In this research, we intend to develop a dynamical system for the magnetohydrodynamic (MHD) flow of an electrically conducting Casson nanofluid on exponentially shrinking and stretching surfaces, in the presence of a velocity and concertation slip effect, with convective boundary conditions. There are three main objectives of this article, specifically, to discuss the heat characteristics of flow, to find multiple solutions on both surfaces, and to do stability analyses. The main equations of flow are governed by the Brownian motion, the Prandtl number, and the thermophoresis parameters, the Schmid and Biot numbers. The shooting method and three-stage Lobatto IIIa formula have been employed to solve the equations. The ranges of the dual solutions are f w c 1 ≤ f w and λ c ≤ λ , while the no solution ranges are f w c 1 > f w and λ c > λ . The results reveal that the temperature of the fluid increases with the extended values of the thermophoresis parameter, the Brownian motion parameter, and the Hartmann and Biot numbers, for both solutions. The presence of dual solutions depends on the suction parameter. In order to indicate that the first solution is physically relevant and stable, a stability analysis has been performed.

中文翻译：

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通过指数收缩/拉伸板传播的磁化卡森纳米流体的稳定性分析：双解

在这项研究中，我们打算开发一个动力学系统，用于导电卡森纳米流体在指数收缩和拉伸表面上的磁流体动力学 (MHD) 流动，在存在速度和协调滑移效应的情况下，具有对流边界条件。本文的主要目标有三个，具体来说，讨论流动的热特性，在两个表面上找到多个解决方案，以及进行稳定性分析。流动的主要方程由布朗运动、普朗特数和热泳参数、施密德数和比奥数控制。采用射击法和三阶段Lobatto IIIa公式求解方程。对偶解的范围是fwc 1 ≤ fw 和λ c ≤ λ ，而无解的范围是fwc 1 > fw 和λ c > λ 。结果表明，对于两种溶液，流体的温度都随着热泳参数、布朗运动参数以及哈特曼数和比奥数的扩展值而增加。双解的存在取决于吸力参数。为了表明第一个解决方案在物理上是相关的和稳定的，已经进行了稳定性分析。