Introduction | 8 | Capillary Flows with Forming Interfaces

ABSTRACT

Free surface ﬂows are ubiquitous in nature and industry. One can say without exaggeration that free surfaces are present, or can appear, in almost every technological process or natural phenomenon where liquids are involved. The characteristic length and time scales associated with these ﬂows vary in a very wide range, from nanometers and nanoseconds in, as one might expect, nanotechnologies to kilometers and hours typical for tidal waves and lava ﬂows. In order to understand and predict natural phenomena, design new tech-

nologies and improve the existing ones, we need to be able to describe, both qualitatively and quantitatively, the behaviour of free-surface ﬂows in diﬀerent situations. From the quantitative point of view, this means to know the distributions of the ﬂow parameters with the required resolution and accuracy. The ultimate way of obtaining this information is, of course, a high-accuracy direct experiment. However, in many cases this is diﬃcult or impossible for one or both of the following two main reasons. The ﬁrst reason is that often the eﬀect to be understood or predicted is

also the one to be avoided. A typical example of this is a critical situation at nuclear power stations where one would prefer theory (and recommendations based on it) to full-scale experiment. Large-scale natural phenomena, such as the interaction of a tsunami wave with the coastline, is another example of ﬂow where experiments, though technically feasible, are undesirable and quantitative predictions have to be based on the results of mathematical modelling. The second reason that often makes one prefer theory to experiment is a

very high cost of performing experiments with the accuracy and resolution that would allow the experimenter to draw unambiguous conclusions from the data and make reliable quantitative predictions. From another perspective, this can be seen as the limited accessibility of the required information to existing experimental techniques. This source of diﬃculties is usually associated with very small length and time scales. The area of ﬂuid mechanics dealing with such scales is called ‘microhydrodynamics’, and its branch addressing ﬂows where interfacial eﬀects are important is known as mechanics

FIGURE 1.1: Impact of a microscopic drop on a solid surface. The initial stage of impact, when the point of contact grows into an area, is the most important one and it is not accessible to experimental observation. (Courtesy of P.M. Suckling.)

of capillary ﬂows. An important aspect of experimentation in the area of capillary ﬂows is that

often it is necessary to investigate the role of material parameters almost invariably involving properties of interfaces. This brings in a multidimensional parameter space which, in most cases, cannot be investigated experimentally since, in order to vary a dimensionless similarity parameter formed by material constants, one has to change the composition/conditions of the medium thus causing variation in other similarity parameters. As a result, an experiment goes along a line in the parameter space. Drawing conclusions from the data obtained along such a line or a set of lines of diﬀerent shapes in a multi-dimensional parameter space becomes a problem diﬃcult even from a methodological, leave alone practical, point of view. An example of ﬂow where all these diﬃculties are wrapped in one is the

impact of a microscopic droplet on a solid surface (Fig. 1.1). This is an element of ink-jet printing as well as of some bioengineering technologies. In order to understand the processes involved in the initial stage of the impact, one would require, ideally, an experiment capable of resolving the ﬂow down to molecular length and time scales. Although this is feasible in principle, the cost of a systematic study of this phenomenon covering the whole parameter space would be prohibitively high. The process of spreading of the liquid over the solid substrate that follows the initial stage of the impact is determined by both hydrodynamic factors and material properties of the gas/liquid/solid system. Experimental investigation of the former in the immediate vicinity of the three-phase-contact line requires a high degree of spatial resolution, whereas the latter are inseparable in experiments and hence their role cannot

FIGURE 1.2: Curtain coating is one of the most ﬂexible methods of depositing liquid ﬁlms on solid substrates. The quality of coating is largely determined by the processes near the moving three-phase-contact line which are virtually impossible to monitor experimentally. (Courtesy of A. Clarke.)

be studied independently. All the previously mentioned factors make it desirable to develop methods

capable of modelling capillary ﬂows mathematically in diﬀerent situations that occur in applications and natural processes. Adequate mathematical models would make it possible to obtain full information with the required accuracy and, if a complex experiment is performed, correctly interpret its results and extend them to the areas which are not accessible to measurements. An important aspect is that the theory would also make it possible to study separately the role of diﬀerent factors inseparable in a real-life experiment and suggest how the composition of materials could be modiﬁed to achieve the set technological objectives. Below, we will brieﬂy describe two broad classes of capillary ﬂows where the issues of modelling are of prime importance and which are the main subject of this book. A wide class of capillary ﬂows have at their core the process of ‘dynamic

wetting’, that is, the spreading of a liquid over a solid substrate. This process is the key element of so-called ‘coating ﬂows’ (Kistler & Schweitzer 1997, Weinstein & Ruschak 2004) deﬁned by the technological need to ‘coat’ the solid surface with a ﬁlm of liquid. The ultimate goal is either to modify properties of the solid surface, like in painting, lamination, etc., or, by using the solid as a temporary support, to produce a ﬁlm of solidiﬁed liquid (e.g., polymer ﬁlms, metal sheets, etc.). The ﬂow conﬁgurations employed in coating technologies include curtain coating (Fig. 1.2), forward and reverse roll coating, slot and knife coating, spin coating, to mention but a few. A relatively new area of application is micro-and nanotechnology, where it is important to describe

FIGURE 1.3: What happens during the ﬁnal stage of breakup of a liquid thread is not accessible to experimental observations and has to be understood from analysing the pre-breakup evolution of the thread and the post-breakup recoil of its pieces. The latter can be seen as a macroscopic ‘summary’ of how the breakup has taken place.