Although viscoelastic flows are characterized by a very low Mach number regime, it involves a weakly compressible liquid phase, which requires a special treatment. This work is devoted to the mathematical modeling and numerical simulation for viscoelastic fluids. It follows directly a previous publication, PhD dissertation of the author. A Unified purely hyperbolic mathematical model for compressible and incompressible viscoelastic fluids is presented. To complete the mathematical model, a complete chapter is devoted to describe a new procedure to determine the correct type and number of boundary conditions for hyperbolic systems. Then a new model describing the non-isothermal viscoelastic flows is introduced while keeping the hyperbolic nature of the system. The main advantage of the proposed model over the existing ones is its hyperbolic nature, which overcomes some of the drawbacks of the available models. The proposed model is then solved numerically using a hybrid finite element/finite difference scheme.

This book presents the theory of waves propagation in a fluid-saturated porous medium (a Biot medium) and its application in Applied Geophysics. In particular, a derivation of absorbing boundary conditions in viscoelastic and poroelastic media is presented, which later is employed in the applications. The partial differential equations describing the propagation of waves in Biot media are solved using the Finite Element Method (FEM). Waves propagating in a Biot medium suffer attenuation and dispersion effects. In particular the fast compressional and shear waves are converted to slow diffusion-type waves at mesoscopic-scale heterogeneities (on the order of centimeters), effect usually occurring in the seismic range of frequencies. In some cases, a Biot medium presents a dense set of fractures oriented in preference directions. When the average distance between fractures is much smaller than the wavelengths of the travelling fast compressional and shear waves, the medium behaves as an effective viscoelastic and anisotropic medium at the macroscale. The book presents a procedure determine the coefficients of the effective medium employing a collection of time-harmonic compressibility and shear experiments, in the context of Numerical Rock Physics. Each experiment is associated with a boundary value problem, that is solved using the FEM. This approach offers an alternative to laboratory observations with the advantages that they are inexpensive, repeatable and essentially free from experimental errors. The different topics are followed by illustrative examples of application in Geophysical Exploration. In particular, the effects caused by mesoscopic-scale heterogeneities or the presence of aligned fractures are taking into account in the seismic wave propagation models at the macroscale. The numerical simulations of wave propagation are presented with sufficient detail as to be easily implemented assuming the knowledge of scientific programming techniques.

This monograph is based on a series of lectures presented at the 1999 NSF-CBMS Regional Research Conference on Mathematical Analysis of Viscoelastic Flows. It begins with an introduction to phenomena observed in viscoelastic flows, the formulation of mathematical equations to model such flows, and the behavior of various models in simple flows. It also discusses the asymptotics of the high Weissenberg limit, the analysis of flow instabilities, the equations of viscoelastic flows, jets and filaments and their breakup, as well as several other topics.

In this translation of the German edition, the authors provide insight into the numerical simulation of fluid flow. Using a simple numerical method as expository example, the individual steps of scientific computing are presented.