## Abstract

It is well known that the inclusion of surface tension in the linear water-wave problem introduces an additional term in the free-surface boundary condition. Furthermore, if the fluid contains one or more partially immersed bodies, edge conditions describing the motion of the fluid at each contact line need to be applied. In this paper, an inverse procedure is used to construct examples of two-dimensional, surface-piercing trapping structures for non-zero values of surface tension. The problem is considered with two separate edge conditions that are appropriate for investigations involving trapped modes. The first edge condition fixes each contact line and the procedure used forces the bodies to be horizontal at the contact points; it is shown that results can be found for all values of surface tension. The second condition forces the free-surface slope to be zero at the contact points, and results are obtained for a restricted range of surface tension values.

## 1. Introduction

The question of whether a given linear water-wave problem admits a unique solution is a long-standing one. It is now known that non-unique solutions can exist in such problems (see McIver 1996 for the first example). In that paper, symmetric pairs of surface-piercing bodies that support trapped modes were provided. However, surface tension was neglected in that analysis.

There have been many investigations into the propagation of time-harmonic free-surface water waves around surface-piercing bodies; for example, the reader is referred to Ursell (1949), Evans & Morris (1972) and Newman (1974). The addition of surface tension to such problems necessitates the introduction of an additional boundary condition at the line of contact between the fluid and the bodies (see for example Rhodes-Robinson 1982). At present, there is no single generally accepted form for this contact-line condition. For these reasons, the majority of investigations involving capillarity focus on simple geometries such as vertical plates. There have been relatively few studies on trapped modes in the presence of surface tension. In fact, to the authors' knowledge, the only investigations of this type have been by Groves (1998) who considered trapped modes in a horizontal channel, and Harter *et al*. (2007) where the studies of McIver (1996, 2000) were extended. However, Harter *et al*. (2007) did not include a contact-line condition for the surface-piercing case; its main focus was on the submerged bodies found by McIver (2000), which does not require an additional condition when capillarity is considered. In the present study, the work of McIver (1996) will be extended to include surface tension while using two physically realistic contact-line conditions.

## 2. Formulation of the problem

In the sections to follow, we consider the inviscid, incompressible, irrotational motion of an unbounded fluid occupying a two-dimensional region , in the presence of a pair of symmetric surface-piercing bodies, the wetted boundaries of which we denote by . To this end, we introduce Cartesian coordinates (*x*, *y*) with the *y*-axis pointing vertically downwards; the undisturbed free surface of the fluid lies along *y*=0. In order to show that a non-unique solution exists, a non-trivial time-harmonic velocity potential *Φ*(*x*, *y*, *t*)=Re{*ϕ*(*x*, *y*)e^{−iωt}} must be found to the following homogeneous boundary-value problem:(2.1)(2.2)(2.3)(2.4)where *ω* is the angular frequency, *K*=*ω*^{2}/*g* is the wavenumber of free-surface waves when capillarity is absent, *T* is the surface tension of the fluid, *ρ* is the fluid density and *g* is the acceleration due to gravity. Note that the surface displacement *Z* is related to the fluid velocity potential via *Z*(*x*, *t*)=Re{*η*(*x*)e^{−iωt}}, where *η*(*x*)=(i/*ω*)*∂ϕ*/*∂y*|_{y=0}. In addition to (2.1)–(2.4) shown above, an additional condition is needed to describe the motion of the fluid at each line of contact between a surface-piercing body and the free surface. However, due to the complicated nature of the flow near such a point, there is no single generally accepted form for this condition. One of the first such conditions used to model the contact-line behaviour was that employed by Benjamin & Scott (1979) while considering the propagation of waves in a brimful channel. They assumed that the free-surface elevation *Z*(*x*, *t*) is pinned (i.e. fixed) at each contact point, arguing that in their situation water waves of sufficiently small amplitude will not displace the contact line. A more general condition has since been described by Hocking (1987) who proposed the following relationship between free-surface speed and slope at the contact line:where *n* is the normal to the solid boundary and points into the fluid, and *λ* is a constant. Hocking considered the effect of varying *λ* on the rate of damping of standing waves between two rigid vertical plates, showing in this case that energy is lost at the contact line for all values of *λ* except 0 and ∞. These special values correspond to setting *η*=0 (the fixed-edge condition of Benjamin & Scott 1979) and *∂η*/*∂x*=0 (the zero-slope condition).

Let us now consider the case where waves (of frequency *ω*) pass through the arbitrary curve whose extrema cut the free surface at *x*=*x*_{1},*x*_{2} with *x*_{1}<*x*_{2}. It can be shown (see Evans 1968) that the energy flux through , over one period, is given bywhere an overbar denotes complex conjugation of a quantity. It can easily be seen therefore that *E*=0 when is a fixed rigid body (i.e. a streamline on which *∂ϕ*/*∂n*=0) and either *η*=0 or *∂η*/*∂x*=0 at each contact point; hence neither of these latter conditions is dissipative, regardless of body geometry. This result can be extended to multiple-body systems, in particular, the case where two fixed surface-piercing bodies are present; can then be chosen so that *E* represents the energy radiated by any single surface-piercing body in the system.

Since the present study concerns the trapping of energy, it is clearly inappropriate to employ a contact-line condition that dissipates energy. Hence, we will use only the non-dissipative conditions, first considering the fixed-edge condition and briefly discussing the zero-slope condition later. The fixed-edge (or pinned) condition can be written in terms of the velocity potential as(2.5)where we have assumed that the four contact points are situated at (±*a*, 0) and (±*b*, 0), with *a*<*b*. The Cauchy–Riemann equations then give *∂ψ*/*∂x*=0 (where *ψ* is a streamfunction) at the contact points, and so in the vicinity of these points the body, which is a streamline, will be horizontal. Such an edge condition is equivalent to a *fully wetted* (hydrophilic) body, where the contact angle between the fluid and the bounding surface is zero. Alternatively, consider a thin layer of flexible material (e.g. *cling film*) lying on top of the fluid and attached to the solid body at the point where the slope is zero.

Before we proceed with the analysis, it is convenient to non-dimensionalize the problem. We first note that (2.1) and (2.2) admit propagating-wave solutions (that decay with depth into the fluid) of the form , where *k*_{0} is the (unique) real positive root of the dispersion relationWe can therefore non-dimensionalize using 1/*k*_{0} as the length-scale. This modifies the free-surface condition (2.2) to(2.6)where is a Bond number which when large indicates that capillary effects are dominant over gravity. All the remaining conditions are left unchanged by this non-dimensionalization, and note that the free-surface wave solutions of (2.6) now take the form e^{±ix−y}.

## 3. Solution of the boundary-value problem

The first non-trivial solution to the above system (without surface tension) was derived by McIver (1996); however, since *s*=0 in her analysis she was able to disregard condition (2.5). She showed that a combination of equal sources placed at (±*π*/2, 0) does not radiate waves to infinity, and that many of the streamlines of the flow can be interpreted as surface-piercing bodies. It is our aim to derive a generalized potential that accounts for surface tension, incorporating the contact-line condition (2.5), and that recovers McIver's (1996) result as *s*→0.

We start by considering two sorts of potential. The first is the source derived in Harter *et al*. (2007); this is given bywhere the source is positioned at (*x*_{0}, *y*_{0}), *r*^{2}=(*x*−*x*_{0})^{2}+(*y*−*y*_{0})^{2}, *r*′^{2}=(*x*−*x*_{0})^{2}+(*y*+*y*_{0})^{2} and the contour is taken under the pole at *m*=1. (In the present work we will let *y*_{0}→0 and so log (*r*/*r*′)→0.) This source potential satisfies (2.6) andFor convenience, we will consider a symmetrically placed pair of sources positioned at (±*x*_{0}, 0), which give rise to the potentialIt is easy to show by contour integration that, as |*x*|→∞,and so the choice *x*_{0}=*π*/2 is wave-free at large distances.

The second potential *ϕ*_{2}(*x*, *y*; *x*_{0}) to be employed satisfies (2.1) and(3.1)It is clear that *ϕ*_{2} has a weaker type of singularity than *ϕ*_{1}. The potential *ϕ*_{2} has a convergent limit as (*x*, *y*)→(*x*_{0}, 0) and hence it is permissible to place it at a contact point. However, integration of (3.1) shows that there is a discontinuity in the mixed second partial derivative of *ϕ*_{2} at *x*=*x*_{0}, and so, while the edge condition (2.5) is well defined there, the zero-slope condition discussed in §6 must be taken as a limiting procedure. It is straightforward to show that a symmetric combination of these potentials is given byand that

The potential we will use to satisfy (2.1) together with boundary conditions (2.3)–(2.6) is given by(3.2)where(3.3)with the parameters *ξ*, *ζ*, *D*, *a*, *A*, *b* and *B* to be deduced. By construction, this potential is symmetric about *x*=0, and so we need only consider the region *x*>0. We therefore have two conditions from (2.5); one further condition comes from (2.4), which requires *ϕ* to be wave-free at large distances, and so *F*(1)=0, which can be satisfied automatically by setting(3.4)A further condition comes from the fact that we require the contact points (*a*, 0) and (*b*, 0) to lie on the surface of the same surface-piercing body, represented by the same streamfunction value. Thus,(3.5)where *ψ*(*x*, *y*) is the streamfunction corresponding to the potential (3.2), namely(3.6)To facilitate numerical computation, we split the integrands in and via partial fractions. We see that(3.7)and(3.8)whereNote that *m*_{+}+*m*_{−}=*q*_{+}+*q*_{−}=*q*_{+}*m*_{+}+*q*_{−}*m*_{−}=−1. The resulting integrals in (3.2) and (3.6) can then be evaluated using(3.9)where *E*_{1} is the exponential integral. In order to apply condition (2.5), we need to evaluateand again partial fraction decompositions are useful for this integrand; we have(3.10)(3.11)As before, the resulting integrals can be computed using (3.9).

## 4. Numerical procedure

For each value of surface tension *s*, we seek a set of parameters (*A*, *a*, *ζ*, *ξ*, *B*, *b*) that satisfy (2.5) and (3.5), with the choice for *D* given in (3.4) ensuring no waves at infinity. The solution must not exhibit singular behaviour in the fluid, and so we impose *a*<*ζ*<*b* and *a*<*ξ*<*b*, i.e. there is a closed streamline cutting the free surface at *x*=*a* and *x*=*b* which encloses the sources at (*ζ*, 0), (*ξ*, 0). A typical arrangement is shown schematically in figure 1, with the singularities along *y*=0 at *x*=*a*, *ζ*, *ξ* and *b* (with magnitudes *A*, *D*, 1 and *B*, respectively) and showing the form of the resulting streamline. A mirror-image streamline connects the points (−*a*, 0) and (−*b*, 0), and the two streamlines can be interpreted as a pair of surface-piercing bodies that support trapped modes.

McIver's (1996) work for *s*=0 corresponds to the choice *ξ*=*π*/2 and *A*=*B*=0; consequently *D*=0, and *ζ* is therefore irrelevant. Even in this case, where the potential contains no free parameters, McIver found that there is a family of streamlines which can represent surface-piercing bodies that support trapped modes. It is expected, therefore, that the present investigation will also reveal a family of surfaces, allowing us to choose values of *ξ*, *ζ* and *a*, say, and find the corresponding values of *A*, *B* and *b* required to satisfy the three conditions (2.5) and (3.5). In practice, what this means is choosing a starting value of *b*, finding the values of *A* and *B* required to satisfy (2.5) and then computing the left-hand side of (3.5); then a root-finding approach is used to iterate to the correct value of *b*, for which (3.5) is satisfied. The numerical results indicate that for *s*>0, solutions can be found only when every singular term in (3.2) is present, i.e. *A*, *D* and *B* are all non-zero.

We first consider the case *ξ*=*π*/2, as in McIver's (1996) solution, and for illustrative purposes set *ζ*=1. Figure 2 shows how the rightmost edge of the body *b* varies as a function of the leftmost point *a*. The graph is labelled with the values of the surface tension parameter *s*. It can be seen that *b* increases with *s* for fixed *a*, and decreases as *a* increases for fixed *s*. This latter point is to be expected from the form of McIver's (1996) streamlines when *s*=0; in her analysis an arbitrarily large pair of bodies could be found by choosing *a* sufficiently small, and this is still the case when *s*≠0.

It is also clear from figure 2 that, for each value of *s*, there is a maximum value of *a*, *a*_{max} say, that will give surface-piercing bodies; beyond this value we have *b*=*π*/2, and the streamlines do not isolate the source at (*π*/2, 0) from the fluid. Thus, for each value of surface tension, trapped modes can be found for 0<*a*≤*a*_{max}. This range widens as *s* increases, and so the maximum range is attained as *s*→∞. For this case we find that *a*_{max} is roughly 0.923 (compared with *a*_{max}=0.321 for *s*=0). Furthermore, it is found numerically that *A* and *B* scale on *s* and so the potential (3.2) reduces towhere *A*=*λs*; *B*=*μs*; and *D*=*O*(1). Hence, the problem becomes independent of *s* as *s*→∞.

If we now let *ζ* vary, then *a*_{max} can be increased further. It turns out that *a*_{max} increases with *ζ*, and so *a* is maximized as *ζ*→*π*/2. For this case *a*_{max}≈1.05. It is interesting to note that in this limit *D*, the strength of the source at (*ζ*, 0), tends to −1. In other words, this latter solution could also have been obtained by replacing the two pairs of sources with a pair of oppositely oriented dipoles at (±*π*/2, 0). For this particular case, (3.2) becomeswhere *ν*, the dipole strength, is independent of *s*.

In general, any solution we can find by letting *ζ*→*ξ* in (3.2) has the property that *∂ϕ*/*∂x*∼0 as *x*→±*a*,±*b*. This means that there is a stagnation point close to each contact point. If these stagnation points coincided with the contact points, then the streamlines need not be horizontal there. However, this is not the case as the value of *∂ϕ*/*∂x* at each contact point never reaches zero.

Returning to the case where *ξ*=*π*/2 and *ζ*=1, it is worth mentioning that as *s*→0, *A*, *B* and *D* all become very small, and so, away from the contact points, the potential becomeswhich is McIver's (1996) solution with *s*=0. Thus, the original result is recovered as *s*→0. This can also be verified by noting that for very small *s*, each choice of *a* results in a single pair of streamlines that are almost indistinguishable from the equivalent pair of streamlines in McIver (1996) that pass through (±*a*, 0). The only difference occurs in a small vicinity around each contact point; this will be investigated more thoroughly in §5.

Figure 3 shows how *b* varies as surface tension is increased, with *a* fixed and *ζ*=1. As previously mentioned, as *s* increases so too does the value of *b* that gives trapped modes, and as expected for each value of *a* there is a limit for *b* as *s*→∞. Furthermore, in accordance with figure 2, for each value of *a* there is a minimum value of *s*, *s*_{min} say, which is required for surface-piercing bodies to exist. It can be verified that *s*_{min}=0 for *a*<*a*^{*}, where *a*^{*} denotes the value of *a*_{max} when *s*=0, and with *ξ*=*π*/2 this is approximately 0.321.

The streamline pattern for a few values of *s* is shown in figure 4. The sources are again located at *x*=1 and *x*=*π*/2 and are marked on the figure. It must be emphasized that in order to produce similarly sized bodies, different values of *a* have been chosen for different *s*-values. It can be seen that, as surface tension increases, the bodies become flatter at the free surface, in view of (2.5). Another example is given in figure 5, along with a plot at a fixed time of *∂ϕ*/*∂y*, which is proportional to the free-surface elevation. It can be seen that additional closed loops of the same streamline exist just above the singular points; however, these lie out of the fluid and therefore have no physical significance. The free-surface plot shows that *∂ϕ*/*∂y*=0 at *x*=*a*,*b*, as expected from (2.5). Since the motion is time-harmonic, the free surface oscillates up and down in the regions |*x*|<*a* and |*x*|>*b* (the region *a*<|*x*|<*b* has no physical significance). Furthermore, the bulk of this motion occurs in the moonpool, and the motion decays as |*x*|→∞. Note that the surface elevation shown in figure 5*b* and the zero contact angles at the pinned edges are compatible within the linear theory employed herein.

It should also be noted that for the case where both *ξ* and *ζ* are allowed to vary, it is possible to obtain multiple surface-piercing bodies. However, only two of these satisfy (2.5) and so the solution is not physically realistic.

We shall now concentrate on the case where dipoles at *x*=±*ξ* replace the pairs of sources at (±*ξ*, 0) and (±*ζ*, 0). We shall fix *s* and allow *ξ* to vary. Numerical investigation shows that for each value of surface tension, there is a cut-off value for *ξ* above which a solution cannot be obtained for any value of *a*. By contrast, if *ξ* is sufficiently small, then any value of *a* in the range (0, *ξ*) will give surface-piercing bodies. For these cases, it is possible to obtain a value for *b* that is very close to the dipole. Such bodies, however, can only be found with values of *s* that are not too large. An example is illustrated in figure 6, which is produced by positioning the left contact point a very small distance from the dipole, situated at 0.7. It can be seen that *a* and *b*, marked as dots in figure 6*b*, are very close to each other, and indeed can be made even closer by moving *a* nearer to *ξ*. The resulting streamlines represent a pair of bodies that barely intersect the free surface. In general, the bodies can be made more elongated by reducing *a*, and such an example is shown in figure 7. The similarity of the structures given in figures 6 and 7 with those exhibited in Harter *et al*. (2007) is striking. In that paper, however, a family of barely *submerged* bodies were constructed by placing a combination of dipoles beneath the free surface. Finally, figure 8 shows a larger body that, together with its mirror image, supports localized oscillations.

## 5. Asymptotic analysis for small *s*

We now briefly consider the case where the source at (*ξ*, 0) is positioned close to (*π*/2, 0) and is sufficiently far from the other source so that 1/cos *ζ* is an *O*(1) quantity. We also assume that *s*≪1. When all these conditions are met, the constants *A*, *B* and *D* become very small and the resulting streamline plots closely resemble those exhibited in McIver (1996). However, one difference is the existence of a region close to each contact point where the slope rapidly tends to zero in order to satisfy (2.5). The size of each of these regions becomes vanishingly small as *s*→0, allowing McIver's (1996) result to be recovered. By considering a small region around a single contact point and rewriting *ψ* in terms of exponential integrals, it can be seen that the dominant terms in the region come from the *O*(1) source terms and the term that has a logarithmic singularity at the contact point. Indeed, it is the logarithmic singularity that gives *ψ* its horizontal slope at the contact point in question. We shall consider a small region in the vicinity of the contact point (*a*, 0); in this region we approximate *ψ* (given by (3.6)) by *ψ*_{a}, whereThe first integral in this expression represents the dominant contribution from the *O*(1) source terms and the logarithmic terms arise from the second integral.

We now make the substitutions *x*−*a*=*ϵ*(*s*)*X*, *y*=*ϵ*(*s*)*Y*, where *X*,*Y*=*O*(1) and *ϵ*(*s*)→0 as *s*→0. This modifies the above expression to(5.1)say, where *z*=*X*+i*Y*. The condition that must be satisfied is(5.2)Thus, we expand (5.1) with the aid of (3.9), differentiate the resulting expression with respect to *z* and then use condition (5.2). The details are given in appendix A. On substituting (A 1) and (A 6) into (5.2), we obtainwhereandClearly, in order to satisfy this condition, we must have(5.3)Similarly, we can expand about the contact point situated at (*b*, 0) to find(5.4)whereThus, in the limit as *s*→0, the constants *A* and *B* scale on *s*^{1/2}, and so does *D* by (3.4). These approximations agree well with the numerical results. For example, with *s*=10^{−10}, *ξ*=*π*/2, *ζ*=1 and *a*=0.3, we find numerically that *b*=1.59961, and that *A*=3.55409×10^{−5} and *B*=−1.96435×10^{−5}, whereas the asymptotic results give *A*=3.55455×10^{−5} and *B*=−1.9645×10^{−5}.

Finally, figure 9 illustrates a pair of streamlines that, due to the positions of the sources and the small value of *s* chosen, closely resemble those provided by McIver (1996). Figures 9*b*,*c* show magnified plots near the contact points. These magnified plots demonstrate that the streamline slope is horizontal at each contact point as required, even though this is not visible in figure 9*a*. Furthermore, the streamline slopes in the inner region match with their corresponding outer region slope. The horizontal slopes are visible in inner regions of length-scale *O*(*s*^{1/2}), in accordance with the asymptotic analysis discussed in appendix A.

## 6. The zero-slope contact-line condition

We have considered in detail the fixed-edge condition, since the resulting boundary-value problem yields a solution that does not lose energy at the contact points. We now consider the case where (2.5) is replaced by the zero-slope condition *∂η*/*∂x*=0. In terms of the velocity potential, the condition takes the form(6.1)Recall, from discussions in §3, that these conditions must be interpreted in a limiting sense from the fluid side, i.e. taking *x* increasing to *a* and decreasing to the point *b*. The streamlines that result from solving this new boundary-value problem are similar to those found in the previous sections, and with *ξ*=*π*/2 and *ζ*−*ξ*=*O*(1) McIver's results are regained as *s*→0. In this limit, however, there is no inner region close to each contact point where the streamline slope differs from the outer region. This can be understood by noting that (6.1) implies *∂*^{2}*ψ*/*∂y*^{2}=0 at the contact points, and this condition gives no information about the contours of *ψ*. A typical result is shown in figure 10. Numerical experimentation suggests that, in contrast to the case considered in §4, no solutions can be found with *ξ*=*π*/2 when *s* exceeds an *O*(1) critical value, i.e. no localized modes exist for large values of surface tension.

## 7. Conclusion

In the present study, the work of Harter *et al*. (2007) for surface-piercing bodies has been extended to include two distinct choices of contact-line conditions, namely the fixed-edge and zero-slope conditions. For problems involving localized oscillations, these are appropriate conditions to apply since in both cases there is no dissipation of energy at the contact points. Both problems are attacked by the construction of a harmonic velocity potential consisting of pairs of singularities whose location and strengths are chosen so that all the conditions of the problem are satisfied. Using the fixed-edge condition (*η*=0), it is found that the results given in Harter *et al*. (2007) are valid, away from the contact points, for small values of *s*. At each contact point the streamline slope is horizontal, but it is shown that the length-scale over which the streamline changes becomes vanishingly small as *s*→0, enabling the results of McIver (1996) to be recovered. Results are also obtained for large values of *s*; thus, using the fixed-edge contact-line condition, trapped modes can be found for all values of surface tension. Examples of surface-piercing bodies that support trapped modes can also be found when the zero-slope (*∂η*/*∂x*=0) condition is applied, and McIver's (1996) results can again be obtained by letting *s*→0. In this case, it is found that trapped modes do not exist for all values of *s* but just for a restricted range.

## Footnotes

- Received February 8, 2008.
- Accepted June 18, 2008.

- © 2008 The Royal Society