Y. Nawaz, M. Arif, Muavia Mansoor, K. Abodayeh, A. Baazeem
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引用次数: 0
Abstract
An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme.
本文提出了一种用于求解分形时变偏微分方程(PDEs)的显式计算方案。该方案是利用分形泰勒级数构建的三阶段方案。该方案的分形时间阶数为三阶。该方案还确保了稳定性。该方法用于模拟非牛顿流体在静止和振荡表面上的时变边界层流动,同时考虑了取决于空间和温度的热量产生的影响。所考虑的不可压缩流体的连续性方程采用一阶反向差分公式离散化,而无量纲纳维-斯托克斯方程、能量和纳米粒子体积分数方程则采用所提出的分形时间方案离散化。不同参数对速度、温度和纳米粒子体积分数的影响以图形显示。速度曲线随着参数 I 的增大而上升。我们主要应用这种计算方法来分析非牛顿流体在平面和振荡片上随时间变化的边界层流动。考虑与空间和温度相关的热量产生是一个关键因素,它为分析带来了额外的复杂性。不可压缩流体的连续性方程采用一阶反向差分公式离散化。另一方面,无量纲纳维-斯托克斯方程、能量方程和控制纳米粒子体积分数的方程则采用所提出的分形时间相关方案进行离散化。
期刊介绍:
Fractal and Fractional is an international, scientific, peer-reviewed, open access journal that focuses on the study of fractals and fractional calculus, as well as their applications across various fields of science and engineering. It is published monthly online by MDPI and offers a cutting-edge platform for research papers, reviews, and short notes in this specialized area. The journal, identified by ISSN 2504-3110, encourages scientists to submit their experimental and theoretical findings in great detail, with no limits on the length of manuscripts to ensure reproducibility. A key objective is to facilitate the publication of detailed research, including experimental procedures and calculations. "Fractal and Fractional" also stands out for its unique offerings: it warmly welcomes manuscripts related to research proposals and innovative ideas, and allows for the deposition of electronic files containing detailed calculations and experimental protocols as supplementary material.