Instability for shearing of an electrified Kelvin fluid sheet with or without supporting surface charges

The present study demonstrates the instability of streaming in a fluid layer sandwiched between two other bounded fluids under the influence of a vertical periodic electric field. The fluids are of a viscoelastic nature where the constitutive equation is Kelvin type. Due to the inclusion of streaming flow and the influence of a periodic force, a mathematical simplification is urged. Equation of motion is solved in light of the weakness effect for the viscoelastic properties. The instabilization of the problem is examined in view of the linearization of the perturbation approach. The boundary value problem is discussed for a charged or uncharged fluid sheet. Both cases are lead to derive linear coupled Mathieu equations with complex coefficients and damping terms. Stability analysis is discussed through a simplified configuration for the system of Mathieu equations. It is found that the elasticity parameters as well as the viscosity parameters have a stabilizing influence. The field frequency plays a destabilizing role in the presence of surface charges and a dual role in the absence of surface charges. The presence of surface charges retards the stabilizing influence of the viscoelastic effects. This calculation shows that the fluid velocity retards the destabilizing influence for the electric field. The increase of the thickness of the fluid sheet plays two different roles. A stabilizing role in the presence of surface charges and a destabilizing influence in their absence.