Discrete Mechanics

Jean-Paul Caltagirone is Professor Emeritus at INP, University of Bordeaux, France. Over the past decade, his research into continuum mechanics, and particularly fluid mechanics, has led him to develop a discrete theoretical approach to mechanics.

• Preface xiIntroduction xiiiList of Symbols xxiChapter 1. Fundamental Principles of Discrete Mechanics 11.1. Definitions of discrete mechanics 11.1.1. Notion of discrete space–time 11.1.2. Notion of a discrete medium 41.2. Properties of discrete operators 61.3. Invariance under translation and rotation 91.4. Weak equivalence principle 111.5. Principle of accumulation of stresses 131.6. Duality-of-action principle 141.7. Physical characteristics of a medium 161.8. Composition of velocities and accelerations 201.9. Discrete curvature 231.10. Axioms of discrete mechanics 28Chapter 2. Conservation of Acceleration 312.1. General principles 312.2. Continuous memory 342.3. Modeling the compression stress 372.3.1. Compression experiment 372.3.2. Modeling the stress in a solid 392.3.3. Modeling the stress in a fluid 392.3.4. Compression with small time constants 412.3.5. Modeling the accumulation of the normal stress 422.3.6. The energy formula, e = mc2 432.4. Modeling the rotation stress 442.4.1. Couette’s experiment 442.4.2. Behavior over time 452.4.3. Rotation stress in solids 462.4.4. Rotation stress in fluids 472.4.5. Stresses in a porous medium, Darcy’s law 472.4.6. Modeling the accumulation of the rotation stress 482.4.7. Rotation in Couette and Poiseuille flows 492.5. Modeling other effects 492.5.1. Gravitational effects 502.5.2. Inertial effects 522.6. Discrete equations of motion 562.6.1. Geometric description 562.6.2. Derivation of the equations of motion 582.6.3. Dissipation of energy 602.7. Coupling conditions 632.8. Formulation of the equations of motion at a discontinuity 652.9. Other forms of the equations of motion 662.9.1. Curl and vector potential formulation 672.9.2. Conservative form of the equations of motion 692.10. Incompressible models derived from the discrete formulation 702.10.1. Kinematic projection methods 702.10.2. Incompressibility in discrete mechanics 742.11. Consequences on the dynamics of the vorticity 74Chapter 3. Conservation of Mass, Flux and Energy 773.1. Conservation of mass in a homogeneous medium 773.1.1. In continuum mechanics 783.1.2. In discrete mechanics 803.2. Transport within multicomponent mixtures 813.2.1. Classical approach 813.2.2. Discrete model for the transport of chemical species 843.2.3. Equilibrium in a binary mixture 863.3. Advection 883.4. Conservation of flux 893.4.1. General remarks 893.4.2. Model 903.5. Conservation of energy 933.5.1. Conservation of total energy 933.5.2. Conservation of kinetic energy 943.5.3. Conservation of internal energy 953.5.4. Monotonically decreasing kinetic energy 973.6. A complete system of equations 983.7. A simple heat conduction problem 993.7.1. Case of anisotropic materials 1013.8. Phase change 1023.8.1. The Stefan problem 1033.8.2. Condensation 108Chapter 4. Properties of the Discrete Formulation 1154.1. Fundamental properties 1154.1.1. Limitations on the velocity 1154.1.2. Inverting the formulas Vφ = ∇φ and Vψ = ∇×ψ 1184.1.3. Material frame-indifference 1214.1.4. Fundamental invariants 1234.2. System of equations 1244.3. Differences from continuum mechanics 1264.3.1. Differences from the Navier-Lamé equations 1264.3.2. Differences from the Navier-Stokes equations 1274.3.3. Dissipation 1314.3.4. Compatibility conditions for the Navier-Stokes equations 1334.4. Examples of analytic solutions of the equations of motion 1364.4.1. Rigid rotational motion 1364.4.2. Planar Couette flow 1384.4.3. Poiseuille flow 1404.4.4. Radial flow 1444.5. Incompressible motion 1454.5.1. The Green-Taylor vortex 1454.5.2. Lid-driven cavity 1484.6. Compressible fluids and perfect fluids 1504.6.1. Generalized Bernoulli equation 1514.6.2. Propagation of linear waves 1524.6.3. Sod shock tube 1554.7. Statics of fluids and solids 1584.8. Conditions for modeling a rigid solid 1594.9. Flows in a porous medium 1604.10. Stretching of space-time and Hugoniot’s theorem 165Chapter 5. Two-Phase Flows, Capillarity and Wetting 1695.1. Formulation of the equations at the interfaces 1695.1.1. Modeling the curvature 1705.1.2. Formulation of the equations of motion 1745.2. Two-phase flows 1795.2.1. Two-phase Poiseuille flow 1795.2.2. Sloshing of two immiscible fluids 1815.3. Capillarity-dominated flows 1875.3.1. The Laplace problem 1875.3.2. Oscillating ellipse 1885.3.3. Marangoni-type flow in a droplet 1905.3.4. Interacting bubbles 1925.3.5. Simulating foam in equilibrium 1945.4. Partial wetting 1955.4.1. Droplet in equilibrium on a plane 1985.4.2. Spreading of a droplet 2005.4.3. Droplet acted upon by gravity 2045.4.4. Flows within a lens 2055.4.5. Capillary ascension in a tube 206Chapter 6. Stresses and Strains in Solids 2096.1. Discrete solid medium 2096.2. Stresses in solids 2116.2.1. Discrete equations 2126.2.2. Material frame-indifference 2146.2.3. Solid statics equations 2156.2.4. Calculating the displacement 2176.3. Properties of solid media 2196.3.1. In continuum mechanics 2206.3.2. In discrete mechanics 2236.4. Boundary conditions 2256.5. Rigid motion 2286.6. Validation of the model on examples 2306.6.1. Simple example of a monolithic fluid–structure interaction 2306.6.2. Mechanical equilibrium of sloshing 2336.6.3. Beam under extension 2356.6.4. Multimaterial compression 2376.6.5. Planar shearing 2386.6.6. Flexing beam 2396.6.7. Settling of a block under gravity 2406.6.8. Mechanical equilibrium of a solid object 2426.6.9. Extension to other constitutive laws 2436.7. Toward a unification of solid and fluid mechanics 246Chapter 7. Multiphysical Extensions 2497.1. Deflection of light 2497.1.1. Description of the physical phenomenon 2507.1.2. Deflection of light by the Sun in Newtonian mechanics 2527.1.3. Deflection of light by the Sun according to the duality-of-action principle 2567.1.4. Deflection of light by the Sun in a one-dimensional approach 2577.2. On a discrete approach to turbulence 2627.2.1. General remarks about the approach 2627.2.2. Dynamics of the vorticity in two spatial dimensions 2647.2.3. Analysis of a turbulent flow in a planar channel 2667.2.4. Model of the turbulence in discrete mechanics 2717.2.5. Application to a flow in a channel with Reτ = 590 2727.3. The lid-driven cavity problem with Re = 5,000 2807.4. Natural convection into the non-Boussinesq approximation 2837.5. Fluid–structure interaction 286References 289Index 295