Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
Sci Rep. 2023 Mar 7;13(1):3764. doi: 10.1038/s41598-023-30749-y.
The major aim of this work is to achieve a mathematical technique to scrutinize the nonlinear instability of a vertical cylindrical boundary separation of two streaming Reiner-Rivlin liquids. The system is portrayed by an unchanged longitudinal electric strength. Furthermore, the action of mass and heat transfer (MHT) and permeable media are also considered. The problem is not only of methodological interest but also of scientific and practical interest. To shorten the mathematical analysis, Hsieh's modulation together with the viscous potential theory (VPT) is employed. The nonlinear diagram is contingent on tackling the governing linear mechanism along with the nonlinear applicable border restrictions. A non-dimensional process produces several non-dimensional physical numbers. A linear dispersion equation is attained and the stability standards are theoretically governed and numerically established. The nonlinear stability procedure reveals a Ginzburg-Landau formula. Consequently, nonlinear stability stipulations are accomplished. Furthermore, by way of the Homotopy perturbation approach, along with the expanded frequency concept, an accurate perturbed technique of surface deflection is attained theoretically and numerically. To validate the theoretical outcomes, the analytical expression is confirmed through the Rung-Kutta of the fourth order. The stable and unstable zones are signified graphically displaying the influences of several non-dimensional numbers.
这项工作的主要目的是实现一种数学技术，以仔细研究两种流动的 Reiner-Rivlin 液体的垂直圆柱边界分离的非线性不稳定性。该系统由不变的纵向电场强度描述。此外，还考虑了质量和热传递（MHT）以及可渗透介质的作用。这个问题不仅具有方法学上的意义，而且具有科学和实际意义。为了简化数学分析，采用了 Hsieh 的调制以及粘性势理论（VPT）。非线性图取决于解决沿非线性适用边界限制的线性控制机制。无量纲处理产生了几个无量纲物理数。获得了线性频散方程，并从理论上和数值上确定了稳定性标准。非线性稳定性过程揭示了一个吉布斯-朗道公式。因此，完成了非线性稳定性规定。此外，通过同伦摄动方法以及扩展频率概念，从理论上和数值上获得了表面挠度的精确摄动技术。为了验证理论结果，通过四阶龙格-库塔方法对解析表达式进行了验证。通过图形显示稳定和不稳定区域，显示了几个无量纲数的影响。
