Book contents
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
1 - The continuity equation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
Summary
Theory: Definition of a geological medium as a continuum. Field variables used for the representation of a continuum. Methods for definition of the field variables. Eulerian and Lagrangian points of view. Continuity equation in Eulerian and Lagrangian forms and their derivation. Advective transport term. Continuity equation for an incompressible fluid.
Exercises: Computing the divergence of velocity field in 2D.
Continuum – what is it?
What we should understand from the very beginning is that geodynamics considers major rock units, such as the Earth's crust and mantle as continuous geological media. Continuity of any medium implies that, on a macroscopic scale, the material under consideration does not contain mass-free voids or gaps (there can indeed be pores or cavities but they are also filled with some continuous substances). Different physical properties of a continuum may vary at every geometrical point and we thus need a continuous description. In continuum mechanics, the physical properties of a continuum (field properties) are described by field variables such as pressure, temperature, density, velocity, etc. There are three major types of field variables:
scalars (e.g., pressure, temperature, density),
vectors (e.g., velocity, mass flux, heat flux),
tensors (e.g. stress, strain, strain rate).
Field variables can be represented in a fully continuous manner (analytical expressions, Fig. 1.1(a)) or in a discrete-continuous way (by arrays of values which characterise selected nodal geometrical points, Fig. 1.1(b–d)).
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- Introduction to Numerical Geodynamic Modelling , pp. 11 - 24Publisher: Cambridge University PressPrint publication year: 2009