This study investigates the role of thermal boundary conditions on natural convection and entropy generation in non-Newtonian power-law fluids confined within a square cavity and a concentric cylindrical annulus. Steady, two-dimensional governing equations based on the incompressible power-law model and the Boussinesq approximation are solved using the Gridap.jl finite element framework. The numerical methodology is validated against benchmark solutions for both Newtonian and non-Newtonian convection, showing good agreement in terms of isotherm fields, streamlines, local Nusselt number distributions, and entropy generation. The effects of fluid rheology and heating mode are examined for shear-thinning, Newtonian, and shear-thickening fluids under uniform and non-uniform thermal boundary conditions. The results show that shear-thinning behavior enhances buoyancy-driven circulation, steepens thermal gradients, and increases heat transfer, whereas shear-thickening behavior suppresses convection and promotes conduction-dominated transport. Thermal boundary conditions are found to play an important role in controlling the intensity and spatial distribution of flow, heat transfer, and irreversibility. In both geometries, uniform heating produces stronger and more distributed convective structures, while non-uniform sinusoidal heating localizes thermal forcing and consistently reduces total entropy generation. An entropy analysis further reveals that viscous dissipation dominates irreversibility in shear-thinning fluids, whereas heat-transfer irreversibility becomes dominant as the power-law index increases. The study demonstrates that appropriate thermal boundary design, together with fluid rheology, provides an effective route for controlling heat transfer and minimizing thermodynamic losses in non-Newtonian convection systems. The source code and metadata are publicly available.

翻译：本研究探讨了热边界条件对方形空腔和同心圆柱环形区域内非牛顿幂律流体自然对流与熵产的作用。基于不可压缩幂律模型和Boussinesq近似建立的稳态二维控制方程，采用Gridap.jl有限元框架求解。数值方法通过牛顿流体和非牛顿流体对流的基准解验证，在等温场、流线、局部努塞尔数分布及熵产方面均吻合良好。针对剪切稀化、牛顿和剪切增稠流体，研究了均匀与非均匀热边界条件下流体流变性与加热模式的影响。结果表明：剪切稀化行为增强浮力驱动循环、加剧热梯度并提升传热效率，而剪切增稠行为抑制对流并促进导热带主导的输运。热边界条件在控制流动强度与空间分布、传热及不可逆性方面具有重要作用。两种几何结构中，均匀加热产生更强且更分散的对流结构，而非均匀正弦加热会局域化热驱动力并持续降低总熵产。熵分析进一步揭示：剪切稀化流体中黏性耗散主导不可逆性，而随幂律指数增大，传热不可逆性逐渐占优。研究表明，结合流体流变性的合理热边界设计，可为非牛顿对流系统传热控制与热力学损耗最小化提供有效途径。源代码及元数据已公开。