{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T23:48:28Z","timestamp":1774482508660,"version":"3.50.1"},"reference-count":64,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T00:00:00Z","timestamp":1697846400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T00:00:00Z","timestamp":1697846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DGE-1840340"],"award-info":[{"award-number":["DGE-1840340"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1908267"],"award-info":[{"award-number":["DMS-1908267"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1908267"],"award-info":[{"award-number":["DMS-1908267"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s00332-023-09970-6","type":"journal-article","created":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T12:02:10Z","timestamp":1697889730000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Hydrodynamical Model of Nematic Liquid Crystal Films with a General State of Orientational Order"],"prefix":"10.1007","volume":"34","author":[{"given":"Lucas","family":"Bouck","sequence":"first","affiliation":[]},{"given":"Ricardo H.","family":"Nochetto","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Yushutin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,21]]},"reference":[{"issue":"4","key":"9970_CR1","doi-asserted-by":"crossref","first-page":"3050","DOI":"10.1137\/130945405","volume":"46","author":"H Abels","year":"2014","unstructured":"Abels, H., Dolzmann, G., Liu, Y.: Well-posedness of a fully coupled Navier-Stokes\/Q-tensor system with inhomogeneous boundary data. SIAM J. Math. Anal. 46(4), 3050\u20133077 (2014)","journal-title":"SIAM J. Math. Anal."},{"key":"9970_CR2","doi-asserted-by":"crossref","unstructured":"Abels, Helmut, Dolzmann, Georg, Liu, Yuning: Strong solutions for the beris-edwards model for nematic liquid crystals with homogeneous dirichlet boundary conditions. Adv. Differ. Equ., (2015)","DOI":"10.57262\/ade\/1448323166"},{"issue":"3","key":"9970_CR3","volume":"79","author":"M Arroyo","year":"2009","unstructured":"Arroyo, M., DeSimone, A.: Relaxation dynamics of fluid membranes. Phys. Rev. E 79(3), 031915 (2009)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"9970_CR4","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1039\/B917180K","volume":"6","author":"MA Bates","year":"2010","unstructured":"Bates, M.A., Ska\u010dej, G., Zannoni, C.: Defects and ordering in nematic coatings on uniaxial and biaxial colloids. Soft Matter 6(3), 655\u2013663 (2010)","journal-title":"Soft Matter"},{"key":"9970_CR5","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195076943.001.0001","volume-title":"Thermodynamics of Flowing Systems: with Internal Microstructure","author":"AN Beris","year":"1994","unstructured":"Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems: with Internal Microstructure, vol. 36. Oxford University Press, Oxford (1994)"},{"key":"9970_CR6","doi-asserted-by":"crossref","unstructured":"Borthagaray, Juan\u00a0Pablo, Walker, Shawn\u00a0W.: The Q-tensor model with uniaxial constraint. In Handbook of Numerical Analysis, volume\u00a022, pages 313\u2013382. Elsevier, (2021)","DOI":"10.1016\/bs.hna.2020.09.001"},{"issue":"4","key":"9970_CR7","doi-asserted-by":"crossref","first-page":"A1807","DOI":"10.1137\/21M1403126","volume":"44","author":"P Brandner","year":"2022","unstructured":"Brandner, P., Jankuhn, T., Praetorius, S., Reusken, A., Voigt, A.: Finite element discretization methods for velocity-pressure and stream function formulations of surface stokes equations. SIAM J. Sci. Comput. 44(4), A1807\u2013A1832 (2022)","journal-title":"SIAM J. Sci. Comput."},{"issue":"08","key":"9970_CR8","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1142\/S0218202517500245","volume":"27","author":"Y Cai","year":"2017","unstructured":"Cai, Y., Shen, J., Xiang, X.: A stable scheme and its convergence analysis for a 2d dynamic Q-tensor model of nematic liquid crystals. Math. Models Methods Appl. Sci. 27(08), 1459\u20131488 (2017)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"9970_CR9","doi-asserted-by":"crossref","first-page":"030701","DOI":"10.1103\/PhysRevE.66.030701","volume":"66","author":"C Chiccoli","year":"2002","unstructured":"Chiccoli, C., Feruli, I., Lavrentovich, O.D., Pasini, P., Shiyanovskii, S.V., Zannoni, C.: Topological defects in schlieren textures of biaxial and uniaxial nematics. Phys. Rev. E 66(3), 030701 (2002)","journal-title":"Phys. Rev. E"},{"issue":"2","key":"9970_CR10","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1103\/PhysRevA.43.835","volume":"43","author":"GP Crawford","year":"1991","unstructured":"Crawford, G.P., Vilfan, M., Doane, J.W., Vilfan, I.: Escaped-radial nematic configuration in submicrometer-size cylindrical cavities: Deuterium nuclear-magnetic-resonance study. Phys. Rev. A 43(2), 835 (1991)","journal-title":"Phys. Rev. A"},{"key":"9970_CR11","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198520245.001.0001","volume-title":"The Physics of Liquid Crystals","author":"P-G De Gennes","year":"1993","unstructured":"De Gennes, P.-G., Prost, J.: The Physics of Liquid Crystals. Oxford University Press, Oxford (1993)"},{"issue":"28","key":"9970_CR12","volume":"23","author":"M Doi","year":"2011","unstructured":"Doi, M.: Onsager\u2019s variational principle in soft matter. J. Phys.: Condens. Matter 23(28), 284118 (2011)","journal-title":"J. Phys.: Condens. Matter"},{"issue":"2","key":"9970_CR13","volume":"24","author":"M Doi","year":"2015","unstructured":"Doi, M.: Onsager principle as a tool for approximation. Chin. Phys. B 24(2), 020505 (2015)","journal-title":"Chin. Phys. B"},{"issue":"2","key":"9970_CR14","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1093\/imanum\/drl023","volume":"27","author":"G Dziuk","year":"2007","unstructured":"Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262\u2013292 (2007)","journal-title":"IMA J. Numer. Anal."},{"issue":"2","key":"9970_CR15","doi-asserted-by":"crossref","first-page":"259","DOI":"10.4171\/IFB\/211","volume":"11","author":"C Eck","year":"2009","unstructured":"Eck, C., Fontelos, M.A., Gr\u00fcn, G., Klingbeil, F., Vantzos, O.: On a phase-field model for electrowetting. Interfaces Free Bound. 11(2), 259\u2013290 (2009)","journal-title":"Interfaces Free Bound."},{"issue":"2","key":"9970_CR16","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1002\/fld.4510","volume":"88","author":"T-P Fries","year":"2018","unstructured":"Fries, T.-P.: Higher-order surface fem for incompressible navier-stokes flows on manifolds. Int. J. Numer. Meth. Fluids 88(2), 55\u201378 (2018)","journal-title":"Int. J. Numer. Meth. Fluids"},{"issue":"29","key":"9970_CR17","doi-asserted-by":"crossref","first-page":"6911","DOI":"10.1039\/c3sm00126a","volume":"9","author":"MA Gharbi","year":"2013","unstructured":"Gharbi, M.A., Se\u010d, D., Lopez-Leon, T., Nobili, M., Ravnik, M., \u017dumer, S., Blanc, C.: Microparticles confined to a nematic liquid crystal shell. Soft Matter 9(29), 6911\u20136920 (2013)","journal-title":"Soft Matter"},{"issue":"6","key":"9970_CR18","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1007\/s00332-015-9264-7","volume":"25","author":"D Golovaty","year":"2015","unstructured":"Golovaty, D., Montero, J.A., Sternberg, P.: Dimension reduction for the Landau-de Gennes model in planar nematic thin films. J. Nonlinear Sci. 25(6), 1431\u20131451 (2015)","journal-title":"J. Nonlinear Sci."},{"issue":"6","key":"9970_CR19","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1007\/s00332-017-9390-5","volume":"27","author":"D Golovaty","year":"2017","unstructured":"Golovaty, D., Montero, J.A., Sternberg, P.: Dimension reduction for the Landau-de Gennes model on curved nematic thin films. J. Nonlinear Sci. 27(6), 1905\u20131932 (2017)","journal-title":"J. Nonlinear Sci."},{"key":"9970_CR20","unstructured":"Gudibanda, Varun\u00a0M., Weber, Franziska, Yue, Yukun: Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals. arXiv preprintarXiv:2012.00278, (2020)"},{"issue":"4","key":"9970_CR21","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF00261375","volume":"57","author":"ME Gurtin","year":"1975","unstructured":"Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291\u2013323 (1975)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"9970_CR22","doi-asserted-by":"crossref","first-page":"354","DOI":"10.4208\/cicp.210115.180515a","volume":"19","author":"H Yucheng","year":"2016","unstructured":"Yucheng, H., Yang, Q., Zhang, P.: On the disclination lines of nematic liquid crystals. Commun. Comput. Phys. 19(2), 354\u2013379 (2016)","journal-title":"Commun. Comput. Phys."},{"issue":"5","key":"9970_CR23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-016-1051-2","volume":"55","author":"R Ignat","year":"2016","unstructured":"Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of point defects of degree $$\\pm \\frac{1}{2}$$ in a two-dimensional nematic liquid crystal model. Calc. Var. Partial. Differ. Equ. 55(5), 1\u201333 (2016)","journal-title":"Calc. Var. Partial. Differ. Equ."},{"issue":"3","key":"9970_CR24","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1007\/s00205-020-01539-x","volume":"237","author":"R Ignat","year":"2020","unstructured":"Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals. Arch. Ration. Mech. Anal. 237(3), 1421\u20131473 (2020)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3","key":"9970_CR25","doi-asserted-by":"crossref","first-page":"353","DOI":"10.4171\/IFB\/405","volume":"20","author":"T Jankuhn","year":"2018","unstructured":"Jankuhn, T., Olshanskii, M.A., Reusken, A.: Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound. 20(3), 353\u2013377 (2018)","journal-title":"Interfaces Free Bound."},{"issue":"2","key":"9970_CR26","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1090\/qam\/1452","volume":"75","author":"H Koba","year":"2017","unstructured":"Koba, H., Liu, C., Giga, Y.: Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Math. 75(2), 359\u2013389 (2017)","journal-title":"Q. Appl. Math."},{"issue":"1","key":"9970_CR27","doi-asserted-by":"crossref","first-page":"012703","DOI":"10.1103\/PhysRevE.94.012703","volume":"94","author":"V Koning","year":"2016","unstructured":"Koning, V., Lopez-Leon, T., Darmon, A., Fernandez-Nieves, A., Vitelli, V.: Spherical nematic shells with a threefold valence. Phys. Rev. E 94(1), 012703 (2016)","journal-title":"Phys. Rev. E"},{"issue":"20","key":"9970_CR28","doi-asserted-by":"crossref","first-page":"4993","DOI":"10.1039\/c3sm27671f","volume":"9","author":"V Koning","year":"2013","unstructured":"Koning, V., Lopez-Leon, T., Fernandez-Nieves, A., Vitelli, V.: Bivalent defect configurations in inhomogeneous nematic shells. Soft Matter 9(20), 4993\u20135003 (2013)","journal-title":"Soft Matter"},{"issue":"2","key":"9970_CR29","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1039\/C0SM00378F","volume":"7","author":"S Kralj","year":"2011","unstructured":"Kralj, S., Rosso, R., Virga, E.G.: Curvature control of valence on nematic shells. Soft Matter 7(2), 670\u2013683 (2011)","journal-title":"Soft Matter"},{"issue":"5","key":"9970_CR30","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1038\/nphys1920","volume":"7","author":"T Lopez-Leon","year":"2011","unstructured":"Lopez-Leon, T., Koning, V., Devaiah, K.B.S., Vitelli, V., Fernandez-Nieves, A.: Frustrated nematic order in spherical geometries. Nat. Phys. 7(5), 391\u2013394 (2011)","journal-title":"Nat. Phys."},{"issue":"2","key":"9970_CR31","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1017\/S0956792509990210","volume":"21","author":"A Majumdar","year":"2010","unstructured":"Majumdar, A.: Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21(2), 181\u2013203 (2010)","journal-title":"Eur. J. Appl. Math."},{"issue":"2","key":"9970_CR32","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1080\/14786437308227417","volume":"27","author":"RB Meyer","year":"1973","unstructured":"Meyer, R.B.: On the existence of even indexed disclinations in nematic liquid crystals. Philosophical Magazine: J. Theor. Exp. Appl. Phys. 27(2), 405\u2013424 (1973)","journal-title":"Philosophical Magazine: J. Theor. Exp. Appl. Phys."},{"issue":"1","key":"9970_CR33","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1073\/pnas.1810896115","volume":"116","author":"A Mietke","year":"2019","unstructured":"Mietke, A., J\u00fclicher, F., Sbalzarini, I.F.: Self-organized shape dynamics of active surfaces. Proc. Natl. Acad. Sci. 116(1), 29\u201334 (2019)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9970_CR34","unstructured":"Mottram, Nigel\u00a0J., Newton, Christopher\u00a0J.P.: Introduction to Q-tensor theory. arXiv preprintarXiv:1409.3542, (2014)"},{"issue":"45","key":"9970_CR35","doi-asserted-by":"crossref","first-page":"8442","DOI":"10.1039\/C7SM01954H","volume":"13","author":"BS Murray","year":"2017","unstructured":"Murray, B.S., Kralj, S., Rosenblatt, C.: Decomposition vs. escape of topological defects in a nematic liquid crystal. Soft Matter 13(45), 8442\u20138450 (2017)","journal-title":"Soft Matter"},{"issue":"6","key":"9970_CR36","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.85.061701","volume":"85","author":"G Napoli","year":"2012","unstructured":"Napoli, G., Vergori, L.: Surface free energies for nematic shells. Phys. Rev. E 85(6), 061701 (2012)","journal-title":"Phys. Rev. E"},{"issue":"2","key":"9970_CR37","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.94.020701","volume":"94","author":"G Napoli","year":"2016","unstructured":"Napoli, G., Vergori, L.: Hydrodynamic theory for nematic shells: The interplay among curvature, flow, and alignment. Phys. Rev. E 94(2), 020701 (2016)","journal-title":"Phys. Rev. E"},{"issue":"10","key":"9970_CR38","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1021\/nl0202096","volume":"2","author":"DR Nelson","year":"2002","unstructured":"Nelson, D.R.: Toward a tetravalent chemistry of colloids. Nano Lett. 2(10), 1125\u20131129 (2002)","journal-title":"Nano Lett."},{"issue":"16","key":"9970_CR39","doi-asserted-by":"crossref","first-page":"4032","DOI":"10.1039\/C9SM02475A","volume":"16","author":"M Nestler","year":"2020","unstructured":"Nestler, M., Nitschke, I., L\u00f6wen, H., Voigt, A.: Properties of surface Landau-de Gennes Q-tensor models. Soft Matter 16(16), 4032\u20134042 (2020)","journal-title":"Soft Matter"},{"key":"9970_CR40","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.jcp.2019.03.006","volume":"389","author":"M Nestler","year":"2019","unstructured":"Nestler, M., Nitschke, I., Voigt, A.: A finite element approach for vector-and tensor-valued surface pdes. J. Comput. Phys. 389, 48\u201361 (2019)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9970_CR41","doi-asserted-by":"crossref","first-page":"947","DOI":"10.4208\/cicp.OA-2021-0206","volume":"31","author":"M Nestler","year":"2022","unstructured":"Nestler, M., Voigt, A.: Active nematodynamics on curved surfaces - the influence of geometric forces on motion patterns of topological defects. Commun. Comput. Phys. 31(3), 947\u2013965 (2022)","journal-title":"Commun. Comput. Phys."},{"issue":"2214","key":"9970_CR42","first-page":"20170686","volume":"474","author":"I Nitschke","year":"2018","unstructured":"Nitschke, I., Nestler, M., Praetorius, S., L\u00f6wen, H., Voigt, A.: Nematic liquid crystals on curved surfaces: a thin film limit. Proc. Royal Soc. Math. Phys. Eng. Sci. 474(2214), 20170686 (2018)","journal-title":"Proc. Royal Soc. Math. Phys. Eng. Sci."},{"issue":"4","key":"9970_CR43","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevFluids.4.044002","volume":"4","author":"I Nitschke","year":"2019","unstructured":"Nitschke, I., Reuther, S., Voigt, A.: Hydrodynamic interactions in polar liquid crystals on evolving surfaces. Phys. Rev. Fluids 4(4), 044002 (2019)","journal-title":"Phys. Rev. Fluids"},{"key":"9970_CR44","doi-asserted-by":"crossref","DOI":"10.1016\/j.geomphys.2021.104428","volume":"173","author":"I Nitschke","year":"2022","unstructured":"Nitschke, I., Voigt, A.: Observer-invariant time derivatives on moving surfaces. J. Geom. Phys. 173, 104428 (2022)","journal-title":"J. Geom. Phys."},{"key":"9970_CR45","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1017\/jfm.2012.317","volume":"708","author":"I Nitschke","year":"2012","unstructured":"Nitschke, I., Voigt, A., Wensch, J.: A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418\u2013438 (2012)","journal-title":"J. Fluid Mech."},{"issue":"01","key":"9970_CR46","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S0218202513500474","volume":"24","author":"RH Nochetto","year":"2014","unstructured":"Nochetto, R.H., Salgado, A.J., Walker, S.W.: A diffuse interface model for electrowetting with moving contact lines. Math. Models Methods Appl. Sci. 24(01), 67\u2013111 (2014)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"6","key":"9970_CR47","doi-asserted-by":"crossref","first-page":"6007","DOI":"10.1137\/18M1165189","volume":"50","author":"MR Novack","year":"2018","unstructured":"Novack, M.R.: Dimension reduction for the Landau-de Gennes model: The vanishing nematic correlation length limit. SIAM J. Math. Anal. 50(6), 6007\u20136048 (2018)","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"9970_CR48","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRev.37.405","volume":"37","author":"L Onsager","year":"1931","unstructured":"Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37(4), 405 (1931)","journal-title":"Phys. Rev."},{"issue":"12","key":"9970_CR49","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1103\/PhysRev.38.2265","volume":"38","author":"L Onsager","year":"1931","unstructured":"Onsager, L.: Reciprocal relations in irreversible processes. ii. Phys. Rev. 38(12), 2265 (1931)","journal-title":"Phys. Rev."},{"issue":"5","key":"9970_CR50","doi-asserted-by":"crossref","first-page":"4181","DOI":"10.1063\/1.466115","volume":"99","author":"MA Osipov","year":"1993","unstructured":"Osipov, M.A., Hess, S.: Density functional approach to the theory of interfacial properties of nematic liquid crystals. J. Chem. Phys. 99(5), 4181\u20134190 (1993)","journal-title":"J. Chem. Phys."},{"issue":"1","key":"9970_CR51","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00205-011-0443-x","volume":"203","author":"M Paicu","year":"2012","unstructured":"Paicu, M., Zarnescu, A.: Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203(1), 45\u201367 (2012)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"25","key":"9970_CR52","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.123.258101","volume":"123","author":"P Pearce","year":"2019","unstructured":"Pearce, P., Song, B., Skinner, D.J., Mok, R., Hartmann, R., Singh, P.K., Jeckel, H., Oishi, J.S., Drescher, K., Dunkel, J.: Flow-induced symmetry breaking in growing bacterial biofilms. Phys. Rev. Lett. 123(25), 258101 (2019)","journal-title":"Phys. Rev. Lett."},{"key":"9970_CR53","doi-asserted-by":"crossref","first-page":"R8","DOI":"10.1017\/jfm.2020.564","volume":"900","author":"S Reuther","year":"2020","unstructured":"Reuther, S., Nitschke, I., Voigt, A.: A numerical approach for fluid deformable surfaces. J. Fluid Mech. 900, R8 (2020)","journal-title":"J. Fluid Mech."},{"issue":"2","key":"9970_CR54","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1137\/140971798","volume":"13","author":"S Reuther","year":"2015","unstructured":"Reuther, S., Voigt, A.: The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simulat. 13(2), 632\u2013643 (2015)","journal-title":"Multiscale Model. Simulat."},{"key":"9970_CR55","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-87815-7","volume-title":"Dissipative Ordered Fluids: Theories for Liquid Crystals","author":"AM Sonnet","year":"2012","unstructured":"Sonnet, A.M., Virga, E.G.: Dissipative Ordered Fluids: Theories for Liquid Crystals. Springer Science & Business Media, Berlin (2012)"},{"issue":"13","key":"9970_CR56","doi-asserted-by":"crossref","first-page":"3634","DOI":"10.1039\/D0SM02076A","volume":"17","author":"H Wang","year":"2021","unstructured":"Wang, H., Qian, T., Xinpeng, X.: Onsager\u2019s variational principle in active soft matter. Soft Matter 17(13), 3634\u20133653 (2021)","journal-title":"Soft Matter"},{"key":"9970_CR57","doi-asserted-by":"crossref","unstructured":"Wang, Qi: Generalized Onsager principle and it applications. In: Frontiers and Progress of Current Soft Matter Research, pages 101\u2013132. Springer, (2021)","DOI":"10.1007\/978-981-15-9297-3_3"},{"issue":"2","key":"9970_CR58","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1103\/PhysRevLett.29.90","volume":"29","author":"C Williams","year":"1972","unstructured":"Williams, C., Piera\u0144ski, P., Cladis, P.E.: Nonsingular s=+ 1 screw disclination lines in nematics. Phys. Rev. Lett. 29(2), 90 (1972)","journal-title":"Phys. Rev. Lett."},{"issue":"2","key":"9970_CR59","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1007\/s00205-018-1297-2","volume":"231","author":"W Hao","year":"2019","unstructured":"Hao, W., Xiang, X., Zarnescu, A.: Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals. Arch. Ration. Mech. Anal. 231(2), 1217\u20131267 (2019)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"30","key":"9970_CR60","doi-asserted-by":"crossref","first-page":"4001","DOI":"10.1016\/S0020-7683(97)00267-9","volume":"35","author":"H Xiao","year":"1998","unstructured":"Xiao, H., Bruhns, O.T., Meyers, A.: On objective corotational rates and their defining spin tensors. Int. J. Solids Struct. 35(30), 4001\u20134014 (1998)","journal-title":"Int. J. Solids Struct."},{"issue":"6","key":"9970_CR61","doi-asserted-by":"crossref","first-page":"202","DOI":"10.3390\/e18060202","volume":"18","author":"X Yang","year":"2016","unstructured":"Yang, X., Li, J., Forest, M.G., Wang, Q.: Hydrodynamic theories for flows of active liquid crystals and the generalized Onsager principle. Entropy 18(6), 202 (2016)","journal-title":"Entropy"},{"key":"9970_CR62","unstructured":"Yang, Xiaogang, Li, Jun, Wang, Qi: Continuum models of active liquid crystals on surfaces. Unpublished"},{"key":"9970_CR63","first-page":"187","volume":"7","author":"A Zarnescu","year":"2012","unstructured":"Zarnescu, A.: Topics in the Q-tensor theory of liquid crystals. Topics Math. Model. Anal. 7, 187\u2013252 (2012)","journal-title":"Topics Math. Model. Anal."},{"key":"9970_CR64","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1016\/j.cma.2017.01.031","volume":"318","author":"J Zhao","year":"2017","unstructured":"Zhao, J., Yang, X., Gong, Y., Wang, Q.: A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals. Comput. Methods Appl. Mech. Eng. 318, 803\u2013825 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09970-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-023-09970-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09970-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,15]],"date-time":"2024-02-15T21:26:12Z","timestamp":1708032372000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-023-09970-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,21]]},"references-count":64,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["9970"],"URL":"https:\/\/doi.org\/10.1007\/s00332-023-09970-6","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,21]]},"assertion":[{"value":"27 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"5"}}