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1.1 MATHMATICS, Applied mathematics
A model for thermally induced phase transitions in materials with thermal memory was recently proposed, where the equations determining heatflow were assumed to be the same in both phases. In this work, the model is generalized to the case of phase dependent heatflow relations. The temperature (or coldness) gradient is decomposed into two parts, each zero on one phase and equal to the temperature (or coldness) gradient on the other. However, they vary smoothly over the transition zone. These are treated as separate independent quantities in the derivation of field equations from thermodynamics. Heat flux is given by an integral over the history of the temperature gradient, with different kernels on each phase. Asymptotic analysis is carried out to obtain generalizations of previous results. These involve the jump in temperature across the transition zone and the normal derivatives of the temperature on each phase boundary, which are related to the velocity of the transition zone and a latent heat dependent on this velocity, as well as the speeds of thermal disturbances in the two phases.
Golden, J.M. (2009). Phase transitions in materials with thermal memory: the case of unequal conductivities. Physica D: Nonlinear Phenomena, vol. 238, no. 4, pg. 428–448. doi:10.1016/j.physd.2008.11.004