Nonlinear electrorheological instability of two Rivlin-Ericksen elastico-viscous fluids

The behaviour of surface waves propagating between two Rivlin-Ericksen elastico-viscous fluids is examined. The investigation is made in the presence of a vertical electric field and a relative horizontal constant velocity. The influence of both surface tension and gravity force is taken into account. Due to the inclusion of streaming flow a mathematical simplification is considered. The viscoelastic contribution is demonstrated in the boundary conditions. From this point of view the approximation equations of motion are solved in the absence of viscoelastic effects. The solutions of the linearized equations of motion under nonlinear boundary conditions lead to derivation of a nonlinear equation governing the interfacial displacement and having damping terms with complex coefficients. This equation is accomplished by utilizing the cubic nonlinearity. The use of the Gardner-Morikawa transformation yields a simplified linear dispersion relation so that the periodic solution for the linear form is utilized. The perturbation analysis, in the light of the multiple scales in both space and time, leads to imposing the well-known nonlinear Schroedinger equation having complex coefficients. 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 fields versus the wavenumber for the wavetrain of the disturbance. Numerical calculations showed that the ratio of the dielectric constant plays a dual role in the stability criteria. The damping role for the viscosity coefficient is observed. The viscoelasticity coefficient plays two different roles. A stabilizing influence is observed through the linear scope and a destabilizing role in the nonlinear stability picture is seen