Asymptotic Behaviour of Capillary Problems governed by Disjoining Pressure Potentials

Introduction Capillarity describes the effects caused by the surface tension on liquids. When considering small amounts ofliquid,thesurfacetension becomes the dominating parameter. In this situation the arising mathematical task is to determine the occurring capillary surface. At the beginning of the research on this topic, problems such as the ascent of fluids in a circular tube, on a vertical wall or on a wedge were some of the first problems scientists were concerned with. At the beginning of the 19th century, scientists like Young1, Laplace2, Taylor 3 and Gauß 4 established the mathematical foundations of this field. For the capillary tube5 they found, by applying variational methods, the so called mean curvature equation or capillary equation with the associated boundary condition. As Finn in [Fin86, Chapter 1] describes, this leads to the following boundary value problem: divTu = u + in , · Tu = cos on @ where Tu = ∇u p 1 + |∇u|2 . is called the Lagrange6 multiplier and is the contact angle, established between the capillary surface and the container wall. In the past, one tried to solve the problem by linearisation – with more or less satisfying results. In the last decades, expedited by the developing of micromechanics and the arising space-technology, capillary effects became more and more significant. Thereby the observed results differed from the predicted. The reason is the strong non-linearity of the problem. Interior molecular forces are responsible for the establishing of equilibrium surfaces. The force, operating between two materials, is called adhesion and cohesion is the molecular force within a medium. Under some specifications there arises a non-negligible force, called disjoining pressure. This pressure causes an additional term in the capillary equation, which 1Thomas Young (*13 June 1773, Milverton; †10 May 1829, London); Englisch polymath; made notable contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony and Egyptology, found the Young–Laplace equation 2Pierre-Simon (Marquis de) Laplace (28 March 1749, Beaumont-en-Auge; †5 March 1827, Paris); French mathematician and astronomer; found the Young–Laplace equation 3Brook Taylor (*18 August 1685, Edmonton; †29 December 1731, Somerset House/London); English mathematician; experiments in capillary attraction 4Johann Carl Friedlich Gauß (*30 April 1777, Braunschweig; †23 February 1855, G¨ottingen); German mathematician and scientist; contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics 5A capillary tube is a container with cross-section and perpendicular container walls, which contains an amount of liquid. 6Joseph-Louis de Lagrange (*25 January 1736, Turin; †10 April 1813, Paris); Italian mathematician and astronomer. 7 is called the disjoining pressure potential, denoted by P(x, u(x)). That is, we are led to the following modified capillary equation, see [MMS08]: divTu = u + P + in , with a similar boundary condition (see Section 1.3 for more details). The main task of this paper is to examine the behaviour of the capillary problem, considering the disturbance P. A generic example for such configurations is vapour nitrogen//liquid nitrogen//quartz, see also [Isr92, Chapter 11] or [MMS08]. The present work with regard to contents is divided in three parts. In the first part, inspired by the work of Concus and Finn [CF74], [FH89], we prove a Comparison Principle. As in the classical context, this principle is a powerful tool to find solutions of the boundary problem. Thus we can see that the disjoining pressure potential is the key for the asymptotic of the solutions. The second part is concerned with the asymptotic behaviour of the solutions for some classical cases. In particular for the capillary tube with circular cross-section (see [Mie93b], [Mie94], [Mie96] for the classical setting) the ascent on a horizontal wall and between two parallel horizontal plates, results are presented. There we are able to specify the asymptotic behaviour up to a constant term. In the last part we observe the solution of the problem on a corner. There it is more difficult to obtain a result. But in return, we gain a better result near the cusp of the edge. In the articles of Miersemann [Mie88], [Mie89], [Mie90] or Scholz [Sch04] some results for the classical setting are given. The formal arrangement is divided into three main chapters. The first of them is a summary of some notations which will be needed in the following chapters and also the physical background is illuminated. The main part, where asymptotic results are presented, is contained in Chapter 2. To afford a better reading, most of the proofs are given in Chapter 3.8

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