Philippe Angot

**Five selected articles as the most significative ones** 1. **Philippe Angot**: Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows, Math. Meth. Appl. Sci. **22**(16), 1395--1412, 1999.This work studies the asymptotic behaviours of the generalized Darcy-Brinkman equation with variable effective porosity and permeability to model incompressible viscous flows in complex fluid-porous-solid systems within volume penalty and fictitious domain methods. These original techniques were early proposed in *Angot and Caltagirone (WCCM2, 1990)*. For fluid-porous systems with an infinite permeability in the pure fluid, the resulting error estimate is optimal with respect to the penalization parameter. For fluid-solid systems, both the $L^{2}$ and $H^{1}$ volume penalty method are investigated with a permeability going to zero in the solid region and/or an effective viscosity going to infinite in the solid obstacle. The latter can be used to penalize the strain rate tensor to get the rigidity of a solid particle transported in the flow. An optimal error estimate is obtained for the $H^{1}$ volume penalty method. Moreover, a simple estimate of the resulting force applied to the solid obstacle is provided. These results are generalized in *Angot et al. (1999)* for the unsteady inertial flows with the Navier-Stokes/Brinkman equations. The numerical validation of this fictitious domain approach is carried out in *Khadra et al. (2000)* for isothermal or convective heat transfer flow problems with non body-fitted obstacles. Moreover, the volume penalty method is extended for hyperbolic systems of conservation laws in *Angot et al. (2014)*. Due to their simplicity, these techniques that have now become standard, are widely used in Computational Fluid Dynamics and incorporated in many numerical simulation codes including commercial software. 2. **Philippe Angot, Franck Boyer and Florence Hubert**: Asymptotic and numerical modelling of flows in fractured porous media, ESAIM: Math. Model. Numer. Anal. **43**(2), 239--275, 2009.This paper follows the ideas proposed in *Angot (2003)* and *Angot et al. (FVCA4, 2005)* to derive dimensionally-reduced models by asymptotic modelling for the flows in fractured porous media. The developed models are shown to be well-posed. Moreover, an original finite volume method is detailed for the numerical solution and the convergence of this scheme is proved. Several numerical validations are provided including benchmark problems. Up to our knowledge, this study is the first in the literature considering fully immersed fractures in the porous media. This topic is now the subject of intense research activity because of its numerous applications to real-world problems. 3. **Philippe Angot, Jean-Paul Caltagirone and Pierre Fabrie**: A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems, Appl. Math. Lett. **25**(11), 1681--1688, 2012. This paper details and validates an original time-splitting scheme, the so-called vector penalty-projection method, briefly proposed in *Angot et al. (FVCA6, 2011)* for the numerical solution of variable-density or multiphase Navier-Stokes equations with strong stresses. This is an efficient alternative to the well-known Chorin-Temam projection method with a scalar pressure correction made by the solution of a Poisson equation. The method is based on new fast discrete Helmholtz-Hodge decompositions in bounded domains studied in *Angot et al. (2013)*. Unlike all scalar projection methods, the vector projection method is shown to be fast and accurate for multiphase flows with large density (up to $10^{6}$) or viscosity (up to $10^{17}$) ratios, as demonstrated in the benchmark problem of free fall of an heavy ball. In a more recent paper *Angot et al. (2016)*, an original kinematic version of this method is proposed. Unlike to usual methods which fail for density ratios of several hundreds, our method remains stable and accurate for multiphase flow benchmark problems of bubble dynamics (with surface tension) with a density ratio up to $10^{4}$ (for air bubles in a liquid melted steel). Moreover, there is no spurious eddies up to machine precision close to the interface between the fluid phases that is one of the common drawbacks of all numerical methods in this topic. In *Angot and Cheaytou (2019)*, the vector penalty-projection method is also shown to be fast and accurate for open boundary conditions. Indeed, the optimal second-order precision is fully recovered unlike to usual time-splitting schemes. 4. **Philippe Angot**: Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions, ESAIM: Math. Model. Numer. Anal. **52**(5), 1875--1911, 2018.This work proves the global solvability in time of the unsteady Stokes/Brinkman and Stokes/Darcy coupled problems with no restriction on the size of the data for the macroscale modelling of viscous flows in fluid-porous systems. The work follows the ideas and the mathematical framework briefly proposed for the steady problems in *Angot (2010, 2011)* to deal with jump interface conditions of both the tangential velocity and stress vectors. The new jump interface conditions considered here are theoretically derived in *Angot et al. (2017)* for the non-inertial fluid-porous flow using the volume averaging method and a suitable asymptotic analysis. Moreover, they are shown to be physically relevant for arbitrary flow directions and they are inherently able to take account of anisotropic effects unlike *ad-hoc* extended Beavers-Joseph interface conditions that are most often used in the literature. In particular, the well-posedness is proved whatever the size of the slip coefficient $\\alpha\\geq0$ for the full Beavers-Joseph jump condition. The previous results only proved the solvability either for the approximate and simplified Beavers-Joseph-Saffman condition (where the effective velocity jump is no more taken into account) or for complete Beavers-Joseph's condition such that $\\alpha^2$ is sufficiently small. 5. **Philippe Angot, Benoît Goyeau and J. Alberto Ochoa-Tapia**: A nonlinear asymptotic model for the inertial flow at a fluid-porous interface, Adv. Water Res. **149**, 103798, 2021 (online 30 October 2020).In this theoretical study of mathematical modelling, we develop an asymptotic analysis of the homogenized Navier-Stokes equations in the thin transition porous layer between the pure fluid and the homogeneous porous medium. This yields an original nonlinear and multi-dimensional interface model for the macroscale inertial flow over a permeable medium. These new jump interface conditions that inherently include the anisotropic effects are shown to be physically meaningful for arbitrary inertial flow directions. The present theory also enables us to analyze the dependence on porosity of the slip and friction coefficients included in the proposed jump interface conditions. Moreover, the coupled Navier-Stokes/Darcy-Forchheimer fluid-porous model supplemented with the present set of interface conditions is shown to be globally dissipative. This is the first nonlinear and multi-dimensional globally dissipative model proposed in the literature for the inertial flow at a permeable interface with arbitrary flow directions.