Fractional order model of electroviscoelasticity of liquid-liquid interfaces: nonlinear case

Abstract

A number of theories that describe the behavior of liquid-liquid interfaces have been developed and applied to various dispersed systems e.g., Stokes, Reiner-Rivelin, Ericksen, Einstein, Smoluchowski, Kinch, etc. A new theory of electroviscoelasticity describes the behavior of electrified liquid-liquid interfaces in fine dispersed systems, and is based on a new constitutive model of liquids. According to this model liquid-liquid droplet or droplet-film structure (collective of particles) is considered as a macroscopic system with internal structure determined by the way the molecules (ions) are tuned (structured) into the primary components of a cluster configuration. How the tuning/structuring occurs depends on the physical fields involved, both potential (elastic forces) and nonpotential (resistance forces). Ali these microelements of the primary structure can be considered as electromechanical oscillators assembled into groups, so that excitation by an external physical field may cause oscillations at the resonant/characteristic frequency of the system itself (coupling at the characteristic frequency). Up to day, there are three possible mathematical formalisms discussed related to the theory of electroviscoelasticity. The first is tension tensor model, where the normal and tangential forces are considered, only in mathematical formalism, regardless to their origin (mechanical and/or electrical). The second is Van der Pol derivative model. Finally, the third, here presented model comprise an effort to generalize the previous Van der Pol differential equations, nonlinear; i.e. the ordinary time derivatives and integrals are now replaced by corresponding fractional-order time derivatives and integrals. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. The theory of electroviscoelasticity, i.e., its physical and mathematical formalism, using fractional calculus become more realistic and compatible, and can be helpful in solving entrainment problems in solvent extraction and in studies of fine dispersed systems (micro, nano and atto, suspensions, emulsions, fluosols, foams and beams/streams of entities), the physics of liquids and biological systems.