Inspection of nonlinear instability of two cylindrical interfaces between viscoelastic magneto-rheological fluids

The study examines the dynamics of sinking viscoelastic magnetic fluids in porous media saturated within three concentric cylindrical structures. The cylindrical interfaces delineate two viscoelastic Powell-Eyring liquids, which are interposed by a viscoelastic media. Each cylindrical layer has an axial direction that extends infinitely in both vertical orientations. Tangentially orientated homogeneous magnetic fields impose stress on the system and affect surface tension. The calculations are simplified through the application of viscous potential theory aimed at improved clarity. Consequently, the viscoelastic effects are analyzed to illustrate the contribution of the nonlinear boundary conditions. The Maxwell equations are utilized for the magnetic field, whereas the Brinkman-Darcy equations describe fluid dynamics. The introduction of a uniform magnetic field increases the system’s complexity, rendering it suitable for validating and improving models in Magnetohydrodynamics. Accordingly, two nonlinear characteristic ordinary differential equations that dictate the surface displacements are formulated. A thorough examination of the pertinent nonlinear stability requirements is provided. A unique methodology termed the non-perturbative approach is employed, mostly based on He’s frequency formula. The purpose of this technology is to convert nonlinear ordinary differential equations into linear ones. The Mathematica Software is utilized to confirm the accurate alignment between the coupled systems of linear and nonlinear ordinary differential equations. It is found that the Darcy number enhances stability in the stability schematic, whereas the Powell-Eyring fluids and Ohnesorge numbers lead to destabilizing impact. A series of PolarPlot configurations, to guarantee stable solutions, are analyzed for various physical factors.