Evaporation of Interacting Droplets on an Inclined Surface

该项目旨在利用数值模拟和渐近理论相结合的方法来解决倾斜平面上蒸发液滴的新问题。这个问题是为了研究可能含有悬浮溶质的固着物的蒸发而设立的。在时间t=0时，液滴的横截面是圆形的，具有最初规定的接触角。液滴蒸发，将液体转化为蒸汽，结果液滴失去质量并收缩。局部质量通量简单地由蒸气浓度的表面梯度给出。假定主体中的蒸汽浓度满足拉普拉斯方程，即它简单地扩散。通过做一些简化的假设，可以取得分析上的进展。假设液滴很薄，即它始终保持扁平的圆盘形状。由此，利用渐近性得到了水蒸气浓度问题的解析表达式。同样，薄滴假设使得使用成熟的润滑理论来求解液滴的流体动力学成为可能。最后，颗粒浓度满足对流扩散方程，并将其耦合到数值求解的流动问题中。因此，有三个实体相互作用是非线性的，蒸气浓度、流速和溶质颗粒浓度产生了非常丰富的动力学。有实际意义的模型的最终结果是液滴完全蒸发后留下的溶质浓度。最近，人们利用Surface Evolver等黑盒求解器通过最小化Helmholtz自由能来获得稳定的液滴形状，然后通过有限体积法求解蒸汽浓度场的拉普拉斯方程来计算蒸发率。虽然边界元方法最适合于求解拉普拉斯方程和斯托克斯方程，是数值模拟的必然选择，但目前还没有利用边界元方法来求解液滴蒸发问题的模拟。我们首先提出通过对蒸发的固着液滴进行边界元模拟来填补这一空白。计算结果将与上述润滑理论进行比较。溶质颗粒浓度满足平流方程，使得标准边界元不再适用，因此必须采用双互易边界元方法。然而，对于简单地与液滴内部产生的流场平流的颗粒，这个问题可以很容易地通过朗格朗日描述来规避。在许多工业环境中，倾斜和垂直平面上的液滴蒸发问题具有重要的现实意义。倾斜平面可以使重力在最终的沉积模式中发挥核心作用。最近，这个问题分别在二维和三维得到了数值求解。然而，模拟的自然选择，边界元，到目前为止还没有用来研究这个问题，瞬时液滴动力学仍然没有得到解决。我们建议通过考虑墙倾斜的影响来扩展结果。最后，我们将讨论一对液滴在斜面上通过蒸汽场相互作用这一最实际的情况。这同样可以使用边界元很容易地实现。蒸发液滴的相互作用问题到目前为止还没有得到数值研究。

The proposed project aims to tackle novel aspects of evaporating drops on an inclined plane by using a combination of numerical simulations and asymptotic theory. The problem set up for studying evaporation of a sessile, possibly containing suspended solutes. The cross-section of the drop at time t = 0 is circular with an initially prescribed contact angle. The droplet evaporates, converting liquid to vapour, as a result of which the drop loses mass and shrinks. The local mass flux is simply given by the surface gradient of the vapour concentration. The vapour concentration in the bulk is assumed to satisfy Laplace's equation, i.e. it simply diffuses. Analytical progress can be made by making some simplifying assumptions. The drop is assumed to be thin, i.e. it always maintains a flat disc shape. Consequently, analytical expressions for the vapour concentration problem are obtained by using asymptotics. Similarly, the thin drop assumption makes it possible to solve the hydrodynamics of the drop using well-established lubrication theory. Finally, the particles concentration satisfies advection-diffusion equation which is coupled to the flow problem which is solved numerically. Hence, there are three entities that interact with each other nonlinearly, the vapour concentration, flow velocity, and solute particles concentration giving rise to very rich dynamics. The final outcome of the model of practical interest is the solute concentration that is left behind after the drop has completely evaporated. Recently, black-box solvers like Surface Evolver have been used to obtain steady drop shapes by minimising the Helmholtz free energy and then the evaporation rate is computed by solving Laplace's equation for the vapor concentration field via a finite-volume method. To the best of our knowledge, there are no simulations that make use of boundary element method (BEM) to solve the problem of evaporating droplets even though it is the natural choice for numerical simulations as it is most suitable for solving Laplace's and Stokes equation. We first propose to fill this gap in literature by performing BEM simulations of an evaporating sessile droplet. The results will be compared against lubrication theory described above. The solute particles concentration satisfy advection equation making standard BEM inapplicable and hence, a dual-reciprocity BEM method will have to be employed. However, this problem can be circumvented easily with a Langrangian description for the particles that are simply advected with the flow field generated inside the droplet. The problem of evaporating drops on inclined and vertical planes is of practical significance in many industrial settings. Inclining the plane allows gravitational forces to take a central role in the final deposit patterns. This problem has been solved numerically recently in two- and three-dimensions, respectively. However, the natural choice for simulations, BEM, has not employed to study this problem so far and transient drop dynamics remains unaddressed. We propose to extend the results by including the effect of wall inclination. Finally, the most practical situation where a pair drops on an inclined plane are interacting with each other through the vapour field will be tackled. This again can be easily achieved using BEM. The problem of interacting evaporating drops has not been studied numerically so far.