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The discrete vision of mechanics is based on the founding ideas of Galileo and the principles of relativity and equivalence, which postulate the equality between gravitational mass and inertial mass. To these principles are added the Hodge-Helmholtz decomposition, the principle of accumulation of constraints and the hypothesis of the duality of physical actions. These principles make it possible to establish the equation of motion based on the conservation of acceleration considered as an absolute quantity in a local frame of reference, in the form of a sum of the gradient of the scalar…mehr
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- Produktdetails
- Verlag: John Wiley & Sons
- Seitenzahl: 336
- Erscheinungstermin: 24. Januar 2019
- Englisch
- ISBN-13: 9781119575160
- Artikelnr.: 55288303
- Verlag: John Wiley & Sons
- Seitenzahl: 336
- Erscheinungstermin: 24. Januar 2019
- Englisch
- ISBN-13: 9781119575160
- Artikelnr.: 55288303
- Herstellerkennzeichnung Die Herstellerinformationen sind derzeit nicht verfügbar.
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118 4.1.3. Material frame-indifference 121 4.1.4. Fundamental invariants 123 4.2. System of equations 124 4.3. Differences from continuum mechanics 126 4.3.1. Differences from the Navier-Lamé equations 126 4.3.2. Differences from the Navier-Stokes equations 127 4.3.3. Dissipation 131 4.3.4. Compatibility conditions for the Navier-Stokes equations 133 4.4. Examples of analytic solutions of the equations of motion 136 4.4.1. Rigid rotational motion 136 4.4.2. Planar Couette flow 138 4.4.3. Poiseuille flow 140 4.4.4. Radial flow 144 4.5. Incompressible motion 145 4.5.1. The Green-Taylor vortex 145 4.5.2. Lid-driven cavity 148 4.6. Compressible fluids and perfect fluids 150 4.6.1. Generalized Bernoulli equation 151 4.6.2. Propagation of linear waves 152 4.6.3. Sod shock tube 155 4.7. Statics of fluids and solids 158 4.8. Conditions for modeling a rigid solid 159 4.9. Flows in a porous medium 160 4.10. Stretching of space-time and Hugoniot's theorem 165 Chapter 5. Two-Phase Flows, Capillarity and Wetting 169 5.1. Formulation of the equations at the interfaces 169 5.1.1. Modeling the curvature 170 5.1.2. Formulation of the equations of motion 174 5.2. Two-phase flows 179 5.2.1. Two-phase Poiseuille flow 179 5.2.2. Sloshing of two immiscible fluids 181 5.3. Capillarity-dominated flows 187 5.3.1. The Laplace problem 187 5.3.2. Oscillating ellipse 188 5.3.3. Marangoni-type flow in a droplet 190 5.3.4. Interacting bubbles 192 5.3.5. Simulating foam in equilibrium 194 5.4. Partial wetting 195 5.4.1. Droplet in equilibrium on a plane 198 5.4.2. Spreading of a droplet 200 5.4.3. Droplet acted upon by gravity 204 5.4.4. Flows within a lens 205 5.4.5. Capillary ascension in a tube 206 Chapter 6. Stresses and Strains in Solids 209 6.1. Discrete solid medium 209 6.2. Stresses in solids 211 6.2.1. Discrete equations 212 6.2.2. Material frame-indifference 214 6.2.3. Solid statics equations 215 6.2.4. Calculating the displacement 217 6.3. Properties of solid media 219 6.3.1. In continuum mechanics 220 6.3.2. In discrete mechanics 223 6.4. Boundary conditions 225 6.5. Rigid motion 228 6.6. Validation of the model on examples 230 6.6.1. Simple example of a monolithic fluid-structure interaction 230 6.6.2. Mechanical equilibrium of sloshing 233 6.6.3. Beam under extension 235 6.6.4. Multimaterial compression 237 6.6.5. Planar shearing 238 6.6.6. Flexing beam 239 6.6.7. Settling of a block under gravity 240 6.6.8. Mechanical equilibrium of a solid object 242 6.6.9. Extension to other constitutive laws 243 6.7. Toward a unification of solid and fluid mechanics 246 Chapter 7. Multiphysical Extensions 249 7.1. Deflection of light 249 7.1.1. Description of the physical phenomenon 250 7.1.2. Deflection of light by the Sun in Newtonian mechanics 252 7.1.3. Deflection of light by the Sun according to the duality-of-action principle 256 7.1.4. Deflection of light by the Sun in a one-dimensional approach 257 7.2. On a discrete approach to turbulence 262 7.2.1. General remarks about the approach 262 7.2.2. Dynamics of the vorticity in two spatial dimensions 264 7.2.3. Analysis of a turbulent flow in a planar channel 266 7.2.4. Model of the turbulence in discrete mechanics 271 7.2.5. Application to a flow in a channel with Re
= 590 272 7.3. The lid-driven cavity problem with Re = 5,000 280 7.4. Natural convection into the non-Boussinesq approximation 283 7.5. Fluid-structure interaction 286 References 289 Index 295
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and V
=
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118 4.1.3. Material frame-indifference 121 4.1.4. Fundamental invariants 123 4.2. System of equations 124 4.3. Differences from continuum mechanics 126 4.3.1. Differences from the Navier-Lamé equations 126 4.3.2. Differences from the Navier-Stokes equations 127 4.3.3. Dissipation 131 4.3.4. Compatibility conditions for the Navier-Stokes equations 133 4.4. Examples of analytic solutions of the equations of motion 136 4.4.1. Rigid rotational motion 136 4.4.2. Planar Couette flow 138 4.4.3. Poiseuille flow 140 4.4.4. Radial flow 144 4.5. Incompressible motion 145 4.5.1. The Green-Taylor vortex 145 4.5.2. Lid-driven cavity 148 4.6. Compressible fluids and perfect fluids 150 4.6.1. Generalized Bernoulli equation 151 4.6.2. Propagation of linear waves 152 4.6.3. Sod shock tube 155 4.7. Statics of fluids and solids 158 4.8. Conditions for modeling a rigid solid 159 4.9. Flows in a porous medium 160 4.10. Stretching of space-time and Hugoniot's theorem 165 Chapter 5. Two-Phase Flows, Capillarity and Wetting 169 5.1. Formulation of the equations at the interfaces 169 5.1.1. Modeling the curvature 170 5.1.2. Formulation of the equations of motion 174 5.2. Two-phase flows 179 5.2.1. Two-phase Poiseuille flow 179 5.2.2. Sloshing of two immiscible fluids 181 5.3. Capillarity-dominated flows 187 5.3.1. The Laplace problem 187 5.3.2. Oscillating ellipse 188 5.3.3. Marangoni-type flow in a droplet 190 5.3.4. Interacting bubbles 192 5.3.5. Simulating foam in equilibrium 194 5.4. Partial wetting 195 5.4.1. Droplet in equilibrium on a plane 198 5.4.2. Spreading of a droplet 200 5.4.3. Droplet acted upon by gravity 204 5.4.4. Flows within a lens 205 5.4.5. Capillary ascension in a tube 206 Chapter 6. Stresses and Strains in Solids 209 6.1. Discrete solid medium 209 6.2. Stresses in solids 211 6.2.1. Discrete equations 212 6.2.2. Material frame-indifference 214 6.2.3. Solid statics equations 215 6.2.4. Calculating the displacement 217 6.3. Properties of solid media 219 6.3.1. In continuum mechanics 220 6.3.2. In discrete mechanics 223 6.4. Boundary conditions 225 6.5. Rigid motion 228 6.6. Validation of the model on examples 230 6.6.1. Simple example of a monolithic fluid-structure interaction 230 6.6.2. Mechanical equilibrium of sloshing 233 6.6.3. Beam under extension 235 6.6.4. Multimaterial compression 237 6.6.5. Planar shearing 238 6.6.6. Flexing beam 239 6.6.7. Settling of a block under gravity 240 6.6.8. Mechanical equilibrium of a solid object 242 6.6.9. Extension to other constitutive laws 243 6.7. Toward a unification of solid and fluid mechanics 246 Chapter 7. Multiphysical Extensions 249 7.1. Deflection of light 249 7.1.1. Description of the physical phenomenon 250 7.1.2. Deflection of light by the Sun in Newtonian mechanics 252 7.1.3. Deflection of light by the Sun according to the duality-of-action principle 256 7.1.4. Deflection of light by the Sun in a one-dimensional approach 257 7.2. On a discrete approach to turbulence 262 7.2.1. General remarks about the approach 262 7.2.2. Dynamics of the vorticity in two spatial dimensions 264 7.2.3. Analysis of a turbulent flow in a planar channel 266 7.2.4. Model of the turbulence in discrete mechanics 271 7.2.5. Application to a flow in a channel with Re
= 590 272 7.3. The lid-driven cavity problem with Re = 5,000 280 7.4. Natural convection into the non-Boussinesq approximation 283 7.5. Fluid-structure interaction 286 References 289 Index 295