A non-isothermal compressible Cahn-Hilliard fluid model for air pollution phenomena

Air pollution is an evident and dangerous contamination form. In particular in this paper, we describe a mathematical model for the study of pollution due to the burning of fuels, close to the earth's surface. So, following the phase-field approach, we consider a mixture of two non-isothermal compressible Euler fluids whose mixing is studied by a generalized Cahn-Hilliard theory under suitable constitutive equations. For this model we show the compatibility with the Laws of Thermodynamics, working with the internal (mechanical, heat and chemical) powers. Finally, we implement our model with non-homogeneous boundary conditions, describing the effect of pollutant emission sources like plumes of smoke from the stacks on the earth surface.