An innovative solution to the Mathieu-Van der Pol-Helmholtz-Duffing equation for the stability of superposed electrified Rivlin-Ericksen fluids

Abstract

This paper presents a novel way to solve the Mathieu-Van der Pol-Helmholtz-Duffing equation to investigate the stability of superposed electrified Rivlin-Ericksen fluids. It focuses on the dynamics of surface waves propagating between two Rivlin-Ericksen elastic-viscous fluids under a vertical periodic electric field and relative streaming flow, revealing information about fluid interface stability and interactions. The analysis illustrates the delicate balance of forces that govern the system's behavior by combining surface tension, gravitational forces, and viscoelastic effects. Given the complexities of streaming flow, a mathematical simplification is used to increase analytical tractability. The boundary conditions account for viscoelasticity's effect, which greatly adds to the system's complexity. To assist analysis, the approximate equations of motion are first solved while ignoring viscoelastic effects, allowing for a preliminary understanding of the fundamental interactions between the periodic electric field, surface tension, and gravity. Once this baseline is created, viscoelastic effects are applied to capture the entire dynamics. The resulting nonlinear equation governing interfacial displacement, which is constructed as a nonlinear Mathieu-type equation and then simplified into a linearized form, allows for a detailed examination of stability and reveals areas of stability and instability. Numerical results show that characteristics such as the dielectric constant ratio, viscosity ratio, and viscoelasticity ratio all affect stability differently. Notably, the viscoelasticity coefficient causes destabilization in the nonlinear regime, emphasizing the complex interplay of forces in elastic-viscous fluid systems. This work contributes to a better understanding of complicated fluid interfaces, particularly under periodic electric excitation and viscoelastic effects.