This study deals with the influence of a rotating cylinder inside the square cavity and multiwalled carbon nanotube–iron oxide (${\mathrm{Fe}}_3{\mathrm{O}}_4$) hybrid nanofluid with water. A two-dimensional system of partial differential equations has been discretized by employing Galerkin finite element method. A finite element method involving the cubic polynomials $({\mathbb{P}}_3)$ has been implemented to compute for velocity, temperature, and concentration fields, while the pressure is approximated by quadratic $({\mathbb{P}}_2)$ finite element space of functions. The adaptive Newton method is implemented to solve the discrete system of nonlinear equations. Numerical simulations are performed for different ranges of pertinent parameters such as the nanoparticles volume fraction (between 0 and 0.3%), angular velocity (between $-50$ and $+50$), Richardson number (between 0.01 and 3), and rotating cylinder sizes (between 0.2 and 0.4), and three different thermal conditions on the cylinder are investigated: namely, the adiabatic cylinder, cold cylinder, and hot cylinder. It is inferred that the average Nusselt number increases, when the cylinder rotates clockwise. On the contrary, the average Nusselt temperature increases, when the cylinder rotates counterclockwise. Moreover, the local Nusselt number increasing instead of enhancing the rotational speed of the circular cylinder toward the clockwise rotation. The flow philosophy is presented in the form of isotherms, streamlines, and some appropriate plots.

本研究涉及方腔内旋转圆柱体和多壁碳纳米管-氧化铁的影响。${铁}_3{\mathrm{哦}}_4$) 混合纳米流体与水。采用伽辽金有限元方法离散化了二维偏微分方程组。涉及三次多项式的有限元方法$({\mathbb{磷}}_3)$ 已实现计算速度、温度和浓度场，而压力由二次方近似 $({\mathbb{磷}}_2)$函数的有限元空间。自适应牛顿法用于求解非线性方程的离散系统。对不同范围的相关参数进行数值模拟，例如纳米颗粒的体积分数（介于 0 和 0.3% 之间）、角速度（介于$-50$ 和 $+50$）、理查森数（0.01 和 3 之间）和旋转圆柱体尺寸（0.2 和 0.4 之间）以及圆柱体上的三种不同热条件：即绝热圆柱体、冷圆柱体和热圆柱体。推断当圆柱顺时针旋转时，平均努塞尔数增加。相反，当圆柱逆时针旋转时，平均努塞尔温度升高。此外，局部努塞尔数增加而不是增加圆柱体的旋转速度向顺时针方向旋转。流动原理以等温线、流线和一些适当的图的形式呈现。