Purpose

This study aims to investigate the flow and heat transfer of a hybrid nanofluid through an exponentially stretching/shrinking sheet along with mixed convection and Joule heating. The nanoparticles alumina (Al₂O₃) and copper (Cu) are suspended into a base fluid (water) to form a new kind of hybrid nanofluid (Al₂O₃-Cu/water). Also, the effects of constant mixed convection parameter and Joule heating are considered.

Design/methodology/approach

The governing partial differential equations are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The transformed nonlinear ODEs are solves using the bvp4c solver available in MATLAB software. A comparison of the present results shows a good agreement with the published results.

Findings

Dual solutions for hybrid nanofluid flow obtained for a specific range of the stretching/shrinking parameter values. The values of the skin friction coefficient increases but the local Nusselt number decreases for the first solution with the increasing of the magnetic parameter. Enhancing copper volume fraction and Eckert number reduces the surface temperature, which intimates the decrement of heat transfer rate for the first and second solutions for the stretching/shrinking sheet. In detail, the first solution results show that when the Eckert number increases as 0.1, 0.4 and 0.7 at λ = 1.5, the temperature variations reduced to 10.686840, 10.671419 and 10.655996. While in the second solution, keeping the same parameters temperature variation reduced to 9.750777, 9.557349 and 9.364489, respectively. On the other hand, the results indicate that the skin friction coefficient increases with copper volume fraction. This study shows that the thermal boundary layer thickness rises due to the rise in the solid volume fraction. It is also observed that the magnetic parameter, copper volume fraction and Eckert number widen the range of the stretching/shrinking parameter for which the solution exists.

Practical implications

In practice, the investigation on the flow and heat transfer of a hybrid nanofluid past an exponentially stretching/shrinking sheet with mixed convection and Joule heating is crucial and useful. The problems related to hybrid nanofluid have numerous real-life and industrial applications, such as microelectronics, manufacturing, naval structures, nuclear system cooling, biomedical and drug reduction.

Originality/value

In specific, this study focuses on increasing thermal conductivity using a hybrid nanofluid mathematical model. The novelty of this study is the use of natural mixed convection and Joule heating in a hybrid nanofluid. This paper can obtain dual solutions. The authors declare that this study is new, and there is no previous published work similar to the present study.

中文翻译：

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混合的纳米流体流过具有对流和焦耳加热的指数拉伸/收缩片材

目的

这项研究旨在研究混合纳米流体通过指数拉伸/收缩片材以及混合对流和焦耳热的流动和传热。将纳米粒子氧化铝（Al ₂ O ₃）和铜（Cu）悬浮在基础流体（水）中，以形成一种新型的杂化纳米流体（Al ₂ O ₃ -Cu /水）。此外，还考虑了恒定的混合对流参数和焦耳热的影响。

设计/方法/方法

使用适当的相似性变换，将控制的偏微分方程转换为常微分方程（ODE）。使用MATLAB软件中提供的bvp4c求解器求解转换后的非线性ODE。本结果的比较表明已与已发布的结果很好地吻合。

发现

针对特定范围的拉伸/收缩参数值，获得了混合纳米流体流动的双重解决方案。对于第一解决方案，随着磁参数的增加，皮肤摩擦系数的值增大，但是局部努塞尔数减小。铜体积分数和埃克特数的增加降低了表面温度，这增加了用于拉伸/收缩片材的第一和第二溶液的传热速率的降低。详细地，第一解决方案结果表明，当在λ= 1.5时Eckert数增加为0.1、0.4和0.7时，温度变化减小到10.686840、10.671419和10.655996。在第二种解决方案中，保持相同的参数，温度变化分别降低到9.750777、9.557349和9.364489。另一方面，结果表明，皮肤摩擦系数随铜体积分数的增加而增加。这项研究表明，由于固体体积分数的增加，热边界层的厚度也会增加。还观察到，磁性参数，铜体积分数和埃克特数扩大了溶液存在的拉伸/收缩参数的范围。

实际影响

在实践中，研究混合纳米流体通过具有对流和焦耳加热的指数拉伸/收缩片材的流动和传热是至关重要和有用的。与混合纳米流体有关的问题在现实生活和工业应用中有很多，例如微电子学，制造，海军结构，核系统冷却，生物医学和药物减少。

创意/价值

具体而言，本研究着重使用混合纳米流体数学模型提高热导率。这项研究的新颖之处在于在混合纳米流体中使用自然混合对流和焦耳加热。本文可以获得对偶解。作者宣称，这项研究是新的，并且以前没有发表过与本研究相似的工作。