The meeting is held at The University of Warwick, Mathematics Institute
Organisers: P. Plechac (Warwick), J.W. Barrett (Imperial), C.M. Elliott (Sussex), E. Suli (Oxford)

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The modelling of polymeric fluids by kinetic equations raises interesting and original mathematical and numerical questions, since it implies the coupling of two different mathematical objects : partial differential equations (PDEs) and stochastic differential equations (SDEs). Indeed, the evolution of the velocity is given by the momentum equation, and the motion of the polymer chains into the fluid is modelled by Langevin equations. The stress tensor then depends on the conformation of the polymer chains, so that the equations at the macroscopic and at the microscopic levels are coupled. We are interested in the well-posedness of such system, and also in the analysis of classical numerical schemes used to discretize these equations. These numerical schemes couple methods of discretization of PDEs (finite element methods) with methods of discretization of probabilistic equations (Monte Carlo methods and Euler schemes on SDEs). We will in particular explain how these methods interact in the variance of the results in a simple case. Finally some questions related to the long-time behaviour of the system will be raised.

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A phase field model is considered for the solidification of a binary alloy. A parabolic, strongly nonlinear system has to be solved, the unknowns being the solid phase and the alloy concentration. Existence, a priori and a posteriori error estimates can be proved when the phase anisotropy is small. In order to reduce the number of vertices, adaptive finite elements with high aspect ratio are used. Numerical results are presented. The numerical procedure is then extended to a multiphase field model pertaining to the coalescence of two dendrites.

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The multi-phase field approach originally proposed by Steinbach's group [1] can handle in principle the formation of several phases during phase transformations. However, when the derivation of the energy functional is done in an appropriate way using a Lagrange multiplier, as proposed by Nestler and Wheeler [2], a few problems arise at the junction between more than two phases (triple points in 2 dimensions). This has been observed in particular when two grains of the same phase, $\alpha$, which solidify from the liquid meet (coalescence). When the grain boundary energy is larger than twice the solid-liquid interfacial energy, the situation is "repulsive" if one uses a sharp interface approach. This means that a certain undercooling is required to make the grain boundary dry. However, when using a multi-phase field approach with the Lagrange multiplier technique, an initial attractive stage occurs due to the double-well potential formulation. The present talk will first present the general problem of coalescence and its importance with respect to solidification. It will then introduce the sharp interface and the multi-phase field approaches, with emphasis on the problems associated with this later technique. A few possible improvements will be outlined.
[1] J. Tiaden, B. Nestler, H.-J. Diepers, I. Steinbach: The Multiphase-Field Model with an Integrated Concept for Modelling Solute Diffusion. Physica D 115 (1998) 73-86.
[2] B. Nestler, A. A. Wheeler, A Multi-phase-field Model of Eutectic and Peritectic Alloys: Numerical Simulation of Growth Structure, Physica D 138 (2000) 114-133.

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Movies: Coal Intra [AVI], Coal Inter [AVI]

Full Talk:Part 1 [Powerpoint], Part 2 [Powerpoint], Part 3 [Powerpoint], Part 4 [Powerpoint].

In [Nonlinearity 17, 477(2004)], a diffuse-interface approximation of a step flow model for epitaxial growth which reproduces an arbitrary Ehrlich-Schwoebel barrier has been proposed. It is a version of the Cahn-Hilliard equation with variable mobility. In this talk, we propose a discretisation for this diffuse-interface approximation. Our approach is guided by the fact that the diffuse-interface approximation has a conserved quantity and a Liapunov functional. We obtain an implicit finite volume discretisation of symmetric structure.
We test the discretisation by comparison with the results from the matched asymptotic analysis. We also test the diffuse-interface approximation itself by comparison with theoretically known features of the original free boundary problem. More precisely, we investigate quantitatively the phenomena of step-bunching and the Bales-Zangwill instability.

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Exact Langevin equations for the fluctuations of the occupation numbers in driven lattice models are derived from their master equations. The asymptotic equivalence of the Langevin description and the lattice models is demonstrated by direct comparison with kinetic Monte Carlo simulations. The passage to the continuum limit is then considered for models with analytic and with non-analytic transition rules. As an example of the first case, we show how the stochastic equation of motion for the asymmetric exclusion process yields Burgers' equation in the continuum limit. For the second case, we consider a model of random deposition followed by instantaneous local relaxation based on (i) height minimization and and (ii) coordination maximization to obtain the Edwards-Wilkinson equation in the continuum limit in both cases. We conclude with a discussion about the application of our results to address several issues that have been uncovered by recent simulations of lattice models.

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We use the lattice Boltzmann algorithm as a tool to investigate the hydrodynamic properties of complex fluids. As a first example we consider a strongly non-Newtonian fluid, cholesteric liquid crystals. When these are forced along a direction parallel to the helical axis the apparent viscosity increases by several orders of magnitude. We explain some puzzles in the Helfrich theory of this effect. We then investigate the dynamics of micron-scale droplets spreading on chemically and topologically patterned substrates. For a substrate with lyophobic and lyophilic stripes of the same dimensions as the drop the final droplet shape is determined by the dynamic evolution of the fluid. For a substrate patterned with posts the substrate becomes superhydrophobic -- like the back of the Nambian desert beetle.

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