Coupled Fluid Flow in Energy, Biology and Environmental Research

Book Series: Progress in Computaional Physics (PiCP)

Volume 2


Matthias Ehrhardt

DOI: 10.2174/97816080525471120101
eISBN: 978-1-60805-254-7, 2012
ISBN: 978-1-60805-691-0
ISSN: 1879-4661 (Print)

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Progress in Computational Physics is a new e-book series devoted to recent research trends in computational physics. It contains chapt...[view complete introduction]
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Table of Contents


- Pp. i
Kambiz Vafai
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- Pp. ii-iii (2)
Matthias Ehrhardt
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- Pp. iv
Matthias Ehrhardt
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An Introduction to Fluid-Porous Interface Coupling

- Pp. 3-12 (10)
M. Ehrhardt
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Modeling of Transfers at a Fluid-Porous Interface: A Multi-Scale Approach

- Pp. 13-31 (19)
M. Chandesris and D. Jamet
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Numerical Methods for Subsurface Flows and Coupling with Surface Runoff

- Pp. 32-41 (10)
P. Sochala and A. Ern
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Spectral Discretization of the Stokes Problem with Mixed Boundary Conditions

- Pp. 42-61 (20)
K. Amoura, C. Bernardi, N. Chorfi and S. Saadi
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Decoupled Algorithms for the Coupled Surface /Subsurface Flow Interaction Problems

- Pp. 62-86 (25)
M. Cai
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Integrated Multi-Model Description of the Human Lungs

- Pp. 87-103 (17)
C. Grandmont and B. Maury
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Multiscale and Multiphysics Aspects in Modeling and Simulation of Surface Acoustic Wave Driven Microfluidic Biochips

- Pp. 104-130 (27)
H. Antil, R.H.W. Hoppe, C H. Linsenmann and A. Wixforth
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- Pp. 131-134 (4)
Matthias Ehrhardt
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The mathematical analysis and numerical simulation of coupled flows in plain and porous media is of paramount importance for many current industrial and environmental problems such as proton exchange membrane (PEM) fuel cells, flow through oil filters, contaminant transport from lakes by groundwater, flow in bioreactors, contaminant gas leaking from atomic waste containers in deep rock depositories, CO sequestration in the subsurface, salt water intrusion and cancer therapy.

When using Navier-Stokes and Darcy’s equations to model flow in the two regions, finding effective coupling conditions at the interface between the fluid and the porous layer poses a challenge since the structures of the corresponding differential operators are different. However, when using the Generalized Model (Vafai and Tien, 1981) for the porous media, this difficulty does not occur, i.e. continuity of velocity and stress at the interface can be satisfied. Comprehensive modeling, conceptual set-up and assessment and applications of this approach has been shown in Vafai and Thiyagaraja (1987), Alazmi and Vafai (2001), Vafai and Kim (1990), Vafai and Huang (1994) and Huang and Vafai (1994).

There exists some well-known models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid-porous interface, like the Stokes/Brinkman problem with Ochoa-Tapia & Whitaker (1995) interface conditions and the Stokes/Darcy problem with Beavers & Joseph (1967) or Saffman (1971) conditions that have been shown to be well-posed. More recently, new coupling strategies were constructed using a transition region instead of a sharp interface, where the physical properties of the medium have a strong but still continuous variations. This approach was discussed by Nield (1983) and was developed e.g. by Goyeau et al. (2003), Chandesris & Jamet (2006), Hill & Straughan (2008) and Nield & Kuznetsov (2009).

At present, this research area is very attractive as shown by the active recent literature within several different journals. The book edited by Prof. Matthias Ehrhardt provides some particularly interesting keys to enter in this vast and exciting research domain of coupled fluid flow. By focusing on specific advanced subjects related to coupled flow problems from the different viewpoints of analysis, numerical techniques and practical applications with chapters written by distinguished experts in their respective fields, the reader will find an overview of state-of-the-art research results over a wide range of approaches from theory and concepts to real-world applications. As such, the volume will be very useful to researchers in applied mathematics, computational physics and mechanical engineering, as well as scientists working in coupled fluid flow models in biology or environmental applications.

Kambiz Vafai, Professor
University of California, Riverside
Department of Mechanical Enigineering
Riverside, California


This is the second volume of a new e-book series that is devoted to very recent research trends in computational physics. Hereby, it focus on the computational perspectives of current physical challenges, publishing new numerical techniques for the solution of mathematical equations including chapters describing certain real-world applications concisely.

The goal of this series is to emphasize especially approaches that are of interdisciplinary nature. The scientific topics in the fields of modeling, numerical methods and practical applications include e.g. the coupling between free and porous media flow, coupling of flow and transport models, coupling of atmospheric and ground water models, etc.

This volume contains both, the mathematical analysis of the coupling between fluid flow and porous media flow and state-of-the art numerical techniques, like tailor-made finite element and finite volume methods. Finally, a couple of chapters are devoted to concrete applications in the field of energy, biology and environmental research.

This book consists of 7 invited chapters that are structured in the three parts modeling, numerical techniques and finally practical applications. In the first part modeling we first present a troughout introduction into the general topic of coupled flows in plain and porous media and their numerical simulation. Hereby, we shortly review the coupling conditions between the pure liquid flow and the flow in a porous medium. The two well-known cases of normal and parallel flow over a porous layer are used as examples.

In Chapter 2 the authors describe the modelling of transfers at a fluid-porous interface by using a multi-scale approach. This method allows to determine the form of the boundary conditions and to determine the value of the jump parameters that appear in these boundary conditions. Also, it allows to analyze the physical nature of the jump coefficients and to determine whether these jump coefficients are intrinsic parameters or not.

This topic is a good bridge to the second part, the numerical techniques, consisting of Chapters 3–5. In Chapter 3 the authors propose accurate and robust numerical schemes to simulate subsurface flows and their coupling with surface runoff. These subsurface flows are modelled by the (unsteady) Richards’ equation discretized by a BDF in time and a discontinuous Galerkin method in space. The schemes for the subsurface flow and for the overland flow are mass conservative and can be coupled within a multiple time step procedure. To ensure total water mass conservation for the whole system, interface fluxes must be carefully designed. Accuracy and robustness of their schemes are assessed on several test cases (e.g. drainage, exfiltration and hortonian runoff). Finally, the methodology is applied to investigate the hydrological behaviour of a small-scale drained watershed. The authors of Chapter 4 describe a spectral discretization of an extended Stokes problem with mixed boundary conditions in a three-dimensional domain. They present a detailed numerical analysis that leads to error estimates for the three unknowns vorticity, velocity and pressure.

In Chapter 5 the author explains the coupling of incompressible surface flow with subsurface porous media flow that is typically a multi-domain and multi-physics problem. He studies the coupled (Navier-) Stokes/Darcy models, which are typical models for stationary surface/subsurface flow interactions and proposes several decoupled preconditioning techniques and two grid algorithms so that numerical computations can be run in parallel. In the last part we present some practical application, illustrating the impact of the mathematical ideas. In Chapter 6 gives an application to biology; the authors present an integrated multi-model description of the human lungs. An advection diffusion equation is used to describe the transfer of oxygen to the blood, where this advection is triggered by inflation-deflation cycles of the paremchyma. The mechanical part of the lungs can then be seen as a tree-like domain (conducting airways) embedded in an elastic medium. They address the difficult issues in terms of theory, numerics, and modelling, raised by the coupling of those models (Navier-Stokes, Darcy equations on a network, elasticity equations).

Finally, Chapter 7 considers a most modern application: microfluidic biochips are devices that are designed for high throughput screening and hybridization in genomics, protein profiling in proteomics, and cell analysis in cytometry. They are used in clinical diagnostics, pharmaceutics and forensics. The biochips consist of a lithographically produced network of channels and reservoirs on top of a glass or plastic plate. The fluid and surface acoustic waves interaction can be described by a mathematical model which consists of a coupling of the piezoelectric equations and the compressible Navier-Stokes equations featuring processes that occur on vastly different timescales. The authors use a homogenization approach in order to cope with the multiscale behavior of the coupled system that enables a separate treatment of the fast and slowly varying processes. In particular, the challenge to deal with the resulting large scale optimal control and optimization problems can be met by the application of projection based model reduction techniques.

We would like to thank Prof. Kambiz Vafai for writing the foreword, Prof. Ronald Hoppe for providing the figures for the title page and Bentham Science Publishers, particularly Manager Bushra Siddiqui, for their support and efforts.

List of Contributors

Matthias Ehrhardt
Bergische Universitat Wuppertal

Karima Amoura
Assistant Professor, Université Badji-Mokhtar
Faculté des Sciences, Département de Mathématiques
B.P. 12, 23000

Harbir Antil
Associate Professor, Department of Computational and Applied Mathematics
Rice University

Christine Bernardi
Research Director, Laboratoire Jacques-Louis Lions, C.N.R.S.
& Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu
Cedex 05, 75252

Mingchao Cai
Associate Professor, Institute of Applied Mathematics
TU Dortmund

Marion Chandesris
Associate Professor, CEA
French Atomic Energy Commission (Commissariat à l’énergie atomique)

Nejmeddine Chorfi
Full Professor, Department of Mathematics
College of Science, King Saud University
Riyadh, 11451
Saudi Arabia

Matthia Ehrhardt
Full Professor, Bergische UniversitätWuppertal
Fachbereich C Mathematik und Naturwissenschaften
Lehrstuhl für Angewandte Mathematik und Numerische Analysis

Alexandre Ern
Full Professor, Université Paris-Est, CERMICS, Ecole des Ponts
Champs sur Marne
7455 Marne la Vallée Cedex 2

Didier Jamet
Associate Professor, CEA
French Atomic Energy Commission (Commissariat à l’énergie atomique)

Céline Grandmont
Full Professor, INRIA, REO project-team
Le Chesnay Cedex

Ronald H.W. Hoppe
Full Professor, Department of Mathematics
University of Houston
Faculty of Mathematics and Natural Sciences
University of Augsburg

Christopher Linsenmann
Assistant Researcher, Institute of Mathematics
University of Augsburg

Bertrand Maury
Full Professor, Laboratoire de Mathématiques
Université Paris-Sud

Samira Saadi
Assistant Professor, Université Badji-Mokhtar
Faculté des Sciences, Département de Mathématiques, B.P. 12
Annaba, 23000

Pierre Sochala
Associate Professor, BRGM (French Geological Survey)
3 avenue Claude-Guillemin - BP 36009 - 45060 Orléans Cedex 2

Achim Wixforth
Full Professor, Institute of Physics
University of Augsburg


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