Computational Rheology

"Computational Rheology" is the first text book dedicated entirely to the field after the 1984 book "Numerical Simulation of Non-Newtonian Flow" by Crochet, Davis, and Walters and numerous review articles which have been published since the mid 1980s. Since much work has been done during the past 20 years and computational rheology has experienced enormous progress since then, the expectations in a new book are high. The objective of this new book is to give a treatment of the physical modeling of viscoelastic fluids and of the main challenges concerning computational aspects in theoretical rheology. According to the authors the book is written for researchers and students in applied mathematics and engineering. The first part of the book comprises chapters one to seven and it gives an introduction to several issues which are essential prior to solve problems in computational non-Newtonian fluid mechanics. This part of the book deals with the derivation of viscoelastic flow equations, the construction of stress constitutive models, the mathematical theory of partial differential equations, and the essential elements of discretization techniques and numerical algorithms for solving viscoelastic flow equations. This first part of the book has been written to be as self-contained as possible. In the first chapter the reader is introduced to the Maxwell model, the Kelvin model, and to the behavior of non-Newtonian liquids in shear and uniaxial elongational flow. Furthermore, Chap. 1 explores some of the reasons for the high Weissenberg number problem a key issue in computational rheology and how innovative techniques have been devised to solve/circumvent it. The second chapter consists of two main parts: First the definition of several kinematic quantities are given and some fundamental results in fluid mechanics are derived, then constitutive relationships for the stress tensor are introduced. Readers being interested in a more in-depth study of the mathematics of viscoelasticity, the description of experimental studies of viscoelasticity viz. rheometry, and the microstructural modeling of polymeric liquids (cf. also Chap. 11) are referred to alternative textbooks. The subsequent Chap. 3 deals with the mathematical properties of the underlying viscoelastic flow equations: existence and uniqueness of the solutions of the flow equations, specification of boundary conditions, and the presence of singularities at some locations of the flow domain (such as corners or edges). The following Chap. 4 addresses the difficulties in choosing appropriate constitutive equations for the extra stress tensor to achieve agreement of computer simulations with experimentally observed properties of viscoelastic liquids. Furthermore, the characterization of viscoelastic fluids in terms of basic material functions and the problem of phenomenological parameter entering the flow equations are addressed. Chap. 5 and 6 deal with the discretization of viscoelastic flow equations and the numerical solution of the discretized equations, respectively. Chap. 6 treats a variety of different solution methods such as finite difference methods, Galerkin finite element methods, finite volume methods, and specApplied Rheology Vol.12/6.qxd 10.01. 2003 13:58 Uhr Seite 280