Purely elastic instabilities and turbulence in flows of polymer solutions

复杂流体（如聚合物、胶体、乳液、膏体等）的流动在日常生活中是很丰富的。人们可以想到从瓶子里倒糖浆或从管子里挤牙膏，也可以想到纤维纺丝和挤出-用于生产塑料袋、光纤、电线涂层、吉他弦等的工艺。复杂的流体，特别是聚合物溶液，通常会表现出意想不到的行为-它们不像水一样流动。例如，如果一个人在一杯茶中旋转勺子，茶被推向杯壁。相反，当用聚合物液体重复这个实验时，聚合物向勺子移动，产生所谓的爬杆效应。更令人惊讶的是，聚合物的流动可能变得不稳定。其中一个著名的例子是在挤出浓缩聚合物溶液或熔体时观察到的熔体破裂现象，这是工业中聚合物加工的主要要素之一。在那里，液体被压过一根细毛细管，产生一股规则的聚合物射流。在低挤出速度下，射流保持笔直和均匀，而在较大的速度下，流动开始波动，变得混乱并最终破裂。这些不稳定性是主要的生产限制因素之一，多年来一直困扰着技术和工业。它们的存在是令人惊讶的，因为在牛顿流中，不稳定性和向湍流的转变是由惯性驱动的，并且预计在雷诺数超过某个临界值时发生。雷诺数表征惯性效应与粘性效应的比率，并且与流体粘度成反比。对于极粘的聚合物流体，典型的雷诺数非常小，远低于临界值。因此，惯性驱动的转变是不存在的，不能解释聚合物溶液中的不稳定性。聚合物溶液和熔体的显著特性源于它们的微观结构和流动之间的相互作用：长的聚合物分子被流动拉伸和定向。在过去的20年里，我们已经开始了解到，聚合物的流动诱导拉伸和取向不仅可以使聚合物的流动不同于水，而且还可以使流动不稳定，导致漩涡和随机流动。这种混沌运动看起来类似于牛顿湍流，但不是惯性起源。这种新的湍流类型，即所谓的弹性湍流，人们对其了解甚少，对其结构和可能出现的条件也知之甚少，本研究方案的目的是通过计算机模拟和最近发展的半分析方法来研究这种新的湍流类型，以描述接近转捩的牛顿湍流的结构。进行这项研究的动机有三个方面。首先，这是一种全新类型的湍流，我们在牛顿流体（如水）中从未遇到过。由于许多日常流体是非牛顿和粘弹性的，因此理解弹性湍流可能比理解牛顿湍流更重要。其次，了解弹性湍流的起源可能会为熔体破裂等工业问题提供解决方案。最后，通过对两者的比较，我们可以对牛顿湍流的机制有所了解。

Flows of complex fluids (such as polymers, colloids, emulsions, pastes etc.) are abundant in everyday life. One can think of pouring syrup from a bottle or squeezing toothpaste from a tube, but also of fibre-spinning and extrusion - processes used to produce plastic bags, optical fibres, wire coatings, guitar strings etc. Complex fluids, and polymer solutions in particular, often exhibit unexpected behaviour - they do not flow like water. For example, if one rotates a spoon in a cup of tea, the tea is pushed towards the walls of the cup. When, instead, this experiment is repeated with a polymeric liquid, the polymers move towards the spoon producing the so-called rod-climbing effect.What is even more surprising is that flows of polymers can become unstable. One of the famous examples is the melt-fracture phenomenon observed in extrusion of concentrated polymer solutions or melts, which is one of the main elements of polymer processing in industry. There the liquid is pressed through a thin capillary to produce a regular jet of polymers. At low extrusion speeds the jet remains straight and homogeneous, while at larger speeds the flow starts undulating, becomes chaotic and eventually breaks up. These instabilities are one of the main production-limiting factors and have been plaguing technology and industry for years. Their presence is surprising since in Newtonian flows, instabilities and the transition to turbulence are inertia-driven, and are expected to occur when the Reynolds number exceeds some critical value. The Reynolds number characterises the ratio of inertial to viscous effects and is inversely proportional to the fluid viscosity. For extremely viscous polymeric fluids typical Reynolds numbers are very small, far below the critical value. The inertia-driven transition is thus absent and can not explain instabilities in polymeric solutions. Instead, some other mechanism causes destabilisation.The striking properties of polymer solutions and melts arise from the interactions between their microstructure and the flow: long polymer molecules are stretched and oriented by the flow. In the past 20 years, we have begun to understand that flow-induced stretching and orientation of polymers can not only make polymers flow differently than water, but can also destabilise the flow, leading to vortices and random flows. This chaotic motion looks similar to Newtonian turbulence but is not inertial in origin. This new type of turbulence, the so-called elastic turbulence, is poorly understood and little is known about its structure and conditions at which it might appear.The aim of this research programme is to study this new type of turbulence by means of computer simulations and semi-analytical methods recently developed to describe the structure of Newtonian turbulence close to the transition. The motivation to perform this study is three-fold. First, this is a completely new type of turbulence which we have not encountered in Newtonian fluids like water. Since many every-day fluids are non-Newtonian and viscoelastic, it might be that understanding elastic turbulence is even more important than understanding Newtonian one. Secondly, understanding the origin of elastic turbulence might provide a solution to the industrial problems like melt-fracture. Finally, by comparing the two, we may learn something about the mechanism of Newtonian turbulence.