NCMW - Modelling, Analysis and New Mathematical Perspectives for Complex Fluids and Liquid Crystals (2022)

Speakers and Syllabus

Synopsis

Research in materials and complex fluids has witnessed unprecedented growth in recent years with the advent of metamaterials, nano-materials, biomaterials, polymers,microfluidics, smart fluids etc. In particular, soft materials that are intermediate in character between conventional solids and liquids, have attracted huge academic interest and in fact, several soft materials are also classified as complex fluids with unusual mechanical, optical and rheological properties. Nematic liquid crystals are paradigm examples of soft materials and complex fluids. Nematics combine the fluidity of liquids with the orientational order of conventional solids i.e. they have distinguished special directions, referred to as “directors”.Consequently, they have a direction-dependent response to external fields and light, making them the working material of choice for the multi-billion liquid crystal display industry, along with new applications in microfluidics, smart devices, photonics, actuators etc.

The mathematics of complex fluids and nematics is broad and rich, spanning multiple branches of mathematics such as calculus of variations, nonlinear partial differential equations, numerical analysis, topology, stochastic analysis and scientific computation. In this training school, we will deliver six different lecture courses touching different mathematical aspects of this challenging field. Each course comprises four core lectures with one discussion session. The target audience are postgraduate students in analysis, numerical methods, stochastics and probability theory and applied mathematics. The training school is also suitable for postdoctoral researchers, early-career academics and indeed any researcher wishing to learn about current cutting-edge research in complex fluids and nematic liquid crystals in a mathematical framework. The training school will also offer excellent networking opportunities and foster research connections between academics in the UK and in India.

Course Descriptions

The training school is structured to be a set of introductory courses to the mathematics, modelling, analysis and applications of complex fluids and nematic liquid crystals. Stephen Wilson and Apala Majumdar will give broad overview courses of complex fluids and nematic liquid crystals respectively. Michael Grinfeld will discuss powerful ODE tools for different types of liquid crystals, Utpal Manna will lecture on stochastic approaches that complement and supplement deterministic approaches, Misha Osipov will describe parallel approaches from statistical mechanics and Neela Nataraj will teach the basics of numerical analysis for systems of nonlinear partial differential equations, with nematic liquid crystals as an illustrative concept. We provide brief course descriptions below.

There are six speakers in this proposed workshop and each will be taking four lectures and one discussion session. Details of the speakers are given below.

 sr.n Name and affiliation of the speakers  Course details
1 Dr Michael Grinfeld (University of Strathclyde) is a leading expert in the theory of differential equations, particularly ordinary differential equations and has extensively used ODE techniques in materials science, economics and biomathematics. He has lectured internationally in India, Laos, Ghana, South Africa. Course Title: ODE techniques in liquid crystals.
Summary: Many problems in liquid crystals lead to genuinely important one dimensional PDES, and hence ODE techniques are of use when dealing with stationary solutions of these PDEs and with travelling waves. In this course we will discuss the Freedericksz transition with weak anchoring in nematic liquid crystals and a travelling wave problem in smectic C* liquid crystals.
  • Lecture 1: The physical background for the Freedericksz transition and the formulation of the boundary value problem;
  • Lecture 2: Time maps, the Hamilton-Jacobi equation and convexity methods;
  • Lecture 3: The physical set-up of smectic C* liquid crystals, travelling waves in general and classic results about minimality;
  • Lecture 4: Variational methods for minimality, the anisotropic case and the exact solvability puzzle.

Discussion session: Suggestions for research projects.
2 Professor Utpal Manna (IISER Thiruvananthapuram) works in the areas of stochastic partial differential equations with applications to hydrodynamic models, liquid crystals, micromagnetism etc. He is the recipient of Kerala Young Scientist Award (2014) and various international grants, e.g. Royal Society Grant, DUO-India Professor Fellowship Grant. He is a visiting professor at University of Sydney and University of York. Course Title: Stochastic Analysis of Nematic Liquid Crystals
Summary : In this course, we will discuss about the description of Ericksen-Leslie model for nematic liquid crystals and analyse the need of a stochastic model in order to understand the Freedericksz transition in a better way. We shall also draw motivation and connection with various other geometric PDEs where similar phase transition phenomena are observed.
  • Lecture 1: Ericksen-Leslie model and solvability theory;
  • Lecture 2: Motivation and description of stochastic models;
  • Lecture 3: Basics of stochastic calculus; stochastic differential equations etc;
  • Lecture 4: Solvability theory of stochastic models; large deviation theory towards understanding of phase transition phenomena.

Discussion session: Suggestions for research projects.
3 Professor Apala Majumdar (University of Strathclyde) is a leading expert in the mathematics and applications of nematic liquid crystals, having being awarded the British Liquid Crystal Society Young Scientist Prize in 2012 and the Cyril Hilsum Medal in 2020. She specializes in the calculus of variations and theory of nonlinear partial differential equations, with emphasis on variational theories in materials science, with over fifty publications to her credit and multiple international projects. She is also a Visiting Professor at IIT Bombay and the University of Bath. Course Title: Introduction to Mathematics of Liquid Crystals.
Summary: This course will be an introductory course to the powerful Landau-de Gennes theory for nematic liquid crystals, the governing systems of nonlinear coupled partial differential equations, phase transitions and some special solutions of physical relevance.
  • Lecture 1: Introduction to Nematic Liquid Crystals
  • Lecture 2: The Landau-de Gennes Theory for Nematic Liquid Crystals
  • Lecture 3: The Isotropic-Nematic Phase Transition
  • Lecture 4: The Landau-de Gennes Euler-Lagrange Equations – existence, regularity and some exact solutions.

Discussion session: Suggestions for research projects.
4 Professor Neela Nataraj (IIT Bombay) works as a Professor in Department of Mathematics, Indian Institute of Technology Bombay is currently the Professor-in-Charge of the IIT Bombay Monash Research Academy. Some of her areas of research interest are finite element methods, finite volume methods and discontinuous Galerkin methods for linear and nonlinear elliptic problems. She has more than 50 research publications in international journals. She is also the Convener, Executive Committee, Indian Women and Mathematics and a Member of the International Mathematics Union Committee for Women in Mathematics. Course Title: Numerical Analysis of Partial Differential Equations
Summary: This course will be an introductory course to finite element analysis of systems of elliptic partial differential equations, with the Landau-de Gennes Euler-Lagrange equations as a benchmark example.
  • Lecture 1: Linear elliptic problems, Weak formulation, Well-posedness – inf-sup condition;
  • Lecture 2: Conforming Finite Element Method (FEM) and a priori error estimates;
  • Lecture 3: Discontinuous Galerkin methods: consistency, symmetry, penalty terms, energy norm error estimate, lower order estimates using duality arguments;   
  • Lecture 4: FEM formulation for Landau de Gennes equilibrium equations.

Discussion session: Suggestions for research projects.
5 Professor Misha Osipov (University of Strathclyde) is an internationally recognized expert in the molecular theories for liquid crystals and related materials. He has published more than 150 papers including 8 reviews, and has received the Merkator Guest Professor Award from German Science Foundation, Hilsum medal from British Liquid Crystal Society and Frederiksz Medal from Russian Liquid Crystal Society. Course Title: Statistical Mechanics approaches to Nematic Liquid Crystals
Summary: In this short course we focus into the molecular statistical theory of the nematic phase which only possesses the orientational order. However, even in this case the phase transitions may be highly nontrivial because the nematic phase may be either uniaxial or biaxial. The nematic phase may also be composed of chiral molecules which results in spontaneous formation of a macroscopic helical structure. The theory of such helical ordering will also be considered.
  • Lecture 1: Interaction potentials including multipolar, dispersion and induction interactions;
  • Lecture 2: Molecular theory of isotropic-uniaxial nematic phase transition. Mean-field and density functional approaches;
  • Lecture 3: Molecular models for biaxial nematic LCs;
  • Lecture 4: Chirality effects in nematic LCs including a molecular theory of helical twisting and the effect of temperature and concentration on the helical pitch.

Discussion session: Suggestions for research projects.
6 Professor Stephen Wilson (University of Strathclyde) holds the 1984 Chair in Mathematics at the University of Strathclyde in Glasgow and has made notable contributions to fluid mechanics, including thin-film flows, microfluidics, liquid crystals, non-Newtonian fluids (including viscoplastic fluids, thixotropic fluids and nanofluids) etc. The common theme is the use of mathematical (namely asymptotic, analytical and numerical) methods to bring new insights into a wide range of “real world” problems and he is joint Editor-in-Chief of the Journal of Engineering Mathematics published by Springer Nature. Course Title: Complex Fluids: Theory and Applications
Summary: The aim of this course is to give an overview of the theory of complex fluids and to give examples of its application to a number of real-world situations, including both industrial and geophysical contexts. Particular attention will be paid to both viscoplastic and thixotropic fluids, and examples of recent research on both will be described.
  • Lecture 1: Overview of continuum modelling of complex fluids, including a variety of standard non-Newtonian models;
  • Lecture 2: Overview of models describing shear thinning and shear thickening behaviour, and illustrative examples of flows described by these models;
  • Lecture 3: Overview of models describing viscoplastic behaviour, and illustrative examples of flows described by these models (including channel flow and viscoplastic rivulet flow);
  • Lecture 4: Overview of models describing thixotropic behaviour, and illustrative examples of flows described by these models (oscillatory boundary layer flow and channel flow).

Discussion session: Suggestions for research projects.

 Course Descriptions

The training school is structured to be a set of introductory courses to the mathematics, modelling, analysis and applications of complex fluids and nematic liquid crystals. Stephen Wilson and Apala Majumdar will give broad overview courses of complex fluids and nematic liquid crystals respectively. Michael Grinfeld will discuss powerful ODE tools for different types of liquid crystals, Utpal Manna will lecture on stochastic approaches that complement and supplement deterministic approaches, Misha Osipov will describe parallel approaches from statistical mechanics and Neela Nataraj will teach the basics of numerical analysis for systems of nonlinear partial differential equations, with nematic liquid crystals as an illustrative concept. We provide brief course descriptions below.

 

Time Table


Day

Date

 

 

 

Lecture 1

(13.30-14.30)

 

Lecture 2

(14.30–15.30)

Discussion/ Tutorial (Optional)

Lecture 3
(16.30 -17.30)

Mon

18.07.2022

 

 

 

AM

 

MO

Open

UM

Tues

19.07.2022

 

 

 

UM

 

 

Open

AM

Wed

20.07.2022

 

 

 

UM

 

MO

AM

AM

Thu

21.07.2022

 

 

 

UM

 

MO

UM

AM

Fri

22.07.2022

 

 

 

NN

 

MG

SW

SW

Mon

25.07.2022

 

 

 

NN

 

MG

NN

SW

Tues

26.07.2022

 

 

 

NN

 

MG

 

SW

Wed

27.07.2022

 

 

 

NN

 

MG

MG

SW

 

  •  Dr Michael Grinfeld - MG   
  •  Prof. Utpal Manna - UM
  •  Prof. Apala Majumdar - AM
  •  Prof. Neela Nataraj - NN
  •  Prof. Misha Osipov - MO
  •  Prof. Stephen Wilson - SW
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