Particulate flows, Mixture flows
Pollutant transport and dispersion, and the dynamic of sprays appear as complex inhomogeneous flows,where a disperse phase interacts with a dense phase. There are many possible modeling of such flows:microscopic models where the two phases occupy distinct domains and coupling arises through intricate interface conditions; macroscopic models which are of hydrodynamic (multiphasic) type, involving nonstandard state laws, possibly with non conservative terms, and the so–called mesoscopic models. The latter are based on Eulerian–Lagrangian description where the disperse phase is described by a particle distribution function in phase space. Following this path we are led to a Vlasov-like equation coupled to a system describing the evolution of the dense phase that is either the Euler or the Navier-Stokes equations. It turns out that the leading effect in such models is the drag force. Hydrodynamic regimes can be identified but the nature of the coupled hydrodynamic system could be quite intriguing and original so that it should also be discussedcarefully. For instance we can derive that way models for mixtures which looks like the Navier–Stokes equation but where the divergence free condition is replaced by a constraint involving derivatives of the density. The mathematical analysis of these models, known as the Graffi and Smagulov-Kazhikov equations, remains in its infancy. The constraint induces new and nontrivial difficulties. The treatment of these difficulties certainly requires a clear understanding of the underlying modeling issues (for miscible flows modeling it relies on the distinction between the mean mass velocity and the mean volume velocity). Definitely, the constraint cannot be treated by a mere adaptation from the incompressible case. For instance it induces a complex interplay between the regularity of the approximation to be used for the density and the velocity so that the naive approaches rapidly exhibit instability phenomena. Another example is given by models accounting for energy exchanges because unusual coupling terms appear in the hydrodynamic regime, that can be interpreted as convection-diffusion terms, with coefficients depending in a complex way on density and temperature. For such limit models, there is no “natural” scheme; hence coming back to the microscopic modeling and designing an Asymptotic Preserving scheme can be an efficient way to simulate the macroscopic model.
Flows in Porous media
We are interested in modeling issues and the development of specific numerical methods, based on Finite Volume discretization for treating multiphases flows in porous media. This topics is highly motivated by applications related to CO2 storage, oil recovery and nuclear waste disposal.
Low Mach flows, Radiative Transfer
Low Mach flows arise in many combustion phenomena. The interesting fact for our purpose relies on the non
standard constraint that appears in such regime so that the limit model shares some features of the Mixtures
models. Furthermore, for many applications, like for instance the description of fire in tunnels, the validation
of fire prevention strategies, the design of industrial furnaces… it makes sense to couple the Zero Mach system
to a kinetic equation describing energy exchanges by radiation. Hence this subject is appealing for the project,
both in terms of potential applications and of interesting technical developments.
Members of the team have developed an original research devoted to the modeling and simulation of biological
damage on monuments and algae proliferation in the Mediterranean Sea. First of all, one phenomenon
responsible for biodegradation of monuments is the formation of biofilms, namely a colony of bacteria embedded
within an extra-cellular matrix. A proposed model comes from the mixture theory and leads to a complex
multi-dimensional hydrodynamic-type system. Moreover, the algae proliferation in the Mediterranean Sea
contains two phases : a development one on the sea bed as a biofilm and a spreading one in water which can
be described thanks to coagulation-fragmentation equations. The coupling of these two types of equations is
of great interest and suits perfectly the current project. Let us also notice that the bacteria yet studied, the
cyanobacteria, are deeply considered in order to produce energy as bio-fuel. Another question studied for the
moment in the biological setting but which can be linked to the other points of the project is the analysis and
simulation of hyperbolic type equations in inhomogeneous media, like porous media. This is a direction to
improve the existing models in biology and it can give rise in analytical and numerical viewpoints to fruitful
exchanges between the biological domain and the environmental one.
Plasmas physics is a privileged topic because of the development of magnetic and inertial confinement fusion.
Again, it naturally leads to multiscale, non linear problems, with unusual coupling. Compared to other research
teams already working on this subject, our main purpose will be to work on the derivation, analysis and
simulation of reduced models, that can be used for instance for routine computations, and to address questions
related to the coupling of different models through domain decomposition.