In this paper, the laminar natural convection of non-Newtonian Carreau fluid in a square cavity under uniform magnetic field in different directions is investigated numerically. Based on the projection method, a new finite-difference algorithm on a staggered grid is employed to solve the laminar magnetohydrodynamic natural convection problems, which involves the second-order central scheme for the discretization of non-Newtonian viscous terms. To assess numerical capability of the newly proposed algorithm, the calculation results for Newtonian fluid in the square cavity show excellent agreement with results available in the literature. Research work has been performed for the certain pertinent parameters of Rayleigh number (\(10^{4}\) and \(10^{5}\)) and Prandtl number (\(Pr=0.065\)) using the new numerical algorithm. The computed results show that the natural convection of Carreau fluid is not only determined by the strength of the magnetic field, but also influenced by the inclination angle. In particular, when the Carreau fluid describes a non-Newtonian fluid, it is found that the inclination angle plays a large role on flow and heat transfer.

中文翻译：

---

磁场在不同方向上充满Carreau流体的方腔中MHD自然对流换热的数值模拟

本文研究了非牛顿Carreau流体在不同方向上均匀磁场作用下方腔内的层流自然对流。基于投影方法，在交错网格上采用了一种新的有限差分算法来解决层流磁流体动力学自然对流问题，该算法涉及非牛顿粘性项离散化的二阶中心方案。为了评估新提出的算法的数值能力，方腔内牛顿流体的计算结果与文献中的结果具有极好的一致性。已经对瑞利数（\（10 ^ {4} \）和\（10 ^ {5} \））和普朗特数（\（Pr = 0.065 \））使用新的数值算法。计算结果表明，Carreau流体的自然对流不仅取决于磁场强度，而且还受倾角的影响。特别地，当卡洛流体描述为非牛顿流体时，发现倾斜角对流动和热传递起很大作用。