Insights into fractal space features in nonlinear electrohydrodynamic Rayleigh–Taylor instability of viscous fluids

This paper describes a unique method for detecting and evaluating nonlinear Rayleigh–Taylor instability (RTI) in electro-viscous fluids exposed to an external vertical electric field. The governing equations are based on a linearized Navier–Stokes framework with nonlinear boundary conditions, capturing the system's complexity. Using a traveling wave transformation, the analysis reduces the system's complicated dynamics to a nonlinear characteristic equation in the elevation function that includes quadratic and cubic nonlinearities. The strategy utilizes El-Dib's frequency formula, which allows for the derivation of an equivalent linearized form of the characteristic equation, simplifying the nonlinear equation and making it more tractable for analytical investigation. The study emphasizes the critical function of the electric field in the system's stability. Smaller electric fields improve stability and equilibrium, resulting in damped oscillations that maintain the fluid–fluid interface. Larger electric fields, on the other hand, enhance instabilities, causing the system to behave nonlinearly, which might lead to chaotic motion if the oscillations are severe. The analysis is extended to convert the characteristic equation into a fractal space description. The fractal derivative form enables the modeling and study of complicated, nonlinear, and chaotic processes commonly encountered in fluid dynamics problems. This methodology is especially well-suited to handling multi-scale dynamics and nonlinear growth in RTI. The influence of fractal factors on system behavior is examined. Increasing the fractal order consistently has a stabilizing effect, lowering the oscillation amplitude and increasing damping, hence improving stability. In contrast, raising the fractalness parameter introduces a destabilizing influence, resulting in bigger oscillations and lower damping, destabilizing the system over time. This study sheds light on the behavior of nonlinear RTI in electro-viscous fluids in the presence of electric fields and fractal dynamics.