在斯托克斯振荡流下具有时间分数阶加速度的牛顿流体自由对流稳定性数值模拟

Numerical Simulation of Free Convection Stability of Newtonian Fluids with Time-Fractional Acceleration under Stokes Oscillatory Flow

摘要: 为揭示时间分数阶动力学对牛顿流体自由对流行为的调控机制，本研究将时间分数阶导数纳入流体自由对流控制方程，结合相平面图分析，通过数值模拟与Mathematica参数化算法(每对控制参数生成10⁴个样本点)，系统探究分数阶特性对流体稳定性及关键物理量的影响。结果表明：时间分数阶导数的非局部特性与记忆效应可显著增强系统不稳定性，催生以不稳定振荡为特征的复杂对流模式；牛顿流体的壁面剪切应力与努塞尔数随分数阶参数($\alpha$ )减小而降低，随幂律指数增大而升高；所建算法与分析方法实现了稳定/不稳定区域的精准界定。本研究完善了复杂流体系统的理论框架，深化了记忆效应与反常扩散对流体稳定性及混沌转变的认知，为涉及牛顿流体的工程与工业应用提供了控制优化的理论支撑与新视角。

Abstract: To reveal the regulatory mechanism of time-fractional dynamics on the natural convection behavior of Newtonian fluids, this study incorporates the time-fractional derivative into the governing equations of fluid free convection. Combined with phase plane analysis, numerical simulations and a Mathematica-based parameterized algorithm (generating 10⁴ sample points for each pair of control parameters) are employed to systematically investigate the effects of fractional characteristics on fluid stability and key physical quantities. The results show that the nonlocal property and memory effect of the time-fractional derivative can significantly enhance system instability, leading to complex convection patterns characterized by unstable oscillations; the wall shear stress and Nusselt number of Newtonian fluids decrease with the reduction of the fractional parameter ($\alpha$ ) and increase with the rise of the power-law index; the established algorithm and analytical method realize the accurate delineation of stable/unstable regions. This study improves the theoretical framework of complex fluid systems, deepens the understanding of the impacts of memory effect and anomalous diffusion on fluid stability and chaos transition, and provides theoretical support and new perspectives for the control and optimization in various engineering and industrial applications involving Newtonian fluids.