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20/11/2006, 14:00 — 15:00 — Room P3.10, Mathematics Building

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Workshop on Shallow-Water Equations

14.00-14.40 **Carine Lucas** (LMC/IMAG - Université Joseph Fourier)

*Shallow water type models derivations: Boundary conditions and stress tensors effects*

**Abstract:** Shallow water type models derivations are not well understood. In this talk, we will focus on various setting around boundary conditions and stress tensors effects during the modeling process. We will try to present various models and describe some of their mathematical properties.

14.45-15.25 **Guillaume Riflet** (MARETEC - IST)

*A simple numerical model for shallow-water equations*

**Abstract:** The shallow water equations are introduced and discretized in a standard C-grid with a leapfrog, FCTS numerical scheme combined with simple Asselin-Roberts filtering as presented in Kantha and Clayson "Numerical models of Oceans and oceanic processes". Plain Dirichlet conditions were implemented at the boundaries. Simple testing were performed with a gaussian level elevation. The geostrophic equilibrium of a gaussian level elevation is presented and an analytical solution of the steady-state is obtained. Results show that Dirichlet boundary conditions reflect all surface waves back inside the domain and multiple linear superpositions occur.

15.30-16.00 **Coffee Break**

16.30-17.10 **Christian Bourdarias** (Université de Savoie)

*A finite volume scheme for a model coupling free surface and pressurised flows in pipes*

**Abstract:** A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a well-balanced Roe-like finite volume scheme adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.

17.15-17.55 **Marguerite Gisclon** (Université de Savoie)

*A kinetic formulation for a model coupling free surface and pressurised flows in pipes*

**Abstract:** We are interesting in flows occuring in closed pipes. It may happen that some parts of the flow are free-surface and the other parts are pressurised. The phenomenom of transition from free surface to pressurised flow occur in many situations as storm sewers, waste or supply pipes in hydorelectric installations. We propose a kinetic approach for a model coupling transient free-surface and pressurised flows in closed pipes. For each type of flow, we build a kinetic fomulation and verify (in some general case) that it leads to a Gibbs equilibirum that minimises an energy and preserves the still water steady state. Despite some limitations due to geometric constraints, the corresponding kinetic scheme shows a very good behaviour.

18.00-18.40 **Juha H. Videman** (CEMAT-IST)

*A nonlinear Reynolds equation for thin viscous flows*

**Abstract:** We derive a nonlinear second-order differential equation for the pressure approximation in hydrodynamic lubrication. This equation, in contrast to the classical Reynolds equation, takes into account both the inertial and the curvature effects and its solution corresponds to the first two terms in the asymptotic pressure expansion. The equation is rigorously justified with optimal error estimates in parameter dependent Sobolev norms and in Hölder norms. It is also applied to the classical problem of journal bearing.