The continuity equation

Abstract

Listen

Translate article

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)).