Partial Differential Equations in Applied Mathematics (Sep 2024)
Effect of a uniform magnetic field on the nonlinear instability of double cylindrical interfaces concerning three magnetic liquids
Abstract
The present manuscript relates to the movement of three immersed viscous magnetic liquids in porous media that are positioned between three concentric cylindrical interfaces. Every cylindrical layer has an axial extension that extends in both vertical directions to infinity. Moreover, the pressure gradients are the driving force behind the uniform motion of all fluids in a similar upward motion at varying velocities. Permanent magnetic fields are tangentially oriented exert stress on the system. The computations are rendered simpler by applying the viscous potential theory. The Maxwell equations are utilized for the magnetic field, while the Brinkman-Darcy equations define the fluid mobility. Moreover, the rise of the disturbance and the subsequent extraction of fuel from the tiny cracks of the reservoir stone can be managed through the implementation of engineering techniques such as electromagnetic field impacts and oil extraction engineering. Typically, the nonlinear strategy involves incorporating the applicable nonlinear boundary conditions and analyzing the linearized equations of motion. This research offers a framework for verifying theoretical models and simulations based on experimental observations. The inclusion of a homogeneous magnetic field introduces complexity to the system, rendering it a suitable candidate for validating and improving models in the field Magneto hydrodynamics. The originality of the problem lies in the dual nonlinear stability of cylindrical interfaces when subjected to uniform magnetic field. Accordingly, two nonlinear characteristic differential equations controlling the surface displacements are produced. The nonlinear stability prerequisite is met by applying the matrix concept and the multiple scale technique in conjunction with a theoretical analysis of stability. Additionally, the Routh-Hrutwitz criterion is encompassed to judge the stability cofiguration. A detailed examination of the associated nonlinear stability requirements is showed. Meanwhile, the estimated limited solutions of perturbed surfaces are achieved. For the cylindrical middle layer, it is found that the outer cylindrical interface has a more stabilizing effect than the inner one. The approximate solutions for displacements at the interface are calculated. The influence of Weber numeral of the problem on the stability profile is investigated.
