Analysis of principle frame-indifferent formulas for two-phase and nanofluid flows and the effect of spin and shear lift forces on suspended particles

Multiphase flows because of their increasing importance in many industries, particularly in recent decades, turn to a significant subject. Many articles have been published in regards to multiphase flows which one of the two continuum theories has been developed to describe such situations: Mixture Theory (or interacting of continua theory) or Averaging Theory [1]. We consider the mixture of fluid and solid particles to be a purely mechanical system in whichthermal, electromagnetic, and chemical reactions are ignored. In this paper, first, the importance of the principle of Frame-Indifference Material, in multiphase flows, is discussed. Because of the ever-increasing use of two-phase flows and especiallynanofluids, the principle formulas for two-phase mixtures and nanofluids are obtained which are in agreement with the literature