Effect of Dielectric Viscoelastic Interface on Nonlinear Kelvin-Helmholtz Instability

We consider the nonlinear stability of two unbounded fluids that are separated by a plane interface and stressed by an initially perpendicular uniform electric field. On each side of the interface there is a Maxwell fluid shearing with constant relative motion. The fluids have different viscoelasticities, densities and electrical properties and surface tension acts at the interface. Due to the including of streaming flow a mathematical simplification is considered. The viscoelastic contribution is demonstrated in the boundary conditions. In 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 a nonlinear equation governing the interfacial displacement and having complex coefficients. This equation is accomplished by utilizing the cubic nonlinearity. Taylor theory is used to expand the governing nonlinear equation in the light of the multiple scales in both space and time. The perturbation analysis lead to imposing two levels of the solvability conditions, which are used to construct the well-known nonlinear Schrödinger equation with 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 stratified electric fields versus the wavenumber for the wavetrain of the disturbance. Numerical calculations showed that the difference between the vertical electric fields or the difference between the dielectric constants for both fluids produced a destabilizing influence on the stability criteria. It is found that the relaxation time and the fluid viscosity have played different roles in the stability profile. These several roles depend on the wavenumber. A consequence of destabilizing influence and stabilizing effect on the stability behavior are observed for increasing fluid velocities.