Purpose

This paper aims to explore dual solutions for the flow of a hybrid nanofluid over a permeable melting stretching/shrinking sheet with nanoparticle shape factor, second-order velocity slip conditions and viscous dissipation. The hybrid nanofluid is formulated by dispersing alumina (Al₂O₃) and copper (Cu) nanoparticles into water (H₂O).

Design/methodology/approach

The governing partial differential equations (PDEs) are first reduced to a system of ordinary differential equations (ODEs) using a mathematical method of similarity transformation technique. These ODEs are then numerically solved through MATLAB’s bvp4c solver.

Findings

Key parameters such as slip parameter, melting parameter, suction parameter, shrinking parameter and Eckert number are examined. The results reveal the existence of two distinct solutions (upper and lower branches) for the transformed ODEs when considering the shrinking parameter. Increasing value of Cu-volume fraction and the second-order velocity slip enhances boundary layer thicknesses, whereas the heat transfer rate diminishes with rising melting and suction parameters. These numerical results are illustrated through various figures and tables. Additionally, a stability analysis is performed and confirms the upper branch is stable and practical, while the lower branch is unstable.

Practical implications

The analysis of hybrid nanofluid flow over a shrinking surface has practical significance with applications in processes such as solar thermal management systems, automotive cooling systems, sedimentation, microelectronic cooling or centrifugal separation of particles. Both steady and unsteady hybrid nanofluid flows are relevant in these contexts.

Originality/value

While the study of hybrid nanofluid flow is well-documented, research focusing on the shrinking flow case with specific parameters in our study is still relatively scarce. This paper contributes to obtaining dual solutions specifically for the shrinking case, which has been less frequently addressed.

中文翻译：

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混合纳米流体的双解流经具有更高阶滑移、形状因子和粘性耗散效应的可渗透熔融收缩片

 目的

本文旨在探索混合纳米流体在具有纳米颗粒形状因子、二阶速度滑移条件和粘性耗散的可渗透熔融拉伸/收缩片上流动的双重解决方案。杂化纳米流体是通过将氧化铝 （Al₂O₃） 和铜 （Cu） 纳米颗粒分散到水中 （H₂O） 中配制而成的。

设计/方法/方法

首先使用相似性变换技术的数学方法将控制偏微分方程 （PDE） 简化为常微分方程组 （ODE）。然后通过 MATLAB 的 bvp4c 求解器对这些 ODE 进行数值求解。

 发现

检查关键参数，如滑移参数、熔融参数、吸力参数、收缩参数和埃克特数。结果表明，在考虑收缩参数时，变换的 ODE 存在两个不同的解（上分支和下分支）。Cu 体积分数和二阶速度滑移值的增加增加了边界层厚度，而传热速率随着熔化和吸力参数的增加而降低。这些数值结果通过各种数字和表格进行说明。此外，还进行了稳定性分析，确认上支管稳定实用，而下支管不稳定。

 实际意义

分析收缩表面上的混合纳米流体流动具有实际意义，可应用于太阳能热管理系统、汽车冷却系统、沉降、微电子冷却或颗粒离心分离等过程。稳态和非稳态混合纳米流体流动在这些情况下都是相关的。

 原创性/价值

虽然对混合纳米流体流动的研究有据可查，但在我们的研究中，专注于具有特定参数的收缩流动情况的研究仍然相对较少。本文有助于获得专门针对收缩情况的双重解决方案，而这种情况很少得到解决。