## Abstract

The solution of the Laplace–Young equation determines the equilibrium height of the free surface of a liquid contained in a vessel under the action of gravity and surface tension. There are only two non-trivial exact solutions known; one corresponds to a liquid occupying a semi-infinite domain bounded by a vertical plane wall while the other relates to the case when the liquid is constrained between parallel walls. A technique called *boundary tracing* is introduced; this procedure allows one to modify the geometry of the domain so that both the Laplace–Young equation continues to be satisfied while the necessary contact condition on the boundary remains fulfilled. In this way, new solutions of the equation are derived and such solutions can be found for certain boundaries with one or more sharp corners and for others that possess small-scale irregularities that can be thought of as a model for roughness. The method can be extended to construct new solutions for a variety of other physically significant partial differential equations.

## 1. Introduction

Solutions of the Laplace–Young equation determine the equilibrium height of the free surface of a liquid contained in a vessel and subject to the action of gravity and surface tension. If the fluid overlies a domain in the *x*–*y* plane and the free surface is located at then the Laplace–Young equation arises by demanding that the total energy of the liquid is minimized over all possible free surface distributions. After non-dimensionalization using the capillary number as the length scale, it follows that *η* satisfies,(1.1)

If the walls of the container rise vertically from the boundary of , , and if they have an external normal ** n** (which, of course, lies in the

*x*–

*y*plane) then a contact condition holds which can be expressed as(1.2)where

*γ*represents the angle that the free surface makes with the wall at the point of contact. Fuller derivations of the system (1.1) and (1.2) may be found in the texts by Landau & Lifshitz (1987) or Finn (1986).

Despite the passages of two centuries since the original derivation of the Laplace–Young equation, only two non-trivial exact solutions are known. Laplace and Young independently obtained the solution for the height rise of a liquid against a vertical plane wall, which was subsequently extended to account for the situation when the liquid is contained between two parallel walls (hereafter referred to as the channel problem). In the absence of further exact results, work has concentrated on determining bounds on, and estimates for, the capillary rise in various shaped domains , for example, Concus & Finn (1970); Siegel (1980) or Finn & Hwang (1989) to mention only a few.

Domains with corners have received special attention, both because of the interesting theoretical issues raised and the practical importance of the results in applications like industrial coating problems. The ubiquitous example of this type is the infinite wedge problem in which fluid is contained between two (supposedly semi-infinite) vertical walls that meet at some angle 2*α* (<2*π*). It is known that if *α* lies in the medium-value range , where(1.3)the capillary rise is bounded close to the corner and the free-surface is locally planar (Concus & Finn 1970). On the other hand, at wedge angles outside this range other behaviours are possible. When the surface height *η* becomes unbounded as the corner is approached and formulae for this singular behaviour have been given by Concus & Finn (1970) and Miersemann (1993). While such singular behaviour might seem unlikely, we remark that the photographs in Finn (1986) show that the height remains uniformly bounded as the included angle decreases down to , and that the height jumps discontinuously to infinity as *α* reduces further. In the large-angle regime, while solutions remain bounded the slope at the corner does not. King *et al*. (1999) demonstrated that in this latter case a full asymptotic analysis requires some knowledge of far-field behaviour. Other work has looked at establishing formal power series solutions near the corner of a wedge (Norbury *et al*. 2005) and asymptotic expressions valid far from the corner have been derived by Fowkes & Hood (1998). Some recent numerical simulations using finite volume methods on an unstructured mesh have given further information on the detailed form of the free surface in a case of a medium-angled wedge, see Scott *et al*. (2005).

Given the sparcity of exact solutions of the Laplace–Young system, in this work our aim is to derive some more exact results and we tackle the task using a method known as boundary tracing. This is a conceptually elementary technique for generating new solutions of a system of partial differential equations from known ones. Put simply, given a solution in some domain , one asks how the boundary of can be deformed so that the boundary contact condition remains satisfied. Thus one can anticipate that given a solution in a relatively ‘simple’ domain, boundary tracing could generate solutions relevant to more complicated geometries. Boundary tracing is not new in the sense that it has been used numerically in an *ad hoc* way to obtain various results. Anderssen *et al*. (1969) used the idea to predict the evolution of the shape of growing plant roots while McNabb *et al*. (1991) were interested in estimating the cooling times of pseudo-ellipsoidal objects. Here we illustrate the method in a more analytical context and emphasize that our presentation is aimed principally at showing the types of new solution possible. It is not intended to be comprehensive and completely exhaustive of all the possibilities (nor could it ever be) and a thorough description of the theory underpinning the method is deferred to a future paper. However, it should be appreciated that the techniques described in the coming sections are by no means restricted to the Laplace–Young problem and have applications in many other contexts. Indeed the lead author of this work has implemented boundary tracing to derive new exact solutions of Helmholtz's equation, Poisson's equation and the constant mean curvature equation (Anderson 2002). His thesis shows how new solutions of physically significant partial differential systems can be obtained and how the ‘sharp’ corner solutions can be modified and made relevant to geometries with rounded smooth boundaries. Furthermore, boundary tracing is not restricted to two-dimensions and Anderson (2002) outlines how the procedure can be implemented for higher order and higher-dimensional equations.

In order to apply boundary tracing it is necessary to start with some known solution, so in §2 we describe the classical wall and channel solutions of the Laplace–Young equation in a manner convenient for the further development. Once these basic solutions are explained, in §3 we set out the basis of the boundary tracing and describe new geometries for which exact solutions of the Laplace–Young equation can be obtained. There we also show how our results relate to some of the asymptotic theories for wedge problems and aid understanding as to how free surfaces behaves close to rough boundaries. Finally, we close with a short discussion and suggest how the methods introduced in this paper could find wider application.

## 2. The classical solutions

For simplicity suppose that the wall(s) of the domain are given by *y*= constant so that for both the half-plane and channel solutions of the Laplace–Young equation we have . Then equation (1.1) reduces to(2.1)while the contact condition (1.2) becomes(2.2)where we have supposed that the wall is located at . (The choice of the sign in (2.2) is fixed depending on whether the liquid lies on the or the side of the wall.) Equation (2.1) can be integrated once to give(2.3)for some constant *A*.

### (a) The half-plane solution

In the case of a semi-infinite region of fluid occupying the integration constant *A* in (2.3) is chosen so that as . Then *A*=1 and the height rise at the wall is given by(2.4)which follows directly from (2.2) and (2.3). Further, with this value of *A*, (2.3) can be integrated to give an implicit form for the surface displacement,(2.5)for some . Usually is determined by requiring that the contact condition (2.2) holds at *y*=0, so that the physical domain of relevance is simply *y*>0. However, for ease in using the boundary tracing ideas later, we find it more convenient to arrange so that at *y*=0. This value of *η* is special, for with it the function has infinite slope at *y*=0 although this point lies outside the physical domain. The resulting solution is given by(2.6)displayed in figure 1. With this choice of integration constant the location of the wall is determined using (2.6) with as in (2.4). Note that the graph of has two branches in *y*>0 which meet at (labelled on figure 1); the lower of these branches is the physically important one and we shall henceforth refer to it as the universal curve. It merits this designation in the sense that for any given contact angle *γ* the corresponding surface elevation can be obtained directly merely by identifying the part of the universal curve such that at (labelled ). Across this range is a monotonically decaying function which changes from at the wall to zero as . It will be shown presently that the part of the universal curve that is of interest for boundary tracing purposes is that lying in where , i.e. the portion on figure 1.

### (b) The channel solution

The channel solution is characterized by an additional parameter *d* which measures its width in terms of the capillary length scale. For convenience, suppose the origin of coordinates is positioned so that the physical domain of interest is . The first integral (2.3) still holds although the constant *A* can no longer be fixed as quickly as for the half-plane problem. More knowledge of the precise physical situation is required now; for example, if the channel has a fixed solid base then *A* is tied down by a constraint on the total volume of liquid in the channel while, on the other hand, if the channel has an open bottom and overlies a relatively massive reservoir of fluid then global energy arguments determine *A*. Fortunately our primary interest here is not dependent on the exact value taken by *A* for we are rather more concerned with the shape of the contact line. However, *A* can be inferred from an observation of the minimum height rise across the channel; from (2.3) it follows immediately that . If this minimum occurs at say, then (2.3) integrates to give(2.7)as an implicit description for the solution . If desired this expression can be evaluated in terms of elliptic integrals but this is not necessary for our purposes. Although the physical domain is merely by design, just as for the half-plane case for boundary tracing purposes we extend the solution (2.7) out to where becomes unbounded.

## 3. Boundary tracing

For any given the contact condition (1.2) can be thought of as a prescription governing the local orientation of . That is, rather than the usual process of regarding (1.2) as providing a constraint on *η* for a given normal ** n** and contact angle

*γ*, the thinking can be turned around and the following question posed. For known

*η*and

*γ*, what are the possibilities for the associated normal? As both

**and lie in the**

*n**x*–

*y*plane, it follows from the dot product constraint (1.2) that if is sufficiently large then at any point P on there are two possible directions for

**which are symmetrical about . On the other hand, for sufficiently small () no such directions are possible. Let us call the region in which directions exist the**

*n**viable tracing domain*. Through each point of this domain there are two boundary curves on which the contact condition will be satisfied and the whole of the viable region is separated from the remainder of space by a terminal curve on which . We note that for the half-plane solution discussed in §2

*a*the terminal curve is the wall (by construction) and the viable tracing domain is . Thus, boundaries can be traced within the region contained between the singularity in the Laplace–Young solution at

*y*=0 and the wall itself, that is the section in figure 1.

To obtain explicit expressions for possible traced boundaries we need to probe a little deeper and we do so in the case of the half-plane. If we parametrize the boundary curve so that a general point P on it has coordinates , then the contact condition (1.2) can be used to derive an ordinary differential equation for the slope of at P. For the classical half-plane solution, we find that(3.1)where dots denote differentiation with respect to *t* and *η* is as given implicitly by (2.6). Simplification is achieved in this instance if the parameter *t* is taken to be *η* itself for then equation (2.3) (with *A*=1) and (3.1) reduce to(3.2)where is as defined in (2.4) and . Recall that we earlier noted that traced boundaries exist where which, in terms of *η*, corresponds to . Across this range the expressions in (3.2) are real and these two equations can be integrated directly to give(3.3)(3.4)where and are the standard elliptic integrals of the first and third kinds (Gradsteyn & Ryzhik 1980) and the arguments satisfy

Upon integrating equations (3.2) there is clearly some flexibility in choosing the associated constants; although the *y*-solution is tied down by the requirement that *y*=0 when , there is no equivalent constraint on the *x*-component. Thus, the traced curves defined by (3.3) and (3.4) can be freely translated in the *x*-direction or reflected in the *y*-axis and thus lead to a whole family of possible traced curves. A simple example of a composite curve is given in figure 2. Here we see two curves that meet at the point labelled as C and which touch the original wall at B and D. Thus we conclude that the original half-plane solution which holds in the domain continues to hold in the domain lying above the composite curve ABCDE of figure 2 and, moreover, also satisfies the contact condition along the entirety of the new boundary.

Some important properties of the traced curves can be deduced from the equations above. First, there is the question of the behaviours of the curves as they approach the wall (i.e. at points B or D in figure 2). It is immediate from dividing the two parts of (3.2) that as so ; hence the curves smoothly match onto the wall and assume a quadratic profile in the neighbourhood of the meeting point. In the simple example shown in figure 2, the original domain can be modified slightly by arranging the curves so that B and D are relatively close together so that the meeting point C is near to . A more extreme modification arises if B and D are more widely separated so that C lies near *y*=0. The two parts of the traced curve are locally linear at C so they meet at some angle . Naturally, in the case of the original plane wall, but as *C* nears *y*=0 then , where this critical value is precisely that identified in (1.3) in the context of the wedge problem.

It should be noted that the important solution properties of the Laplace–Young problem for any domain akin to that in figure 2 can be deduced with minimal effort. The surface displacement around the traced curves can be obtained directly from the classical solution for the half-plane form (3.4), which also yields the size of the elevation in the corner. It is also worth noting that the inclusion of the additional region in has no effect whatsoever on the half-plane solution in . At first sight this might appear to be surprising, for the ellipticity of the Laplace–Young equation means that the solution within the domain should depend on the details of the entire boundary. Of course this does not occur here because the traced boundaries are very special indeed and for any other deformations the effect of the change in the boundary shape would, in general, be felt everywhere.

It is not difficult to generalize our ideas so as to construct more complex boundary shapes; a small gallery is shown in figure 3. We can construct domains with any finite number of indentations although their depths and angles are restricted in much the same way as for the simpler example. Note that, domains with re-entrant corners can be acheived by judicious choosing of curves although the corner angles thus constructed are all restricted to the medium range interval , see (1.3). Naturally, all the domains sketched in figure 3 have associated Laplace–Young solutions that are the one and the same in which is some indication of how sensitive calculations of the inverse problem for the Laplace–Young equation can be.

The procedure for generating new boundaries for the half-plane solution can, of course, be adapted in the case of the channel problem. The traced boundary equation (3.1) can again be integrated to givewhich needs to be combined with the solution (2.7). These solutions can be extended outside the channel to the regions . A typical form of traced boundary is displayed in figure 4, though, of course, extension to much more involved indentations is feasible. Just as in the half-plane situation the presence of the deformations is of no consequence for the pre-existing channel solution (2.7) which continues to apply across the bulk of the domain.

Our new exact solutions of the Laplace–Young equation in suitable domains with sharp corners are, to the best of our knowledge, the first of their type. As such they provide a yardstick against which some of the established wedge-domain asymptotic results outlined in the introduction can be compared. First, it should be remarked that our traced boundaries have corner angles which coincide precisely with the medium range of wedge angles (1.3) for which the corresponding surface elevation is locally planar. Previous derivations of this elevation, although able to find the slope of the surface, were unable to determine the size of the surface elevation right in the corner and which remains an unknown. Our results do not suffer from this drawback and provide a quick and easy recipe for determining the elevation all the way along the boundary of our domain. Note that as the indentations in our domain become larger and approach their maximum possible size, and at the cusp. Asymptotic results obtained by King *et al*. (1999) describe the behaviour near the corner of a wedge as and predict that the fall-off in height from the corner occurs in a square-root fashion. An exact analytic solution in a domain with a corner of angle at *y*=0 can be created by patching together parts of our half-plane solution and this leads to the prediction thatwhich incorporates the expected square root behaviour. Note, however, that the coefficient of the square-root term differs from that given in King *et al*. (1999), but this can be attributed to the fact that our boundaries are curved rather than planar.

It is of interest to note that our piecing together of elementary solutions lead to forms that are continuous at the corners. This is to be expected as they arise from an inscribed rectangular domain as introduced by Concus & Finn (1996); see fig. 5 of that paper. All solutions derived from such domains have the height of the fluid near the vertex varying in the square-root manner predicted by King *et al*. (1999). On the other hand, Lancaster & Seigel (1996*a*) have proved rigorously that if the solution arises from other-shaped domains (in particular a domain in the notation of Concus & Finn) and if the solution admits a ‘fan’ of width *π* of constant radial limits at the vertex, then for any approach to the vertex within the fan the height varies Hölder continuously with exponent 2/3. Thus it is not inevitable that the square-root behaviour must occur; other possibilities can arise corresponding to a qualitative change in the boundary data.

Wedge domains in which the contact angles on the two faces of the wedge differ are of significant practical interest. Lancaster & Siegel (1996*b*) (with printers errors corrected in Lancaster & Siegel 1997) considered the situation in which the two contact angles and together with the wedge angle are related by and . They were able to prove that in this case *η* is both continuous up to the vertex and locally planar. Sharp-cornered domains that have different contact angles either side of a corner can be constructed using the boundary tracing technique as this is merely a matter of matching up traced domains across the corner. By judicious choice of curves it is possible to cover the entire angle range considered by Lancaster & Siegel (1996*b*) and, appealing to the ideas used earlier, we find that at the corner the surface height and slope satisfywhere .

An important application of the idea of varying contact angles arises in a discussion of roughness. Roughness is a phenomenon that often plagues experimentalists, especially in the surface tension context, where exceptionally flat and clean surfaces are frequently the ideal. The presence of roughness also has a number of undesirable theoretical and numerical consequences. Solutions of the Laplace–Young equation are typically sensitive to small-scale surface features and numerical simulations are often fraught with difficulty due to the wide disparity of scales present; to properly account for the entire range from the (typically) large size of the overall situation down to the minutia of the roughness is often far from straightforward. Additionally, real surfaces, whether man-made or naturally occurring, are often ‘dirty’ in the sense that the variation in contact angle can be significant. For a very clean and flat glass surface in contact with pure water the effective contact angle can be virtually zero (so that the water spreads to cover the entire surface) but in more usual circumstances the contact angle is typically about 25°. This appreciable range of contact angles has inevitably led to the question as to whether it is possible to determine some kind of effective (or macroscopic) contact angle as a function of the microscopic contact angle *γ* and the local micro-geometry. This raises formidable issues, not least of which is how one might properly define an effective contact angle. We will not make any attempt to address these issues in detail here. Instead, we merely point out how boundary tracing can generate particular rough surfaces with exactly determined contact characteristics.

Figure 5 illustrates how the repeated use of suitably chosen traced curves can produce a rough geometry composed of a series of arbitrarily-sized teeth-like pieces which are almost triangular in profile. The major benefit of this strategy is that we can determine the exact solution of the Laplace–Young equation with this particular form of boundary, using the universal curve of figure 1. The height rise will vary with location around an individual tooth, being smallest at the exposed outer edge of the indentation and greatest at the innermost point. Conventional wisdom for modelling the height rise at a rough surface is that the surface behaves as if it were flat but with a macroscopic contact angle determined experimentally. Here we see a perfect application of that thinking; for the properties of the traced curves mean that (in terms of the notation in figure 5) the solution of the Laplace–Young equation above the virtual wall at is unchanged from that were the actual boundary of the domain. Of course for our traced boundary walls we can also determine the surface shape within the wall indentations and determine the effective location of the wall. Also a macroscopic contact angle can be ascribed, see figure 6. Of course the above properties are a consequence of the particular characteristics of the traced curves, but it could be hoped that the results using these arcs might not be completely atypical. As a partial verification of this idea, we solved the Laplace–Young equation in a semi-infinite domain with a sinusoidal bounding wall using a nonlinear finite element technique with triangular elements. The results of the simulations were compared with simple predictions arising by approximating the boundary with traced curves and over an appreciable range of parameters the agreement between the two techniques is very good; more details can be obtained from Anderson (2002).

## 4. Closing remarks

In this work we have considered how the idea of boundary tracing can be used to generate a whole family of new exact solutions of the Laplace–Young equation. Exact solutions of physically significant nonlinear equations are of interest, both for their intrinsic value, but perhaps more importantly in the advent of increasingly sophisticated numerical schemes, as test cases for the reliability of such methods. Although it is impossible to give a comprehensive overview of the entirety of possible solutions, the key results (3.3) and (3.4) are the building blocks from which complicated solutions can be constructed. It is of interest how the results we have obtained dovetail with previous asymptotic results concerning the surface shape in wedge-shaped domains and reinforce conventional wisdom concerning the likely behaviour of the free surface in the neighbourhood of rough walls.

There are a number of directions in which this work should be extended. First is the observation that the ideas used here are not restricted to plane wall problems. For instance, although exact radially symmetric solutions are not expressible in simple form, it is not difficult to derive accurate numerical solutions of the resulting ordinary differential equations and thereby use tracing techniques to obtain new domains for both cylindrical and annular regions. Interesting results arising from boundary tracing are by no means restricted to the Laplace–Young equation in relatively simple geometries and future work will include both a careful explanation of the rigorous theory lying behind boundary tracing and a snapshot of the spectrum of equations for which it has application.

## Acknowledgments

The authors are grateful to the referees whose suggestions have led to a much improved paper.

## Footnotes

↵† Present address: Rising Sun Research, 133 Gouger Street, Adelaide, South Australia 5000, Australia.

- Received January 27, 2006.
- Accepted May 18, 2006.

- © 2006 The Royal Society