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Three-dimensional coating flow of nematic liquid crystal on an inclined substrate†

Published online by Cambridge University Press:  22 April 2015

M. A. LAM ,
L. J. CUMMINGS ,
T.-S. LIN  and
L. KONDIC
M. A. LAM
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ, 07102 email: [email protected], [email protected], [email protected]
L. J. CUMMINGS
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ, 07102 email: [email protected], [email protected], [email protected]
T.-S. LIN
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Heueh Road, Hsinchu 300, Taiwan email: [email protected]
L. KONDIC
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ, 07102 email: [email protected], [email protected], [email protected]
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Abstract

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

Keywords

Thin filmsliquid crystalscontact linesand gravity driven flow
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 647 - 669
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

This work was supported by NSF grant DMS-1211713.

References

[1]Carou, J., Mottram, N., Wilson, S. & Duffy, B. (2007) A mathematical model for blade coating of a nematic liquid crystal. Liq. Cryst. 35, 621.CrossRefGoogle Scholar
[2]Cazabat, A. M., Delabre, U., Richard, C. & Sang, Y. Y. C. (2011) Experimental study of hybrid nematic wetting films. Adv. Colloid Interface Sci. 168, 29.CrossRefGoogle ScholarPubMed
[3]Cummings, L. J., Lin, T.-S. & Kondic, L. (2011) Modeling and simulations of the spreading and destabilization of nematic droplets. Phys. Fluids 23, 043102.CrossRefGoogle Scholar
[4]Delabre, U., Richard, C. & Cazabat, A. M. (2009) Thin nematic films on liquid substrates. J. Phys. Chem. B 113, 3647.CrossRefGoogle ScholarPubMed
[5]Diez, J. A., González, A. G. & Kondic, L. (2009) On the breakup of fluid rivulets. Phys. Fluids 21, 082105.CrossRefGoogle Scholar
[6]Diez, J. A., Kondic, L. & Bertozzi, A. (2000) Global models for moving contact lines. Phys. Rev. E 63, 011208.CrossRefGoogle ScholarPubMed
[7]Ehrhand, P. & Davis, S. H. (1991) Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365.CrossRefGoogle Scholar
[8]Kondic, L. (2003) Instabilities in gravity driven flow of thin fluid films. SIAM Rev. 45, 95.CrossRefGoogle Scholar
[9]Kondic, L. & Diez, J. (2005) On nontrivial traveling waves in thin film flows including contact lines. Physica D 209, 135–144.CrossRefGoogle Scholar
[10]Lam, M. A., Cummings, L. J., Lin, T.-S. & Kondic, L. (2014) Modeling flow of nematic liquid crystal down an incline. J. Eng. Math. doi: 10.1007/s10665-014-9697-2.CrossRefGoogle Scholar
[11]Leslie, F. M. (1979) Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1.CrossRefGoogle Scholar
[12]Lin, T.-S., Cummings, L. J., Archer, A. J., Kondic, L. & Thiele, U. (2013) Note on the hydrodynamic description of thin nematic films: Strong anchoring model. Phys. Fluids 25, 082102.CrossRefGoogle Scholar
[13]Lin, T.-S. & Kondic, L. (2010) Thin films flowing down inverted substrates: Two dimensional flow. Phys. Fluids 22, 052105.CrossRefGoogle Scholar
[14]Lin, T.-S., Kondic, L. & Filippov, A. (2012) Thin films flowing down inverted substrates: Three-dimensional flow. Phys. Fluids 24, 022105.CrossRefGoogle Scholar
[15]Lin, T.-S., Kondic, L., Thiele, U. & Cummings, L. J. (2013) Modeling spreading dynamics of liquid crystals in three spatial dimensions. J. Fluid Mech. 729, 214.CrossRefGoogle Scholar
[16]Manyuhina, O. V. & Ben Amar, M. (2013) Thin nematic films: Anchoring effects and stripe instability revisited. Phys. Lett. A 377, 1003.CrossRefGoogle Scholar
[17]Mayo, L. C., McCue, S. W. & Moroney, T. J. (2013) Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder. Phys. Rev. E 87, 053018.CrossRefGoogle ScholarPubMed
[18]Naughton, S. P., Patel, N. K., Seric, I., Kondic, L., Lin, T.-S. & Cummings, L. J. (2013) Instability of gravity driven flow of liquid crystal films. SIAM Undergrad. Res. Online 5, 56.CrossRefGoogle Scholar
[19]Poulard, C. & Cazabat, A. M. (2003) Spontaneous spreading of nematic liquid crystals. Langmuir 21, 6270.CrossRefGoogle Scholar
[20]Rapini, A. & Papoular, M. (1969) Distorsion d'une lamelle nèmatique sous champ magnètique conditions d'ancrage aux parios. J. Phys. Colloques 30, C4–54.CrossRefGoogle Scholar
[21]Rey, A. D. (2008) Generalized young-laplace equation for nematic liquid crystal interfaces and its application to free-surface defects. Mol. Cryst. Liq. Cryst. 369, 63.CrossRefGoogle Scholar
[22]Thiele, U., Archer, A. J. & Plapp, M. (2012) Thermodynamically consistent description of the hydrodynamics of free surfaces covered by insoluble surfactants of high concentration. Phys. Fluids 24, 102107.CrossRefGoogle Scholar
[23]Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. (1989) Fingering instabilities of driven spreading films. Europhys. Lett. 10, 25.CrossRefGoogle Scholar
[24]van Effenterre, D. & Valignat, M. P. (2003) Stability of thin nematic films. Eur. Phys. J. E 12, 367.Google ScholarPubMed
[25]Witelski, T. & Bowen, M. (2003) ADI schemes for higher-order nonlinear diffusion equations. Appl. Num. Math. 45, 331.CrossRefGoogle Scholar
[26]Yang, L. & Homsy, G. M. (2007) Capillary instabilities of liquid films inside a wedge. Phys. Fluids 19, 044101.CrossRefGoogle Scholar
[27]Ziherl, P. & Zumer, S. (2003) Morphology and structure of thin liquid-crystalline films at nematic-isotropic transition. Eur. Phys. J. E 12, 361.CrossRefGoogle ScholarPubMed