Jalili Payam, Ahmadi Azar Ali, Jalili Bahram, Domiri Ganji Davood
Department of Mechanical Engineering, North Tehran Branch, Islamic Azad University, Tehran, Iran.
Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran.
Sci Rep. 2023 Aug 15;13(1):13241. doi: 10.1038/s41598-023-40410-3.
The motion of the fluid due to the swirling of a disk/sheet has many applications in engineering and industry. Investigating these types of problems is very difficult due to the non-linearity of the governing equations, especially when the governing equations are to be solved analytically. Time is also considered a challenge in problems, and times dependent problems are rare. This study aims to investigate the problem related to a transient rotating angled plate through two analytical techniques for the three-dimensional thin film nanomaterials flow. The geometry of research is a swirling sheet with a three-dimensional unsteady nanomaterial thin-film moment. The problem's governing equations of the conservation of mass, momentum, energy, and concentration are partial differential equations (PDEs). Solving PDEs, especially their analytical solution, is considered a serious challenge, but by using similar variables, they can be converted into ordinary differential equations (ODEs). The derived ODEs are still nonlinear, but it is possible to approximate them analytically with semi-analytical methods. This study transformed the governing PDEs into a set of nonlinear ODEs using appropriate similarity variables. The dimensionless parameters such as Prandtl number, Schmidt number, Brownian motion parameter, thermophoretic parameter, Nusselt, and Sherwood numbers are presented in ODEs, and the impact of these dimensionless parameters was considered in four cases. Every case that is considered in this problem was demonstrated with graphs. This study used modified AGM (Akbari-Ganji Method) and HAN (Hybrid analytical and numerical) methods to solve the ODEs, which are the novelty of the current study. The modified AGM is novel and has made the former AGM more complete. The second semi-analytical technique is the HAN method, and because it has been solved numerically in previous articles, this method has also been used. The new results were obtained using the modified AGM and HAN solutions. The validity of these two analytical solutions was proved when compared with the Runge-Kutta fourth-order (RK4) numerical solutions.
由于圆盘/薄片的旋转而导致的流体运动在工程和工业中有许多应用。由于控制方程的非线性，研究这类问题非常困难，尤其是当要解析求解控制方程时。时间在这类问题中也被认为是一个挑战，与时间相关的问题很少见。本研究旨在通过两种解析技术研究与瞬态旋转角板相关的问题，用于三维薄膜纳米材料流动。研究的几何形状是一个具有三维非稳态纳米材料薄膜矩的旋转薄片。该问题的质量、动量、能量和浓度守恒的控制方程是偏微分方程（PDEs）。求解偏微分方程，尤其是其解析解，被认为是一项严峻的挑战，但通过使用相似变量，可以将它们转换为常微分方程（ODEs）。导出的常微分方程仍然是非线性的，但可以用半解析方法进行解析近似。本研究使用适当的相似变量将控制偏微分方程转换为一组非线性常微分方程。常微分方程中出现了诸如普朗特数、施密特数、布朗运动参数、热泳参数、努塞尔数和舍伍德数等无量纲参数，并在四种情况下考虑了这些无量纲参数的影响。该问题中考虑的每种情况都用图表进行了展示。本研究使用改进的AGM（阿克巴里 - 甘吉方法）和HAN（混合解析和数值）方法来求解常微分方程，这是当前研究的新颖之处。改进的AGM是新颖的，使原来的AGM更加完善。第二种半解析技术是HAN方法，由于之前的文章中已经对其进行了数值求解，所以也使用了该方法。使用改进的AGM和HAN解获得了新的结果。与四阶龙格 - 库塔（RK4）数值解相比，证明了这两种解析解的有效性。
