Kickback in nematic liquid crystals
Authors:
F. P. da Costa, M. Grinfeld, M. Langer, N. J. Mottram and J. T. Pinto
Journal:
Quart. Appl. Math. 70 (2012), 99-110
MSC (2000):
Primary 34D15, 35Q72; Secondary 76A15, 82D30
DOI:
https://doi.org/10.1090/S0033-569X-2011-01265-5
Published electronically:
September 7, 2011
MathSciNet review:
2920618
Full-text PDF Free Access
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Additional Information
Abstract: We describe a nonlocal linear partial differential equation arising in the analysis of dynamics of a nematic liquid crystal. We confirm that it accounts for the kickback phenomenon by decoupling the director dynamics from the flow. We also analyse some of the mathematical properties of the decoupled director equation.
References
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- I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: a Mathematical Introduction, The Liquid Crystals Book Series, Taylor and Francis, New York, 2004.
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References
- D. W. Berreman, Liquid-crystal twist cell dynamics with backflow, J. Appl. Phys. 46 (1975), 3746–3751.
- M. G. Clark and F. M. Leslie, A calculation of orientational relaxation in nematic liquid crystals, Proc. Royal Soc. Lond. A, 361 (1978), 463–485.
- S. L. Cornford, T. S. Taphouse, C. J. P. Newton, and J. R. Sambles, Determination of the director profile in a nematic cell from guided wave data: an inverse problem, New J. Phys. 9 (2007), 166–170.
- F. P. da Costa, M. Grinfeld, N. J. Mottram, and J. T. Pinto, Uniqueness in the Freedericksz transition with weak anchoring, J. Differ. Eqns. 246 (2009), 2590–2600. MR 2503014 (2009m:34029)
- R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1982. MR 696935 (84j:35142)
- C. J. Gerritsma, C. Z. van Doorn, and P. van Zanten, Transient effects in the electrically controlled light transmission of a twisted nematic layer, Phys. Lett A 48 (1974), 263–264.
- I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Providence, RI, 1969. MR 0246142 (39:7447)
- M. Grinfeld, M. Langer and N. J. Mottram, Nematic viscosity estimation using director kickback dynamics, to appear in Liquid Crystals (2011).
- J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981. MR 610244 (83j:35084)
- M. H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics, Vol. 20, Springer-Verlag, New York, 1995. MR 1351250 (96j:34095)
- J. R. Sambles and L. Z. Ruan, Testing the dynamic theory of nematics using fully-leaky guided modes and a convergent beam system J. Non-Newtonian Fluid Mech. 119 (2004), 39–49.
- I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: a Mathematical Introduction, The Liquid Crystals Book Series, Taylor and Francis, New York, 2004.
- C. Z. van Doorn, Dynamic behavior of twisted nematic liquid crystal layers in switched fields, J. Appl. Phys. 46 (1975), 3738-3745.
- M. D. van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, 1975. MR 0416240 (54:4315)
- F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Texts in Applied Mathematics, Vol. 50, Springer-Verlag, New York, 2005. MR 2148856 (2006k:34001)
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Additional Information
F. P. da Costa
Affiliation:
Departamento de Ciências e Tecnologia, Universidade Aberta, Rua da Escola Politécnica, 141, P-1269-001 Lisboa, Portugal, and Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais, 1, P-1049-001 Lisboa, Portugal
Email:
fcosta@univ-ab.pt
M. Grinfeld
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
Email:
m.grinfeld@strath.ac.uk
M. Langer
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
Email:
m.langer@strath.ac.uk
N. J. Mottram
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
Email:
n.j.mottram@strath.ac.uk
J. T. Pinto
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais, 1, P-1049-001 Lisboa, Portugal, and Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais, 1, P-1049-001 Lisboa, Portugal
Email:
jpinto@math.ist.utl.pt
Keywords:
Nematic liquid crystals,
kickback,
singular perturbations,
nonlocal operators
Received by editor(s):
June 10, 2010
Published electronically:
September 7, 2011
Article copyright:
© Copyright 2011
Brown University
