The problem of an enforced fluid flow in a flat channel with the counter motion of one of the walls was considered. The fluid was characterized by a nonmonotonic flow curve consisting of three segments; a left segment (an ascending branch), a middle segment (a descending branch), and a right segment (an ascending branch). The rheological properties of the fluid were described by the modified Vmogradov-Pokrovsky model. The constants of the model were determined using the results of rheological tests of a high-density polyethylene melt performed with a laser Doppler viscometer. All exact analytical solutions of this problem were obtained in a parametric form for the one-dimensional case. The profiles of the velocity, effective viscosity, and velocity gradient along the channel height are constructed for different values of the parameters of the rheological model. Three solutions exist for the same given stress field in the range of shear rates corresponding to the middle branch of the flow curve. One of them is unstable and physically unrealizable, while the other two solutions are stable; however, the loading prehistory determines which of them is observed. The solution corresponding to the left branch is monotonic, while the solution corresponding to the right branch of the curve demonstrates the flow stratification into “bands” with different strain rates and different physical and mechanical properties. At the same time, the dependence of the effective viscosity on the strain rate, which is a monotonically decreasing function, allows its representation in the form of an exponential series. The same pressure-flow problem is solved in the two-dimensional formulation by the finite-element method using the semiweak Galerkin formulation and an approximation function for the viscosity. The comparison of the numerical results and the analytical solution shows that they are similar with a sufficient degree of accuracy. In both cases, as the counter pressure difference approaches zero, a limiting transition to the Couette flow is impossible.

中文翻译：

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具有流动应力对应变率的非单调依赖性的流体流动的剪切带

考虑了在平坦通道中强制流体流动与其中一个壁的反向运动的问题。流体的特点是由三段组成的非单调流动曲线；左段（升支）、中段（降支）和右段​​（升支）。流体的流变特性由修正的 Vmogradov-Pokrovsky 模型描述。模型的常数是使用激光多普勒粘度计对高密度聚乙烯熔体进行流变测试的结果确定的。该问题的所有精确解析解都是在一维情况下以参数形式获得的。速度、有效粘度、沿通道高度的速度梯度和速度梯度是针对流变模型参数的不同值构建的。在对应于流动曲线中间分支的剪切速率范围内，对于相同的给定应力场，存在三种解。其中一个是不稳定且物理上无法实现的，而另外两个解是稳定的；然而，加载史前史决定了它们中的哪些被观察到。对应于左分支的解是单调的，而对应于曲线右分支的解表明流动分层为具有不同应变率和不同物理和机械性能的“带”。同时，有效粘度对应变率的依赖性，这是一个单调递减的函数，允许以指数级数的形式表示。使用半弱伽辽金公式和粘度近似函数，通过有限元方法在二维公式中解决了相同的压力-流动问题。数值结果与解析解的比较表明它们具有足够的精度相似。在这两种情况下，当反压差接近零时，向 Couette 流的限制过渡是不可能的。