Electroviscoelasticity of liquid/liquid interfaces: Fractional order van der Pol model -nonlinear and linearized 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. A new, recently developed, 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. 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 of their origin (mechanical and/or electrical). The second is van der Pol derivative model presented by the linear and non-linear differential equations. Finally, the third model presents an effort to generalize the previous van der Pol equation, i.e. the ordinary time derivatives and integrals are now replaced with corresponding fractional-order time derivatives and integrals of order p lt 1. Central parts of this paper are: proposed algorithm, using suitable numerical method, to obtain a homogeneous solution of a previous nonlinear equation of van der Pol type, and numerical solution of linearized fractional equation of the van der Pol type, using Grunwald definition of fractional differ-integral forms.