Phil. Trans. R. Soc. A (2010) 368, 285–300 doi:10.1098/rsta.2009.0204 Time-dependent, irreversible entropy production and geodynamics BY KLAUS REGENAUER-LIEB 1,2,4, *, ALI KARRECH 4 ,HUI TONG CHUA 1,3 , FRANKLIN G. HOROWITZ 1,2 AND DAVE YUEN 5 1 Western Australian Geothermal Centre of Excellence, 2 Earth and Environment, and 3 School of Mechanical Engineering, University of Western Australia, Western Australia 6009, Australia 4 CSIRO Exploration and Mining, PO Box 1130, Bentley, Western Australia 6102, Australia 5 Department of Geology and Geophysics and Supercomputer Institute, University of Minnesota Minneapolis, MN 55455, USA We present an application of entropy production as an abstraction tool for complex processes in geodynamics. Geodynamic theories are generally based on the principle of maximum dissipation being equivalent to the maximum entropy production. This represents a restriction of the second law of thermodynamics to its upper bound. In this paper, starting from the equation of motion, the first law of thermodynamics and decomposition of the entropy into reversible and irreversible terms, 1 we come up with an entropy balance equation in an integral form. We propose that the extrema of this equation give upper and lower bounds that can be used to constrain geodynamics solutions. This procedure represents an extension of the classical limit analysis theory of continuum mechanics, which considers only stress and strain rates. The new approach, however, extends the analysis to temperature-dependent problems where thermal feedbacks can play a significant role. We apply the proposed procedure to a simple convective/conductive heat transfer problem such as in a planetary system. The results show that it is not necessary to have a detailed knowledge of the material parameters inside the planet to derive upper and lower bounds for self-driven heat transfer processes. The analysis can be refined by considering precise dissipation processes such as plasticity and viscous creep. Keywords: limit theorems; plasticity theory; finite-time thermodynamics; irreversible entropy; Carnot efficiency; endoreversible engine *Author for correspondence (klaus.regenauer-lieb@csiro.au). 1 The equality form of the second law and the decomposition of entropy are due to Tolman & Fine (1948). One contribution of 17 to a Theme Issue ‘Patterns in our planet: applications of multi-scale non-equilibrium thermodynamics to Earth-system science’. This journal is © 2010 The Royal Society 285 Downloaded from https://royalsocietypublishing.org/ on 21 January 2022