Abstract In this paper, we provide simple expressions for the permanent solutions corresponding to some oscillatory motions of two classes of Newtonian fluids with power-law dependence of viscosity on the pressure between two infinite horizontal parallel plates. The fluid motion is generated by the lower plate that applies an oscillatory shear stress to the fluid. Such solutions, which are lack in the existing literature, can be useful both for those who want to eliminate the transients from their experiments and as tests to verify numerical schemes that are developed to study complex unsteady flow problems of these fluids. The similar solutions corresponding to the motion due to a constant shear stress on the boundary are also determined and, contrary to our expectations, the shear stresses are constant on the whole flow domain although the associated velocity fields depend both of the spatial variable and the dimensionless pressure-viscosity coefficient. Finally, for validation, some comparative graphical illustrations are included and the convergence of starting solutions to the permanent solutions is graphically proved. Spatial profiles of starting solutions are also provided.

中文翻译：

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流体的某些振荡运动的永久解具有粘度对边界压力和剪切应力的幂律依赖性

摘要 在本文中，我们提供了对应于两类牛顿流体的一些振荡运动的永久解的简单表达式，其粘度对两个无限水平平行板之间的压力具有幂律依赖性。流体运动由对流体施加振荡剪切应力的下板产生。这种现有文献中缺乏的解决方案对于那些想要从实验中消除瞬变的人以及作为验证数值方案的测试都是有用的，这些方案是为研究这些流体的复杂非定常流动问题而开发的。与由于边界上的恒定剪应力引起的运动相对应的类似解也被确定，与我们的预期相反，尽管相关的速度场取决于空间变量和无量纲压力粘度系数，但剪切应力在整个流域上是恒定的。最后，为了验证，包含了一些比较图形说明，并以图形方式证明了起始解决方案与永久解决方案的收敛性。还提供了起始溶液的空间分布。