Melt nucleating and the three-dimension steady model of the temperature fluctuation with convection

A three-dimensional steady model of temperature fluctuation with melt convection is studied. It is proved that there exists a unique and stable solution in the model and the solution is expressed in a Fourier series form. It theoretically confirms the mechanism of melt nucleating: as long as the convection with transverse directions exists, the melt temperature on the front of the solid-liquid interface would be not only periodical along the direction which is perpendicular to the direction of crystal growth, but also oscillatory and exponential decay along the direction of crystal growth; this oscillatory property, i.e. temperature fluctuation, leads to local supercooling, accelerates local temperature fluctuation and then results in a large number of nuclei.