Stability of Streaming in an Electrified Maxwell Fluid Sheet Influenced by a Vertical Periodic Field in the Absence of Surface Charges

Abstract

The problem of electroviscoelastic Kelvin–Helmholtz waves of Maxwellian fluids under the influence of a vertical periodic electric field is studied in the absence of surface charges. The system is composed of a streaming dielectric fluid sheet of finite thickness embedded between two different streaming semi-infinite dielectric fluids. Due to the streaming flow and the influence of a periodic force, a mathematical simplification is considered. The weak viscoelastic effects are taken into account so that their contributions are demonstrated in the boundary conditions. The approximate equations of motion are solved in the absence of viscoelastic effects. The solutions of the linearized equations of motion and boundary conditions lead to two simultaneous Mathieu equations of damping terms having complex coefficients. Symmetric or antisymmetric deformation that relaxes the coupled Mathieu equations and yields a single Mathieu equation is considered. Stability criteria are discussed and numerical estimation shows that the increase in the sheet thickness plays a destabilizing effect in the presence or in the absence of the field frequency as well as the field intensity. In the absence of the field frequency the velocity ratio between the upper fluid velocity and the sheet velocity has a destabilizing influence, while that between the velocity of the lower fluid and the velocity of the sheet has a stabilizing influence. Moreover, the viscosity ratios have a damping influence while the elasticity ratios have a destabilizing influence. Furthermore, a range of general deformations of the surface deflections is studied. Moreover, the stability behavior for the resonance cases is studied and discussed. The coupled Mathieu equations are analyzed by the multiple scale method. The numerical examination for stability yields some changes in the stability behavior. The fluid sheet thickness plays a stabilizing role in the presence of a constant field while the damping role is observed for the resonance case. Similar results are found for both the stratified velocities and the stratified relaxation times. The dual role of the stratified viscosities is observed in the presence or the absence of the field frequency.