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8 - Continuum Equations and Boundary Conditions

from PART I - CONTINUUM MECHANICS IN GEOPHYSICS

Published online by Cambridge University Press:  17 March 2011

B. L. N. Kennett
Affiliation:
Australian National University, Canberra
H.-P. Bunge
Affiliation:
Universität Munchen
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Summary

We summarise here the main equations for continuum behaviour that need to be employed in a broad range of applications, by drawing on the development in earlier chapters. We start with the conservation laws for mass, momentum and energy expressed in differential form, that need to be supplemented by the appropriate constitutive laws to express the rheological state of the medium. We then consider the boundary conditions that prevail at the surface of a continuum or an interface between two different continua. The normal components of velocity and the stress tensor are required to be continuous across a general interface. The tangential components of velocity are also continuous for solid–solid and fluid–fluid boundaries and also for a solid–viscous fluid interface under a “no-slip” condition. Heat flux is continuous across an interface, but because of variations in thermal conductivity temperature gradients can have a jump. At a phase boundary, additional thermodynamic constraints need to be applied to describe the equilibrium scenario along the interface.

Hitherto, we have concentrated on the way in which a continuum responds to deformation or imposed stress, but we also need to take into account electromagnetic phenomena. The iron-rich core of the Earth is a conducting fluid where a complex interaction of motion and electromagnetic effects leads to dynamo action and creates the Earth's internal magnetic field. We therefore provide a brief development of the topic of continuum electrodynamics and show how the continuum equations need to be modified to accommodate magnetic effects.

Type
Chapter
Information
Geophysical Continua
Deformation in the Earth's Interior
, pp. 131 - 150
Publisher: Cambridge University Press
Print publication year: 2008

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