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This article reviews the mathematical and computer modeling of the process of thermal phase transition in two-phase fluid flows. The nucleate boiling process is investigated in the presence of a constant heat source on a solid wall. Bubble formation and phase transition are taken into account. The flow characteristics and temperature distribution during nucleate boiling are obtained. The results of the numerical study were obtained using a 2D numerical algorithm implemented on the basis of the D2Q9 model of the Lattice Boltzmann method (LBM-Lattice Boltzmann method) and the phase field method. The calculations show that first the nucleation of a bubble is formed, then the bubble grows, breaks away from the boundary with the heat source, then, rising upward, undergoes deformation under the action of buoyancy forces. The effect of gravity and surface wettability on the bubble diameter during ascent is also numerically investigated. The results obtained are in good agreement with the experimental and numerical results of other authors. [ABSTRACT FROM AUTHOR]