The thermal entrance problem (also known as the classical Graetz problem) is studied for the complex rheological Carreau fluid model. The solution of two-dimensional energy equation in the form of an infinite series is obtained by employing the separation of variables method. The ensuing eigenvalue problem (S–L problem) is solved for eigenvalues and corresponding eigenfunctions through MATLAB routine bvp5c. Numerical integration via Simpson’s rule is carried out to compute the coefficient of series solution. Current problem is also tackled by an alternative approach where numerical solution of eigenvalue problem is evaluated via the Runge–Kutta fourth order method. This problem is solved for both flat and circular confinements with two types of boundary conditions: (i) constant wall temperature and (ii) prescribed wall heat flux. The obtained results of both local and mean Nusselt numbers, fully developed temperature profile and average temperature are discussed for different values of Weissenberg number and power-law index through graphs and tables. This study is valid for typical range of Weissenberg number We≤1$\left(We\le 1\right)$ and power-law index n<1$\left(n{< }1\right)$ for shear-thinning trend while n>1$\left(n{ >}1\right)$ for shear-thickening behaviour. The scope of the present study is broad in the context that the solution of the said problem is achieved by using two different approaches namely, the traditional Graetz approach and the solution procedure documented in M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.

中文翻译：

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使用 Carreau 流体模型对轴对称管和扁平通道的经典 Graetz-Nusselt 问题进行数学建模：数值基准研究

研究了复杂流变 Carreau 流体模型的热入口问题（也称为经典 Graetz 问题）。采用分离变量的方法得到了无穷级数形式的二维能量方程的解。随后的特征值问题（S-L 问题）通过 MATLAB 例程 bvp5c 求解特征值和相应的特征函数。通过辛普森规则进行数值积分来计算级数解的系数。当前问题也通过另一种方法解决，其中通过 Runge-Kutta 四阶方法评估特征值问题的数值解。对于具有两种类型边界条件的平面和圆形限制，此问题已得到解决：(i) 壁面温度恒定和 (ii) 规定的壁面热通量。通过图表和表格讨论了对于不同的魏森伯格数和幂律指数的局部和平均努塞尔数、完全发展的温度剖面和平均温度的所得结果。本研究适用于典型的Weissenberg数We≤1$\left(We\le 1\right)$和幂律指数n<1$\left(n{< }1\right)$剪切稀化趋势而 n>1$\left(n{ >}1\right)$ 用于剪切增稠行为。本研究的范围很广，因为通过使用两种不同的方法来解决上述问题，即传统的 Graetz 方法和 MD Mikhailov 和 MN Ozisik 中记录的求解过程，Unified Analysis and Solutions of Heat 和质量扩散，纽约，多佛，1994。通过图表和表格讨论了针对不同的魏森伯格数和幂律指数值的完全发展的温度曲线和平均温度。本研究适用于典型的Weissenberg数We≤1$\left(We\le 1\right)$和幂律指数n<1$\left(n{< }1\right)$剪切稀化趋势而 n>1$\left(n{ >}1\right)$ 用于剪切增稠行为。本研究的范围很广，因为通过使用两种不同的方法来解决上述问题，即传统的 Graetz 方法和 MD Mikhailov 和 MN Ozisik、Unified Analysis and Solutions of Heat 和质量扩散，纽约，多佛，1994。通过图表和表格讨论了针对不同的魏森伯格数和幂律指数值的完全发展的温度曲线和平均温度。本研究适用于典型的Weissenberg数We≤1$\left(We\le 1\right)$和幂律指数n<1$\left(n{< }1\right)$剪切稀化趋势而 n>1$\left(n{ >}1\right)$ 用于剪切增稠行为。本研究的范围很广，因为通过使用两种不同的方法来解决上述问题，即传统的 Graetz 方法和 MD Mikhailov 和 MN Ozisik、Unified Analysis and Solutions of Heat 和质量扩散，纽约，多佛，1994 年。