Discrete Mechanics

Jean-Paul Caltagirone is Professor at the University of Bordeaux in France. He conducts research into fluid mechanics, heat transfers and porous media in which he develops original numerical methodologies to resolve equations.

"This book develops a new and original approach to mechanics."  (Zentralblatt MATH, 1 June 2015)

• PREFACE ixLIST OF SYMBOLS xvINTRODUCTION xxiCHAPTER 1. FRAMEWORK OF DISCRETE MECHANICS 11.1. Frames of reference and uniform motions 11.2. Concept of a Discrete Medium 41.2.1. Vectors and components 61.2.2. Physical meaning of the differential operators 81.2.3. Use of the theorems of differential geometry 101.2.4. Two essential properties 121.2.5. Tensorial values 171.2.6. The scalar and vectorial potentials 191.3. The physical characteristics 201.4. Equilibrium stress state 221.4.1. Two examples of mechanical equilibrium 251.5. Thermodynamic non-equilibrium 261.5.1. Forces and fluxes 291.6. Conservation of mass 30CHAPTER 2. MOMENTUM CONSERVATION 332.1. Classification of forces 332.2. Three fundamental experiments 352.2.1. Equilibrium in a glass of water 352.2.2. Couette flow 442.2.3. Poiseuille flow 472.3. Postulates 512.4. Modeling of the pressure forces 522.5. Modeling of the viscous forces 572.5.1. Modeling of the viscous effects of volume 572.5.2. Modeling of the viscous surface effects 592.5.3. Stress state 622.6. Objectivity 642.7. Discrete motion balance equation 672.7.1. Fundamental law of dynamics 672.7.2. Eulerian step 732.7.3. Mechanical equilibrium 742.8. Formulation in terms of density and temperature 782.9. Similitude parameters 812.9.1. Impact on the surface of a liquid 852.10. Hypercompressible media 88CHAPTER 3. CONSERVATION OF HEAT FLUX AND ENERGY 913.1. Introduction 913.2. Conservation of flux 923.3. Conservation of energy 953.3.1. Conservation of total energy 953.3.2. Conservation of kinetic energy 973.3.3. Conservation of the internal energy 983.4. Discrete equations for the flux and the energy 993.5. A simple heat-conduction problem 1003.5.1. Case of anisotropic materials 102CHAPTER 4. PROPERTIES OF DISCRETE EQUATIONS 1054.1. A system of equations and potentials 1054.2. Physics represented 1074.2.1. Poiseuille flow and potentials 1104.2.2. Celerity and maximum velocity 1124.2.3. Remarks about turbulence 1134.3. Boundary conditions 1144.3.1. Contact surface 1144.3.2. Shockwaves 1174.3.3. Edge conditions 1184.3.4. Slip condition 1194.3.5. Capillary effects 1204.3.6. Thermal boundary conditions 1244.4. Penalization of the potentials 1254.5. Continua and discrete mediums 1294.5.1. Differences with the Navier–Stokes equation 1294.5.2. Dissipation 1334.5.3. Case of rigidifying motions 1354.5.4. An example of the dissipation of energy 1374.6. Hodge–Helmholtz decomposition 1394.7. Approximations 1414.7.1. Bernoulli’s law 1414.7.2. Irrotational flow 1434.7.3. Inviscid fluid 1444.7.4. Incompressible flow 1454.8. Gravitational waves 1474.9. Linear visco-elasticity 1504.9.1. Viscous dissipation in a visco-elastic medium 1534.9.2. Dissipation of longitudinal waves in a visco-elastic medium 1554.9.3. Consistency with Continuum Mechanics 1564.9.4. Pure compression 1594.9.5. Pure shear stress 1604.9.6. Bingham fluid 162CHAPTER 5. MULTIPHYSICS 1655.1. Extensions to other branches of physics 1655.1.1. Coupling between a fluid and a porous medium 1675.2. Flow around a cylinder in an infinite medium 1695.2.1. Darcian model 1705.2.2. Stokes model 1745.2.3. Model of an ideal fluid 1755.2.4. Brinkman model 1765.3. Fluid statics 1785.3.1. Perfect gas in isothermal evolution 1795.3.2. Perfect gas in adiabatic evolution 1815.4. Injection of a gas into a cavity 1835.4.1. Isothermal injection 1845.4.2. Adiabatic injection 1855.5. Nonlinear wave propagation 1885.5.1. Sod shock tube 1905.6. Thermo-acoustics 1925.6.1. Heating of a cavity filled with air 1935.7. Natural convection in an enclosed cavity 1985.8. Multi-component transport 2005.9. Modeling of phase change 2035.10. Critical opalescence 2075.11. Conclusions regarding the multiphysics approach 209APPENDIX 211BIBLIOGRAPHY 215INDEX 219