Moatimid Galal M, Mohamed Yasmeen M
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
Sci Rep. 2024 Nov 21;14(1):28843. doi: 10.1038/s41598-024-78848-8.
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.
现有研究考察了在多孔介质中完全饱和的、分隔两种宾汉流体的平面界面的非线性稳定性。两种流体受到法向磁场的压缩。两种流体具有不同的粘弹性、密度、磁性和孔隙率介质，界面处存在表面张力。应用物理与工程关系的研究动机促使了本文的讨论。由于数学行为相当复杂，粘弹性仅在分离表面体现，这就是著名的粘性势理论。因此，运动方程以线性形式进行研究，同时假设了一组非线性边界条件。这一过程产生了关于界面位移的非线性表达的非线性偏微分方程。采用基于何氏频率公式的非微扰方法，将具有复系数的非线性特征常微分方程转化为线性方程。利用一种基于非微扰方法的新过程来研究非线性稳定性并审视界面表现。无量纲分析产生了几个无量纲物理数值。为验证新方法，通过一组图表描述并解释了非微扰方法与其对应的线性常微分方程在Mathematica软件中的比较。此外，还阐述了极坐标图。结果发现，在实系数和复系数情况下，稳定性机制不变。
