Couette–Poiseuille (C–P) flow, which is driven by drag from a moving wall and a pressure gradient, can exist in different states depending on the relative strengths of the two above-mentioned factors. Of particular interest is the onset of flow reversal, which is characterized kinematically by a zero shear rate on the stationary wall. This study presents two different methods for obtaining the critical conditions for the onset of flow reversal in C–P flows. In the first method, exact values of the critical flow rate and pressure gradient are computed by solving a pair of algebraic equations derived from the Weissenberg–Rabinowitsch relation. Using this method, the difficulty in solving the nonlinear differential equation is avoided. In the second method, estimates of the critical conditions are obtained analytically by locally approximating the given fluid as a power-law fluid. To evaluate the prediction accuracy, the methods are applied to the C–P flows of Carreau–Yasuda and Bingham–Carreau–Yasuda fluids. It is demonstrated that the relative errors remained reasonably low in most system parameter ranges, except in cases where the flow curve in the log–log scale is highly nonlinear.

中文翻译：

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在 Couette-Poiseuille 流中获得流动反转条件的简单方法

库埃特-泊肃叶 (C-P) 流由移动壁的阻力和压力梯度驱动，可以根据上述两个因素的相对强度以不同的状态存在。特别令人感兴趣的是流动逆转的开始，其运动学特征是静止壁上的零剪切速率。本研究提出了两种不同的方法来获得 C-P 流中流动逆转开始的临界条件。在第一种方法中，临界流速和压力梯度的精确值是通过求解从 Weissenberg-Rabinowitsch 关系导出的一对代数方程来计算的。使用这种方法，避免了求解非线性微分方程的困难。在第二种方法中，通过将给定的流体局部近似为幂律流体，通过分析获得临界条件的估计值。为了评估预测精度，将这些方法应用于 Carreau-Yasuda 和 Bingham-Carreau-Yasuda 流体的 C-P 流。结果表明，在大多数系统参数范围内，相对误差保持相当低，但对数-对数标度中的流量曲线高度非线性的情况除外。