Mathematical approaches to complex fluids: a twoweek summer school
Monday 22nd July 2013 to Friday 2nd August 2013
09:00 to 09:25  Registration  
09:25 to 09:30  Welcome from John Toland (INI Director)  INI 1  
09:30 to 11:00 
E Knobloch (University of California, Berkeley) Localized structures in fluid flows
In these two lectures I will show examples of spatially localized structures arising in different types of fluid flows and will describe physical and mathematical approaches that have proved useful to understanding the origin of localization and the properties of the localized structures that result. I will illustrate these ideas using model equations such as the SwiftHohenberg equation and the conserved SwiftHohenberg equation and apply the results to systems modeled by the NavierStokes equation on the one hand and dynamical density function theory on the other. I will also remark on the motion of fronts separating different but structured phases and relate the growth of localized structures to the problem of structure formation on growing domains.

INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00  Surfactants and Thin Liquid Layers 1  INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45  Surfactants and Thin Liquid Layers 1  INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
R Kamien (University of Pennsylvania) Topological defects in crystals and liquid crystals 1 
INI 1 
09:30 to 10:15 
Bridging the scales near the contact line 1
Interface is where the macroscopic meets the microscopic; even a simple fluid becomes there a complex fluid. The origins of surface tension and disjoining pressure lie in nanoscale density gradients governed by molecular interactions. We shall see how the paradox of the moving contact line is resolved on the molecular scale when it is viewed as a physicochemical problem dependent on fluid–substrate interactions.
There is enormous scale separation between molecular and hydrodynamic scales, which makes computation difficult but facilitates analytical theory. We ascend from molecular to macroscopic scales  from density functional theory to lubrication equations  by the approximation ladder. Multiscale perturbation theory elucidates dynamics of the contact line and provides tools for the study of various instabilities, as demonstrated taking as an example the motion of droplets driven by surface forces. 
INI 1  
10:15 to 11:00 
Gradient dynamics formulations of thin film equations
The course starts with a brief review of a number of experiments on
dewetting and evaporating thin films/drops of simple and complex
liquids. Then the concept of a gradient dynamics description of the
evolution of interfacedominated films and drops on solid substrates
is introduced starting with the case of a single layer of simple
nonvolatile liquid, and advancing towards the formulation for films
of mixtures. The second part of the course uses the obtained models to investigate depinning transitions and deposition patterns in a number of different settings that can all be described by the introduced evolution equations. An extended abstract and a reference list may be found in the attached .txt file. 
INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00 
D Holm (Imperial College London) Geometric approach to modelling complex fluids 1 
INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
D Holm (Imperial College London) Geometric approach to modelling complex fluids 1 
INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
I Stewart (University of Strathclyde) Introduction to liquid crystal continuum theory: statics and dynamics
An introduction to the mathematical modelling of liquid crystals will be presented. A brief review of the static theory will lead in to a presentation of the EricksenLeslie theory for the dynamics of nematic liquid crystals. Recent developments related to smectic and other liquid crystals phases will also be discussed. Applications to model the influence of flow in 'switching phenomena' (e.g., the time taken to switch a pixel 'on' or 'off') in flat panel liquid crystal displays (LCDs and LEDs) will also be discussed.

INI 1  
16:30 to 17:30  Poster Session / Drinks Reception 
09:30 to 11:00 
E Knobloch (University of California, Berkeley) Localized structures in fluid flows
In these two lectures I will show examples of spatially localized structures arising in different types of fluid flows and will describe physical and mathematical approaches that have proved useful to understanding the origin of localization and the properties of the localized structures that result. I will illustrate these ideas using model equations such as the SwiftHohenberg equation and the conserved SwiftHohenberg equation and apply the results to systems modeled by the NavierStokes equation on the one hand and dynamical density function theory on the other. I will also remark on the motion of fronts separating different but structured phases and relate the growth of localized structures to the problem of structure formation on growing domains.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 13:00 
Lattice Boltzmann Methods for multiphase turbulent flows
I will review recent results obtained by numerical simulations of multiphase and/or multicomponent flows using Lattice Boltzmann Methods. In particular, I will discuss limitations and potentialities of the numerical method to study boiling systems and droplet dispersion under strong turbulent conditions.

INI 1  
13:00 to 13:30  Lunch at Wolfson Court  
19:30 to 22:00  Conference Dinner at Emmanuel College 
09:30 to 11:00 
I Stewart (University of Strathclyde) Introduction to liquid crystal continuum theory: statics and dynamics
An introduction to the mathematical modelling of liquid crystals will be presented. A brief review of the static theory will lead in to a presentation of the EricksenLeslie theory for the dynamics of nematic liquid crystals. Recent developments related to smectic and other liquid crystals phases will also be discussed. Applications to model the influence of flow in 'switching phenomena' (e.g., the time taken to switch a pixel 'on' or 'off') in flat panel liquid crystal displays (LCDs and LEDs) will also be discussed.

INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00 
D Holm (Imperial College London) Geometric approach to modelling complex fluids 2 
INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
D Holm (Imperial College London) Geometric approach to modelling complex fluids 2 
INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
Fluid dynamic approaches to modelling bacterial biofilms growth
Biofilms are slimy colonies of bacteria that have settled on a fluidsolid interface. They are ubiquitous and often undesirable and present numerous challenges in medicine and industry. They consist of bacteria, polymeric substances and water to form a porous structure that changes as the biofilm grows and matures. In this talk, mathematical models will be presented to describe biofilm growth as an expanding viscous fluid. The talk will broadly be in two parts. Firstly, a model for the early stages of biofilm development using thinfilm approaches, where Depending on the strength of interaction between bacteria and the substratum two limits naturally arise. Secondly, a model that uses ideas from mixture theory to describe mature biofilm development, enabling prediction of fluid flow regimes within the biofilm structure.

INI 1 
09:30 to 11:00 
R Kamien (University of Pennsylvania) Topological defects in crystals and liquid crystals 2 
INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00 
Bridging the scales near the contact line 2
Interface is where the macroscopic meets the microscopic; even a simple fluid becomes there a complex fluid. The origins of surface tension and disjoining pressure lie in nanoscale density gradients governed by molecular interactions. We shall see how the paradox of the moving contact line is resolved on the molecular scale when it is viewed as a physicochemical problem dependent on fluid–substrate interactions.
There is enormous scale separation between molecular and hydrodynamic scales, which makes computation difficult but facilitates analytical theory. We ascend from molecular to macroscopic scales  from density functional theory to lubrication equations  by the approximation ladder. Multiscale perturbation theory elucidates dynamics of the contact line and provides tools for the study of various instabilities, as demonstrated taking as an example the motion of droplets driven by surface forces. 
INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
Bridging the scales near the contact line 3
Interface is where the macroscopic meets the microscopic; even a simple fluid becomes there a complex fluid. The origins of surface tension and disjoining pressure lie in nanoscale density gradients governed by molecular interactions. We shall see how the paradox of the moving contact line is resolved on the molecular scale when it is viewed as a physicochemical problem dependent on fluid–substrate interactions.
There is enormous scale separation between molecular and hydrodynamic scales, which makes computation difficult but facilitates analytical theory. We ascend from molecular to macroscopic scales  from density functional theory to lubrication equations  by the approximation ladder. Multiscale perturbation theory elucidates dynamics of the contact line and provides tools for the study of various instabilities, as demonstrated taking as an example the motion of droplets driven by surface forces. 
INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
Depinning transitions and deposition patterns
The course starts with a brief review of a number of experiments on
dewetting and evaporating thin films/drops of simple and complex
liquids. Then the concept of a gradient dynamics description of the
evolution of interfacedominated films and drops on solid substrates
is introduced starting with the case of a single layer of simple
nonvolatile liquid, and advancing towards the formulation for films
of mixtures. The second part of the course uses the obtained models to investigate depinning transitions and deposition patterns in a number of different settings that can all be described by the introduced evolution equations. An extended abstract and a reference list may be found in the attached .txt file. 
INI 1 
09:00 to 17:00  Excursion 
09:30 to 11:00  Mechanics and Thermodynamics of Nematic Shape Equations  INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00 
F GayBalmaz (École Normale Supérieure) An introduction to the variational principles and Poisson brackets for complex fluids 
INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
F GayBalmaz (École Normale Supérieure) Equivalent theories of liquid crystals 
INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
Recent progress on the moving contact line problem
The moving contact line problem is a longstanding and fundamental challenge in the field of fluid dynamics, occurring when one fluid replaces another as it moves along a solid surface. Moving contact lines occur in a vast range of applications, where an apparent paradox of motion of a fluidfluid interface, yet static fluid velocity at the solid satisfying the noslip boundary condition arises. In this talk we will review recent progress on the problem made by our group.
The motion of a contact line is examined, and comparisons drawn, for a variety of proposed models in the literature. We first scrutinise a number of models in the classic testbed system of spreading of a thin twodimensional droplet on a planar substrate, showing that slip, precursor film and interface formation models effectively reduce to the same spreading behaviour. This latter model, developed by Shikhmurzaev a few years ago, is a complex and somewhat controversial one, differentiating itself by accounting for a variation in surface layer quantities and having finitetime surface tension relaxation. Extensions to consider substrate heterogeneities in this prototype system for slip models are also considered, such as for surface roughness and fluctuations in wetting properties through chemical variability. Analysis of a solidliquidgas diffuseinterface model is then presented, with noslip at the solid and where the fluid phase is specified by a continuous density field. We first obtain a wetting boundary condition on the solid that allows us to consider the motion without any additional physics, i.e. without density gradients at the wall away from the contact line associated with precursor films. Careful examination of the asymptotic behaviour as the contact line is approached is then shown to resolve the singularities associated with the moving contact line problem. Various features of the model are scrutinised alongside extensions to incorporate slip, finitetime relaxation of the chemical potential, or a precursor film at the wall. But these are not necessary to resolve the moving contact line problem. Ongoing work to rigorously include nonlocal terms into models for contact line motion based on density functional theory will be discussed, with work analysing the contact line in equilibrium presented. **Joint work with David Sibley, Andreas Nold & Nikos Savva 
INI 1 
09:30 to 11:00 
T Ratiu (École Polytechnique) Geometric approach to the Hamiltonian and Lagrangian formulation of complex fluids 
INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00 
J Toner (University of Oregon) Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 1 (Part 1) 
INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
J Toner (University of Oregon) Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 1 (Part 2) 
INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
A Bertozzi (University of California, Los Angeles) Particle laden thin films: theory and experiment 
INI 1  
17:00 to 18:00  Poster Session / Drinks Reception 
09:30 to 11:00  Computation of complex fluid flows  INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 13:00 
Hydrodynamic Coordination at Low Reynolds Number
Microorganisms and the mechanical components of the cell motility machinery such as cilia and flagella operate in low Reynolds number conditions where hydrodynamics is dominated by viscous forces. The medium thus induces a longranged hydrodynamic interaction between these active objects, which could lead to synchronization, coordination and other emergent manybody behaviors. In my talk, I will examine these effects using minimal models that are simple enough to allow extensive analysis that sheds light on the underlying mechanisms for the emergent phenomena.

INI 1  
13:00 to 13:30  Lunch at Wolfson Court 
09:30 to 11:00  Equationfree approach to deriving effective macroscopic equations for complex interacting systems  INI 1  
11:00 to 11:15  Morning Coffee  
11:15 to 12:00  Capillary Models for Liquid Crystal Fibers, Membranes, Films, and Drops  INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45  Capillary Models for Liquid Crystal Fibers, Membranes, Films, and Drops (Part 2)  INI 1  
14:45 to 15:00  Afternoon Tea  
15:00 to 16:30 
J Yeomans (University of Oxford) Swimming at low Reynolds number
I shall introduce the hydrodynamics that underlies the way in which microorganisms, such as bacteria and algae, and fabricated microswimmers, swim. For such tiny entities the governing equations are the Stokes equations, the zero Reynolds number limit of the NavierStokes equations. This implies the wellknown Scallop Theorem, that swimming strokes must be noninvariant under time reversal to allow a net motion. Moreover, biological swimmers move autonomously, free from any net external force or torque. As a result the leading order term in the multipole expansion of the Stokes equations vanishes and microswimmers generically have dipolar far flow fields. I shall introduce the multipole expansion and describe physical examples where the dipolar nature of the bacterial flow field has significant consequences, the velocity statistics of a dilute bacterial suspension and tracer diffusion in a swimmer suspension.

INI 1 
09:30 to 11:00 
T Ratiu (École Polytechnique) Geometric approach to the Hamiltonian and Lagrangian formulation of complex fluids 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 13:00 
J Toner (University of Oregon) Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 2 ¯ 
INI 1  
13:00 to 13:30  Lunch at Wolfson Court 