# Workshop Programme

## for period 22 July - 2 August 2013

### Mathematical approaches to complex fluids: a two-week summer school

22 July - 2 August 2013

Timetable

Monday 22 July | ||||

09:00-09:25 | Registration | |||

09:25-09:30 | Welcome from John Toland (INI Director) | |||

09:30-11:00 | Knobloch, E (University of California, Berkeley) |
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Localized structures in fluid flows | Sem 1 | |||

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 Swift-Hohenberg equation and the conserved Swift-Hohenberg equation and apply the results to systems modeled by the Navier-Stokes 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. |
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11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Shearer, M (North Carolina State University) |
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Surfactants and Thin Liquid Layers 1 | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Shearer, M (North Carolina State University) |
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Surfactants and Thin Liquid Layers 1 | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Kamien, R (University of Pennsylvania) |
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Topological defects in crystals and liquid crystals 1 | Sem 1 |

Tuesday 23 July | ||||

09:30-10:15 | Pismen, L (Technion - Israel Institute of Technology) |
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Bridging the scales near the contact line 1 | Sem 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 physico-chemical 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. |
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10:15-11:00 | Thiele, U (Loughborough University) |
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Gradient dynamics formulations of thin film equations | Sem 1 | |||

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 interface-dominated films and drops on solid substrates is introduced starting with the case of a single layer of simple non-volatile 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. |
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11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Holm, D (Imperial College London) |
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Geometric approach to modelling complex fluids 1 | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Holm, D (Imperial College London) |
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Geometric approach to modelling complex fluids 1 | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Stewart, I (University of Strathclyde) |
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Introduction to liquid crystal continuum theory: statics and dynamics | Sem 1 | |||

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 Ericksen-Leslie 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. |
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16:30-17:30 | Poster Session / Drinks Reception |

Wednesday 24 July | ||||

09:30-11:00 | Knobloch, E (University of California, Berkeley) |
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Localized structures in fluid flows | Sem 1 | |||

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 Swift-Hohenberg equation and the conserved Swift-Hohenberg equation and apply the results to systems modeled by the Navier-Stokes 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. |
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11:00-11:30 | Morning Coffee | |||

11:30-13:00 | Biferale, L (Università degli Studi di Roma Tor Vergata) |
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Lattice Boltzmann Methods for multi-phase turbulent flows | Sem 1 | |||

I will review recent results obtained by numerical simulations of multi-phase and/or multi-component 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. |
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13:00-13:30 | Lunch at Wolfson Court | |||

19:30-22:00 | Conference Dinner at Emmanuel College |

Thursday 25 July | ||||

09:30-11:00 | Stewart, I (University of Strathclyde) |
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Introduction to liquid crystal continuum theory: statics and dynamics | Sem 1 | |||

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 Ericksen-Leslie 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. |
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11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Holm, D (Imperial College London) |
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Geometric approach to modelling complex fluids 2 | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Holm, D (Imperial College London) |
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Geometric approach to modelling complex fluids 2 | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Ward, J (Loughborough University) |
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Fluid dynamic approaches to modelling bacterial biofilms growth | Sem 1 | |||

Biofilms are slimy colonies of bacteria that have settled on a fluid-solid 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 thin-film 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. |

Friday 26 July | ||||

09:30-11:00 | Kamien, R (University of Pennsylvania) |
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Topological defects in crystals and liquid crystals 2 | Sem 1 | |||

11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Pismen, L (Technion - Israel Institute of Technology) |
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Bridging the scales near the contact line 2 | Sem 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 physico-chemical 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. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Pismen, L (Technion - Israel Institute of Technology) |
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Bridging the scales near the contact line 3 | Sem 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 physico-chemical 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. |
||||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Thiele, U (Loughborough University) |
|||

Depinning transitions and deposition patterns | Sem 1 | |||

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 interface-dominated films and drops on solid substrates is introduced starting with the case of a single layer of simple non-volatile 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. |

Saturday 27 July | ||||

09:00-17:00 | Excursion |

Monday 29 July | ||||

09:30-11:00 | Rey, AD (McGill University) |
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Mechanics and Thermodynamics of Nematic Shape Equations | Sem 1 | |||

11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Gay-Balmaz, F (École Normale Supérieure) |
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An introduction to the variational principles and Poisson brackets for complex fluids | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Gay-Balmaz, F (École Normale Supérieure) |
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Equivalent theories of liquid crystals | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Kalliadasis, S (Imperial College London) |
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Recent progress on the moving contact line problem | Sem 1 | |||

The moving contact line problem is a long-standing 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 fluid-fluid interface, yet static fluid velocity at the solid satisfying the no-slip 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 test-bed system of spreading of a thin two-dimensional 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 finite-time 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 solid-liquid-gas diffuse-interface model is then presented, with no-slip 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, finite-time 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 non-local 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 |

Tuesday 30 July | ||||

09:30-11:00 | Ratiu, T (École Polytechnique) |
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Geometric approach to the Hamiltonian and Lagrangian formulation of complex fluids | Sem 1 | |||

11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Toner, J (University of Oregon) |
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Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 1 (Part 1) | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Toner, J (University of Oregon) |
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Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 1 (Part 2) | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Bertozzi, A (University of California, Los Angeles) |
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Particle laden thin films: theory and experiment | Sem 1 | |||

17:00-18:00 | Poster Session / Drinks Reception |

Wednesday 31 July | ||||

09:30-11:00 | Feng, J (University of British Columbia) |
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Computation of complex fluid flows | Sem 1 | |||

11:00-11:30 | Morning Coffee | |||

11:30-13:00 | Golestanian, R (University of Oxford) |
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Hydrodynamic Coordination at Low Reynolds Number | Sem 1 | |||

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 long-ranged hydrodynamic interaction between these active objects, which could lead to synchronization, coordination and other emergent many-body 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. |
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13:00-13:30 | Lunch at Wolfson Court |

Thursday 1 August | ||||

09:30-11:00 | Kevrekidis, Y (Princeton University) |
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Equation-free approach to deriving effective macroscopic equations for complex interacting systems | Sem 1 | |||

11:00-11:15 | Morning Coffee | |||

11:15-12:00 | Rey, AD (McGill University) |
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Capillary Models for Liquid Crystal Fibers, Membranes, Films, and Drops | Sem 1 | |||

12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Rey, AD (McGill University) |
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Capillary Models for Liquid Crystal Fibers, Membranes, Films, and Drops (Part 2) | Sem 1 | |||

14:45-15:00 | Afternoon Tea | |||

15:00-16:30 | Yeomans, J (University of Oxford) |
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Swimming at low Reynolds number | Sem 1 | |||

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 Navier-Stokes equations. This implies the well-known Scallop Theorem, that swimming strokes must be non-invariant 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. |

Friday 2 August | ||||

09:30-11:00 | Ratiu, T (École Polytechnique) |
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Geometric approach to the Hamiltonian and Lagrangian formulation of complex fluids | Sem 1 | |||

11:00-11:30 | Morning Coffee | |||

11:30-13:00 | Toner, J (University of Oregon) |
|||

Fish gotta swim, birds gotta fly, I gotta do Feynmann graphs 'till I die: A hydrodynamic theory of flocking 2 ¯ | Sem 1 | |||

13:00-13:30 | Lunch at Wolfson Court |