# Advanced analysis of nonlinear stability of two horizontal interfaces separating three-stratified non-Newtonian liquids

## TL;DR

This paper studies how three layers of non-Newtonian fluids behave under electric fields and surface tension, aiming to improve engineering applications like coatings and microfluidics.

## Contribution

The study introduces a novel non-perturbative approach using He’s frequency formula to analyze nonlinear stability in three-layered non-Newtonian fluid systems.

## Key findings

- Stability improves with the orientation of the tangential electric field relative to the horizontal wavenumber.
- Nonlinear boundary conditions and dimensionless parameters help simplify and characterize fluid behavior.
- PolarPlots visualize parameter effects, offering insights into interfacial stability mechanisms.

## Abstract

The nonlinear stability of two horizontal interfaces of three-layered stratified non-Newtonian fluids plays a pivotal role in advanced engineering applications. This phenomenon encompasses temperature management systems, microfluidic devices, and precise coating technologies. In an existing study, a multilayer system is considered wherein a central Casson liquid (CL) layer is bounded above and below by Powell–Eyring liquids (PELs). The impact of a uniform tangential electric field (EF) and surface tension is explored within a porous medium. To avoid the mathematical complexity, the viscous potential flow (VPF) is used to simplify the governing hydrodynamic formulations. The model involves Navier–Stokes and Maxwell equations under the quasi-static assumption. To obtain a nonlinear formulation, the linearized regulator equations are derived subject to appropriate nonlinear boundary conditions. The plan interfaces are presumed to propagate horizontally. To handle the nonlinear ordinary differential equations (ODEs) arising from the analysis, He’s frequency formula (HFF) is applied, transforming the problem into linear forms suitable for a non-perturbative approach (NPA). A non-dimensional analysis introduces key dimensionless collections, which help to characterize underlying fluid behavior and reduce system intricacy. A brief methodological summary of NPA is included to support reproducibility and clarity. The numerical calculations indicate that the stability can be evidently improved by the orientation of the tangential EF in relation to the horizontal wavenumber. PolarPlots are employed to imagine the influence of varying parameters, offering valuable insights into the mechanisms of the governing interfacial stability.

The online version contains supplementary material available at 10.1038/s41598-025-24182-6.