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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:
[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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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Z. XiangQ. Chen and C. Mu, Critical curves for degenerate parabolic equations coupled via non-linear boundary flux, Appl. Math. Comput., 189 (2007), 549-559. doi: 10.1016/j.amc.2006.11.130.

[44]

Z. XiangY. Wang and H. Yang, Global existence and nonexistence for degenerate parabolic equations with nonlinear boundary flux, Comput. Math. Appl., 62 (2011), 3056-3065. doi: 10.1016/j.camwa.2011.08.017.

[45]

Z. Xiang and H. Yang, On the regularity criteria for the 3d magneto-micropolar fluids in terms of one directional derivative, Boundary Value Problems, 2012 (2012), 139, 14PP. doi: 10.1186/1687-2770-2012-139.

[46]

X. Yang, Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316. doi: 10.1016/j.jcp.2016.09.029.

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X. Yang and D. Han, Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal equation, J. Comput. Phys., 330 (2017), 1116-1134. doi: 10.1016/j.jcp.2016.10.020.

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X. Yang and L. Ju, Efficient linear schemes with unconditionally energy stability for the phase field elastic bending energy model, Comput. Meth. Appl. Mech. Engrg., 315 (2017), 691-712. doi: 10.1016/j.cma.2016.10.041.

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X. Yang and L. Ju, Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model, Comput. Meth. Appl. Mech. Engrg., 318 (2017), 1005-1029. doi: 10.1016/j.cma.2017.02.011.

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X. YangJ. Zhao and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys., 333 (2017), 104-127. doi: 10.1016/j.jcp.2016.12.025.

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X. YangJ. ZhaoQ. Wang and J. Shen, Numerical approximations for a three components cahn-hilliard phase-field model based on the invariant energy quadratization method, M3AS: Mathematical Models and Methods in Applied Sciences, 27 (2017), 1993-2030. doi: 10.1142/S0218202517500373.

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H. Yu and X. Yang, Numerical approximations for a phase-field moving contact line model with variable densities and viscosities, J. Comput. Phys., 334 (2017), 665-686. doi: 10.1016/j.jcp.2017.01.026.

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Y. ZhangM. Feng and Y. He, Subgrid model for the stationary incompressible navier-stokes equations based on the high order polynomial interpolation, Int. J. Num. Anal. Modelling., 7 (2010), 734-748.

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J. ZhaoH. LiQ. Wang and X. Yang, A linearly decoupled energy stable scheme for phase-field models of three-phase incompressible flows, J. Sci. Comput., 70 (2017), 1367-1389. doi: 10.1007/s10915-016-0283-9.

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J. ZhaoX. YangY. Gong and Q. Wang, A Novel Linear Second Order Unconditionally Energy stable Scheme for a Hydrodynamic Q-tensor Model of Liquid Crystals, Comput. Meth. Appl. Mech. Engrg., 318 (2017), 803-825. doi: 10.1016/j.cma.2017.01.031.

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J. ZhaoX. YangJ. Li and Q. Wang, Energy stable numerical schemes for a hydrodynamic model of nematic liquid crystals, SIAM. J. Sci. Comput., 38 (2016), A3264-A3290. doi: 10.1137/15M1024093.

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J. ZhaoX. YangJ. Shen and Q. Wang, A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids, J. Comput. Phys., 305 (2016), 539-556. doi: 10.1016/j.jcp.2015.09.044.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

Z. XiangQ. Chen and C. Mu, Critical curves for degenerate parabolic equations coupled via non-linear boundary flux, Appl. Math. Comput., 189 (2007), 549-559. doi: 10.1016/j.amc.2006.11.130.

[44]

Z. XiangY. Wang and H. Yang, Global existence and nonexistence for degenerate parabolic equations with nonlinear boundary flux, Comput. Math. Appl., 62 (2011), 3056-3065. doi: 10.1016/j.camwa.2011.08.017.

[45]

Z. Xiang and H. Yang, On the regularity criteria for the 3d magneto-micropolar fluids in terms of one directional derivative, Boundary Value Problems, 2012 (2012), 139, 14PP. doi: 10.1186/1687-2770-2012-139.

[46]

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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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