Novel analytical perspectives on nonlinear instabilities of viscoelastic Bingham fluids in MHD flow fields

The nonlinear stability of a plane interface separating two Bingham fluids and fully saturated in porous media is inspected in the existing work. The two fluids are compressed by a normal magnetic field. The two fluids have diverse viscoelasticity, densities, magnetic, and porosity medium, with the existence of surface tension at the interface. The motivation of applied physics and engineering relations has encouraged the discussion of the current paper. Because the mathematical behavior is rather complex, the viscoelasticity involvement is reproduced only at the surface of separation, which is well-known as the viscous potential theory. Thereby, the equations of movement are scrutinized in a linear form, whereas a set of nonlinear boundary conditions are supposed. This procedure produces a nonlinear expressive nonlinear partial differential equation of the interface displacement. The non-perturbative approach which is based on the He's frequency formula is employed to transform the nonlinear distinguishing ordinary differential equation with complex coefficients into a linear one. A novel process relying on the non-perturbative approach is utilized to examine the nonlinear stability and scrutinize the interface presentation. A non-dimensional analysis produces several dimensionless physical numerals. To validate the new approach, a comparison between the non-perturbative approach and its corresponding linear ordinary differential equation via the Mathematica Software is described and interpreted through a set of diagrams. Additionally, the Polar graphs have been elucidated. It is found that the mechanism of the stability does not change in the cases of real and complex coefficients.