Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis

Reaktive Strömungen, Diffusion und Transport

Vivette Girault, Paris


• The stability of discrete schemes approximating grade-two (and related) fluid models
  
  
  
A grade-two fluid is a fluid of differential type, a theoretical model introduced by Rivlin and Ericksen for describing non-Newtonian behaviour. Its equations generalize the Navier-Stokes equations and it is believed to describe the motion of a water solution of polymers. Interestingly, some years ago, its equations were interpreted by Camassa, Holm, Marsden, Ratiu and Shkoller as a model of turbulence .

The numerical analysis of schemes approximating a grade-two fluid model is difficult and interesting because its difficulties are common to other non-Newtonian fluid models, such as Oldroyd-B models. So far, nothing is really known about their numerical analysis because in particular, they require the stability of the discrete Stokes projection in W^1,\infty and of the discrete transport projection in L^p for p>2. These problems are open, but there is one case where these stabilities are not required: the two-dimensional grade-two fluid model.

In a two-dimensional Lipschitz domain, a grade-two fluid model has a global solution without restriction on the size of the data, exactly as for the Navier-Stokes equations. This remarkable property is due to the fact that, when the problem is well-formulated, solutions can be constructed without requiring that the velocity be bounded in W^1,\infty. This is crucial because when the good formulation is discretized, and the finite-element spaces well-chosen, we can establish that the resulting schemes are stable and hence perform successfully their numerical analysis.

I shall present here several stable numerical schemes discretizing the two-dimensional grade-two fluid model, show convergence of simple algorithms for computing the discrete solution and compare with other schemes for which no stability has yet been established.

This is common work with R. Scott, University of Chicago.



Josef Malek, Prag


• On Analysis of Flows of Fluids with Pressure and Shear Dependent Viscosities
  
  
  
There is clear and incontrovertible evidence that the viscosity of many liquids depends on the pressure. While the density, as the pressure is increased by orders of magnitude, suffers small changes in its value, the viscosity changes dramatically. It can increase exponentially with pressure. In many fluids, there is also considerable evidence for the viscosity to depend on the rate of deformation through the symmetric part of the velocity gradient, and most fluids shear thin, i.e., viscosity decreases with an increase in the rate of shear. In the talk, we study the flow of fluids whose viscosity depends on both the pressure and the symmetric part of the velocity gradient. We find that the shear thinning nature of the fluid can be gainfully exploited to obtain global existence of solution, which would not be possible otherwise. Previous studies of fluids with pressure dependent viscosity require strong restrictions to all data, or assume forms that are clearly contrary to experiments, namely that the viscosity decreases with the pressure. We are able to establish:

(i) existence of spatially periodic solutions that are global in time for both the two and three dimensional problem, without restricting ourselves to small data (joint result with J. Necas and K.R. Rajagopal);

(ii) existence of steady flows subjected to no-slip boundary conditions, again without restricting ourselves to small data (the result obtained jointly with M. Franta and K.R. Rajagopal);

(iii) a partial regularity result for weak solutions of stationary problems (joint work with G.R. Mingione and J. Stara).

Within the lecture, the result (ii) will be discussed in detail.



Adelia Sequeira, Lissabon


• Mathematical and Numerical Modelling of Fluid-Structure Interaction Problems:
  Applications to Blood Rheology
  
  
  
The complexities of the human vascular system are far from being completely understood from either the pysiological or mathematical perspective. Blood flow interacts both mechanically and chemically with vessel walls giving rise to complex fluid-structure interactions whose mathematical analysis is still incomplete and which are difficult to simulate in an efficient manner.
At a macroscopic level, the arterial wall is a complex multi-layer structure that deforms under blood pressure. Even though sophisticated constitutive equations have been proposed for the structural behaviour of the vessel wall, its elastic characteristics in-vivo are still very difficult to determine and are usually inferred from pulse propagation data. Blood is a suspension of highly flexible particles (red blood cells, white blood cells and platelets) in a complex aqueous polymer solution, the plasma, containing inorganic and organic salts, proteins and transported substances. It is commonly accepted that in the framework of large and medium vessels, blood can be modelled as a Newtonian incompressible fluid, at a first level of approximation. However, depending on the size of the blood vessels relative to those of the cells, blood behaves as a shear-thinning fluid and also exhibits viscoelastic properties that cannot be neglected, at least in small arteries where the vessel diameters are comparable with the one of blood cells. In particular the high viscosity behaviour of blood at low shear rates is due to red blood cells aggregation and low viscosity at high shear rates is a consequence of deformability of red blood cells. Also stretching of the elastic red blood cells and their consequent storage of elastic energy account for the memory effects in blood. The complex rheological behaviour of blood and its interaction with the vascular walls play an important role in the physiology of blood circulation. Mathematical and numerical models in haemod ynamics together with the development of computer simulations, being directed to a better understanding of vascular diseases can play a relevant role in clinical research and, in a longer term in medical diagnosis and therapy.

In this talk we address the fluid-structure interaction problem of an incompress ible generalized Newtonian fluid flowing inside a thin compliant vessel whose walls undergo small deformations under the action of the fluid. The numerical approach is based on the coupling of the fluid equations in a moving domain, described in an Arbitrary Lagrangian Eulerian (ALE) frame, with a simple structural model for the vessel wall. The system is completed with suitable matching condittions which play the role of boundary conditions for the submodels.
In the second part of the talk, we focus our attention on some preliminary issues related to specific shear-thinning viscoelastic models for blood flow.



Noel Walkington, Pittsburgh


• Simulating Fluids Exhibiting Microstructure
  
  
  
Liquid crystals, fluids containing elastic particles, and polymer fluids, all exhibit non trivial macroscopic behavior due to interactions occurring on micro/mesoscopic scales. The derivation of rational models for these materials is a non-trivial task and results in formidable systems of partial differential equations. Of these problems the most complete theory is for Ericksen's model of liquid crystals, and I will begin with a discussion of this theory and the corresponding numerical analysis. Multi-component fluid systems will then be considered where elastic effects may enter by virtue of surface tension or elasticity of embedded particles. I will discuss existence results and convergence of numerical approximations for some specific multi-component models.

This is joint work with C. Liu (Pennsylvania State University, USA)