Electroviscoelasticity of Liquid/Liquid Interfaces: Fractional Order Van der Pol Model – New Results

Abstract

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. The behavior of e.g., liquid/liquid interfaces is based on three forms of «instabilities», these are rigid,elastic and plastic. Three possible mathematical formalisms have been considered related to this physical formalism,i.e. to the theory of electroviscoelasticity: the first is tension tensor model, the second is Van der Pol derivative model and the third model is fractional order Van der Pol derivative model. The third model presents an effort to generalize the previous Van der Pol equation using fractional integro-differential operators (fractional calculus). In this paper, central parts of presentation will be solution using Caputo definition of fractional operators to obtain a (non)homogeneous solution of a (nonlinear)/linearized equation of fractional order Van der Pol model that represents the initial electromagnetic oscillation at the electroviscoelastic liquid/liquid interface.Specially, it will be discussed a case of low fractionality i.e the order of fractional derivative p slightly deviates from an integer value n. Finally, it is proposed numerical algorithm using a predictor-corrector approach for calculating corresponding homogeneous solution of nonlinear differential equation of the Van der Pol type.