Purpose

This paper aims to examine the Cu-Al2O3/water hybrid nanofluid flow over a shrinking sheet in the presence of the magnetic field and dust particles.

Design/methodology/approach

The governing partial differential equations for the two-phase flow of the hybrid nanofluid and the dust particles are reduced to ordinary differential equations using a similarity transformation. Then, these equations are solved using bvp4c in MATLAB software. The bvp4c solver is a finite-difference code that implements the three-stage Lobatto IIIa formula. The numerical results are gained for several values of the physical parameters. The effects of these parameters on the flow and the thermal characteristics of the hybrid nanofluid and the dust particles are analyzed and discussed. Later, the temporal stability analysis is used to determine the stability of the dual solutions obtained as time evolves.

Findings

The outcome shows that the flow is unlikely to exist unless satisfactory suction strength is imposed on the shrinking sheet. Besides, the heat transfer rate on the shrinking sheet decreases with the increase of . However, the increase in and lead to enhance the heat transfer rate. Two solutions are found, where the domain of the solutions is expanded with the rising of, and. Consequently, the boundary layer separation on the surface is delayed in the presence of these parameters. Implementing the temporal stability analysis, it is found that only one of the solutions is stable as time evolves.

Originality/value

The dusty fluid problem has been widely studied for the flow over a stretching sheet, but only limited findings can be found for the shrinking counterpart. Therefore, this study considers the problem of the dusty fluid flow over a shrinking sheet containing Cu-Al₂O₃/water hybrid nanofluid with the effect of the magnetic field. In fact, this is the first study to discover the dual solutions of the dusty hybrid nanofluid flow over a shrinking sheet. Also, further analysis shows that only one of the solutions is stable as time evolves.

中文翻译：

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尘土飞扬的混合纳米流体在具有磁场效应的收缩片上流动

目的

本文旨在研究在磁场和灰尘颗粒存在的情况下，Cu-Al2O3/水混合纳米流体在收缩片材上的流动。

设计/方法/方法

使用相似变换将混合纳米流体和灰尘颗粒的两相流的控制偏微分方程简化为常微分方程。然后，在 MATLAB 软件中使用 bvp4c 求解这些方程。bvp4c 求解器是实现三阶段 Lobatto IIIa 公式的有限差分代码。对于物理参数的几个值，获得了数值结果。分析和讨论了这些参数对混合纳米流体和粉尘颗粒的流动和热特性的影响。后来，时间稳定性分析用于确定随着时间的推移获得的对偶解的稳定性。

发现

结果表明，除非在收缩片材上施加令人满意的吸力，否则不太可能存在流动。此外，收缩片上的传热率随着 的增加而减小。但是，增加并导致传热率的提高。找到了两个解，其中解的域随着和的上升而扩大。因此，在这些参数的存在下，表面上的边界层分离被延迟。实施时间稳定性分析，发现随着时间的推移，只有一个解是稳定的。

原创性/价值

尘土飞扬的流体问题已被广泛研究用于拉伸片材上的流动，但对于收缩的对应物只能找到有限的发现。因此，本研究考虑了含尘流体在磁场作用下流过含有Cu-Al ₂ O _{3 /水混合纳米流体的收缩片的问题。}事实上，这是第一项发现尘土飞扬的混合纳米流体在收缩片材上流动的双重解决方案的研究。此外，进一步的分析表明，随着时间的推移，只有一种解是稳定的。