Effects of uniform and periodic magnetic fields at the nonlinear stability of three magnetic fluids in porous media

Abstract

The current manuscript tackles the interaction between three viscous magnetic fluids placed on three layers and saturated in porous media. Two of them fill half of the spaces above and below a thin layer that lies in the middle region. All layers are laterally extended to infinity in both horizontal directions. All fluids move in the same horizontal direction with different uniform velocities and are driven by pressure gradients. The system is stressed by tangential stationary/periodic magnetic fields. The viscous potential theory is used to simplify the mathematical procedure. The motion of the fluids is described by the Brinkman–Darcy equations, and Maxwell equations are used for the magnetic field. The work has been motivated in light of the engineering applications like petroleum products manufacturing and the electromagnetic field effect that can be used to control the growth of the perturbation and then the recovery of crude oil from the pores of reservoir rocks. The nonlinear technique typically relies on solving linear equations of motion and presenting the nonlinear boundary conditions. The novelty of the problem concerns the nonlinear stability of the double interface under the impact of periodic magnetic fields. Therefore, the approach has resulted in two nonlinear characteristic differential equations governing the surface displacements. Accordingly, the development amplitudes of surface waves are designated by two nonlinear Schrödinger equations. Stability is theoretically analyzed; the nonlinear stability criteria are derived, and the corresponding nonlinear stability conditions are explored in detail. The analysis reveals the resonance as well as the non-resonance situations. Approximate bounded solutions of the perturbed interfaces are estimated. It is found that the thickness of the intermediate layer is represented as a function of time and plotted. The impact of different parameters on the stability profile is investigated. For the middle layer, it is observed that magnetic permeability and viscosity have a stabilizing effect. By contrast, basic streaming and permeability have a destabilizing influence. Due to different saturation amplitude at each case, the analysis of the periodic case shows that the lower interface is much more stable than the upper one.