A general formulation of phase-field models for nonisothermal solidification in multicomponent and multiphase alloy systems is derived from an entropy functional in a thermodynamically consistent way. General expressions for the free energy densities, for multicomponent diffusion coefficients, and for both weak and faceted types of surface energy and kinetic anisotropy are possible. A three-dimensional simulator is developed to show the capability of the model to describe phase transitions, complex microstructure formation, and grain growth in polycrystalline textures.

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L 2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen-Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H 1 spaces, and we employ triangulated surfaces and H 1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction -diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/ maskae/CV_Warwick/Chemotaxis.html.

We present an abstract framework for treating the theory of wellposedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.

Abstract. Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are related to sharp interface models. Both cases of dynamic as well as instantaneous adsorption are covered. Flexibility with respect to the choice of bulk and surface free energies allows us to realise various isotherms and relations of state between surface tension and surfactant. Some numerical simulations display the effectiveness of the presented approach.Key words. Two-phase flow, surfactant, phase field model, adsorption isotherm subject classifications. 35R35, 35R01, 76T99, 76D45, 35C20, 35Q351. Introduction Surface active agents (surfactants) reduce the surface tension of fluid interfaces and, via surface tension gradients, can lead to tangential forces resulting in the Marangoni effect. Biological systems take advantage of their impact on fluids with interfaces, but surfactants are also important for industrial applications such as processes of emulsification or mixing. While often much experience and knowledge is available on how surfactants influence the rheology of multi-phase fluids, the goal is to understand how exactly the presence of a surfactant influences coalescence and segregation of droplets.Surfactants can be soluble in at least one of the fluid phases and the exchange of surfactants between the bulk phases and the fluid interfaces is governed by the process of adsorption and desorption. Ward and Tordai [55] derived a time-dependent relation for the surfactant density at the interface and the surfactant density at the adjacent bulk phase (known as the sub-layer or sub-surface). To compute the interfacial density, a closure relation between the two quantities has been proposed in the form of several different equilibrium isotherms [18,33,32], where the underlying assumption is that the interface is in equilibrium with the sub-layer at all times. This corresponds to the case of diffusion-limited adsorption studied in Diamant and Andelman [16], where the process of adsorption to the interface is fast compared to the kinetics in the bulk phases. However, instantaneous adsorption is not valid in the context of ionic surfactant systems [16] or when the diffusion is not limited to a thin layer [12,13,14]. Therefore, we would like to be able to account for non-instantaneous adsorption in our models.Two-phase flow with surfactant is classically modelled with moving hypersurfaces describing the interfaces separating the two fluids. We will derive the following sharp interface model for a domain Ω containing two fluids of different mass densities. We denote by Ω(1) (t), Ω (2) (t) the domains of the fluids which are separated by an interface