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We consider the governing equations for the motion of the viscous fluids intwo moving domains and an evolving surface from both energetic andthermodynamic points of view. We make mathematical models for multiphase flowwith surface flow by our energetic variational and thermodynamic approaches.More precisely, we apply our energy densities, the first law of thermodynamics,and the law of conservation of total energy to derive our multiphase flowsystem with surface tension and flow. We study the conservative forms andconservation laws of our system by using the surface transport theorem andintegration by parts. Moreover, we investigate the enthalpy, the entropy, theHelmholtz free energy, and the Gibbs free energy of our model by applying thethermodynamic identity. The key idea of deriving surface tension andviscosities is to make use of both the first law of thermodynamics and ourenergy densities.

Thermodynamical Modeling of Multiphase Flow System with Surface Tension and Flow