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Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

On various aspects of behaviour of polymers
Piotr Gwiazda (MIM University of Warsaw), Agnieszka Swierczewska-Gwiazda(MIM University of Warsaw) in cooperation with Willi Jager (Institute of Applied Mathematics, University of Heidelberg), Josef Malek (Faculty of Mathematics and Physics Charles University in Prague), Miroslav Bulicek (Faculty of Mathematics and Physics Charles University in Prague)
Our interest is directed to polymers. These are large molecules, which are made of repeating structures, like chains or more complicated structures. The primary examples of polymers appear in nature, like amber, cellulose or proteins. However, because of its interesting properties, synthetic polymers are widely produced. Among others we mention here paints, synthetic rubber, neoprene, nylon, PVC, polyester, teflon and silicone. The synthetic polymers are formed in the process of polymerization. The behaviour and properties of polymers strongly depend on its mictrostructure (configuration of monomers, their length and shape). The inverse process, namely converting a polymer into a monomer or a mixture of monomers is called a depolymerization. In any analysis of behaviour of synthetic polymers it is important to consider both of these processes. Indeed, lots of polymerization processes take place in the state close to equilibrium, hence polymerization will often happen in parallel with depolymerization.

The research goal of the project will have two directions:
1. analysis of the flow of polymers (described by non-newtonian rheology) with taking into account the behaviour on microscopic level
2. analysis on the microscopic level in a non-moving material including the diffusion among the particles

We describe both points in more details. The analysis in 1. will concern the system of equations describing the evolution of macroscopic quantities like the velocity and the pressure (eventually also the density and the temperature). The influence of the processes of polymerization and depolymerization will be accounted through the dependence of viscosity on the level of polymerization. The evolution of the material internal parameter (the level of polymerization) will be modelled by additional equation describing the polymerization process. The candidate for such a equation is a nonlinear size structure equation, cf. [6]. We have considerable in the field of nonlinear structured models, in the direction of existence, dependence on initial data, see [3][4][5]. Moreover, a Ph.D. student could benefit from the conducive environment. Two of the already realized projects concern the analysis of structure models (Project 1 and Project 2). None of them however overlaps with a project at hand, since the combination of structured equations and flow equations is a significant novelty here. The current project associates the problems of biology and fluid dynamics.

The primary question that arises in point 1. is the existence of solutions and then searching for an answer to the problem of uniqueness of solutions. In further step, the scientific computations can be conducted and numerical aspects of such constructed model will be examined. An interesting aspect is the analysis of singular limit of such a system. In particular, the limiting system is the description of flow with a rheology given by an implicit constitutive relation. We mention here, that our analytical background in such problems is confirmed by a series of common papers with our Czech partners, see [1][2]

In the second research goal we assume that the macroscopic velocity of the fluid is equal to zero. However, since the internal energy is nonzero, particles diffuse. Hence the analysis will focus on nonlinear diffusion equation (the diffusion coefficient depends on the integral term of the level of polymerization). On the microscopic level, the diffusion of particles is meaningful, contrary to the first point, where its influence on the flow is minor and hence neglected.

At the University of Warsaw the project will be supervised by Piotr Gwiazda and Agnieszka Swierczewska-Gwiazda. The exchange plan will include two parts, namely the project will be realized with two foreign partners: Charles University and University of Heidelberg. A Ph.D. student will spend 12 months in Prague (hosted by Josef Malek and Miroslav Bulicek). This stay will focus on the part of the project concerning the complex description of the flow (1st. point). The second part of the exchange programme will be realized at the University of Heidelberg (hosted by Willi Jager), lasting 6 months, and will mostly concern the problems related to diffusion equation.

[1] Bulicek, M., Gwiazda, P., Malek, J., Swierczewska, A. (2009) On steady flows of an incompressible uids with implicit power-law-like rheology, Adv. Calc. Var. 2, No. 2, 109-136, 2009
[2] Bulicek, M., Gwiazda, P., Malek, J., Swierczewska, A. (2010) On unsteady flows of implicitly constituted incompressible uids, submitted
[3] Gwiazda P., Lorenz T. and Marciniak-Czochra A. (2010) Flat metric and structural stability of a nonlinear population model. J. Di . Eq. 248 2703-2735
[4] Gwiazda P. and Marciniak-Czochra A. Structured population equations in metric spaces (2010) accepted, J. Hyp. Di . Eq
[5] Carrillo, J., Colombo, R., Gwiazda P., Ulikowska, A.
[6] Perthame B. (2007) Transport equations in biology. Birkhauser.