DescriptionBoth naturally-occurring and man-made materials are often heterogeneous materials formed of various constituents with different properties and behaviours. Studies are usually carried out on volumes of materials that contain a large number of heterogeneities. Describing these media by using appropriate mathematical models to describe each constituent turns out to be an intractable problem. Instead they are generally investigated by using an equivalent macroscopic description - relative to the microscopic heterogeneity scale - which describes the overall behaviour of the media.
Fundamental questions then arise: Is such an equivalent macroscopic description possible? What is the domain of validity of this macroscopic description? The homogenization technique provides complete and rigorous answers to these questions.
This book aims to summarize the homogenization technique and its contribution to engineering sciences. Researchers, graduate students and engineers will find here a unified and concise presentation.
The book is divided into four parts whose main topics are
- Introduction to the homogenization technique for periodic or random media, with emphasis on the physics involved in the mathematical process and the applications to real materials.
- Heat and mass transfers in porous media
- Newtonian fluid flow in rigid porous media under different regimes
- Quasi-statics and dynamics of saturated deformable porous media
Each part is illustrated by numerical or analytical applications as well as comparison with the self-consistent approach.
PART ONE. UPSCALING METHODS 27
Chapter 1. An Introduction to Upscaling Methods 29
Chapter 2. Heterogenous Medium: Is an Equivalent Macroscopic Description Possible? 55
Chapter 3. Homogenization by Multiple Scale Asymptotic Expansions 75
PART TWO. HEAT AND MASS TRANSFER 107
Chapter 4. Heat Transfer in Composite Materials 109
Chapter 5. Diffusion/Advection in Porous Media 143
Chapter 6. Numerical and Analytical Estimates for the Effective Diffusion Coefficient 161
PART THREE. NEWTONIAN FLUID FLOW THROUGH RIGID POROUS MEDIA 195
Chapter 7. Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium 197
Chapter 8. Compressible Newtonian Fluid Flow Though a Rigid Porous Medium 229
Chapter 9. Numerical Estimation of the Permeability of Some Periodic Porous Media 257
Chapter 10. Self-consistent Estimates and Bounds for Permeability 275
PART FOUR. SATURATED DEFORMABLE POROUS MEDIA 337
Chapter 11. Quasi-statics of Saturated Deformable Porous Media 339
Chapter 12. Dynamics of Saturated Deformable Porous Media 367
Chapter 13. Estimates and Bounds for Effective Poroelastic Coefficients 389
Chapter 14. Wave Propagation in Isotropic Saturated Poroelastic Media 407
Bibliography . 453