Two-dimensional interface tracking of unsteady MHD two-layer immiscible Casson micropolar and micropolar fluids using the MCB-DQM approach

The present study investigates the unsteady magnetohydrodynamic (MHD) flow of two immiscible, incompressible micropolar fluids (Casson micropolar and micropolar fluids) confined within a horizontal channel. Both fluids are considered electrically conducting, with Hall current and ion-slip effects incorporated. The novelty of the present work lies in the combined investigation of yield-stress rheology, microrotation, and electromagnetic effects on the evolution and stability of interface dynamics in a two-dimensional immiscible MHD flow involving Hall and ion-slip effects. The governing momentum, microrotation and interface tracking partial differential equations are solved using the Modified Cubic B-Spline Differential Quadrature Method (MCB-DQM) along with a strong stability-preserving Runge–Kutta (RK43) scheme for temporal and spatial discretisation. The Volume-of-Fluid (VOF) method is used to capture the evolving interface. The numerical model is validated in terms of reliability, accuracy, grid independence, and stability in the present study. The study demonstrates that interface evolution is profoundly affected by dimensionless fluid parameters, such as time, pressure gradient, Froude number, Reynolds number, initial amplitude, wavelength, and non-Newtonian characteristics. The balance between shear-induced energy input and damping caused by viscous, magnetic, and surface tension forces has a significant influence on interface dynamics for variation in these fluid parameters. The absence of a magnetic field is observed to have a minor effect on the interface dynamics for low Stuart numbers. Overall, the proposed numerical framework accurately resolves interfacial characteristics across a broad spectrum of wave numbers, demonstrating strong robustness and reliability in modeling complex magnetohydrodynamic (MHD) two-layer immiscible flows with non-Newtonian rheological behaviour.