New Challenges in Fluid Flow Simulations

Fayssal Benkhaldoun

Laboratoire Analyse, Géométrie et Application (LAGA),

Université Sorbonne Paris Nord, France

In this presentation, an overview will be proposed about the use of efficient methods for solving fluid flow problems governed by systems of nonlinear partial differential equations, and applied to problems of the industry and the environment. In particular, we will present a family of Eulerian-Lagrangian schemes dedicated to the approximation of hyperbolic systems in the context of finite volumes and avoiding the use of Riemann solvers. We will show the contribution of convergence analysis, error estimates and dynamic mesh refinement methods. Recent applications in particular in Non-Newtonian fluid flows will be presented.

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Optimal ship forms based on Michell's wave resistance

Morgan Pierre

Université de Poitiers (France)

joint work with Julien Dambrine and Germain Rousseaux

In 1898, J.H. Michell provided a formula for the wave resistance of a thin ship moving at constant speed in calm water. By adding a simple viscous resistance term, we obtain a model for the total resistance of water to the motion of the ship. This total resistance functional allows us to seek optimal ship hulls. We investigate some mathematical and numerical aspects of the optimal hull, first when the domain of parameters is fixed, and second when this domain varies.

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A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity

Rim Aldbaissy (1), Frédéric Hecht (2), Gihane Mansour (1), Toni Sayah (1), Pierre Henri Tournier (2)

1.  Laboratoire de Mathématiques et Applications, URMM, CAR, Faculté des Sciences, Université Saint-Joseph de Beyrouth, Lebanon.

2. UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Sorbonne, Universités, France.

In this work, we study the time dependent Boussinesq (buoyancy) model with nonlinear viscosity depending on the temperature. We propose and analyze first order numerical scheme based on finite element methods. An optimal a priori error estimate is then derived for the numerical scheme. For the numerical experiment, we study the thermal instability that appears from time to time while printing using a 3D printer. To solve the semi-discretized problem at each time-step, we use a scalable parallel algorithm based on two-level Optimized Restricted Additive Schwarz (ORAS) domain decomposition preconditioner for GMRES. Parallel scalability tests are conducted with comparison against the parallel direct solver MUMPS and the one-level Schwarz method, which show lack of robustness for larger number of processors. 2D numerical tests illustrate that the number of iterations to reach GMRES convergence depends on the state of the physical simulation during time, and that the second level of preconditioning is needed to achieve robustness.

Keywords: Boussinesq, Buoyancy, Navier-Stokes equations, heat equation, finite elementmethod, a priori error estimates, Parallel computation, domain decomposition method, preconditioner, high-performance computing, two-level method.

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Numerical analysis for the shallow water model with two velocities

Emmanuel Audusse (2,3), Nelly Boulos Al Makary (2,3), Nina Aguillon (1,3), Martin Parisot (4)

1. Sorbonne Universités, UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France

2. USPN Univ Sorbonne Paris Nord, CNRS UMR 7539, Institut Galilée, LAGA, Villetaneuse, France.
3. Inria, ANGE team, 2 rue Simone Iff, F-75012 Paris, France
4.Inria, CARDAMOM team, INRIA Bordeaux-Sud-Ouest. 200 Avenue de la Vieille Tour. 33405 Talence cedex, France

The Shallow water equations (also called Saint-Venant's equations) are the usual model governing fluid flow in the rivers, channels or the oceans. They are used, for example, for the protection of the environment, the prediction of tides and storm urges, the transport of the sediment or the study of floods. Some references in the literature propose an improvement of the Shallow water equations to take into account the vertical profile of the horizontal velocity. The objective of this work is to develop a scheme of the model with two velocities in the vertical profil based on an analysis of the Riemann problem. We look for a scheme able to exactly recover any subcritical steady solution in 1D over arbitrary topography. To do so, first we analyse the steady solutions following the Bernouilli's principle, even for supercritical regime. We then propose a well- balanced Riemann solver following a strategy proposed in a previous study. Finally, we validate our results with numerical simulations.

Keywords: Shallow water, Riemann problem, Well-balanced, Steady states, Critical flows

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A posteriori error estimates and adaptivity taking into account algebraic errors

Martin Vohralik (1,2)

1. SERENA, Inria Paris, France
2. CERMICS, Ecole des Ponts, 77455 Marne-la-Vall´ee, France

This talk addresses the derivation of a posteriori error estimates and of adaptive strategies for numerical discretizations of partial differential equations when the solution of the underlying (large sparse) systems of linear algebraic equations is taken into account. I will start with the model Laplace equation and show how guaranteed upper and lower bounds on the total error can be obtained, while also developing guaranteed upper and lower bounds on both the discretiza-tion and algebraic error components, following [4, 5, 6]. These results enable to recover the mass balance in any situation. Moreover, they lead to safe stopping criteria for algebraic solvers that guarantee that the algebraic error does not dominate the total error, while avoiding un-necessary iterations. An hp-refinement strategy in presence of inexact solvers is then described, following [3]. It specifically leads to a computable guaranteed bound on the error reduction fac-tor between the consecutive hp-refinement steps. A generic framework for arbitrary (residual) functionals is then presented following [1]. Concrete applications are considered finally, namely adaptive inexact iterative algorithms for the Stokes problem following [2], and an industrial application to multi-phase multi-compositional porous media Darcy flows following [7].

References

[1] J. Blechta, J. M´alek, and M. Vohral´ık, Localization of the W −1,q norm for local a posteriori efficiency, IMA J. Numer. Anal.40(2020), 914-950, DOI10.1093/imanu/drz002

.[2] M. Cerm´ak, F. Hecht, Z. Tang, and M. Vohral´ık, Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem, Numer. Math. 138 (2018), 1027-1065, DOI 10.1007/s00211-017-0925-3.

[3] P. Daniel, A. Ern, and M. Vohral´ık, An adaptive hp-refinement strategy with inexact solvers and computable guaranteed bound on the error reduction factor, Comput. Methods Appl. Mech. Engrg. 359 (2020), 112607, DOI 10.1016/j.cma.2019.112607.

[4] A. Mira¸ci, J. Papeˇz, and M. Vohral´ık, A multilevel algebraic error estimator and the cor-responding iterative solver with p-robust behavior, SIAM J. Numer. Anal.58(2020), 2856-2884, DOI 10.1137/19M1275929.

[5] J. Papeˇz, U. R¨ude, M. Vohral´ık, and B. Wohlmuth, Sharp algebraic and total a pos-teriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation, Comput. Methods Appl. Mech. Engrg.371(2020), 113243, DOI 10.1016/j.cma.2020.113243.

[6] J. Papeˇz, Z. Strakoˇs, and M. Vohral´ık, Estimating and localizing the algebraic and to-tal numerical errors using flux reconstructions, Numer. Math.138(2018), 681-721, DOI 10.1007/s00211-017-0915-5.

[7] M. Vohral´ık, S. Yousef, A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows, Comput. Methods Appl. Mech. Engrg.331 (2018), 728-760, DOI 10.1016/j.cma.2017.11.027.

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Numerical analysis of a chemotaxis-swimming bacteria model on a general triangular mesh

George Chamoun (1), Mazen Saad (2), Raafat Talhouk (1)

1. Laboratoire de Mathématiques, ESDT and Faculty of Sciences, Lebanese University, Lebanon. rtalhouk@ul.edu.lb, georges.chamoun.1@ul.edu.lb

2. Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, 1 rue de la Noé, Nantes, France. mazen.saad@ec-nantes.fr

This work is devoted to the numerical study of a model arising from biology, consisting of chemotaxis equations coupled to incompressible fluid equations through transport and external forcing. A detailed convergence analysis of this chemotaxis-fluid model by means of a suitable combination of the finite volume method and the nonconforming finite element method is investigated. In the case of nonpositive transmissibilities, a correction of the diffusive fluxes is necessary to maintain the monotonicity of the numerical scheme. Finally, many numerical tests are given to illustrate the behavior of swimming bacteria in a fluid.

Keywords: Degenerate parabolic equation; Navier-Stokes equations;heterogeneous and anisotropic diffusion; combined scheme

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Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

Alessio Falocchi(1) and Filippo Gazzola (2)

1. Politecnico di Torino, Italy

2. Politecnico di Milano, Italy

For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least C^2). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest, that we call sectors.

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A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids

Pardeep Kumar, Kush Kinra, Manil T. Mohan

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA.

Abstract: In this talk, we consider an inverse problem for three dimensional viscoelastic fluidflow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.

Keywords: Kelvin-Voigt fluids equation, inverse problem, memory kernel, integral overdetermination condition, contraction mapping principle.

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Nonlinear and dispersive waves in a basin

Dimitrios Mitsotakis

School of Mathematics and Statistics, Victoria University of Wellington, New Zealand

Surface water waves of significant interest such as tsunamis and solitary waves are nonlinear and dispersive waves. Unluckily, the equations describing the propagation of surface water waves known as Euler's equations are immensely hard to solve. In this presentation we show that among the so many simplified systems of PDEs proposed as alternative approximations to Euler's equations there is only one proven to be well-posed (in Hadamard's sense) in bounded domains with slip-wall boundary conditions. We also show that the system obeys most of the physical laws that acceptable water waves equations must obey. Validation with laboratory data is also presented.

Keywords: Boussinesq systems, initial-boundary value problems, well-posedness, solitarywaves, dispersive waves

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Relative energy approach to a diffuse interface model of a compressible two-phase flow

Madalina Pectu

Université de Poitiers, France.

We propose a simple model for a two-phase flow with a diffuse interface. The model couples the compressible Navier-Stokes system governing the evolution of the fluid density and the velocity field with the Allen-Cahn equation for the order parameter. We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a consequence, we show the weak-strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter. Such a result plays a crucial role in the analysis of the associated numerical schemes. The weak-strong uniqueness principle allows us also to perform the low Mach number limit obtaining the standard incompressible model.

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Modeling and numerical simulation of flows in closed conduits

Stéphane Gerbi

Laboratoire de Mathématiques

UMR 5127 - CNRS et Université Savoie Mont Blanc,

73376 Le Bourget-du-Lac Cedex

France.

In this talk, I will present a modeling of flows in closed conduits such as penstock of a hydroelectric dam, conduit of a sewer network, channel of supply to a dam. These pipes are either partially filled : in this case the water is considered to be incompressible, the flow is at a free surface and we then make the Saint-Venant approximation of small thickness compared to the length of the pipe. The pressure is then hydrostatic. When they are completely filled, the pipe is said to be in charge, the water is considered to be weakly compressible and we derive a model in the same spirit as that of Saint-Venant. The pressure is then an "acoustic" pressure We couple the two "very close" models by unique variables. The pressure terms and the source terms are also defined in a unified way. We show the mathematical properties of the model : hyperbolicity of the system, existence of an entropy....

We develop a mixed Finite Volumes-Kinetic Method numerical scheme which preserves the mathe-matical properties of the problem.

Finally, I will show some simulations of mixed flows in pipes and flows on a dry bottom.

This is a joint work with Christian Bourdarias, LAMA, USMB.

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Stabilized Gauge Uzawa Scheme for an incompressible micropolar fluid flow

Sarah Slayi, Toufic El Arwadi and Séréna Dib

Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Lebanon.

Abstract: In this talk, we introduce and analyze a second order Gauge Uzawa scheme for the governing equations of an incompressible micropolar fluid flow. The derivation of this scheme is obtained using the second order backward difference approximation. The unconditional stability of the GUM scheme for the micropolar equations will be shown. We establish an a priori error estimate to prove the convergence. Finally, we present some numerical simulations that confirm the theoretical results.

Keywords: Micropolar Problem, Projection Method, Theory Error Estimate, Gauge UzawaScheme.

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Global solvability of convective Brinkman-Forchheimer equations

Manil T. Mohan

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India.

Abstract: The convective Brinkman -Forchheimer (CBF) equations describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. In this talk, we examine the existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality for CBF equations in bounded domains. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the global in time strong solutions to such systems.

Keywords: convective Brinkman-Forchheimer equations, weak solution, strong solution,monotonicity, Minty-Browder technique.

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Analysis of some asymptotic water waves models with surface tension

Mohammad Haidar (1) - Toufic El Arwadi (1) - Samer Israwi (2)

1. Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University,P.O. Box: 11-5020, Beirut, Lebanon.

2. Department of Mathematics, Faculty of Science 1, Lebanese University, Beirut, Lebanon

Abstract: The aim of this study is to analyze some asymptotic water waves models under the influence of surface tension. For that, we consider two asymptotic models of the 1D Green-Naghdi equations in KdV and Camassa-Holm scales and derive in a formal way by using the Whitham technique the KdV and Camassa-Holm equation under the influence of surface tension. An Hs-consistent solution for the obtained KdV equation has been obtained and the well-posdeness of the obtained Camassa-Holm equation is proved by using the Picard iterative scheme. Also, an Hs-consistent solution for the Boussinesq system has been established, as well as for 1D Green-Naghdi model in Camassa-Holm scale.

Finally, the aspect of breaking wave for the Camassa-Holm equation is discussed in the presence of surface tension and we confirmed the obtained theoretical results for KdV scale numerically.

Keywords: Green-Naghdi equations, KdV scale, Camassa-Holm scale, surface tension.

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Full discretization of time dependent convection-diffusion-reaction equation coupled with the Darcy system

Rebecca El Zahlaniyeh (1,2), Nancy Chalhoub (1), Pascal Omnes (2,3), Toni Sayah (1)

1. Laboratoire de Mathématiques et Applications, URMM, CAR, Faculté des Sciences, Université Saint-Joseph de Beyrouth, Lebanon.

2. UNSP Université Sorbonne Paris Nord, CNRS UMR 7539, Institut Galilée, LAGA, Villetaneuse, France.
3. Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, Gif-sur-Yvette, France.

In this presentation, we study the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

Keywords: Darcy's equations · Convection-diffusion-reaction equation · Finiteelement method · A priori error estimates

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Convergence of random attractors towards deterministic singleton attractor for convective Brinkman-Forchheimer equations

Kush Kinra and Manil T. Mohan

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India.

Abstract: This work is concerned about the long time behavior of the convective Brinkman-Forchheimer (CBF) equations in periodic domains. We prove that the global attractor of the above system is a singleton under small forcing intensity. After perturbing the above system with white noise, the random attractor does not have a singleton structure. But we obtain that the random attractor for stochastic CBF equations with white noise converges towards the deterministic singleton attractor, when the coefficient of random perturbation converges to zero.

Keywords: Deterministic and stochastic convective Brinkman-Forchheimer equations, smallforcing intensity, singleton attractor, upper semicontinuity, lower semicontinuity.

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Fully dispersive nonlinear model equations for surface water waves

Henrik Kalisch

University of Bergen, Norway

In 1967, G. Whitham put forward a simple nonlinear nonlocal model equation for the study of gravity waves at the free surface of an inviscid fluid. The advantage of this equation was that it described the propagation of small amplitude waves nearly perfectly, and in addition featured some nonlinear effects such as wave steepening and peaking.

In this lecture we review Whitham's idea and present recent developments on formal asymptotics and on the mathematical proof of some of Whitham's conjectures. We then present some new models of Whitham type, in particular for capillary and hydro-elastic waves. The models are tested in the case of wave--ice interaction and are used to predict the wave response to a moving load on an ice sheet.

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Global well-posedness for the solution of the Jordan-Moore-Gibson-Thompson equation with arbitrary large higher-order Sobolev norms

Belkacem Saïd-Houari

University of Sharjah, United Arab Emirates

Abstract. In this talk, we consider the nonlinear Jordan-Moore-Gibson-Thompson equation(JMGT) arising in acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This improves some recent result. Second, we prove a new decay estimate for the linearized model and removing the 1 -assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

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A new class of higher-ordered/extended Boussinesq system for efficient numerical simulations by splitting operators

Ralph Lteif

Lebanese American University, Graduate Studies and Research Office

Abstract

In this talk, we present the numerical study of the higher-ordered/extended Boussinesq system describing the propagation of water-waves over flat topography. An equivalent suitable reformulation is proposed, makingthemodelmoreappropriateforthenumericalimplementationandsignificantlyimprovedinterms of linear dispersive properties in high frequency regimes due to the suitable adjustment of a dispersion correction parameter. Moreover, we show that a significant interest is behind the derivation of a new formulation of the higher-ordered/extended Boussinesq system that avoids the calculation of high order derivativesexistinginthemodel. Weshowthatthisformulationenjoysanextendedrangeofapplicability while remaining stable. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and the dispersive part is treated with a finite difference approach. Numerical simulations are then performed to validate the model and the numerical methods.

Key Words: Water waves, Boussinesq system, higher-order asymptotic model, splitting scheme, hy-brid finite volume/finite difference scheme.

Email: ralph.lteif @lau.edu.lb

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Mechanical Balance Laws for Two-Dimensional Boussinesq Systems

Chourouk El Hassanie ∗

Abstract

Most of the asymptotically derived Boussinesq systems modeling long waves of small amplitude fail to satisfy exact mechanical conservation laws for mass, momentum and energy (see [3]). It is thus only fair to consider approximate conservation laws that hold in the context of these systems. Although such approximate mass, momentum and energy conservation laws can be derived (see [1]), the question of a rigorous mathematical justification still remains unanswered. Our aim is to justify the formally derived mechanical balance laws for Boussinesq systems [2]. The reader is referred to [4] for background on the water waves problem.

References

1. Alfatih Ali and Henrik Kalisch. Mechanical balance laws for boussinesq models of surface water waves. Journal of Nonlinear Science, 22, 06 2012.

2. Chourouk El Hassanieh, Samer Israwi, Henrik Kalisch, Dimitrios Mitsotakis, and Amutha Senthilkumar. Mechanical balance laws for two dimensional boussinesq systems.

3. Samer Israwi and Henrik Kalisch. Approximate conservation laws in the kdv equation. Physics Letters A, 383(9):854-858, 2019

4. David Lannes. The water waves problem, volume 188 of mathematical sur-veys and monographs. American Mathematical Society, Providence, RI, 2013.

∗Laboratory of Mathematics EDST, Lebanese University, Lebanon & Laboratoire Jacques Louis-Lions, Sorbonne University, Paris-France

Email: chourouk.el-hassanieh@inria.fr

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Active scalars with non-local drift

Diego Alonso Orán

Institute for Applied Mathematics, University of Bonn, Germany.

Active scalars are a wide class of transport equations where the velocity is determined from the transported quantity through a certain operator. Typically, the operator has a non-local nature, making the analysis much more subtle. In particular, the problem of global regularity versus finite time blow up for active scalar equation with non-local velocities has received a lot of attention in recent years. In this talk, we will review different results in that direction and present recent ideas in the context of compact Riemannian manifolds.

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A combined finite volume- nonconforming finite element scheme for anisotropic Darcy-Brinkman's model of two-phase flows in porous media.

Nasser El Dine (1), M. Saad (2), R. Talhouk (1)

1. Lebanese University
2. Ecole Centrale de Nantes

houssein.nd90@hotmail.com, mazen.saad@ec-nantes.fr, rtalhouk@ul.edu.lb

Two phases flow in a porous media are of the utmost importance in a wide range of applications such as enhanced oil recovery, carbon dioxide storage in underground aquifers and proton exchange membrane (PEM) fuel cells. However, modeling two phase flow in porous media is a challenging task, as it concerns forces acting at different scales in the flow domain. Viscous forces are responsible for the dissipation of energy of the fluid system at larger scales, in the bulk of individual fluids. Interfacial tension governs the shape and movement of the phase boundaries.

Different empirical laws are used to describe the filtration of a fluid through porous media [1, 2]. We mention the Darcy law which states that the filtration velocity of the fluid is proportional to the pressure gradient. The Darcy law cannot sustain the no-slip condition on an impermeable wall or a transmission condition on the contact with free flow. That motivated H. Brinkmann in 1947 to modify the Darcy law in order to be able to impose the no-slip boundary condition on an obstacle submerged in porous medium. He assumed large permeability to compare his law with experimental data and assumed that the second viscosity μ equals the physical viscosity of the fluid in the case of monophasic flow. Also, an up-scaling of the Stokes equations with non-slip boundary condition describing the flow in a porous medium, leads to the Darcy-Brinkman equations [3].

We are interested in the displacement of two incompressible phases in a Darcy-Brinkman flow in a porous media [4, 5, 6]. This model is treated in its general form with the whole nonlinear terms. The main purpose of this work is to propose and analyze a finite volume-nonconforming finite element scheme on general meshes to simulate the two incompressible phases flow in porous media.Where the diffusion term, can be anisotropic and heterogeneous, and is discretized by piecewise linear noncon-forming triangular finite elements. The convection term, is discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh.

REFERENCES

1. Salinger, Andrew G and Aris, Rutherford and Derby, Jeffrey J, Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains, International Journal for Numerical Methods in Fluids 18.12 (1994): 1185-1209.

2. Marusiˇc´-Paloka, Eduard, Igor Pazanin,ˇ and Sanja Marusiˇc,´ "Comparison between Darcy and Brinkman laws in a frac-ture.", Applied mathematics and computation 218.14 (2012), 7538-7545.

3. Auriault, Jean-Louis, Christian Geindreau, and Claude Boutin. "Filtration law in porous media with poor separation of scales." Transport in porous media 60.1 (2005): 89-108.

4. Coclite, G. M., et al. "Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media." Computational Geosciences 18.5 (2014): 637-659.

5. Hannukainen, Antti, Mika Juntunen, and Rolf Stenberg. "Computations with finite element methods for the Brinkman problem.", Computational Geosciences 15.1 (2011): 155-166.

6. Nasser El Dine, H., Saad, M., and Talhouk, R. "A convergent finite volume scheme of Darcy-Brinkman's velocities of two-phase flows in porous media." In 19th European Conference on Mathematics for Industry, p. 478. 2016.
   

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4-Equation Model for Thin liquid films

Khawla Msheik, University Claude Bernard Lyon1

Christian Ruyer Quil, University of Savoie Mont Blanc

Abstract:

After [1], the use of Weighted Residual Method (WRM) to derive shallow water models for free surfaces has been a true refinement in modeling thin liquid films. An advantage in using WRM is that there is no need to use asymptotic expansions to close the system, and also we have an exact velocity profile which would eventually lead to obtain good results for the relaxation terms compatible with the eigenmodes (damping coefficients) of a perturbed viscous film. On the other hand, in [2], the authors have successfully introduced a new variable equivalent to the enstrophy to derive a consistent shallow water model. In the following approach, we introduce two new variables that express an easier expression for the enstrophy and thus enable an easier extension to 2D models (3D flows). We have tried to derive a refined 4-equation shallow water model using WRM and the velocity profile defined in terms of these two variables which are related to the second moment of the deviation of the shear velocity from its depth average. The obtained model is consistent at first order, and good numerical results are obtained for the second order model when tested for solitary wave solutions. Our derivation is done starting from the incompressible Navier Stokes system for a free surface fluid driven by gravity and flowing down an inclined plane.

R´ef´erences

1. Ruyer-Quil C, Manneville P. Improved modeling of flows down inclined planes. The European Physical Journal B-Condensed Matter and Complex Systems.2000 May;15(2):357-69.

2. Richard GL, Ruyer-Quil C, Vila JP. A three-equation model for thin films down an inclined plane. Journal of Fluid Mechanics. 2016 Oct 10;804:162.

Khawla Msheik, ICJ UMR 5208 CNRS, University Claude Bernard Lyon1, Institut Camille Jordan, 21 avenue Claude Bernard, 69622 Villeurbanne

msheik@math.univ-lyon1.fr

Christian Ruyer Quil, LOCIE UMR 5271 CNRS, University of Savoie Mont Blanc, b^at. Helios, 60 rue du lac L´eman, Savoie Technolac, 73370 Le Bourget du Lac

ruyerquc@univ-smb.fr

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A POSTERIORI ERROR ESTIMATES FOR THE TIME DEPENDENT CONVECTION-DIFFUSION-REACTION EQUATION COUPLED WITH THE NAVIER-STOKES SYSTEM

JAD DAKROUB (1), JOANNA FADDOUL (2;3) , PASCAL OMNES (3;4), TONI SAYAH (2)

1. Faculté des ingénieurs, Université Saint-Joseph de Beyrouth, Libanon.

2. Laboratoire de Mathématiques et Applications, Unité de recherche Mathématiques et Modélisation, CAR, Faculté des Sciences, Université Saint-Joseph, B.P 11-514 Riad El Solh, Beyrouth 1107 2050, Libanon.
3. UNSP Univ Sorbonne Paris Nord, CNRS UMR 7539, Institut Galilée, LAGA, Villetaneuse, France.
4. Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, Gif-sur-Yvette, France.

In this presentation, we study the a posteriori error estimates for the time dependent convection-diffusion-reaction equation coupled with Navier-Stokes system. The problem is discretized in space using the implicit Euler method and in time using the finite element method. We establish a posteriori error estimates with two types of computable error indicators, the first one linked to the space discretization and the second one to the time discretization. Finally, numerical investigations are performed and presented.

Keywords: A posteriori error estimation, Navier-Stokes problem, convection-diffusion-reaction equation, finite element method, adaptive methods.