Non-linear electrorheological instability of two streaming cylindrical fluids

Abstract

A weakly non-linear instability of surface waves propagating through two viscoelastic cylindrical dielectric fluids is investigated. The examination is conducted in the presence of a tangential electric field and uniform axial relative streaming. The influence of the surface tension is taken into account, while the gravitational forces are ignored. Weak viscoelastic effects on the interface are considered, so that their contributions are demonstrated through the boundary conditions. Therefore, the equations of motion are solved in the absence of the viscoelastic effects. The solutions of the linearized equations of motion under the non-linear boundary conditions lead to derivation of a non-linear equation governing the interfacial displacement. This characteristic equation has damping terms and complex coefficients, where the nonlinearity is kept up to the third order. The linear state leads to a dispersion relation, where the stability is analysed. Taylor's theory is adopted to expand the governing non-linear equation in the light of the multiple scale technique, to impose the well-known Schrodinger equation. Several special cases are reported upon appropriate data choices. The stability criteria are discussed theoretically and illustrated graphically in which stability diagrams are obtained. Regions of stability and instability are identified for the electric field intensity versus the wave number for the wave train of the disturbance.