Scales and scalings in nematic liquid crystals and beyond (1 ECTS, MAT/05)

Prof. Arghir D. Zarnescu - Basque Centre for Applied Mathematics (Bilbao)

Date and time
Monday, November 16, 2020

Contact person
Giacomo Canevari

Publication date
August 26, 2020

Computer Science  


The aim of this course is to introduce some simple yet fundamental and powerful physical ideas related to physical dimensions and scales, dimensional analysis and nondimensionalisation.  These are  concepts universally relevant to physical systems and we will demonstrate this on a couple of toy physical models, as well  as a on a more realistic model, used in describing nematic liquid crystals. 

Furthermore we will look into the mathematical aspects related to the presence of small scales exploring ideas related to regular/singular perturbations and homogenisation. The course will mix standard introductory examples and ideas from the current  research (reformulated at a technically accessible level).

Please note: All lectures are now available in Panopto, as pre-recorderd videos. Lecture notes will be found on the Moodle page of the mini-course.

Tentative plan of lectures:
1. Units of measurements and physical scales-dimensional analysis
2. Nondimensionalisations and various scalings
3. Small scales and regular perturbations
4. Singular perturbations-boundary layers
5. Singular perturbations-vortices
6. Multiple scales expansions 
7. Homogenisation
1. Alama, S., Bronsard, L., & Lamy, X. (2016). Minimizers of the Landau-de Gennes energy around a spherical colloid particle. Archive for Rational Mechanics and Analysis, 222(1), 427-450.
2. Cioranescu, D., & Donato, P. (1999). An introduction to homogenization (Vol. 17, pp. x+-262). Oxford: Oxford University Press.
3. Barenblatt, G. I. (2003). Scaling (Vol. 34). Cambridge University Press.
4. Canevari, G., & Zarnescu, A. (2020). Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation. Mathematical Models and Methods in Applied Sciences, 30(02), 309-342
5. Gartland, Jr., E. C. (2018). Scalings and limits of Landau-de Gennes models for liquid crystals: a comment on some recent analytical papers. Mathematical Modelling and Analysis, 23(3), 414-432.
6. Holmes, M. H. (2009). Introduction to the foundations of applied mathematics.
7. de Jager, E. M., & Furu, J. F. (1996). The theory of singular perturbations. Elsevier.
8. Rusconi, S., Dutykh, D., Zarnescu, A., Sokolovski, D., & Akhmatskaya, E. (2020). An optimal scaling to computationally tractable dimensionless models: Study of latex particles morphology formation. Computer Physics Communications, 247, 106944.

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