This paper focuses primarily on the thermal analysis of the steady, three-dimensional flow of CMC (sodium carboxymethyl cellulose) base fluid with Cu nanoparticles through the implantation plate of Riga in a porous Darcy–Forchheimer material with internal heat generation (absorption) effects. Thermophysical feature of CMC is taken into account with Al₂O₃ nanoparticle in it. Also, thermal radiation and convective boundary constraint are considered into account. Adequate transformations of the issue controlling formulas produce a system of nonlinear ordinary differential equations. The optimal homotopy analysis method is used for calculating the parameters values impacts. The expressions of local Nusselt number and skin friction coefficient are studied and debated. The physical influences of all emerging parameters are debated on graphically forms. The findings suggest that with higher values of Hartmann number, the skin friction in y-direction decreases on the contrary in the case of x-direction, while the Hartmann number increases the rate of heat transportation. Besides, the addition values of thermal radiation and Biot number enhance the rate of heat transport. Comparisons have been made with published literature, and an excellent agreement is clear. This underlines the importance of the study in potential medical and industrial cooling applications.

本文主要关注带有纳米粒子的CMC（羧甲基纤维素钠）的稳态三维流与Cu纳米粒子通过里加注入板的作用，这种注入是在具有内部生热（吸收）效应的多孔Darcy-Forchheimer材料中进行的。Al ₂ O ₃考虑到了CMC的热物理特征纳米粒子。此外，考虑了热辐射和对流边界约束。问题控制公式的充分变换产生了非线性常微分方程组。最优同伦分析方法用于计算参数值的影响。对局部Nusselt数和皮肤摩擦系数的表达式进行了研究和讨论。所有新兴参数的物理影响都以图形形式讨论。研究结果表明，随着哈特曼数的增加，与x相反，y方向的皮肤摩擦减小。方向，而哈特曼数增加了热传输率。此外，热辐射和比奥数的相加值提高了热传递的速率。与已发表的文献进行了比较，很明显的一致。这强调了这项研究在潜在的医疗和工业制冷应用中的重要性。