The Mathematics of Multilayer Microfluidics: analysis, hybrid modelling and novel simulations underpinning new technologies at the microscale

目前正在发生的最广泛的科学革命之一是追求小型化和制造能够在微观尺度上执行任务（如流体处理和处理）的微型设备。在许多情况下，操作可以快速而准确地完成，并且这些过程的自动化预计将在药物开发和交付（例如药物）等领域产生巨大影响。台“芯片实验室”技术)。小体积的流体意味着大的表面体积比，这样的几何形状增强了大型装置中不存在的机制的效果。许多应用涉及使用多种不混相流体的过程——这些流体不会混合（例如水和油），更重要的是，它们具有分离界面，可以在表面张力、流动和任何其他施加的外部效应（如电场或重力）的作用下自由移动。因此，如果我们能够了解不同流体（或相）之间的界面是如何演变的，就可以使过程成功和健壮。这样的理解为引入流量控制开辟了道路。这些方法可以是被动的，比如在流体流过的表面上建立固定的结构，比如凸起或溪流；也可以是主动的，比如以一种由不断变化的流动特性决定的方式开关电场。影响多层微流体流动的主要机制之一是表面张力。由于在未知移动界面上产生的边界条件具有固有的非线性性质，它的存在使得数学问题在分析和计算上都具有很高的挑战性。界面构型影响流动，而流动反过来又影响界面位置——它们需要一起解决，存在的不稳定机制需要被识别并跟踪到可能出现复杂动力学的非线性区域。在多流体流动中产生界面并控制其结构和时空动力学对最先进的材料科学也非常重要-被称为折纸工程，一个主要的实验研究领域。界面就像织物一样，粒子可以在其中自组装，产生均匀或预先设计的非均匀材料膜，这些膜可以被操纵和折叠，以达到所需的工程目的。我们的目标是识别、控制和操纵多流体流动中的非线性界面不稳定性，以产生理想的表面，这些表面可用于纳米和微颗粒的定向自组装，以创建具有奇异弹性特性的智能薄膜，或者可以承载哺乳动物细胞用于组织工程。为了获得流体表面相互作用的广泛理论知识，我们考虑了三个典型模型来描述目前实验中使用的一些“设计”衬底：(i)地形结构（凸起和压痕），（ii）粘滑超疏水表面，以及（iii）产生非均匀电场的蚀刻电极网络。在由这些表面组成的通道中，我们有多层流动，有几个流体-流体界面。由此产生的不稳定是复杂的，包括共振、剪切诱导的稳定或不稳定，以及电流体动力不稳定。本建议处理的另一个挑战是三维性。计算方面的挑战是巨大的，至少部分会得到解决。我们将通过推导简化模型方程来产生依赖于时间和两个空间变量的非线性偏微分方程的耦合系统，从而取得分析进展。这些将被充分研究，分析和计算，并与直接数值模拟进行比较。重点将给予新的解决方案和数学结构，但也对他们所描述的现象和产生复杂动力学的潜在机制。

One of the widest scientific revolutions currently taking place is the quest towards miniaturization and manufacture of tiny devices that can perform tasks (such as fluid handling and processing) on the micro-scale. In many cases the manipulation can be done rapidly and accurately and the automation of such processes is expected to have a huge impact in areas such as drug development and delivery (e.g. ``lab-on-chip" technologies). Small volumes of fluid imply large surface to volume ratios, and such geometries enhance the effects of mechanisms that are absent in larger scale devices. Many applications involve processes that utilise more than one immiscible fluid - such fluids do not mix (e.g. water and oil) and more importantly they have separating interface(s) that is free to move under the action of surface tension, flow and any other imposed external effects such as electric fields or gravity. Consequently, a process can be made successful and robust if we can understand how the interface between the different fluids (or phases) evolves. Such understanding opens the way for introducing flow controls. These can be either passive, as for example by building fixed structures such as bumps or rivulets on surfaces over which the fluids flow, or, active as in the case of switching an electric field on and off in a way determined by the evolving flow characteristics. One of the main mechanisms affecting multilayer microfluidic flows is surface tension. Its presence makes the mathematical problems highly challenging both analytically and computationally due to the intrinsically nonlinear nature of the resulting boundary conditions on unknown moving interfaces. The interfacial configuration affects the flow and the flow in turn affects the interfacial position - they need to be solved together and the instability mechanisms present need to be identified and followed into the nonlinear regime where complex dynamics can emerge.Producing interfaces in multi-fluid flows and controlling their configurations and spatio-temporal dynamics is also of vast importance to state-of-the-art materials science - known as Origami engineering, a mostly experimental research field. Interfaces act as the fabric where particles can self-assemble to produce homogeneous or pre-designed inhomogeneous material membranes to be manipulated and folded for desired engineering purposes.Our goal is to identify, control and manipulate nonlinear interfacial instabilities in multifluid flows to produce desirable surfaces that could be used forthe directed self-assembly of nano- and micro-particles to create smart films with exotic elastic properties, or that can host mammalian cells for tissue engineering.To achieve an extensive theoretical knowledge of fluid-surface interactions we consider three canonical models to describe some of the "designer" substrates currently used experimentally: (i) topographical structures (bumps and indentations), (ii) stick-slip superhydrophobic surfaces, and (iii) etched electrode networks that produce non-uniform electric fields. Within channels made up of such surfaces we have multilayer flows with several fluid-fluid interfaces. The resulting instabilities are complicated and include resonance, shear-induced stability or instability, and electrohydrodynamic instability to mention some. An additional challenge addressed by the present proposal is three-dimensionality. The computational challenges are enormous and will be addressed at least partially. We will make analytical progress by deriving reduced model equations to produce coupled systems of nonlinear partial differential equations depending on time and two spatial variables. These will be studied fully, both analytically and computationally, and compared with direct numerical simulations. Emphasis will be given to new solutions and mathematical structures but also on the phenomena that they describe and the underlying mechanisms that produce complex dynamics.