×

Second order, linear, and unconditionally energy stable schemes for a hydrodynamic model of smectic-A liquid crystals. (English) Zbl 1388.76201

A model of liquid crystals is developed. It is based on the variational approach for the modified Oseen-Frank energy and the incompressible Navier-Stokes equations coupled with a constitutive equation for the layer variable. Second-order numerical time marching schemes based on the invariant energy quadratization method for nonlinear terms in the constitutive equation, the projection method for the Navier-Stokes equations, and some subtle implicit-explicit treatments for the convective and stress terms are developed. The well-posedness of the linear system and their unconditionally energy stable schemes are established. Various numerical experiments are presented.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76A15 Liquid crystals
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] S. Badia, F. G. Gonzalez, and J. V. Gutierrez-Santacreu, {\it An overview on numerical analyses of nematic liquid crystal flows}, Arch. Comput. Methods Eng., 18 (2011), pp. 285-313. · Zbl 1284.76247
[2] F. Bethue, H. Brezis, and F. Helein, {\it Asymptotics for the minimization of a Ginzburg-Landau functional}, Calc. Var. Partial Differential Equations, 1 (1993), pp. 123-148. · Zbl 0834.35014
[3] D. Brogioli and A. Vailati, {\it Diffusive mass transfer by nonequilibrium fluctuations: Fick’s law revisited}, Phys. Rev. E, 63 (2000), 012105.
[4] J. W. Cahn and J. E. Hillard, {\it Free energy of a nonuniform system. I. Interfacial free energy}, J. Chem. Phys., 28 (1958), pp. 258-267. · Zbl 1431.35066
[5] M. C. Calderer and S. Joo, {\it A continuum theory of chiral smectic C liquid crystals}, SIAM J. Appl. Math., 69 (2008), pp. 787-809, . · Zbl 1170.35497
[6] S. Chandrasekhar, {\it Liquid Crystals}, Vol. 1, 2nd ed., Cambridge University Press, Cambridge, UK, 1992.
[7] J. Chen and T. Lubensky, {\it Landau-ginzburg mean-field theory for the nematic-c and nematic to smectic-A phase transitions}, Phys. Rev. A., 14 (1976), pp. 1202-1207.
[8] R. Chen, G. Ji, X. Yang, and H. Zhang, {\it Decoupled energy stable schemes for phase-field vesicle membrane model}, J. Comput. Phys., 302 (2015), pp. 509-523. · Zbl 1349.76842
[9] Q. Cheng, X. Yang, and J. Shen, {\it Efficient and accurate numerical schemes for a hydrodynamically coupled phase field diblock copolymer model}, J. Comput. Phys., 341 (2017), pp. 44-60. · Zbl 1380.65203
[10] B. Climent-Ezquerra and F. Guillén-González, {\it Global in time solution and time-periodicity for a smectic-A liquid crystal model}, Commun. Pure Appl. Anal., 9 (2010), pp. 1473-1493. · Zbl 1428.76024
[11] B. Climent-Ezquerra and F. Guillén-González, {\it A review of mathematical analysis of nematic and smectic-A liquid crystal models}, European J. Appl. Math., 25 (2014), pp. 133-153. · Zbl 1391.76042
[12] T. A. Davis and E. C. Gartland, {\it Finite element analysis of the Landau-de Gennes minimization problem for liquid crystals}, SIAM J. Numer. Anal., 35 (1998), pp. 336-362, . · Zbl 0908.65120
[13] P. G. de Gennes and J. Prost, {\it The Physics of Liquid Crystals}, Oxford Science, Oxford, UK, 1993.
[14] W. E, {\it Nonlinear continuum theory of smectic-A liquid crystal}, Arch. Rational Mech. Anal., 137 (1997), pp. 1159-175.
[15] W. E and J.-G. Liu, {\it Projection method. I. Convergence and numerical boundary layers}, SIAM J. Numer. Anal., 32 (1995), pp. 1017-1057, . · Zbl 0842.76052
[16] J. L. Ericksen, {\it Anisotropic fluids}, Arch. Rational Mech. Anal., 4 (1960), pp. 231-237. · Zbl 0093.18002
[17] J. L. Ericksen, {\it Hydrostatic theory of liquid crystal}, Arch. Rational Mech. Anal., 9 (1962), pp. 371-378. · Zbl 0105.23403
[18] D. J. Eyre, {\it Unconditionally gradient stable time marching the Cahn-Hilliard equation}, in Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), Mater. Res. Soc. Sympos. Proc. 529, MRS, Warrendale, PA, 1998, pp. 39-46.
[19] X. Feng, Y. He, and C. Liu, {\it Analysis of finite element approximations of a phase field model for two-phase fluids}, Math. Comp., 76 (2007), pp. 539-571. · Zbl 1111.76028
[20] X. Feng and A. Prol, {\it Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows}, Numer. Math., 94 (2003), pp. 33-65.
[21] M. G. Forest, S. Heidenreich, S. Hess, X. Yang, and R. Zhou, {\it Robustness of pulsating jet-like layers in sheared nano-rod dispersions}, J. Non-Newtonian Fluid Mech., 155 (2008), pp. 130-145. · Zbl 1274.76025
[22] M. G. Forest, S. Heidenreich, S. Hess, X. Yang, and R. Zhou, {\it Dynamic texture scaling of sheared nematic polymers in the large Ericksen number limit}, J. Non-Newtonian Fluid Mech., 165 (2010), pp. 687-697. · Zbl 1274.76148
[23] M. G. Forest, Q. Wang, and X. Yang, {\it LCP droplet dispersions: A two-phase, diffuse-interface kinetic theory and global droplet defect predictions}, Soft Matter, 37 (2012), pp. 9642-9660.
[24] C. J. García-Cervera, T. Giorgi, and S. Joo, {\it The phase transitions from chiral nematic toward smectic liquid crystals}, Comm. Math. Phys., 269 (2007), pp. 367-399. · Zbl 1125.82030
[25] C. J. García-Cervera, T. Giorgi, and S. Joo, {\it Sawtooth profile in smectic A liquid crystals}, SIAM J. Appl. Math., 76 (2016), pp. 217-237, . · Zbl 1338.82067
[26] C. J. García-Cervera and S. Joo, {\it Layer undulations in smectic A liquid crystals}, J. Comput. Theor. Nanosci., 7 (2010), pp. 1-7. · Zbl 1318.76002
[27] C. J. García-Cervera and S. Joo, {\it Analytic description of layer undulations in smectic A liquid crystals}, Arch. Rational Mech. Anal. 203, 203 (2012), pp. 1-43. · Zbl 1318.76002
[28] J. L. Guermond, P. Minev, and J. Shen, {\it An overview of projection methods for incompressible flows}, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 6011-6045. · Zbl 1122.76072
[29] J. L. Guermond, J. Shen, and X. Yang, {\it Error analysis of fully discrete velocity-correction methods for incompressible flows}, Math. Comp., 77 (2008), pp. 1387-1405. · Zbl 1285.76029
[30] F. Guillén-González and G. Tierra, {\it Approximation of smectic-A liquid crystals}, Comput. Methods Appl. Mech. Engrg., 290 (2015), pp. 342-361. · Zbl 1423.76041
[31] F. Guillén-González and G. Tierra, {\it On linear schemes for a Cahn-Hilliard diffuse interface model}, J. Comput. Phys., 234 (2013), pp. 140-171, . · Zbl 1284.35025
[32] Y. He, {\it The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data}, Math. Comp., 77 (2008), pp. 2097-2124. · Zbl 1198.65222
[33] Y. He and J. Li, {\it A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations}, Appl. Numer. Math., 58 (2008), pp. 1503-1514. · Zbl 1155.35406
[34] Y. He, Y. Liu, and T. Tang, {\it On large time-stepping methods for the Cahn-Hilliard equation}, Appl. Numer. Math., 57 (2007), pp. 616-628. · Zbl 1118.65109
[35] Y. He and W. Sun, {\it Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations}, SIAM J. Numer. Anal., 45 (2007), pp. 837-869, . · Zbl 1145.35318
[36] R. Ingram, {\it A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations}, Math. Comp., 82 (2013), pp. 1953-1973. · Zbl 1457.65115
[37] D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros, and L. G. Leal, {\it Ericksen number and deborah number cascade predictions of a model for liquid crystalline polymers for simple shear flow}, Phys. Fluids, 19 (2007), 023101. · Zbl 1146.76444
[38] F. M. Leslie, {\it Some constitutive equations for liquid crystals}, Arch. Rational Mech. Anal., 28 (1968), pp. 265-283. · Zbl 0159.57101
[39] F. M. Leslie, I. W. Stewart, and M. Nakagawa, {\it A continuum theory for smectic C liquid crystals}, Molecular Crystals and Liquid Crystals, 198 (1991), pp. 443-454.
[40] J. Li and Y. He, {\it A stabilized finite element method based on two local Gauss integrations for the Stokes equations}, J. Comput. Appl. Math., 214 (2008), pp. 58-65. · Zbl 1132.35436
[41] F. H. Lin, {\it On nematic liquid crystals with variable degree of orientation}, Comm. Pure Appl. Math., 44 (1991), pp. 453-468. · Zbl 0733.49005
[42] P. Lin and C. Liu, {\it Simulations of singularity dynamics in liquid crystal flows: A \(C^0\) finite element approach}, J. Comput. Phys., 215 (2006), pp. 348-362. · Zbl 1101.82039
[43] C. Liu, {\it The dynamic for incompressible smectic-A liquid crystal: Existence and regularity}, Discrete Contin. Dyn. Syst., 6 (2000), pp. 591-608. · Zbl 1021.35083
[44] C. Liu and J. Shen, {\it A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method}, Phys. D, 179 (2003), pp. 211-228. · Zbl 1092.76069
[45] C. Liu, J. Shen, and X. Yang, {\it Dynamics of defect motion in nematic liquid crystal flow: Modeling and numerical simulation}, Comm. Comput. Phys, 2 (2007), pp. 1184-1198. · Zbl 1164.76422
[46] C. Liu and N. J. Walkington, {\it Approximation of liquid crystal flows}, SIAM J. Numer. Anal., 37 (2000), pp. 725-741, . · Zbl 1040.76036
[47] C. S. MacDonald, J. A. Mackenzie, A. Ramage, and C. J. P. Newton, {\it Efficient moving mesh method for Q-tensor models of nematic liquid crystals}, SIAM J. Sci. Comput., 37 (2015), pp. B215-B238. · Zbl 1322.82020
[48] P. Oswald and S. I. Ben-Abraham, {\it Undulation instability under shear in smectic A liquid crystals}, J. de Physique, 43 (1982), pp. 1193-1197.
[49] A. M. Parshin, V. A. Gunyakov, V. Y. Zyryanov, and V. F. Shabanov, {\it Electric and magnetic field-assisted orientational transitions in the ensembles of domains in a nematic liquid crystal on the polymer surface}, Inter. J. Molecular Sci., 15 (2014), pp. 17838-17851.
[50] A. D. Rey, {\it Capillary models for liquid crystal fibers membranes, films, and drops}, Soft Matter, 3 (2007), pp. 1349-1368.
[51] A. Sakamoto, K. Yoshino, U. Kubo, and Y. Inuishi, {\it Effects of the magnetic field on the phase transition temperature between smectic-A and nematic states}, Jpn. J. Appl. Phys., 15 (1976), pp. 545-546.
[52] A. Segatti and H. Wu, {\it Finite dimensional reduction and convergence to equilibrium for incompressible smectic-A liquid crystal flows}, SIAM J. Math. Anal., 43 (2011), pp. 2445-2481, . · Zbl 1252.35079
[53] J. Shen, {\it On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes}, Math. Comp., 65 (1996), pp. 1039-1065. · Zbl 0855.76049
[54] J. Shen and X. Yang, {\it An efficient moving mesh spectral method for the phase-field model of two-phase flows}, J. Comput. Phys., 228 (2009), pp. 2978-2992. · Zbl 1159.76032
[55] J. Shen and X. Yang, {\it Numerical approximations of Allen-Cahn and Cahn-Hilliard equations}, Discrete Contin. Dyn. Syst., 28 (2010), pp. 1669-1691. · Zbl 1201.65184
[56] J. Shen and X. Yang, {\it A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities}, SIAM J. Sci. Comput., 32 (2010), pp. 1159-1179, . · Zbl 1410.76464
[57] J. Shen and X. Yang, {\it Decoupled energy stable schemes for phase-field models of two-phase complex fluids}, SIAM J. Sci. Comput., 36 (2014), pp. B122-B145, . · Zbl 1288.76057
[58] J. Shen and X. Yang, {\it Decoupled energy stable schemes for phase-field models of two-phase incompressible flows}, SIAM J. Numer. Anal., 53 (2015), pp. 279-296, . · Zbl 1327.65178
[59] J. Shen, X. Yang, and H. Yu, {\it Efficient energy stable numerical schemes for a phase field moving contact line model}, J. Comput. Phys., 284 (2015), pp. 617-630. · Zbl 1351.76184
[60] T. Soddemann, G. Auernhammer, H. Guo, B. Dunweg, and K. Kremer, {\it Shear-induced undulation of smectic-A: Molecular dynamics simulations vs. analytical theory}, Eur. Phys. J. E, 13 (2004), pp. 141-151.
[61] M. Tabata and D. Tagamai, {\it Error estimates for finite element approximations of drag and lift in nonstationary navier-stokes flows}, Japan J. Indust. Appl. Math., 17 (2000), pp. 371-389. · Zbl 1306.76026
[62] R. Temam, {\it Navier-Stokes Equations: Theory and Numerical Analysis}, American Mathematical Society, Providence, RI, 2001. · Zbl 0981.35001
[63] G. Tierra and F. Guillén-González, {\it Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models}, Arch. Comput. Methods Eng., 22 (2015), pp. 269-289. · Zbl 1348.82080
[64] J. van Kan, {\it A second-order accurate pressure-correction scheme for viscous incompressible flow}, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 870-891, . · Zbl 0594.76023
[65] C. Xu and T. Tang, {\it Stability analysis of large time-stepping methods for epitaxial growth models}, SIAM. J. Numer. Anal., 44 (2006), pp. 1759-1779, . · Zbl 1127.65069
[66] K. Xu, M. G. Forest, and X. Yang, {\it Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data}, Discrete Contin. Dyn. Syst. Ser. B, 15 (2010), pp. 457-474. · Zbl 1402.82023
[67] X. Xu, G. J. V. Zwieten, and K. G. van der Zee, {\it Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models}, Int. J. Numer. Methods Biomed. Eng., 30 (2014), pp. 180-203.
[68] X. Yang, {\it Error analysis of stabilized semi-implicit method of Allen-Cahn equation}, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), pp. 1057-1070. · Zbl 1201.65170
[69] X. Yang, {\it Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends}, J. Comput. Phys., 327 (2016), pp. 294-316. · Zbl 1373.82106
[70] X. Yang, {\it Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system}, J. Sci. Comput, in press, . · Zbl 1456.65080
[71] X. Yang, Z. Cui, M. G. Forest, Q. Wang, and J. Shen, {\it Dimensional robustness and instability of sheared, semi-dilute, nano-rod dispersions}, Multiscale Model. Simul., 7 (2008), pp. 622-654, . · Zbl 1277.76005
[72] X. Yang, J. J. Feng, C. Liu, and J. Shen, {\it Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method}, J. Comput. Phys., 218 (2006), pp. 417-428. · Zbl 1158.76319
[73] X. Yang, M. G. Forest, H. Li, C. Liu, J. Shen, and Q. Wang, {\it Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immsersed in viscous fluids}, J. Comput. Phys., 236 (2013), pp. 1-14. · Zbl 1286.65112
[74] X. Yang, M. G. Forest, C. Liu, and J. Shen, {\it Shear cell rupture of nematic droplets in viscous fluids}, J. Non-Newtonian Fluid Mech., 166 (2011), pp. 487-499. · Zbl 1282.76061
[75] X. Yang, M. G. Forest, W. Mullins, and Q. Wang, {\it Dynamic defect morphology and hydrodynamics of sheared nematic polymers in two space dimensions}, J. Rheology, 53 (2009), pp. 589-615.
[76] X. Yang, M. G. Forest, W. Mullins, and Q. Wang, {\it Quench sensitivity to defects and shear banding in nematic polymer film flows}, J. Non-Newtonian Fluid Mech., 159 (2009), pp. 115-129. · Zbl 1274.76152
[77] X. Yang, M. G. Forest, W. Mullins, and Q. Wang, {\it \(2\)-d lid-driven cavity flow of nematic polymers: An unsteady sea of defects}, Soft Matter, 6 (2010), pp. 1138-1156.
[78] X. Yang and L. Ju, {\it Efficient linear schemes with unconditionally energy stability for the phase field elsatic bending energy model}, Comput. Methods Appl. Mech. Engrg., 315 (2017), pp. 691-712. · Zbl 1439.74165
[79] X. Yang and L. Ju, {\it Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model}, Comput. Methods Appl. Mech. Engrg., 318 (2017), pp. 1005-1029. · Zbl 1439.76029
[80] X. Yang and H. Yu, {\it Linear, second order and unconditionally energy stable schemes for a phase-field moving contact line model}, preprint, , 2017.
[81] X. Yang, J. Zhao, and Q. Wang, {\it Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method}, J. Comput. Phys., 333 (2017), pp. 104-127. · Zbl 1375.82121
[82] X. Yang, J. Zhao, Q. Wang, and J. Shen, {\it Numerical approximations for a three components Cahn-Hilliard phase-field model based on the invariant energy quadratization method}, Math. Models Methods Appl. Sci., 27 (2017), pp. 1993-2030. · Zbl 1393.80003
[83] H. Yu and X. Yang, {\it Numerical approximations for a phase-field moving contact line model with variable densities and viscosities}, J. Comput. Phys., 35 (2017), pp. 665-686. · Zbl 1375.76201
[84] P. Yue, J. J. Feng, C. Liu, and J. Shen, {\it A diffuse interface method for simulating two phase flows of complex fluids}, J. Fluid Mech., 515 (2004), pp. 293-317. · Zbl 1130.76437
[85] J. Zhao, Q. Wang, and X. Yang, {\it Numerical approximations to a new phase field model for immiscible mixtures of nematic liquid crystals and viscous fluids}, Comput. Meth. Appl. Mech. Eng., 310 (2016), pp. 77-97.
[86] J. Zhao, Q. Wang, and X. Yang, {\it Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach}, Internat. J. Numer. Methods Engrg., 110 (2017), pp. 279-300. · Zbl 1365.74138
[87] J. Zhao, X. Yang, Y. Gong, and Q. Wang, {\it A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals}, Comput. Methods Appl. Mech. Engrg., 318 (2017), pp. 803-825. · Zbl 1439.76124
[88] J. Zhao, X. Yang, J. Li, and Q. Wang, {\it Energy stable numerical schemes for a hydrodynamic model of nematic liquid crystals}, SIAM. J. Sci. Comput., 38 (2016), pp. A3264-A3290, .
[89] J. Zhao, X. Yang, J. Shen, and Q. Wang, {\it 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), pp. 539-556. · Zbl 1349.76019
[90] C. Zhou, P. Yue, and J. J. Feng, {\it Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal}, Langmuir, 24 (2008), pp. 3099-3110.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.