Mechanics of Mixtures

The following sections are included:

• Preface

• Contents

The Oxford English Dictionary [l] defines a mixture as “a product of mixing, a complex unity or aggregate (material or immaterial) composed of various ingredients or constituent parts mixed together.” It goes on to describe a mixture as a “mixed state or condition; co-existence of different ingredients or of different groups or classes of things mutually diffused through each other.” Most bodies, astrophysical, geological, biological or otherwise are, in the spirit of the above definition, mixtures in which two or more constituents co-exist; if one of the constituents is preponderant and the other constituents essentially insignificant, the body is usually assumed to be of the predominant single constituent. However, there are innumerable situations where none of the constituents present in the body can be ignored with respect to the others; plasmas and gaseous mixtures that surround stars, biological tissues and muscles, slurries and suspensions, porous rocks and soil infused with water or oil, and gaseous mixtures that occur naturally are examples that come immediately to mind. Sometimes, the constituents in a mixture undergo chemical reactions, usually resulting in an interconversion between the constituents and the formation of new compounds. However, there is a large class of mixtures whose constituents do not react chemically and it is such mechanical mixtures that will be studied in this book…

The following sections are included:

• Kinematics

• Partial Tractions and Partial Stresses

• Balance of Mass

• Balance of Linear Momentum

• Balance of Angular Momentum

• Conservation of Energy

• Second Law of Thermodynamics

• Volume Additivity Constraint

The studies on the flow of fluids through porous solids are too numerous to document, the books by Bear [165] and Freeze and Cherry [166] contain details pertinent to the problems of flows of fluids through solids undergoing small deformations, and the paper by Raats [167] presents applications pertinent to diffusion through soils and discusses many other papers devoted to this area. The many papers by Biot [168, 169,170], Hassanizadeh and Gray [171,172,173], Hassanizadeh [174,175], and Katsube and Carroll [176] consider the flow through a porous media, but are restricted to the solids undergoing small deformations. In this chapter, we shall be especially interested in constructing a constitutive model for diffusion wherein the solid undergoes large deformations, and we shall study this within the context of the equations documented in the previous chapter…

The following sections are included:

• Boundary Conditions

• Diffusion of a Fluid Through a Rubber Slab

• Diffusion of a Fluid Through a Rubber Annulus

The following sections are included:

• Analysis of a Time-dependent Diffusing Singular Surface

• Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Slab

• Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Cylindrical Annulus

The following sections are included:

• Biot's Equations

• Counterpart of Biot's Equations

• Wave Propagation in Anisotropic Swollen Mixtures

The following sections are included:

• Basic Balance Equations

• Constitutive Variables

• Constitutive Modelling

• Two Special Motions

The following sections are included:

• Turbulence Modelling in Newtonian Fluids

• Balance Equations for the Mixture

• On Constitutive Modelling

• Specific Model

The following sections are included:

• Appendix A SOME RESULTS FROM DIFFERENTIAL GEOMETRY

• Appendix B STATUS OF DARCY'S LAW WITHIN THE CONTEXT OF MIXTURE THEORY

• Bibliography

• Index