdsb.2018230

Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes

1. 

Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China

2. 

Department of Mathematics, University of South Carolina, Columbia, SC, 20208, USA

3. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

4. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

* Corresponding author

Received  June 2016 Revised  January 2018 Published  July 2018

In this paper, we consider numerical approximations for a model of smectic-A liquid crystal flows in its weak flow limit. The model, derived from the variational approach of the de Gennes free energy, is consisted of a highly nonlinear system that couples the incompressible Navier-Stokes equations with two nonlinear order parameter equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop an unconditionally energy stable, linear and decoupled time marching numerical scheme for the reduced model in the weak flow limit. We also rigorously prove that the numerical scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability of the scheme.

Citation: Qiumei Huang, Xiaofeng Yang, Xiaoming He. Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2177-2192. doi: 10.3934/dcdsb.2018230
References:
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C. ChenX. He and J. Huang, Mechanical quadrature methods and their extrapolations for solving the first kind boundary integral equations of stokes equation, Appl. Num. Math., 96 (2015), 165-179. doi: 10.1016/j.apnum.2015.05.004. Google Scholar

[2]

J. Chen and T. C. Lubensky, Landau-ginzburg mean-field theory for the nematic to smectic-c and nematic to smectic-a phase transitions, Phys. Rev. A., 14 (1976), 1202-1207. Google Scholar

[3]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford University Press, 1993.Google Scholar

[4]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-stokes model with nonlinear diffusion, Int. Math. Res. Notices, 7 (2014), 1833-1852. doi: 10.1093/imrn/rns270. Google Scholar

[5]

W. E, Nonlinear continuum theory of smectic-a liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175. doi: 10.1007/s002050050026. Google Scholar

[6]

D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), volume 529 of Mater. Res. Soc. Sympos. Proc., pages 39-46. MRS, Warrendale, PA, 1998. doi: 10.1557/PROC-529-39. Google Scholar

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A. Fick, Poggendorff's annalen, Journal of the American Mathematics Society, (1855), 59-86. Google Scholar

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M. G. ForestS. HeidenreichS. HessX. Yang and R. Zhou, Robustness of pulsating jetlike layers in sheared nano-rod dispersions, J. Non-Newtonian Fluid Mech., 155 (2008), 130-145. Google Scholar

[9]

M. G. ForestS. HeidenreichS. HessX. Yang and R. Zhou, Dynamic texture scaling of sheared nematic polymers in the large ericksen number limit, J. Non-Newtonian Fluid Mech., 165 (2010), 687-697. Google Scholar

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Y. GaoX. HeL. Mei and X. Yang, Fully decoupled, linearized, and energy stable finite element method for cahn-hilliard-navier-stokes-darcy model, SIAM. J. Sci. Comput., 40 (2018), B110-B137. doi: 10.1137/16M1100885. Google Scholar

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Z. GeM. Feng and Y. He, A stabilized nonconfirming finite element method based on multiscale enrichment for the stationary navier-stokes equations, Appl. Math. Comput., 202 (2008), 700-707. doi: 10.1016/j.amc.2008.03.016. Google Scholar

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J. ZhaoY. GongX. Yang and Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic q-tensor model of liquid crystals, in press, Comput. Meth. Appl. Mech. Engrg., 318 (2017), 803-825. doi: 10.1016/j.cma.2017.01.031. Google Scholar

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J. L. GuermondJ. Shen and X. Yang, Error analysis of fully discrete velocity-correction methods for incompressible flows, Math. Comp, 77 (2008), 1387-1405. doi: 10.1090/S0025-5718-08-02109-1. Google Scholar

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F. Guillen-Gonzaleza and G. Tierra, Approximation of smectic-a liquid crystals, Comput. Methods Appl. Mech. Engrg., 290 (2015), 342-361. doi: 10.1016/j.cma.2015.03.015. Google Scholar

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D. HanA. BrylevX. Yang and Z. Tan, Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows, J. Sci. Comput., 70 (2017), 965-989. doi: 10.1007/s10915-016-0279-5. Google Scholar

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Z.-J. HuT.-Z. Huang and N.-B. Tan, A splitting preconditioner for the incompressible Navier-Stokes equations, Mathematical Modelling and Analysis, 18 (2013), 612-630. doi: 10.3846/13926292.2013.868839. Google Scholar

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J. P. Hurault, Static distortions of a cholesteric planar structure induced by magnetic or ac electric fields, The Journal of Chemical Physics, 59 (1973), 2068-2075. doi: 10.1063/1.1680293. Google Scholar

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S. Jiang and Y. Ou, Incompressible limit of the non-isentropic navier-stokes equations with well-prepared initial data in three-dimensional bounded domains, Journal de Mathématiques Pures et Appliquées, 96 (2011), 1-28. doi: 10.1016/j.matpur.2011.01.004. Google Scholar

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S. Joo and D. Phillips, The phase transitions from chiral nematic toward smectic liquid crystals, Communications in Mathematical Physics, 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z. Google Scholar

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F. H. Lin, On nematic liquid crystals with variable degree of orientation, Communications on Pure and Applied Mathematics, 44 (1991), 453-468. doi: 10.1002/cpa.3160440404. Google Scholar

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C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar

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Y. Ou and D. Ren, Incompressible limit of global strong solutions to 3-d barotropic navier-stokes equations with well-prepared initial data and navier's slip boundary conditions, J. Math. Anal. Appl., 420 (2014), 1316-1336. doi: 10.1016/j.jmaa.2014.06.029. Google Scholar

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show all references

References:
[1]

C. ChenX. He and J. Huang, Mechanical quadrature methods and their extrapolations for solving the first kind boundary integral equations of stokes equation, Appl. Num. Math., 96 (2015), 165-179. doi: 10.1016/j.apnum.2015.05.004. Google Scholar

[2]

J. Chen and T. C. Lubensky, Landau-ginzburg mean-field theory for the nematic to smectic-c and nematic to smectic-a phase transitions, Phys. Rev. A., 14 (1976), 1202-1207. Google Scholar

[3]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford University Press, 1993.Google Scholar

[4]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-stokes model with nonlinear diffusion, Int. Math. Res. Notices, 7 (2014), 1833-1852. doi: 10.1093/imrn/rns270. Google Scholar

[5]

W. E, Nonlinear continuum theory of smectic-a liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175. doi: 10.1007/s002050050026. Google Scholar

[6]

D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), volume 529 of Mater. Res. Soc. Sympos. Proc., pages 39-46. MRS, Warrendale, PA, 1998. doi: 10.1557/PROC-529-39. Google Scholar

[7]

A. Fick, Poggendorff's annalen, Journal of the American Mathematics Society, (1855), 59-86. Google Scholar

[8]

M. G. ForestS. HeidenreichS. HessX. Yang and R. Zhou, Robustness of pulsating jetlike layers in sheared nano-rod dispersions, J. Non-Newtonian Fluid Mech., 155 (2008), 130-145. Google Scholar

[9]

M. G. ForestS. HeidenreichS. HessX. Yang and R. Zhou, Dynamic texture scaling of sheared nematic polymers in the large ericksen number limit, J. Non-Newtonian Fluid Mech., 165 (2010), 687-697. Google Scholar

[10]

Y. GaoX. HeL. Mei and X. Yang, Fully decoupled, linearized, and energy stable finite element method for cahn-hilliard-navier-stokes-darcy model, SIAM. J. Sci. Comput., 40 (2018), B110-B137. doi: 10.1137/16M1100885. Google Scholar

[11]

Z. GeM. Feng and Y. He, A stabilized nonconfirming finite element method based on multiscale enrichment for the stationary navier-stokes equations, Appl. Math. Comput., 202 (2008), 700-707. doi: 10.1016/j.amc.2008.03.016. Google Scholar

[12]

Z. GeM. Feng and Y. He, Stabilized multiscale finite element method for the stationary navier-stokes equations, J. Mech. Anal. Appl., 354 (2009), 708-717. doi: 10.1016/j.jmaa.2009.01.039. Google Scholar

[13]

Z. GeM. Feng and Y. He, Stabilized multiscale finite element method for the stationary Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 354 (2009), 708-717. doi: 10.1016/j.jmaa.2009.01.039. Google Scholar

[14]

J. ZhaoY. GongX. Yang and Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic q-tensor model of liquid crystals, in press, Comput. Meth. Appl. Mech. Engrg., 318 (2017), 803-825. doi: 10.1016/j.cma.2017.01.031. Google Scholar

[15]

J. L. GuermondJ. Shen and X. Yang, Error analysis of fully discrete velocity-correction methods for incompressible flows, Math. Comp, 77 (2008), 1387-1405. doi: 10.1090/S0025-5718-08-02109-1. Google Scholar

[16]

F. Guillen-Gonzaleza and G. Tierra, Approximation of smectic-a liquid crystals, Comput. Methods Appl. Mech. Engrg., 290 (2015), 342-361. doi: 10.1016/j.cma.2015.03.015. Google Scholar

[17]

D. HanA. BrylevX. Yang and Z. Tan, Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows, J. Sci. Comput., 70 (2017), 965-989. doi: 10.1007/s10915-016-0279-5. Google Scholar

[18]

W. Helfrich, Electrohydrodynamic and dielectric instabilities of cholesteric liquid crystals, The Journal of Chemical Physics, 55 (1971), 839-842. doi: 10.1063/1.1676151. Google Scholar

[19]

Z.-J. HuT.-Z. Huang and N.-B. Tan, A splitting preconditioner for the incompressible Navier-Stokes equations, Mathematical Modelling and Analysis, 18 (2013), 612-630. doi: 10.3846/13926292.2013.868839. Google Scholar

[20]

J. P. Hurault, Static distortions of a cholesteric planar structure induced by magnetic or ac electric fields, The Journal of Chemical Physics, 59 (1973), 2068-2075. doi: 10.1063/1.1680293. Google Scholar

[21]

S. Jiang and Y. Ou, Incompressible limit of the non-isentropic navier-stokes equations with well-prepared initial data in three-dimensional bounded domains, Journal de Mathématiques Pures et Appliquées, 96 (2011), 1-28. doi: 10.1016/j.matpur.2011.01.004. Google Scholar

[22]

S. Joo and D. Phillips, The phase transitions from chiral nematic toward smectic liquid crystals, Communications in Mathematical Physics, 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z. Google Scholar

[23]

F. H. Lin, On nematic liquid crystals with variable degree of orientation, Communications on Pure and Applied Mathematics, 44 (1991), 453-468. doi: 10.1002/cpa.3160440404. Google Scholar

[24]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar

[25]

Y. Ou, Low mach number limit of viscous polytropic fluid flows, J. Diff. Eqns, 251 (2011), 2037-2065. doi: 10.1016/j.jde.2011.07.009. Google Scholar

[26]

Y. Ou and D. Ren, Incompressible limit of global strong solutions to 3-d barotropic navier-stokes equations with well-prepared initial data and navier's slip boundary conditions, J. Math. Anal. Appl., 420 (2014), 1316-1336. doi: 10.1016/j.jmaa.2014.06.029. Google Scholar

[27]

D. Ren and Y. Ou, Strong solutions to an oldroyd-b model with slip boundary conditions via incompressible limit, Math. Meth. Appl. Sci., 38 (2015), 330-348. doi: 10.1002/mma.3071. Google Scholar

[28]

X. RenJ. WuZ. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-d mhd equations, J. Functional Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020. Google Scholar

[29]

X. RenZ. Xiang and Z. Zhang, Global existence and decay of smooth solutions for the 3-d mhd-type equations without magnetic diffusion, Sci. China. Math, 59 (2016), 1949-1974. doi: 10.1007/s11425-016-5145-2. Google Scholar

[30]

X. RenZ. Xiang and Z. Zhang, Global well-posedness for the 2d mhd equations without magnetic diffusion in a strip domain, Nonlinearity, 29 (2016), 1257-1291. doi: 10.1088/0951-7715/29/4/1257. Google Scholar

[31]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Disc. Conti. Dyn. Sys.-A, 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669. Google Scholar

[32]

J. Shen and X. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscositites, SIAM J. Sci. Comput., 32 (2010), 1159-1179. doi: 10.1137/09075860X. Google Scholar

[33]

J. Shen and X. Yang, Decoupled energy stable schemes for phase filed models of two phase complex fluids, SIAM J. Sci. Comput., 36 (2014), B122-B145. doi: 10.1137/130921593. Google Scholar

[34]

J. Shen and X. Yang, Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows, SIAM J. Num. Anal., 53 (2015), 279-296. doi: 10.1137/140971154. Google Scholar

[35]

N. -B. Tan, T. -Z. Huang and Z. -J. Hu, A relaxed splitting preconditioner for the incompressible navier-stokes equations, Journal of Applied. Mathematics, 2012 (2012), 402490, 12PP. Google Scholar

[36]

N. -B. Tan, T. -Z. Huang and Z. -J. Hu, Incomplete augmented lagrangian preconditioner for steady incompressible navier-stokes equations, The Scientific World Journal, 2013 (2013), 486323.Google Scholar

[37]

Y. Wang, C. Mu and Z. Xiang, Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary, Boundary Value Problems, 207 (2007), Art. ID 64579, 12 pp. Google Scholar

[38]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2d parabolic-parabolic attraction-repulsion chemotaxis system, J. Korean Math. Soc., 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031. Google Scholar

[39]

Y. Wang and Z. Xiang, Global existence and boundedness in a keller-segel-stokes system involving a tensor-valued sensitivity with saturation: The 3d case, J. Diff. Eqn., 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010. Google Scholar

[40]

Z. WangZ. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Diff. Eqn., 260 (2016), 2225-2258. doi: 10.1016/j.jde.2015.09.063. Google Scholar

[41]

Z. Xiang, The regularity criterion of the weak solution to the 3d viscous boussinesq equations in besov spaces, Appl. Math. Comput., 34 (2011), 360-372. doi: 10.1002/mma.1367. Google Scholar

[42]

Z. XiangQ. Chen and C. Mu, Blow-up rate estimates for a system of reaction-diffusion equations with absorption, J. Korean Math. Soc., 44 (2007), 779-786. doi: 10.4134/JKMS.2007.44.4.779. Google Scholar

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Figure 4.1.  The $L^2$ errors of the layer funciton $\phi$, the director field ${\bf d} = (d_1, d_2)$, the velocity ${\bf u} = (u, v)$ and pressure $p$. The slopes show that the scheme is asymptotically first-order accurate in time
Figure 4.2.  The evolution of the free energy functional for three different time steps of $\delta t = 0.0001, 0.001$ and $0.01$
Figure 4.3.  Snapshots of the layer function $\phi$ are taken at $t = 0$, $0.2$, $0.4$ and $0.8$ for Example 4.2
Figure 4.4.  Snapshots of the director field ${\bf d}$ are taken at $t = 0$, $0.2$, $0.4$ and $0.8$ for Example 4.2
Figure 4.5.  Time evolution of the free energy functional of Example 4.2
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