Solidification. The Stefan Condition as a Consequence of the Configurational Force Balance

Abstract

I begin with a summary of the basic results of Section 6c, but with deformation neglected. The basic thermodynamical laws, balance of energy and growth of entropy, have the form

$$ \frac{d} {{dt}}\left\{ {\int\limits_P {\varepsilon d\upsilon } } \right\} = - \int\limits_{\partial P} {h \cdot n da} + \int\limits_P {r d\upsilon } , $$

(23-1a)

$$ \frac{d} {{dt}}\left\{ {\int\limits_P {\eta d\upsilon } } \right\} \geqslant - \int\limits_{\partial P} {(h/T) \cdot n da} + \int\limits_P {(r/T)d\upsilon } , $$

(23-1b)

for P stationary. These are equivalent to the local equations

$$ \dot \varepsilon = - Div h + r, $$

(23-2a)

$$ \dot \eta \geqslant - Div(h/T) + r/T, $$

(23-2b)

which combine to form the free-energy inequality

$$ \dot \Psi + \eta \dot T + T^{ - 1} h \cdot \nabla T \leqslant 0, $$

(23-3)

with

$$ \Psi = \varepsilon - T\eta $$

(23-4)

the free energy.