The Hodograph Equation

When considering transient, nonstationary modes of the front motion, one should have solutions for two important limiting cases. The first case describes the ground state referred to as thermodynamic equilibrium. This case has been considered in the previous Sect. 3.1 devoted to the Gibbs-Thomson equationEquationGibbs-Thomson. Namely, in the framework of Gibbs thermodynamics, the phase equilibrium of the interface with allowance for surface energy is formulated in the form of the Gibbs-Thomson equationEquationGibbs-Thomson. The second case is attributed to the achievement of a state in which the system has entered a quasistationary regimeQuasistationary regime of front propagation. The solutions of the equation of the phase interfaceInterfacephase moving with constant velocity are obtained in the present chapter. Considering the dynamics of phase transformationPhasetransformation in the example of solidification or meltingMelting, an interface condition for the slowly and rapidly moving solid–liquid interface is deduced. At a small driving force, a linear velocity law for the interface is predicted in consistency with the linear irreversible thermodynamics of Onsager and PrigogineThermodynamicsof Onsager and Prigogine. The hodograph equationEquationhodograph as the dynamical interface condition predicts nonlinearity in the behavior of the interface velocityInterfacevelocity appearing at a high driving force. Compared with existing theories, the deduced hodograph equationEquationhodograph presents: (a) a generalization for the well-known velocity-dependent Gibbs-Thomson relation, (b) a generalization for the Born-Infeld equationEquationBorn-Infeld for the hyperbolic motion by mean curvature and under the driving force and (c) the damped Klein-Gordon equationEquationKlein-Gordon known in application to the inflationInflation stage of matter evolution.