Participating Network Members: J.W. Barrett [JWB], (Imperial), P. Houston [PH], (Leicester), D.A Kay [DAK], (Sussex), C.J. Lawrance [CJL], (Imperial), E. Süli [ES], (Oxford).

Non-Newtonian fluids arise in numerous processes commonly employed in manufacturing industry. Due to the nonlinearities which are present in such materials, in industrially relevant geometries one has to resort to computational tools for physically realistic simulations and predictions of flow phenomena. The aim of the present project is to advance the quality and accuracy of existing computational tools by using state-of-the art numerical algorithms, including adaptive and multiscale finite element methods, as well as sparse finite element algorithms for high-dimensional problems.

JWB has made significant contributions to the mathematical analysis of finite element methods for non-Newtonian fluids. In collaboration with Professor Wenbin Liu (Kent), in a series of papers, he developed the regularity theory for partial differential equations modelling shear-thinning power-law type fluids, as well as the error analysis of finite element approximations to such flows. These are the sharpest error bounds to date available in the literature. Simultaneously, ES has supervised a succession of M.Sc. dissertations concerned with the development of adaptive finite element algorithms for power-law type non-Newtonian fluids. He has also secured a 3-year CASE studentship through the Smith Institute, with Unilever as industrial partner, starting in October 2003, in the area of finite element approximation of shear-thinning fluids.

During the Isaac Newton Institute programme on Computational Challenges in PDEs, JWB and ES have started to collaborate on the mathematical analysis of PDE systems modelling polymeric fluids. The focus of their work has been the question of the existence of solutions to a polymeric bead-spring flow model where molecules in the polymer chain are modelled as dumbbells (or beads connected with springs), suspended in a viscous flow.

The PDE system consists of the Navier-Stokes equations for the fluid subjected to an extra stress term, which depends on a probability distribution for the dumbells. This probability distribution in turn satisfies a Fokker-Planck equation, whose drag term depends on the fluid flow. Therefore the overall macroscopic/microscopic system is a coupled Navier Stokes/Fokker-Planck problem. The macroscopic extra stress tensor in the Navier-Stokes equations depends on the microscopic dynamics of the polymer chains.

If the dumbbell springs are assumed to be Hookean, then the above model corresponds exactly to the macroscopic model of an Oldroyd-B fluid, whose physical validity is now in doubt. A more interesting and physically realistic model is the FENE (finitely extensible nonlinear elastic) model, where the elongation of the dumbbell springs cannot exceed an input parameter. In this case there does not exist an equivalent macroscopic model.

In recent years there has been great interest in both the mathematical theory of kinetic polymer models of this type, as well as in the development of accurate computational algorithms for their approximate solution. ([5,6,7,9]). Despite considerable efforts, many relevant questions, such as the existence of global weak solutions, remain open.

In the present project, JWB and ES will collaborate on the development of the mathematical theory of existence and uniqueness of solutions to these equations. Some preliminary results in this directions are promising, and a joint publication in this direction, based on research pursued at the Isaac Newton Institute, is underway.

The high-dimensionality (the number of independent variables of the probability distribution in three space dimensions is 7) of the solution to the Fokker-Planck equation raises the question of efficient computational tools.

In collaboration with Professor Christoph Schwab (ETH Zürich), JWB and ES have embarked on the development and error analysis of sparse finite element methods for high-dimensional Fokker-Planck type equations. The aim in this thread of the project is to extend that work to polymeric flow problems where the Fokker-Planck equation is coupled to the incompressible Navier-Stokes equations through the extra-stress tensor. Together with PH, JWB and ES will also explore the use of adaptive Lagrange-Galerkin and discontinuous Galerkin finite element methods for polymeric flow problems. Deterministic sparse finite element algorithms of this type will serve as viable alternatives to coupled deterministic/stochastic computational algorithms proposed, for example, by Öttinger [8], where, instead of the Fokker-Planck equation, the associated stochastic differential equation is solved numerically; the overall rate of convergence of such coupled deterministic/stochastic algorithms is dominated by the slow convergence of the error in the stochastic component. Our overall objective is to overcome this accuracy limitation by using computationally efficient sparse finite element approximations to the high-dimensional (but deterministic) coupled Navier-Stokes/Fokker-Planck problem.

To meet the objectives in Theme II, we also wish to capitalise on the algorithmic developments discussed in Theme III below.