Purpose

The purpose of this study is to investigate the simultaneous effects of normal wall transpiration, stretching strength parameter, velocity slip and nanoparticles on the flow of a ternary hybrid nanofluid through an elastic surface. The goal is to understand the behavior of the flow field, temperature distribution, skin friction and temperature gradient under these conditions, and to explore the existence and nature of solutions under varying parameter values.

Design/methodology/approach

The analysis involves expressing the flow field, power-law temperature field, skin friction and temperature gradient in closed-form formulas. The study examines both stretching and shrinking surfaces, distinguishing between unique and dual solutions. The methodology includes deriving exact solutions for exponential and algebraic temperature and temperature rate formulas analytically by deriving the system of governing equations into ordinary differential equations.

Findings

The study reveals that for a stretching sheet, the solution is unique, whereas dual solutions are observed for a shrinking surface. Special solutions are provided for various parametric values, showing the behavior of the exponential and algebraic temperature and temperature rate, with a focus on identifying turning points that demarcate the existence and non-existence of single or multiple solutions. The solutions are represented through graphs and tables to facilitate a comprehensive qualitative analysis. The research identifies turning points that determine the presence or absence of single or multiple solutions, uncovering multiple solutions for different parameter sets. These findings are displayed graphically and in tabular form, highlighting the complex interplay between the parameters and the resulting flow behavior.

Originality/value

This analysis contributes to the field by providing new insights into the multiple solution phenomena in ternary hybrid nanofluid flows, particularly under the combined effects of normal wall transpiration, stretching strength, velocity slip and nanoparticle presence. The identification of turning points and the exact solutions for various temperature profiles are of significant value, offering a deeper understanding of the factors influencing the flow and thermal characteristics in such systems. The study’s findings have potential applications in optimizing fluid flow in engineering systems where such conditions are prevalent.

中文翻译：

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三杂化纳米流体流动中的多个精确解：弹性表面效应的研究

 目的

本研究的目的是研究法向壁蒸腾、拉伸强度参数、速度滑移和纳米颗粒对三元杂化纳米流体通过弹性表面的流动的同时影响。目标是了解这些条件场、温度分布、蒙皮摩擦和温度梯度的行为，并探索不同参数值下解的存在和性质。

设计/方法/方法

分析涉及用封闭式公式表示流场、幂律温度场、蒙皮摩擦和温度梯度。该研究检查了拉伸和收缩表面，区分了独特解决方案和双重解决方案。该方法包括通过将控制方程组推导出常微分方程组，以分析方式推导出指数和代数温度以及温度速率公式的精确解。

 发现

研究表明，对于拉伸板，解是唯一的，而对于收缩表面，观察到双重解。为各种参数值提供了特殊解，显示了指数和代数温度以及温度速率的行为，重点是确定划分单个或多个解的存在和不存在的转折点。这些解决方案通过图形和表格表示，以便于进行全面的定性分析。该研究确定了决定是否存在单个或多个解决方案的转折点，揭示了不同参数集的多个解决方案。这些发现以图形和表格形式显示，突出了参数与生成的流动行为之间的复杂相互作用。

 原创性/价值

该分析通过为三元杂化纳米流体流动中的多重解现象提供新的见解，特别是在正常壁蒸腾、拉伸强度、速度滑移和纳米颗粒存在的综合影响下，为该领域做出了贡献。确定转折点和各种温度曲线的精确解决方案具有重要价值，有助于更深入地了解影响此类系统中流动和热特性的因素。该研究结果在优化此类条件普遍存在的工程系统中的流体流动方面具有潜在应用。