The flow of Bingham materials inside rectangular channels is characterized by a plug region located in the central region of the conduit and four dead zones in the vicinity of the corners of the geometry (i.e., the unyielded regions) that grow as the Bingham number is increased. In this paper, a semianalytical technique is presented for solving the flow of Bingham yield-stress materials inside straight rectangular ducts. To capture the topological shapes of the yielded and unyielded regions, a “squircle” mapping approach is proposed and the interfaces between the yielded and unyielded regions are captured based on minimizing the variational formulation for the velocity principle. The nonlinear governing equations are consequently solved using the Ritz method. To find the optimized solution, a genetic algorithm method is implemented, applying bounded constraints and a sufficient number of populations. The presented results are benchmarked against numerical works available in literature, revealing that the presented solution can accurately capture the positions of both the plug and dead regions. The critical Bingham number at which the plug region meets the dead regions is reported for different aspect ratios. The velocity profiles and the friction factor of flow with different Bingham numbers and aspect ratios are investigated in detail. Following this, in the limiting case in which the Bingham number is sufficiently small, an alternative, simpler analytical approach using elliptical mapping is also proposed. The authors believe that the proposed method could be employed as a useful tool to obtain fast, accurate approximate analytical solutions for similar classes of problem.

宾厄姆材料在矩形通道内的流动的特征在于，位于导管中心区域的塞子区域和几何学角附近的四个盲区（即，非屈服区域）随着宾厄姆数的增加而增长。在本文中，提出了一种半解析技术来解决宾厄姆屈服应力材料在矩形直管内的流动。为了捕获屈服区域和非屈服区域的拓扑形状，提出了一种“孢子”映射方法，并基于最小化速度原理的变分公式，捕捉了屈服区域和非屈服区域之间的界面。因此，使用Ritz方法求解非线性控制方程。为了找到优化的解决方案，实施了一种遗传算法方法，应用有限的约束和足够数量的人口。提出的结果与文献中的数字工作进行了基准比较，表明所提出的解决方案可以准确地捕获堵塞区域和死区的位置。对于不同的纵横比，报告了塞子区域与死区相遇的临界宾汉数。详细研究了具有不同宾厄姆数和纵横比的流动的速度分布和摩擦系数。此后，在宾厄姆数足够小的极限情况下，还提出了使用椭圆映射的另一种更简单的分析方法。作者认为，所提出的方法可以用作有用的工具，以获取针对类似问题类别的快速，准确的近似解析解。