A lamellar model for analysis of liquid-liquid mixing

Two quantities, a_ν, intermaterial area per unit volume and s, striation thickness, are defined to describe the state of a system mixed or being mixed. These quantities are related to one another and to the fluid mechanics of mixing in Lagrangian and Eulerian frames of reference. The model considers that mechanical mixing is synonymous with deformation. An equation for rate of generation of intermaterial area is presented and integrated for systems of engineering interest. An upper bound for this rate of generation is obtained for the case of Newtonian incompressible fluids. Special attention is given to the microscale of mixing where reaction and diffusion occur. It is shown that diffusion and reaction can be combined into a single variable due to a particular choice of reference frame. In this way the necessary tools for analyzing liquid-liquid mixing are presented. The restrictions to this analysis are: (i) microscale can be represented by a lamellar structure, (ii) no surface tension effects, (iii) no instabilities at the microscale of mixing.