A common description for surface-tension and diffusion in two-component flows

This report tries to develop a common representation for diffusion and surface tension in two components newtonian flows. First, we make some preliminary considerations: (1) the Navier-Stokes equations of a fluid with two immiscible components is derived from the single components equations.In this case, the surface tension arrives from external consideration; (2) we shortly review the Cahn-Hilliard and Allen-Cahn equations describing a phase separation process, the surface-tension being considered as the terminal part of this process; and (3) we report a conservative representation of the surface tension and some of its variants. Second, we reinterprete the conservative form of the single components as also valid for miscible fluids. Making an hypothesis common in diffusion phenomena, we make the surface tension naturally appear from the single phase equations. An anti-diffusive term connected with the surface tension is put in evidence. Compatibility with diffusion is discussed.

@TechReport{MO10f,
  author       = {Moreau, V.},
  title        = {A common description for surface-tension and diffusion in two-component flows},
  institution  = {CRS4, Center for Advanced Studies, Research and Development in Sardinia},
  number       = {10/100},
  month        = {august},
  year         = {2010},
  address      = {Cagliari, Italy},
  note         = {idxproject: THINS},
  type         = {techreport},
  keywords     = {surface tension, diffusion, Navier-Stokes},
  url          = {https://publications.crs4.it/pubdocs/2010/MO10f},
}