CONTENTS -- PREFACE -- LIST OF SYMBOLS -- 1. INTRODUCTION -- 1.1. The concept of an "interface" -- 1.2. The concept of an "interfacial layer" -- 1.3. Presentation of the volume -- 2. THERMODYNAMICS AND KINEMATICS OF INTERFACES -- 2.1. Definition of surfaces -- 2.2. Interfacial quantities -- 2.3. Thermodynamic relations -- 2.3.1. The bulk -- 2.3.2. The interface -- 2.3.3. Thermodynamic equilibrium between two phases at rest -- 2.3.4. Surface tension out of equilibrium -- 2.4. Velocities and deformation rates of the interface -- 2.4.1. Material velocities in the bulk -- 2.4.2. Interfacial velocities in intrinsic description -- 2.4.3. Velocities in orthogonal curvilinear coordinates -- 2.4.4. Strain rates -- 2.4.5. Transport theorem for a curvilinear integral -- 2.4.6. Transport theorem for a surface integral -- 2.4.7. Divergence theorem on a surface -- 2.4.8. Interfacial fluxes -- 2.5. Examples -- 2.5.1. Effect of curvature on surface integrals -- 2.5.2. Parallel curves -- 2.5.3. Parallel surfaces -- 2.5.4. Effect of curvature on lateral surface integrals in the case of parallel surfaces -- 2.5.5. Effect of curvature on equilibrium surface tension -- 2.5.6. Determination of the mean normal curvature -- 2.5.7. Deformation along a surface -- 2.5.8. Stretch of a moving cylinder -- 2.5.9. Stretch of a planar flame -- 3. INTERFACE BALANCE LAWS -- 3.1. General interface balance law -- 3.1.1. Balance law for the three-dimensional continuum -- 3.1.2. First integration method of the local balance laws for the three-dimensional continuum -- 3.1.3. Second integration method of the balance laws for the three-dimensional continuum -- 3.1.4. Some comments -- 3.2. Interface balance laws for species, mass, momentum and energy -- 3.2.1. Interface balance laws for species -- 3.2.2. Interface balance law for mass.

3.2.3. Momentum interface balance law -- 3.2.4. Energy interface balance law -- 3.3. Interfacial entropy production -- 3.3.1. Interfacial entropy inequality -- 3.3.2. Interface Clausius-Duhem inequality -- 3.3.3. Balance laws for an interface inside one component fluids -- 3.3.4. A remark for the interfaces without mass -- 4. CONSTITUTIVE RELATIONS DEDUCED FROM LINEAR IRREVERSIBLE THERMODYNAMICS FOR TWO-DIMENSIONAL INTERFACES -- 4.1. Analysis of the surface entropy production and possible coupling -- 4.2. Capillarity at equilibrium -- 4.3. Newtonian interface and surface viscosities -- 4.3.1. Bénard-Marangoni effect -- 4.3.2. Surface viscosities -- 4.4. Surface heat transfer -- 4.5. Problems related to evaporation / condensation -- 4.5.1. Plane interface case -- 4.5.2. Curvature effect -- 4.6. Surface chemical reactions -- 4.7 Interfaces without mass -- 5. CLASSICAL THREE-DIMENSIONAL CONSTITUTIVE RELATIONS DEDUCED FROM LINEAR IRREVERSIBLE THERMODYNAMICS AND THEIR CONSEQUENCES FOR INTERFACES -- 5.1. Constitutive relations of three-dimensional classical fluid mixtures -- 5.2. The case of premixed flames with high activation energy -- 5.2.1. The classical theory of planar adiabatic premixed flames -- 5.2.2. Curved premixed flames with high activation energy for Lewis number near unity -- 5.3. The case of shock waves and relaxation zones behind shock waves -- 5.3.1. The gaseous shock wave as an interface -- 5.3.2. Shock wave in a dusty gas -- 6. SECOND GRADIENT THEORY APPLIED TO INTERFACIAL MEDIUM -- 6.1. Description of a fluid with internal capillarity -- 6.1.1. Balance law for the mass -- 6.1.2. Virtual power principle -- 6.1.3. Boundary conditions on a wall -- 6.1.4. Balance energy law -- 6.1.5. Entropy inequality and Clausius-Duhem inequality.

6.1.6. Expression for the specific internal energy -- 6.2. A set of equations for a fluid with capillarity -- 6.2.1. Constitutive laws -- 6.2.2. Equation system for the description of a fluid with capillarity -- 6.3. Asymptotic modelization of an interface without mass -- 6.3.1. Order of magnitude analysis -- 6.3.2. Analysis of the interface balance laws -- 6.4. An expression for the surface tension -- 6.5. Description of a mixture of fluids with internal capillarity -- 6.5.1. Balance law for the mass of the N species -- 6.5.2. Virtual power principle for the mixture -- 6.5.3. Balance internal energy -- 6.5.4. Entropy inequality and Clausius-Duhem inequality -- 6.5.5. Expression for the specific internal energy -- 6.5.6. A set of equations for a mixture of fluids with capillarity -- 7. TYPICAL PROBLEMS INVOLVING SURFACE TENSION AND OTHER SURFACE PROPERTIES -- 7.1. The growth of a bubble -- 7.2. An interface with a surface viscosity -- 7.3. An interface with a variable surface tension -- 7.4. Marangoni instability -- 7.4.1. Planar surface -- 7.4.2. Free surface -- 7.5. Vapor recoil -- 7.5.1. The Palmer analysis of the Hickman instability -- 7.5.2. The case of a mixture -- 7.5.3. Boiling crisis -- 7.6. Diffusive interfaces -- 7.6.1. Rayleigh-Taylor and Kelvin-Helmholtz instabilities -- 7.6.2. Linearized theory of interfacial layer instability between two heavy miscible fluids -- 7.6.3. Interfacial layer instability in a pure heated supercritical fluid -- 7.6.4. Linearized study of shear layers -- A.1. TENSORIAL NOTATIONS -- A.2. GEOMETRY OF INTERFACES AND INTERFACIAL LAYERS IN ORTHOGONAL CURVILINEAR COORDINATES -- A2.1. Definition of the surfaces -- A2.2. Curvilinear coordinates -- A2.3. Curl of a vector field -- A2.4. Divergence of a vector field -- A2.5. Gradient of a scalar field.

A2.6. Laplacian of a scalar field -- A2.7. Differentiation of curvilinear basis -- A2.8. Divergence of a second order tensor field P -- A2.9. Gradient of a vector -- A2.10. Projection operators -- A2.11. Surface gradients of a scalar field -- A2.12. Curvature vector of a curve Ci -- A2.13. Definitions of normal and tangential divergences of a vector field -- A2.14. Surface stretch per unit length -- A2.15. Mean normal curvature of a surface -- A2.16. Decomposition of the divergence of a vector field -- A2.17. Decomposition of the Laplacian of a scalar field -- A2.18. Decomposition of the divergence of a second order tensor field -- A2.19. Parallel divergence of the product V∙P for a particular class of tensors P -- A2.20. Intrinsic definition of a surface -- A2.21. Comparison between the two descriptions -- A2.22. Family of parallel surfaces -- A2.23. Variation of the parallel divergence of a vector across an interfacial layer -- A.3. KINEMATICS OF THE INTERFACE -- A3.1. Velocities in intrinsic description -- A3.2. Velocities in orthogonal curvilinear coordinates -- A3.3. Strain rates -- A3.4. Parallel surfaces -- A3.5. Stretching rate of a surface -- A3.6. Stretch variation across interfacial layer -- BIBLIOGRAPHY.