## Section: Overall Objectives

### Presentation

The aim of this project is to develop modelling tools for problems involving fluid mechanics in order to explain, to control, to simulate and possibly to predict some complex phenomena coming from physics, chemistry, biology or scientific engineering. The complexity may consist of the model itself, of the coupling phenomena, of the geometry or of non-standard applications. The challenges of the scientific team are to develop stable models and efficient adapted numerical methods in order to recover the main physical features of the considered phenomena. The models will be implemented into numerical codes for practical and industrial applications.

We are interested in both high and low Reynolds number flows, interface and control problems in physics and biology.

Our scientific approach may be described as follows. We first determine some reliable models and then we perform a mathematical analysis (including stability). We then develop the efficient numerical methods, which are implemented for specific applications.

In the next paragraphs, we explain our main goals, we describe our project in terms of development of numerical techniques and we present the team with the competence of the members.

#### The main goals

##### Modelling

The first goal of the project consists in modelling some complex phenomena. We combine the term model with the three following adjectives: phenomenological, asymptotical and numerical.

*Phenomenological* : use of ad-hoc models in order to represent some
precise phenomena.
One example of such modelling process is
the construction of nonlinear differential laws for the stress tensor
of visco-elastic fluids or for wormlike micelles.
Another example is the wall law conditions in microfluidics (fluids in micro-channels) that are often taken heuristically in order to
model the slip at the boundary.

In biology, since no fundamental laws are known, the modeling is exclusively phenomenological especially concerning the modeling of tumor growth.

*Asymptotical* : using asymptotic expansions, we derive simpler models containing all
the relevant phenomena.
Examples of such a process are the
penalization method for the simulation of incompressible flows with obstacles or the analysis of riblets in microfluidics that are used to control the mixing of the fluids. Another example is the use of shallow fluid models in order to obtain fast predictions (Hele-Shaw approximation in microfluidics) or the approximation of thin membranes for the modeling of electroporation of cells.

*Numerical* : direct numerical tools are used to simulate the modelized physical phenomena.
A precise analysis of the models is performed to find out the most convenient numerical method in terms of stability, accuracy and
efficiency. A typical example is the POD (proper orthogonal decomposition) and its use
in control theory, or in data assimilation in tumor growth, to obtain fast simulations.

##### Analysis and computation

Once the model has been determined, we perform its mathematical analysis. This analysis includes the effect of boundary conditions (slip conditions in microfluidics, conditions at an interface...) as well as stability issues (stability of a jet, of an interface, of coherent structures). The analysis can often be performed on a reduced model. This is the case for an interface between two inviscid fluids that can be described by a Boussinesq-type system. This analysis of the system clearly determines the numerical methods that will be used. Finally, we implement the numerical method in a realistic framework and provide a feedback to our different partners.

##### Applications

Our methods are used in four areas of **applications**.

1)*Interface problems and complex fluids*:

This concerns microfluidics, complex fluids (bifluid flows, miscible fluids). The challenges are to obtain reliable models that can be used by our partner Rhodia (for microfluidics).

2)*High Reynolds flows and their analysis*:

We want to develop numerical methods in order to address the complexity of high Reynolds flows. The challenges are to find scale factors for turbulent flow cascades, and to develop modern and reliable methods for computing flows in aeronautics in a realistic configuration.

3) *Control and optimization*: the challenges are the drag reduction of a ground vehicle in order to decrease the fuel consummation,
the reduction of turbomachinery noise emissions or the increase of
lift-to-drag ratio in airplanes, the control of flow instabilities to alleviate material fatigue for
pipe lines or off-shore platforms and the detection of embedded defects in materials with industrial and
medical applications.

4) *Tumor growth*: The challenge is to produce patient-specific simulations starting from medical imaging for growth of metastasis to the lung of a distant tumor.

Our main partners on this project will be :

*Industrial:* Renault, IFP, CIRA (Centro italiano ricerche
aerospaziali), Airbus France and Boeing for high Reynolds flows, optimization and control and Rhodia (biggest french company of chemistry) for interface problems and complex fluids.

*Academic:*
CPMOH (Laboratory of Physics, Bordeaux 1 University) for high Reynolds flows, optimization and control,
and Institut Gustave Roussy (Villejuif), University of Alabama at Birmingham and Institut Bergonié (Bordeaux) for tumor growth, optimad (spin-of of the Politecnico de Torino) for simulations
of complex flows.

#### The production of numerical codes

We want to handle the whole process from the modelling part until the simulations. One of the key points is to develop numerical codes in order to simulate the models that are studied with our partners and of course we want to be able to have some feed-back toward the experiments.

i)*Multi-fluid flows and interface problems:*

We perform 2D and 3D simulations of multi-fluid flows using level set methods and mixture models. This includes non newtonian flows such as foams or wormlike miscella. The applications are microfluidics, porous media and complex fluids.

ii) *Modeling of tumor growth:*

Tumor growth in our 3D numerical model includes a cell-cycle, diffusion of oxygen, several population of cells, several enzymes, molecular pathways, angiogenesis, extracellular matrices, non-newtonian effects, membrane, effects of treatments, haptotaxy, acidity. We perform data assimilation processes starting from medical imaging.

iii) *2D and 3D simulations at high Reynolds number:*

We develop various computational methods: multi-grid techniques, vortex methods. The possible applications are turbulence, the flow around a vehicle, the stress on a pipe-line (the penalization method is used in order to take into account the obstacles).

iv) *Fluid structure interactions:*

2D and 3D interaction of a mobile rigid body with a fluid thanks penalty methods.

From a technical point of view, our work is organized as follows. We have build a platform (called **eLYSe**) using only cartesian, regular meshes. This is motivated by the following: we want to address interface problems using level set methods and to take into account obstacles by the penalization method.
For these interface problems,
we will have to compute the curvature of the interface with high precision (in microfluidics, the surface
tension is the leading order phenomenon). The level set technology is now very accurate on structured meshes, we therefore made
this choice. However, we want to address cases with complex geometry and/or obstacles. We will therefore systematically use the penalization method. The idea is to have an uniform format for the whole team that consists of several boxes:

1) Definition of the geometry and of the penalization zones.

2) Specification of the model (bifluid or not, Newtonian or not, mixing or not, presence of membranes etc...)

3) The boundary conditions that have to be imposed by a penalization operator.

4) The solvers.

5) Graphic interface.

As said before, the interface problems and the interaction with a membrane will be handled by level set methods as well as the shape optimization problem. So this platform will be dedicated to direct numerical simulation as well as to shape optimization and control.

The main effort concerning modelling will concern points 2) and 3) (model and boundary conditions). We do not plan for the moment to make special research effort on the solver part and we will use the solvers available in the literature or already developed by the team.

This platform will have two roles: the first one will be to allow a comprehensive treatment for the simulation of complex fluids with interface, membranes, adapted to the world of physical-chemistry and microfluidics and for solving shape optimization problems. The second role will be to keep a set of numerical modules that will be devoted to more specific applications (for example multi-grid methods or vortex methods for the study of turbulence). We therefore need to have some unified standards for the geometry or the graphic interface but it is of course hopeless to consider 3D turbulence and low-Reynolds flows in a micro-channel with the same code !