Exact analytical solutions for start-up and cessation flows are obtained for the affine linear Phan-Thien--Tanner fluid model. They include the results for start-up and cessation of steady shear flows, of steady uniaxial and biaxial extensional flows, and of steady planar extensional flows. The solutions obtained show that at start-up of steady shear flows, the stresses go through quasi-periodic exponentially damped oscillations while approaching their steady-flow values (so that stress overshoots are present); at start-up of steady extensional flows, the stresses grow monotonically, while at cessation of steady shear and extensional flows, the stresses decay quickly and non-exponentially. The steady-flow rheology of the fluid is also reviewed, the exact analytical solutions obtained in this work for steady shear and extensional flows being simpler than the alternative formulas found in the literature. The properties of steady and transient solutions, including their asymptotic behavior at low and high Weissenberg numbers, are investigated in detail. Generalization to the multimode version of the Phan-Thien--Tanner model is also discussed. Thus, this work provides a complete analytical description of the rheology of the affine linear Phan-Thien--Tanner fluid in start-up, cessation, and steady regimes of shear and extensional flows.

中文翻译：

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稳态剪切流和拉伸流的启动和停止：仿射线性 Phan-Thien-Tanner 流体模型的精确解析解

为仿射线性 Phan-Thien-Tanner 流体模型获得启动和停止流的精确解析解。它们包括稳态剪切流、稳态单轴和双轴拉伸流以及稳态平面拉伸流的启动和停止结果。获得的解决方案表明，在稳定剪切流开始时，应力在接近其稳定流值时经历准周期指数阻尼振荡（因此存在应力超调）；在稳定拉伸流开始时，应力单调增长，而在稳定剪切和拉伸流停止时，应力快速且非指数衰减。还回顾了流体的稳态流变学，在这项工作中获得的稳定剪切和拉伸流动的精确解析解比文献中找到的替代公式更简单。详细研究了稳态和瞬态解的性质，包括它们在低和高 Weissenberg 数下的渐近行为。还讨论了对 Phan-Thien--Tanner 模型的多模版本的推广。因此，这项工作为仿射线性 Phan-Thien-Tanner 流体在剪切和拉伸流的启动、停止和稳定状态下的流变学提供了完整的分析描述。还讨论了对 Phan-Thien--Tanner 模型的多模版本的推广。因此，这项工作为仿射线性 Phan-Thien-Tanner 流体在剪切和拉伸流的启动、停止和稳定状态下的流变学提供了完整的分析描述。还讨论了对 Phan-Thien--Tanner 模型的多模版本的推广。因此，这项工作为仿射线性 Phan-Thien-Tanner 流体在剪切和拉伸流的启动、停止和稳定状态下的流变学提供了完整的分析描述。