The response of a spherical droplet on a wall executing small sinusoidal vibrations

Abstract

This study concerns the response of a spherical drop, attached to a sinusoidally vibrating wall. Given that the drop is spherical, the model is more realistic than that of a 2D drop described in [1], with which the results are compared. As in that study, the moving contact line is modelled by a Navier-type wall boundary condition and a prescribed contact angle, $\overline{\theta }$, which can take any of the values $0 < \overline{\theta } < \pi$. Assuming small vibration amplitude, the problem is linearized. In general, wall vibration has components both normal and tangential to the wall. Linearity means that the general response is the sum of decoupled components corresponding to purely normal and purely tangential vibration. The former is axisymmetric, while the latter has either $\cos \phi$ or $\sin \phi$ dependence on the azimuthal angle, $\phi$, depending on the direction of vibration. Analysis of the small Ohnesorge number limit, ${\text{Oh}} \to 0$, brings out two distinct viscous damping mechanisms. One arises from dissipation near the contact line and is characterized by a parameter $\beta$. The other comes from a boundary layer at the wall and is of order ${\text{Oh}}^{1/2}$. Numerical results follow from implementation of the small-${\text{Oh}}$ problem. Overall, and perhaps surprisingly, these results show strong qualitative similarity with the 2D case studied in [1], though they of course differ quantitatively. Since the 2D problem is not realisable experimentally, we hope that the results and methodology described here will lead to quantitative comparisons between theory and experiment.

Appendices

Appendix A: Analytical procedure

This appendix gives analysis of the small-${\text{Oh}}$ problem in preparation for the numerical methods described in appendices B-E. The spherical geometry of the problem suggests the use of toroidal coordinates, $\tau$, $\xi$, $\phi$:

$$ x = b\frac{\sinh \tau \cos \phi }{{\cosh \tau + \cos \xi }} $$

(A.1)

$$ y = b\frac{\sinh \tau \sin \phi }{{\cosh \tau + \cos \xi }} $$

(A.2)

$$ z = b\frac{\sin \xi }{{\cosh \tau + \cos \xi }} $$

(A.3)

where $b = \overline{r}\sin \overline{\theta }$. These coordinates are closely related to the bipolar system used in [1]. The coordinates $\tau > 0$, $0 \le \xi \le \overline{\theta }$, $- \pi \le \phi \le \pi$ form an orthogonal system with scale factors

$$ h_{\tau } = h_{\xi } = \frac{b}{\cosh \tau + \cos \xi },\quad h_{\phi } = \frac{b\sinh \tau }{{\cosh \tau + \cos \xi }} $$

(A.4)

The iso-surfaces of the coordinate $\tau$ are tori: $\tau \to 0$ yields the symmetry axis, $x = y = 0$, while $\tau = \infty$ represents the contact line. Those of the coordinate $\xi$ are spherical, with $\xi = 0$ at the wall and $\xi = \overline{\theta }$ at the equilibrium interface. It should be borne in mind throughout that $\tau$ is restricted to positive values because many of the equations are singular for $\tau = 0$. This does not stop us from considering the limit $\tau \to 0$ in which the symmetry axis is approached. The main technical difficulty in the spherical problem is that, whereas the Laplace equation is fully separable using the bipolar coordinates of the 2D case, it is only partially so for toroidal ones. Here, fully separable means that there are solutions consisting of a product of functions, each of which depends only on just one coordinate. On the other hand, the separated solutions for toroidal coordinates consist of such a product multiplied by $\left( {\cosh \tau + \cos \xi } \right)^{1/2}$.

Equation (2.45) gives

$$ \frac{1}{\sinh \tau }\frac{\partial }{\partial \tau }\left( {\sinh \tau \frac{\partial \Psi }{{\partial \tau }}} \right) + \frac{{\partial^{2} \Psi }}{{\partial \xi^{2} }} - \left( {\frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)\Psi = 0 $$

(A.5)

where, in keeping with separation of variables, $\Psi = \left( {\cosh \tau + \cos \xi } \right)^{ - 1/2} p$. Eq. (2.53) implies

$$ \frac{\partial \Psi }{{\partial \xi }} = - i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \frac{\Phi \left( \tau \right)}{{\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} }} $$

(A.6)

on $\xi = 0$, where

$$ \Phi \left( \tau \right) = \frac{{\left( {\cosh \tau + 1} \right)^{1/2} \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} }}{b\sinh \tau }\left( {\frac{d}{d\tau }\left( {\sinh \tau \frac{d\psi }{{d\tau }}} \right) - \frac{{k^{2} }}{\sinh \tau }\psi } \right) $$

(A.7)

Equation (2.47) yields

$$ \frac{\partial \Psi }{{\partial \xi }} = \frac{\zeta \left( \tau \right)}{{\left( {\cosh \tau + \cos \overline{\theta }} \right)^{3/2} }} $$

(A.8)

on $\xi = \overline{\theta }$, where

$$ \zeta \left( \tau \right) = b\varpi^{2} \eta \left( \tau \right) + \frac{1}{2}p\left( {\tau ,\xi = \overline{\theta }} \right)\sin \overline{\theta } $$

(A.9)

Define Fourier coefficients

$$ \Psi_{n} \left( \tau \right) = \left( { - 1} \right)^{n} \frac{2}{{\overline{\theta }}}\int_{0}^{{\overline{\theta }}} {\Psi \left( {\tau ,\xi } \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}d\xi } . $$

(A.10)

where $n \ge 0$ takes on integer values. The Fourier series

$$ \Psi = \frac{1}{2}\Psi_{0} + \sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} \Psi_{n} \cos \frac{n\pi \xi }{{\overline{\theta }}}} $$

(A.11)

then expresses $\Psi$ in terms of its coefficients. Multiplying (A.5) by $\cos \frac{n\pi \xi }{{\overline{\theta }}}$ and integrating over $\xi$, we use integration by parts to obtain

$$ \int_{0}^{{\overline{\theta }}} {\frac{{\partial^{2} \Psi }}{{\partial \xi^{2} }}\cos \frac{n\pi \xi }{{\overline{\theta }}}d\xi } = \left[ {\frac{\partial \Psi }{{\partial \xi }}\cos \frac{n\pi \xi }{{\overline{\theta }}} + \Psi \frac{n\pi }{{\overline{\theta }}}\sin \frac{n\pi \xi }{{\overline{\theta }}}} \right]_{0}^{{\overline{\theta }}} - \frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }}\int_{0}^{{\overline{\theta }}} {\Psi \cos \frac{n\pi \xi }{{\overline{\theta }}}d\xi } . $$

(A.12)

Using (A.6), (A.8) and (A.10),

$$ \begin{aligned} & \frac{1}{\sinh \tau }\frac{d}{d\tau }\left( {\sinh \tau \frac{{d\Psi_{n} }}{d\tau }} \right) - \left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} + \frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)\Psi_{n} \\ & \quad = - \frac{2}{{\overline{\theta }\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} }} \\ & \quad \left( {\frac{\zeta \left( \tau \right)}{{\cosh \tau + \cos \overline{\theta }}} + \left( { - 1} \right)^{n} i\varpi \left( {i\varpi } \right)^{1/2} Oh^{1/2} \Phi \left( \tau \right)} \right). \\ \end{aligned} $$

(A.13)

1.1 A.1 The leading-order problem

In the leading-order problem, the viscous term in (A.13) is dropped. Furthermore, there is no logarithmic singularity of $p$ at the contact line. Let us denote the leading-order $p$ by $\overline{p}\left( {\tau ,\xi } \right)$ and let $\overline{\zeta }\left( \tau \right) = b\eta \left( \tau \right) + \frac{1}{2}\varpi^{ - 2} \overline{p}\left( {\tau ,\xi = \overline{\theta }} \right)\sin \overline{\theta }$ so that $\zeta = \varpi^{2} \overline{\zeta }$ for the leading-order problem. Define Green’s functions for (A.13) by

$$ \frac{d}{d\tau }\left( {\sinh \tau \frac{{dG_{n} }}{d\tau }} \right) - \sinh \tau \left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} + \frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)G_{n} = \delta \left( {\tau - \tau^{{\prime }} } \right) $$

(A.14)

where $e^{\tau /2} G_{n} \left( {\tau ;\tau^{{\prime }} } \right)$ is bounded as $\tau \to 0$ and $\tau \to \infty$. The solution of (A.13), without the viscous term and with $\zeta$ replaced by $\varpi^{2} \overline{\zeta }$, having bounded $e^{\tau /2} \Psi_{n} \left( \tau \right)$ in these two limits is

$$ \Psi_{n} \left( \tau \right) = - 2\varpi^{2} \overline{\theta }^{ - 1} \int_{0}^{\infty } {G_{n} \left( {\tau ;\tau^{\prime } } \right)\frac{{\overline{\zeta }\left( {\tau^{\prime}} \right)\sinh \tau^{\prime } }}{{\left( {\cosh \tau^{\prime } + \cos \overline{\theta }} \right)^{3/2} }}d\tau^{\prime } } $$

(A.15)

Applying (A.11) with $\xi = \overline{\theta }$ and the definitions of $\Psi$ and $\overline{\zeta }$,

$$ \frac{{\overline{\theta }}}{{\sin \overline{\theta }}}\left( {\overline{\zeta }\left( \tau \right) - b\eta \left( \tau \right)} \right) + \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau ;\tau^{{\prime }} } \right)\frac{{\overline{\zeta }\left( {\tau^{{\prime }} } \right)\sinh \tau^{{\prime }} }}{{\left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{2} }}d\tau^{{\prime }} } = 0 $$

(A.16)

where

$$ \begin{aligned} G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime } } \right) & = \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} \left( {\cosh \tau^{\prime } + \cos \overline{\theta }} \right)^{1/2} \\ & \quad \left( {\frac{1}{2}G_{0} \left( {\tau ;\tau^{\prime } } \right) + \sum\limits_{n = 1}^{\infty } {G_{n} \left( {\tau ;\tau^{\prime } } \right)} } \right). \\ \end{aligned} $$

(A.17)

Section 1 of appendix C gives expressions for $G_{n} \left( {\tau ;\tau^{{\prime }} } \right)$ and describes some of its properties. In particular, as $\tau \to \infty$, (C.1), (C.2) and (C.10) imply

$$ G_{0} \left( {\tau ;\tau^{\prime}} \right) \sim - 2f_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)e^{ - \tau /2} $$

(A.18)

$$ G_{n} \left( {\tau ;\tau^{{\prime }} } \right)\sim - \frac{{\overline{\theta }}}{n\pi }f_{n}^{\left( 1 \right)} \left( {\tau^{{\prime }} } \right)\exp \left( { - \left( {\frac{n\pi }{{\overline{\theta }}} + \frac{1}{2}} \right)\tau } \right)\quad ,n \ne 0 $$

(A.19)

where $f_{n}^{\left( 1 \right)} \left( \tau \right)$ is the solution of

$$ \frac{d}{d\tau }\left( {\sinh \tau \frac{df}{{d\tau }}} \right) - \sinh \tau \left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} + \frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)f = 0 $$

(A.20)

which is bounded as $\tau \to 0$ and normalised according to (C.1) and (C.2) as $\tau \to \infty$. It follows that $G_{0}$ dominates the sum in (A.17) in the limit $\tau \to \infty$. Using (A.17) and (A.18), $G^{\left( 1 \right)} \left( {\tau ;\tau^{{\prime }} } \right) \to - 2^{ - 1/2} \left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( {\tau^{{\prime }} } \right)$ as $\tau \to \infty$. Thus, (A.16) and $\eta \to \eta_{c}$ imply $\overline{\zeta }\left( \tau \right) \to \overline{\zeta }^{\infty }$, where $\overline{\zeta }^{\infty }$ follows from

$$ \frac{{2^{1/2} \overline{\theta }}}{{\sin \overline{\theta }}}\left( {\overline{\zeta }^{\infty } - b\eta_{c} } \right) - \int_{0}^{\infty } {\frac{{\overline{\zeta }\left( {\tau^{{\prime }} } \right)f_{0}^{\left( 1 \right)} \left( {\tau^{{\prime }} } \right)\sinh \tau^{{\prime }} }}{{\left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{3/2} }}d\tau^{{\prime }} } = 0. $$

(A.21)

(A.11) with $\xi = 0$, (A.15) and the definitions of $\Psi$ and $\psi$ imply

$$ \frac{1}{2}\overline{\theta }\psi \left( \tau \right) + \int_{0}^{\infty } {G^{\left( 3 \right)} \left( {\tau ;\tau^{{\prime }} } \right)\frac{{\overline{\zeta }\left( {\tau^{{\prime }} } \right)\sinh \tau^{{\prime }} }}{{\left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{2} }}d\tau^{{\prime }} } = 0 $$

(A.22)

where

$$ \begin{aligned} G^{\left( 2 \right)} \left( {\tau ;\tau^{{\prime }} } \right) = & \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} \left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{1/2} \\ \quad & \left( {\frac{1}{2}G_{0} \left( {\tau ;\tau^{\prime}} \right) + \sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} G_{n} \left( {\tau ;\tau^{{\prime }} } \right)} } \right) \\ \end{aligned} $$

(A.23)

$$ G^{\left( 3 \right)} \left( {\tau ;\tau^{{\prime }} } \right) = \left( {\frac{\cosh \tau + 1}{{\cosh \tau + \cos \overline{\theta }}}} \right)^{1/2} G^{\left( 2 \right)} \left( {\tau ;\tau^{{\prime }} } \right) $$

(A.24)

Differentiating (A.22) and multiplying by $\sinh \tau$,

$$ \frac{1}{2}\overline{\theta }\alpha \left( \tau \right) + \sinh \tau \int_{0}^{\infty } {G^{\left( 4 \right)} \left( {\tau ;\tau^{{\prime }} } \right)\frac{{\overline{\zeta }\left( {\tau^{{\prime }} } \right)\sinh \tau^{{\prime }} }}{{\left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{2} }}d\tau^{{\prime }} } = 0 $$

(A.25)

in which $\alpha \left( \tau \right) = \sinh \tau \frac{d\psi }{{d\tau }}$ and

$$ G^{\left( 4 \right)} \left( {\tau ;\tau^{{\prime }} } \right) = \frac{{\partial G^{\left( 3 \right)} }}{\partial \tau }\left( {\tau ;\tau^{{\prime }} } \right). $$

(A.26)

According to (A.7),

$$ \Phi \left( \tau \right) = \frac{{\left( {\cosh \tau + 1} \right)^{1/2} \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} }}{b\sinh \tau }\left( {\frac{d\alpha }{{d\tau }} - \frac{{k^{2} }}{\sinh \tau }\psi } \right). $$

(A.27)

As shown in section C.1, $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$ satisfy the reciprocity relations $G^{\left( 1 \right)} \left( {\tau^{\prime};\tau } \right) = G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 2 \right)} \left( {\tau^{\prime};\tau } \right) = G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$ (this comes from reciprocity of $G_{n} \left( {\tau ;\tau^{\prime}} \right)$). Among other results, appendix D shows that $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ is a smooth (infinitely differentiable) function of $\tau$ and $\tau^{\prime}$, apart from a logarithmic singularity at $\tau = \tau^{\prime}$. Furthermore, perhaps surprisingly, although, for given $\tau^{\prime}$, the $\tau$-derivative of $G_{n} \left( {\tau ;\tau^{\prime}} \right)$ undergoes a discontinuous jump at $\tau = \tau^{\prime}$, $G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$ is a smooth function for all $\tau$ and $\tau^{\prime}$. Given smoothness of $G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$ are also smooth.

When $k = 0$, the leading-order problem for $\overline{p}$ consists of $\nabla^{2} \overline{p} = 0$, $\partial \overline{p}/\partial z = 0$ at the wall and $\partial \overline{p}/\partial r = \varpi^{2} \eta$ for $r = \overline{r}$. For given $\eta$, the solution for $\overline{p}$ is only unique up to an additive constant. Considering two distinct solutions with the same $\eta$, (A.16) applies to both. Subtracting and recalling the definition of $\overline{\zeta }$,

$$ \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau ;\tau^{{\prime }} } \right)\frac{{\sinh \tau^{{\prime }} }}{{\left( {\cosh \tau^{{\prime }} + \cos \overline{\theta }} \right)^{2} }}d\tau^{{\prime }} } = - \frac{{\overline{\theta }}}{{\sin \overline{\theta }}}. $$

(A.28)

Given this identity, solubility of (A.16) for $\overline{\zeta }\left( \tau \right)$ for given $\eta \left( \tau \right)$ imposes a constraint on possible $\eta \left( \tau \right)$. Multiplying (A.16) by $\sinh \tau \left( {\cosh \tau + \cos \overline{\theta }} \right)^{ - 2}$ and integrating over $\tau$, (A.28) and reciprocity of $G^{\left( 1 \right)}$ make the integral of the left-hand side zero. Thus,

$$ \int_{0}^{\infty } {\frac{\eta \left( \tau \right)\sinh \tau }{{\left( {\cosh \tau + \cos \overline{\theta }} \right)^{2} }}d\tau } = 0 $$

(A.29)

as the constraint on $\eta \left( \tau \right)$. This equation expresses (2.52) after transforming to the integration variable $\tau$, hence (A.29), and thus (2.52), are implied by (A.16) and do not need to be explicitly imposed. As discussed earlier, the condition (2.51) is also unnecessary for $k = 0$. As a result, both (2.51) and (2.52) are dropped when $k = 0$. Another consequence of (A.28) is that the solution of (A.16) for $\overline{\zeta }\left( \tau \right)$ is only unique up to an additive constant. This nonuniqueness can be removed by requiring $\overline{\zeta }\left( 0 \right) = 0$. Using (A.16) and reciprocity of $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, this condition is

$$ \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau^{\prime};0} \right)\frac{{\overline{\zeta }\left( {\tau^{\prime}} \right)\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}d\tau^{\prime}} - \frac{{b\overline{\theta }}}{{\sin \overline{\theta }}}\eta \left( 0 \right) = 0 $$

(A.30)

which is applied when $k = 0$.

1.2 A.2 The full problem

Solutions of (A.13) which are bounded as $\tau \to 0$ have the form

$$ \begin{aligned} \Psi_{n} \left( \tau \right) = & - 2\overline{\theta }^{ - 1} \int_{0}^{\infty } {\frac{{\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} }}G_{n} \left( {\tau ;\tau^{\prime}} \right)} \\ & \quad \left( {\frac{{\zeta \left( {\tau^{\prime}} \right)}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}} + \left( { - 1} \right)^{n} i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \Phi \left( {\tau^{\prime}} \right)} \right)d\tau^{\prime} + Q_{n} f_{n}^{\left( 1 \right)} \left( \tau \right) \\ \end{aligned} $$

(A.31)

in which $Q_{n}$ is any constant. To avoid exponential growth of $\Psi_{n} \left( \tau \right)$ as $\tau \to \infty$, $Q_{n} = 0$ unless $n = 0$. Applying (A.9), (A.11) at $\xi = \overline{\theta }$, (A.31) and the definition of $\Psi$,

$$ \begin{aligned} \frac{{\overline{\theta }}}{{\sin \overline{\theta }}} & \left( {\zeta \left( \tau \right) - b\varpi^{2} \eta \left( \tau \right)} \right) + \int_{0}^{\infty } {\frac{{\sinh \tau^{\prime}}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}}} \\ & \quad \left( {G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\zeta \left( {\tau^{\prime}} \right)}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}} + i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)\Phi \left( {\tau^{\prime}} \right)} \right)d\tau^{\prime} - \\ & \quad \frac{1}{4}\overline{\theta }Q_{0} \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( \tau \right) = 0 \\ \end{aligned} $$

(A.32)

Using (A.24), (A.27) and reciprocity of $G^{\left( 2 \right)}$,

$$ \begin{aligned} & \int_{0}^{\infty } {G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\sinh \tau^{\prime}}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}}\Phi \left( {\tau^{\prime}} \right)d\tau^{\prime}} = \\ & \quad b^{ - 1} \int_{0}^{\infty } {G^{\left( 3 \right)} \left( {\tau^{\prime};\tau } \right)\left( {\frac{d\alpha }{{d\tau^{\prime}}} - \frac{{k^{2} }}{{\sinh \tau^{\prime}}}\psi \left( {\tau^{\prime}} \right)} \right)d\tau^{\prime}} \\ \end{aligned} $$

(A.33)

Integrating by parts,

$$ \begin{aligned} & \int_{0}^{\infty } {G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\sinh \tau^{\prime}}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}}\Phi \left( {\tau^{\prime}} \right)d\tau^{\prime}} = \\ & \quad - 2^{ - 1/2} \frac{{\eta_{c} }}{{\sin \overline{\theta }}}\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( \tau \right) - \\ & \quad b^{ - 1} \int_{0}^{\infty } {\left( {G^{\left( 3 \right)} \left( {\tau^{\prime};\tau } \right)\frac{{k^{2} }}{{\sinh \tau^{\prime}}}\psi \left( {\tau^{\prime}} \right) + G^{\left( 4 \right)} \left( {\tau^{\prime};\tau } \right)\alpha \left( {\tau^{\prime}} \right)} \right)d\tau^{\prime}} \\ \end{aligned} $$

(A.34)

where we have used $\alpha \left( 0 \right) = 0$, (A.26) and the limiting value of $G^{\left( 3 \right)} \left( {\tau^{\prime};\tau } \right)\alpha \left( {\tau^{\prime}} \right)$ as $\tau^{\prime} \to \infty$, which is obtained as follows. Analysis of the leading-order problem as $R \to 0$ shows that $d\psi /dR \to - \eta_{c} /\sin \overline{\theta }$, hence $\alpha \left( {\tau^{\prime}} \right) \to b\eta_{c} /\sin \overline{\theta }$. It follows from (A.18) and (A.19) that (A.23) is dominated by $G_{0}$ in the limit $\tau \to \infty$. Employing (A.18), (A.23) and (A.24), $G^{\left( 3 \right)} \left( {\tau^{\prime};\tau } \right) \to - 2^{ - 1/2} \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( \tau \right)$ as $\tau^{\prime} \to \infty$.

When $k = 0$, (A.28), (A.32) and (A.34) give

$$ \begin{aligned} & \frac{{\overline{\theta }}}{{\sin \overline{\theta }}}\left( {\zeta \left( \tau \right) - \zeta \left( 0 \right) - b\varpi^{2} \eta \left( \tau \right)} \right) + \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\left( {\zeta \left( {\tau^{\prime}} \right) - \zeta \left( 0 \right)} \right)\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}d\tau^{\prime}} - \\ & \quad i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} b^{ - 1} \int_{0}^{\infty } {G^{\left( 4 \right)} \left( {\tau^{\prime};\tau } \right)\alpha \left( {\tau^{\prime}} \right)} d\tau^{\prime} - \\ & \quad \frac{1}{4}\left( {\overline{\theta }Q_{0} + 2^{3/2} i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \frac{{\eta_{c} }}{{\sin \overline{\theta }}}} \right)\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( \tau \right) = 0 \\ \end{aligned} $$

(A.35)

Setting $\tau = 0$ and using reciprocity of $G^{\left( 1 \right)}$,

$$ \begin{aligned} & - \varpi^{2} \frac{{b\overline{\theta }}}{{\sin \overline{\theta }}}\eta \left( 0 \right) + \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau^{\prime};0} \right)\frac{{\left( {\zeta \left( {\tau^{\prime}} \right) - \zeta \left( 0 \right)} \right)\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}d\tau^{\prime}} - \\ & \quad i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} b^{ - 1} \int_{0}^{\infty } {G^{\left( 4 \right)} \left( {\tau^{\prime};0} \right)\alpha \left( {\tau^{\prime}} \right)} d\tau^{\prime} - \\ & \quad \frac{1}{4}\left( {\overline{\theta }Q_{0} + 2^{3/2} i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \frac{{\eta_{c} }}{{\sin \overline{\theta }}}} \right)\left( {1 + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( 0 \right) = 0 \\ \end{aligned} $$

(A.36)

Turning attention to $k = 1$, as the contact line is approached, the final term in (A.31) for $n = 0$ is responsible for the logarithmic singularity of the pressure. Using (A.11), (C.1) and the definition of $\Psi$, this term contributes $2^{ - 3/2} Q_{0} \tau$ to $p$ as $\tau \to \infty$. Since $\tau \sim - \ln R$, agreement with (2.51) implies

$$ Q_{0} = - 2^{3/2} i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \overline{\theta }^{ - 1} \left( {\frac{{\eta_{c} }}{{\sin \overline{\theta }}} + \eta_{*} } \right) $$

(A.37)

(A.32), (A.34) and (A.37) give

$$ \begin{aligned} & \frac{{\overline{\theta }}}{{\sin \overline{\theta }}}\left( {\zeta \left( \tau \right) - b\varpi^{2} \eta \left( \tau \right)} \right) + \int_{0}^{\infty } {G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\zeta \left( {\tau^{\prime}} \right)\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}d\tau^{\prime}} - \\ & \quad i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} b^{ - 1} \int_{0}^{\infty } {\left( {G^{\left( 3 \right)} \left( {\tau^{\prime};\tau } \right)\frac{{\psi \left( {\tau^{\prime}} \right)}}{{\sinh \tau^{\prime}}} + G^{\left( 4 \right)} \left( {\tau^{\prime};\tau } \right)\alpha \left( {\tau^{\prime}} \right)} \right)} d\tau^{\prime} + \\ & \quad 2^{ - 1/2} i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \eta_{*} \left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( \tau \right) = 0 \\ \end{aligned} $$

(A.38)

(A.16), (A.22), (A.25), with (A.35) for $k = 0$ and (A.38) for $k = 1$ are integral equations relating the unknown functions $\overline{\zeta }\left( \tau \right)$, $\zeta \left( \tau \right)$, $\eta \left( \tau \right)$, $\psi \left( \tau \right)$ and $\alpha \left( \tau \right)$. Further equations are derived as follows.

It is shown in section C.2 that bounded solutions of (2.48) have the form

$$ \begin{aligned} \eta \left( \tau \right) = & Ef^{\left( 3 \right)} \left( \tau \right) + b^{2} \int_{0}^{\infty } {G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)\frac{{\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}} \\ & \quad \left( {\frac{2}{{\sin \overline{\theta }}}\left( {b\varpi^{2} \eta \left( {\tau^{\prime}} \right) - \zeta \left( {\tau^{\prime}} \right)} \right) + \frac{{b\left( {1 - \varpi^{2} \eta_{*} } \right)\hat{\sigma }_{k} \left( {\tau^{\prime}} \right)}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}}} \right)d\tau^{\prime} \\ \end{aligned} $$

(A.39)

where $E$ is an unknown constant, $f^{\left( 3 \right)} \left( \tau \right)$ and $G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)$ are functions having explicit analytical expressions, $\hat{\sigma }_{0} \left( \tau \right) = \sin \overline{\theta }$ and $\hat{\sigma }_{1} \left( \tau \right) = \sinh \tau$. It is also shown that (2.49) and (2.50) yield

$$ \begin{aligned} & Ej^{{\prime (3)}} - \frac{1}{2}b^{2} j^{{\prime (4)}} \int_{0}^{\infty } {f^{{\left( 3 \right)}} \left( {\tau ^{\prime } } \right)\frac{{\sinh \tau ^{\prime } }}{{\left( {\cosh \tau ^{\prime } + \cos \bar{\theta }} \right)^{2} }}} \\ & \left( {\frac{2}{{\sin \bar{\theta }}}\left( {b\varpi ^{2} \eta \left( {\tau ^{\prime } } \right) - \zeta \left( {\tau ^{\prime } } \right)} \right) + \frac{{b\left( {1 - \varpi ^{2} \eta _{*} } \right)\hat{\sigma }_{k} \left( {\tau ^{\prime } } \right)}}{{\cosh \tau ^{\prime } + \cos \bar{\theta }}}} \right)d\tau ^{\prime } & = \Omega \\ \end{aligned} $$

(A.40)

$$ \begin{aligned} & Ej^{{\left( 3 \right)}} - \frac{1}{2}b^{2} j^{{\left( 4 \right)}} \int_{0}^{\infty } {f^{{\left( 3 \right)}} \left( {\tau ^{\prime } } \right)\frac{{\sinh \tau ^{\prime } }}{{\left( {\cosh \tau ^{\prime } + \cos \bar{\theta }} \right)^{2} }}} \\ & \left( {\frac{2}{{\sin \bar{\theta }}}\left( {b\varpi ^{2} \eta \left( {\tau ^{\prime } } \right) - \zeta \left( {\tau ^{\prime } } \right)} \right) + \frac{{b\left( {1 - \varpi ^{2} \eta _{*} } \right)\hat{\sigma }_{k} \left( {\tau ^{\prime } } \right)}}{{\cosh \tau ^{\prime } + \cos \bar{\theta }}}} \right)d\tau ^{\prime } & = \eta _{c} \\ \end{aligned} $$

(A.41)

in which $j^{\left( 3 \right)}$, $j^{\prime(3)}$, $j^{\left( 4 \right)}$ and $j^{\prime(4)}$ are known constants and

$$ \Omega = \eta_{c} \left( {\cot \overline{\theta } - \frac{{2i\varpi \beta \overline{r}}}{{\overline{\theta } - \sin \overline{\theta }\cos \overline{\theta }}}} \right) - \frac{{2i\varpi \beta \overline{r}\sin \overline{\theta }}}{{\overline{\theta } - \sin \overline{\theta }\cos \overline{\theta }}}\eta_{*} $$

(A.42)

Note that, according to (2.49), $\Omega$ is the contact-line value of $d\eta /d\chi$. Finally, according to section C.2, (A.39) gives

$$ \begin{aligned} & \eta \left( 0 \right) = E - \frac{1}{2}b^{2} \int_{0}^{\infty } {f^{\left( 4 \right)} \left( {\tau^{\prime}} \right)\frac{{\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}} \\ & \quad \left( {\frac{2}{{\sin \overline{\theta }}}\left( {b\varpi^{2} \eta \left( {\tau^{\prime}} \right) - \zeta \left( {\tau^{\prime}} \right)} \right) + \frac{{b\left( {1 - \varpi^{2} \eta_{*} } \right)\hat{\sigma }_{k} \left( {\tau^{\prime}} \right)}}{{\cosh \tau^{\prime} + \cos \overline{\theta }}}} \right)d\tau^{\prime} \\ \end{aligned} $$

(A.43)

when $k = 0$ and $\tau = 0$, where $f^{\left( 4 \right)} \left( {\tau^{\prime}} \right)$ has an explicit analytical expression.

The full problem determines the unknowns $\overline{\zeta }\left( \tau \right)$, $\zeta \left( \tau \right)$, $\eta \left( \tau \right)$, $\alpha \left( \tau \right)$, $\eta_{c}$, $\eta_{*}$, $\overline{\zeta }^{\infty }$, $E$, $\Omega$, as well as $\zeta \left( 0 \right)$, $\eta \left( 0 \right)$, $Q_{0}$ for $k = 0$ and $\psi \left( \tau \right)$ for $k = 1$. Equations (A.16), (A.21), (A.25) and (A.39)-(A.42) are applied for both values of $k$. When $k = 0$, (A.30), (A.35), (A.36), (A.43) and $\eta_{*} = 0$ are also used. When $k = 1$, (A.22), (A.38) are employed and

$$ \int_{0}^{\infty } {\frac{{\eta \left( {\tau^{\prime}} \right)\sinh \tau^{\prime}}}{{\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{2} }}d\tau^{\prime}} = 0 $$

(A.44)

follows from (2.52). Note that, when $k = 0$, $\psi \left( \tau \right)$ only appears in equation (A.22), which is the reason for dropping $\psi \left( \tau \right)$ and (A.22) unless $k = 1$. The numerical implementation of the above problem is described in appendix B and uses results from appendices C-E.

Appendix B: Numerical method

In [1] there was a numerical problem, apparent in Fig. 5 of that article, for $\overline{\theta }$ near $\pi$. As discussed there, the problem arises from the bipolar coordinate system and interfacial discretisation used. The toroidal coordinates employed here would lead to a similar problem if $\tau$ were uniformly discretised. The problem is that such a discretisation places more and more points near $x = y = z = 0$ as $\overline{\theta } \to \pi$, whereas the liquid/gas interface continues to extend to $x,\;y,\;z = O\left( 1 \right)$. Thus, most of the interface is inadequately covered. Here, we attempt to get around this problem using the transformation of variable $\mu = \ln \left( {\sinh \tau } \right)$. Discretising with uniformly spaced $\mu$, rather than $\tau$, points cluster close to the pole, $\chi = 0$, when $\overline{\theta }$ lies away from $\pi$ and have a much less sparse distribution over $\chi$ when $\overline{\theta }$ is near $\pi$. This does not completely eliminate the problem because as $\overline{\theta } \to \pi$ there still comes a point when the distribution of points over $\chi$ is insufficiently dense. However, it considerably improves matters, making the problematic region of small $\pi - \overline{\theta }$ much smaller.

With the above motivation in mind, let $\mu = \ln \left( {\sinh \tau } \right)$, then functions of $\tau$ can alternatively be regarded as functions of $\mu$. As usual in numerical implementations of continuous problems, discretisation is used. $\overline{\zeta }\left( \tau \right)$, $\zeta \left( \tau \right)$, $\eta \left( \tau \right)$, $\alpha \left( \tau \right)$ and $\psi \left( \tau \right)$ ($k = 1$) are represented by their values at $\mu = m\Delta$, where $\Delta$ is small and $m$ is an integer in the range $- M \le m \le M$, with $M$ an integer and $M\Delta$ large. We denote these values by $\overline{\zeta }^{m}$, $\zeta^{m}$, $\eta^{m}$, $\alpha^{m}$ and $\psi^{m}$ (for $k = 1$). Including $\eta_{c}$, $\eta_{*}$, $\overline{\zeta }^{\infty }$, $E$, $\Omega$, as well as $\zeta \left( 0 \right)$, $\eta \left( 0 \right)$, $Q_{0}$ for $k = 0$, there are $8M + 12$ unknowns when $k = 0$ and $10M + 10$ when $k = 1$. Equations (A.16), (A.22) (for $k = 1$), (A.25), (A.35) (for $k = 0$), (A.38) (for $k = 1$) and (A.39) are applied for $\tau = \tau_{m}$, $- M \le m \le M$, where $\ln \left( {\sinh \tau_{m} } \right) = m\Delta$. Together with (A.21), (A.40)-(A.42), as well as (A.30), (A.36), (A.43) and $\eta_{*} = 0$ when $k = 0$ and (A.44) for $k = 1$, there are the same number of equations as unknowns. The integrals in (A.16), (A.21), (A.22), (A.25), (A.30), (A.35), (A.36), (A.38), (A.39)-(A.41), (A.43) and (A.44) are approximated as follows.

Transforming to the integration variable $\mu^{\prime} = \ln \left( {\sinh \tau^{\prime}} \right)$, each integral is rewritten as

$$ \int_{0}^{\infty } {F\left( {\tau^{\prime}} \right)d\tau^{\prime}} = \int_{ - \infty }^{\infty } {F\left( {\tau^{\prime}} \right)\tanh \tau^{\prime}\,d\mu^{\prime}} $$

(B.1)

prior to numerical integration. The contribution to the integrals from $\mu^{\prime} < - M\Delta$ is neglected. With the exception of the integrals involving $G^{\left( 4 \right)}$, the same is true of the contribution from $\mu^{\prime} > \hat{M}\Delta$, where $\hat{M} - M$ is a positive, even integer and $\left( {\hat{M} - M} \right)\Delta$ is large. Integrals containing $G^{\left( 4 \right)}$ are treated as follows.

Recalling that $\Omega$ is the contact-line value of $d\eta /d\chi$, $d\eta /d\tau \sim 2\Omega e^{ - \tau } \sin \overline{\theta }$ as $\tau \to \infty$. Analysis of the leading-order problem gives $\overline{p}\sim - \varpi^{2} \eta_{c} R\cos \gamma /\sin \overline{\theta }$ as $R \to 0$ to within an additive constant. Thus, $\partial \overline{p}/\partial \tau \sim - 2be^{ - \tau } \partial \overline{p}/\partial R\sim 2\varpi^{2} b\eta_{c} e^{ - \tau } \cos \gamma /\sin \overline{\theta }$ as $\tau \to \infty$. Using these results, the derivative of $\overline{\zeta }\left( \tau \right) = b\eta \left( \tau \right) + \frac{1}{2}\varpi^{ - 2} \overline{p}\left( {\tau ,\xi = \overline{\theta }} \right)\sin \overline{\theta }$ gives $d\overline{\zeta }/d\tau$. Integrating with respect to $\tau$,

$$ \overline{\zeta }\left( \tau \right)\sim \overline{\zeta }^{\infty } - b\left( {2\Omega \sin \overline{\theta } + \eta_{c} \cos \overline{\theta }} \right)e^{ - \tau } $$

(B.2)

as $\tau \to \infty$. This result is used to extend $\overline{\zeta }^{m}$ up to $m = \tilde{M}$, where $\tilde{M} - \hat{M}$ is a positive, even integer and $\left( {\tilde{M} - \hat{M}} \right)\Delta$ is large. (A.25), applied for $\tau = \tau_{m}$, $M < m \le \tilde{M}$, provides a similar extension of $\alpha^{m}$. Thus, in all, (A.25) is used for $\tau = \tau_{m}$, $- M \le m \le \tilde{M}$. The contributions of $\mu^{\prime} < - M\Delta$ and $\mu^{\prime} > \tilde{M}\Delta$ to the integral in (A.25) are neglected, while that from $- M\Delta \le \mu^{\prime} \le \tilde{M}\Delta$ is expressed using Simpson’s rule. The integrals in (A.35), (A.36) and (A.38) involving $G^{\left( 4 \right)}$ are also expressed using Simpson’s rule for $- M\Delta \le \mu^{\prime} \le \tilde{M}\Delta$, the contributions from $\mu^{\prime} < - M\Delta$ and $\mu^{\prime} > \tilde{M}\Delta$ being neglected.

The remaining integrals are expressed as follows. As noted above, the contributions from $\mu^{\prime} < - M\Delta$ and $\mu^{\prime} > \hat{M}\Delta$ are neglected. $\overline{\zeta }^{m}$, $\zeta^{m}$, $\eta^{m}$, $\psi^{m}$ (for $k = 1$) are extended up to $m = \hat{M}$ using the large-$\tau$ approximations $\overline{\zeta }^{m} = \overline{\zeta }^{\infty }$,

$$ \zeta^{m} = \zeta^{M} - \frac{1}{2}i\varpi \left( {i\varpi } \right)^{1/2} {\text{Oh}}^{1/2} \overline{\theta }^{ - 1} \left( {\eta_{c} + \eta_{*} \sin \overline{\theta }} \right)\left( {\tau_{m} - \tau_{M} } \right) $$

(B.3)

$\eta^{m} = \eta_{c}$ and $\psi^{m} = 2\left( {\overline{\zeta }^{\infty } - b\eta_{c} } \right)/\sin \overline{\theta }$. Integrals other than that in (A.39) and those in (A.16), (A.35) and (A.38) involving $G^{\left( 1 \right)}$ are treated using Simpson’s rule applied to the range $- M\Delta \le \mu^{\prime} \le \hat{M}\Delta$.

To represent the integral in (A.39), the range $- M\Delta \le \mu^{\prime} \le \hat{M}\Delta$ is divided into $\left( {M + \hat{M}} \right)/2$ intervals of length $2\Delta$. The values of the integrand are known at the mid- and end-points of the intervals. The location, $\tau$, at which (A.39) is applied corresponds with one of these points. If $\tau$ is an end-point, Simpson’s rule is applied over the whole range. On the other hand, when $\tau$ corresponds to the mid-point of an interval, the resulting discontinuity in the $\tau^{\prime}$-derivative of $G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)$ degrades the accuracy of Simpson’s rule for the given interval, although it is still used for all other intervals. The interval in $\mu^{\prime}$ containing $\tau$ is split into two subintervals of length $\Delta$. A quadratic approximation of the integrand without the factor $G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)$ is carried out using its values at the mid- and end-points of the interval. This approximation provides the values at the mid-points of the subintervals. Multiplying by $G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)$, Simpson’s rule is applied to the subintervals.

The treatment of the integrals in (A.16), (A.35) and (A.38) involving $G^{\left( 1 \right)}$ is complicated by its logarithmic singularity at $\tau^{\prime} = \tau$. As for (A.39), the range $- M\Delta \le \mu^{\prime} \le \hat{M}\Delta$ is divided into $\left( {M + \hat{M}} \right)/2$ intervals of length $2\Delta$. The relation

$$ \tanh \tau_{*} = \frac{{\sinh \tau \sinh \tau^{\prime}}}{{\cosh \tau \cosh \tau^{\prime} - 1}} $$

(B.4)

is solved for $\tau_{*}$ at the mid- and end-points of the intervals apart from $\tau^{\prime} = \tau$, where $\tau_{*} = + \infty$ is used. Intervals are of two types. The first type consists of those for which either $\left| {\tau - \tau^{\prime}} \right| \ge 1$ or $\tau_{*} \le 1$ for the mid- and end-points, all others being classified as of the second type. Intervals of the first type are treated using Simpson’s rule. For the others, the integrand in (B.1) is written

$$ F\left( {\tau^{\prime}} \right)\tanh \tau^{\prime} = F^{\left( 1 \right)} \left( {\mu^{\prime}} \right)\ln \left| {\mu - \mu^{\prime}} \right| + F^{\left( 2 \right)} \left( {\mu^{\prime}} \right) $$

(B.5)

where $F^{\left( 1 \right)} \left( {\mu^{\prime}} \right)$ and $F^{\left( 2 \right)} \left( {\mu^{\prime}} \right)$ are smooth functions, i.e. (B.5) makes the singularity explicit. Quadratic approximations of these functions follow from their values at the mid- and end-points and the contribution of the interval to the required integral is determined analytically.

The above formulation involves the functions $f_{0}^{\left( 1 \right)} \left( \tau \right)$, $f^{\left( 3 \right)} \left( \tau \right)$, $f^{\left( 4 \right)} \left( \tau \right)$, as well as the Green’s functions $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 5 \right)} \left( {\tau ;\tau^{\prime}} \right)$. The determination and properties of these functions is the subject of appendices C-E. As noted earlier, the formulation given here yields the same number of equations as unknowns, namely $8M + 12$ when $k = 0$ and $10M + 10$ when $k = 1$. The numerical versions of (A.16) and (A.25) can be used to eliminate the unknowns $\eta^{m}$ and $\alpha^{m}$, while $\eta_{*} = 0$ eliminates $\eta_{*}$ for $k = 0$, as does (A.22) for $\psi^{m}$ when $k = 1$. As a result, the number of equations and unknowns is reduced to $4M + 9$ for $k = 0$ and $4M + 7$ for $k = 1$. The remaining unknowns consist of $\overline{\zeta }^{m}$, $\zeta^{m}$, $- M \le m \le M$, $\eta_{c}$, $\overline{\zeta }^{\infty }$, $E$, $\Omega$, as well as $\zeta \left( 0 \right)$, $\eta \left( 0 \right)$, $Q_{0}$ for $k = 0$ and $\eta_{*}$ for $k = 1$. The resulting column vector of unknowns is denoted $U$ and the equations have the structure given in (2.57).

As noted in Sect. 2.5, when $k = 1$, the solution of the inviscid problem has $\eta_{*} = \varpi^{ - 2}$ with all other unknowns zero. This solution corresponds to $U = \varpi^{ - 2} V$, where $V$ is a column vector whose components are zero, apart from the $\eta_{*}$ one, which is $1$.

The numerical parameters used to obtain the results given in Sect. 3 are $\Delta = 0.01$, $M = 1000$, $\hat{M} = 2000$ and $\tilde{M} = 3000$. The parameter $N$ occurring in (E.34), (E.35) and (E.39) determines the maximum value of $n$ which is treated and we chose $N = 250$. Finally, the infinite sums in (D.48), (D.54) and (D.55) were truncated to $35$ terms. With these parameters, the accuracy of the numerical approximation was checked for $\overline{\theta } = \pi /2$ by comparison of the numerical and analytical values of $\omega_{\alpha }$, $D_{c}^{\alpha }$, $D_{w}^{\alpha }$, $B^{\alpha }$, $B_{c}^{\alpha }$ and $B_{w}^{\alpha }$. When $k = 0$, the relative errors for the lowest frequency mode are less than $10^{ - 7}$. They tend to increase for higher modes, reaching around $10^{ - 5}$ for the tenth mode and $10^{ - 4}$ for the twentieth. The relative errors of the zero-frequency mode are less than $10^{ - 7}$, while those of the first nonzero-frequency $k = 1$ mode are below $2 \times 10^{ - 7}$. As for the $k = 0$ modes, the errors tend to grow for higher modes, being around $10^{ - 5}$ and $10^{ - 4}$ for the tenth and twentieth nonzero-frequency modes. The degradation of numerical precision for higher-order modes should be no surprise because such modes have small wavelengths, which makes the numerical approximation of integrals, based on discretisation of the integrands, less accurate.

Appendix C: Expression of the Green’s functions and some of their properties

3.1 C.1 Expressions for $G_{n} \left( {\tau ;\tau^{\prime}} \right)$ and reciprocity

Let $f_{n}^{\left( 1 \right)} \left( \tau \right)$ and $f_{n}^{\left( 2 \right)} \left( \tau \right)$ be solutions of (A.20) such that $f_{n}^{\left( 1 \right)} \left( \tau \right)$ is bounded as $\tau \to 0$ and $e^{\tau /2} f_{n}^{\left( 2 \right)} \left( \tau \right)$ is bounded as $\tau \to \infty$. These solutions are normalised by

$$ f_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau e^{ - \tau /2} ,\quad f_{0}^{\left( 2 \right)} \left( \tau \right)\sim e^{ - \tau /2} $$

(C.1)

$$ f_{n}^{\left( 1 \right)} \left( \tau \right)\sim e^{{\left( {n\pi /\overline{\theta } - 1/2} \right)\tau }} ,\quad f_{n}^{\left( 2 \right)} \left( \tau \right)\sim e^{{ - \left( {n\pi /\overline{\theta } + 1/2} \right)\tau }} ,\quad n \ne 0 $$

(C.2)

as $\tau \to \infty$. In the opposite limit, $\tau \to 0$, $f_{n}^{\left( 1 \right)} \left( \tau \right)$ behaves like $\tau^{k}$, whereas $f_{n}^{\left( 2 \right)} \left( \tau \right)$ goes to infinity, like $\log \tau$ when $k = 0$ and $\tau^{ - 1}$ when $k = 1$.

The solution of (A.14) having bounded $e^{\tau /2} G_{n} \left( {\tau ;\tau^{\prime}} \right)$ as $\tau \to 0$ and $\tau \to \infty$ has the form

$$ G_{n} \left( {\tau ;\tau^{\prime}} \right) = Af_{n}^{\left( 2 \right)} \left( \tau \right)\quad \tau \ge \tau^{\prime} $$

(C.3)

$$ G_{n} \left( {\tau ;\tau^{\prime}} \right) = Bf_{n}^{\left( 1 \right)} \left( \tau \right)\quad \tau \le \tau^{\prime} $$

(C.4)

where

$$ Af_{n}^{\left( 2 \right)} \left( {\tau^{\prime}} \right) - Bf_{n}^{\left( 1 \right)} \left( {\tau^{\prime}} \right) = 0 $$

(C.5)

$$ A\frac{{df_{n}^{\left( 2 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) - B\frac{{df_{n}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) = \frac{1}{{\sinh \tau^{\prime}}} $$

(C.6)

Thus,

$$ A = \frac{{f_{n}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)}}{{W_{n} \left( {\tau^{\prime}} \right)}},\quad B = \frac{{f_{n}^{\left( 2 \right)} \left( {\tau^{\prime}} \right)}}{{W_{n} \left( {\tau^{\prime}} \right)}} $$

(C.7)

where

$$ W_{n} \left( \tau \right) = \sinh \tau \left( {f_{n}^{\left( 1 \right)} \left( \tau \right)\frac{{df_{n}^{\left( 2 \right)} }}{d\tau }\left( \tau \right) - f_{n}^{\left( 2 \right)} \left( \tau \right)\frac{{df_{n}^{\left( 1 \right)} }}{d\tau }\left( \tau \right)} \right) $$

(C.8)

Using (A.20),

$$ \frac{{dW_{n} }}{d\tau } = 0 $$

(C.9)

so $W_{n}$ is constant. At large $\tau$, $f_{0}^{\left( 2 \right)} \left( \tau \right)\sim e^{ - \tau /2}$ and $f_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau e^{ - \tau /2}$, the next-order term being proportional to $e^{ - \tau /2}$, hence $W_{0} = - 1/2$. Using (C.2), $W_{n} = - n\pi /\overline{\theta }$ when $n \ne 0$. (C.3), (C.4) and (C.7) give

$$ G_{n} \left( {\tau ;\tau^{\prime}} \right) = \frac{{f_{n}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)f_{n}^{\left( 2 \right)} \left( \tau \right)}}{{W_{n} }}\quad \tau \ge \tau^{\prime} $$

(C.10)

$$ G_{n} \left( {\tau ;\tau^{\prime}} \right) = \frac{{f_{n}^{\left( 1 \right)} \left( \tau \right)f_{n}^{\left( 2 \right)} \left( {\tau^{\prime}} \right)}}{{W_{n} }}\quad \tau \le \tau^{\prime} $$

(C.11)

It follows that $G_{n} \left( {\tau^{\prime};\tau } \right) = G_{n} \left( {\tau ;\tau^{\prime}} \right)$, thus $G^{\left( 1 \right)} \left( {\tau^{\prime};\tau } \right) = G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 2 \right)} \left( {\tau^{\prime};\tau } \right) = G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$.

In principle, $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$ follow from (A.17), (A.23), (A.24), (A.26), (C.10) and (C.11) for all $\tau \ne \tau^{\prime}$. However, when $\tau = \tau^{\prime}$, the sum in (A.17) is logarithmically divergent and that in (A.23) is slowly convergent (for large $n$, $G_{n} \left( {\tau ;\tau } \right)$ behaves like $n^{ - 1}$). These problems are reflected by slow convergence of the infinite sums when $\tau$ is near $\tau^{\prime}$. For this reason, a different approach to the calculation of the Green’s functions is developed in appendix D.

3.2 C.2 Derivation of (A.39)-(A.41) and (A.43)

Defining

$$ f^{\left( 3 \right)} \left( \chi \right) = \cos \chi ,\quad f^{\left( 4 \right)} \left( \chi \right) = \cos \chi \ln \left( {\frac{1 + \cos \chi }{{1 - \cos \chi }}} \right) - 2 $$

(C.12)

for $k = 0$ and

$$ f^{\left( 3 \right)} \left( \chi \right) = \sin \chi ,\quad f^{\left( 4 \right)} \left( \chi \right) = \frac{1}{2}\sin \chi \ln \left( {\frac{1 + \cos \chi }{{1 - \cos \chi }}} \right) + \cot \chi $$

(C.13)

when $k = 1$. These functions satisfy

$$ \frac{d}{d\chi }\left( {\sin \chi \frac{df}{{d\chi }}} \right) + \sin \chi \left( {2 - \frac{{k^{2} }}{{\sin^{2} \chi }}} \right)f = 0 $$

(C.14)

hence

$$ W\left( \chi \right) = \sin \chi \left( {f^{\left( 3 \right)} \left( \chi \right)\frac{{df^{\left( 4 \right)} }}{d\chi }\left( \chi \right) - f^{\left( 4 \right)} \left( \chi \right)\frac{{df^{\left( 3 \right)} }}{d\chi }\left( \chi \right)} \right) $$

(C.15)

is a constant, whose value, $W = - 2$, is determined by examining the limit $\chi \to 0$. Let

$$ G^{\left( 5 \right)} \left( {\chi ;\chi^{\prime}} \right) = - \frac{1}{2}f^{\left( 3 \right)} \left( {\chi^{\prime}} \right)f^{\left( 4 \right)} \left( \chi \right)\quad \chi \ge \chi^{\prime} $$

(C.16)

$$ G^{\left( 5 \right)} \left( {\chi ;\chi^{\prime}} \right) = - \frac{1}{2}f^{\left( 3 \right)} \left( \chi \right)f^{\left( 4 \right)} \left( {\chi^{\prime}} \right)\quad \chi \le \chi^{\prime} $$

(C.17)

then

$$ \frac{d}{d\chi }\left( {\sin \chi \frac{{dG^{\left( 5 \right)} }}{d\chi }} \right) + \sin \chi \left( {2 - \frac{{k^{2} }}{{\sin^{2} \chi }}} \right)G^{\left( 5 \right)} = \delta \left( {\chi - \chi^{\prime}} \right) $$

(C.18)

Thus, $G^{\left( 5 \right)}$ is a Green’s function which is bounded as $\chi \to 0$. Bounded solutions of (2.48) have the form

$$ \begin{aligned} & \eta \left( \chi \right) = Ef^{\left( 3 \right)} \left( \chi \right) + \\ & \quad \overline{r}^{2} \int_{0}^{{\overline{\theta }}} {G^{\left( 5 \right)} \left( {\chi ;\chi^{\prime}} \right)\sin \chi^{\prime}\left( { - p\left( {r = \overline{r},\chi^{\prime}} \right) + \left( {1 - \varpi^{2} \eta_{*} } \right)\overline{r}\sigma_{k} \left( {\chi^{\prime}} \right)} \right)d\chi^{\prime}} \\ \end{aligned} $$

(C.19)

where $E$ is an unknown constant. Using (C.16) and (C.19), (2.49) and (2.50) give

$$ Ej^{\prime(3)} - \frac{1}{2}\overline{r}^{2} j^{\prime(4)} \int_{0}^{{\overline{\theta }}} {f^{\left( 3 \right)} \left( {\chi^{\prime}} \right)\sin \chi^{\prime}\left( { - p\left( {r = \overline{r},\chi^{\prime}} \right) + \left( {1 - \varpi^{2} \eta_{*} } \right)\overline{r}\sigma_{k} \left( {\chi^{\prime}} \right)} \right)d\chi^{\prime}} = \Omega $$

(C.20)

$$ Ej^{\left( 3 \right)} - \frac{1}{2}\overline{r}^{2} j^{\left( 4 \right)} \int_{0}^{{\overline{\theta }}} {f^{\left( 3 \right)} \left( {\chi^{\prime}} \right)\sin \chi^{\prime}\left( { - p\left( {r = \overline{r},\chi^{\prime}} \right) + \left( {1 - \varpi^{2} \eta_{*} } \right)\overline{r}\sigma_{k} \left( {\chi^{\prime}} \right)} \right)d\chi^{\prime}} = \eta_{c} $$

(C.21)

where $j^{\left( 3 \right)} = f^{\left( 3 \right)} \left( {\overline{\theta }} \right)$, $j^{\left( 4 \right)} = f^{\left( 4 \right)} \left( {\overline{\theta }} \right)$, $j^{\prime(3)} = \frac{{df^{\left( 3 \right)} }}{d\chi }\left( {\overline{\theta }} \right)$, $j^{\prime(4)} = \frac{{df^{\left( 4 \right)} }}{d\chi }\left( {\overline{\theta }} \right)$ and $\Omega$ is given by (A.42). Transforming to the variables $\tau$, $\tau^{\prime}$ and using (A.9), (C.19)-(C.21) give (A.39)-(A.41). Note that $G^{\left( 5 \right)} \left( {\chi ;\chi^{\prime}} \right)$ and $f^{\left( 3 \right)} \left( \chi \right)$ are there regarded as functions of $\tau$ and $\tau^{\prime}$. This is achieved using (C.12), (C.13), (C.16), (C.17) and

$$ \sin \chi = \frac{{\sin \overline{\theta }\sinh \tau }}{{\cosh \tau + \cos \overline{\theta }}},\quad \cos \chi = \frac{{\cos \overline{\theta }\cosh \tau + 1}}{{\cosh \tau + \cos \overline{\theta }}} $$

(C.22)

as well as the equivalent of (C.22) for $\chi^{\prime}$.

When $k = 0$, setting $\chi = 0$ in (C.19) and using (C.17) and $f^{\left( 3 \right)} \left( 0 \right) = 1$,

$$ \eta \left( 0 \right) = E - \frac{1}{2}\overline{r}^{2} \int_{0}^{{\overline{\theta }}} {f^{\left( 4 \right)} \left( {\chi^{\prime}} \right)\sin \chi^{\prime}\left( { - p\left( {r = \overline{r},\chi^{\prime}} \right) + \left( {1 - \varpi^{2} \eta_{*} } \right)\overline{r}\sigma_{k} \left( {\chi^{\prime}} \right)} \right)d\chi^{\prime}} $$

(C.23)

Transforming to the variable $\tau^{\prime}$ and using (A.9) gives (A.43), with $f^{\left( 4 \right)} \left( {\chi^{\prime}} \right)$ regarded as a function of $\tau^{\prime}$.

D. Expression of $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$

4.1 D.1 Definitions of $h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$, $H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ and $g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$

Introduce a Green’s function for the entire half-space $z \ge 0$ representing a ring source at $\tau = \tau^{\prime}$, $\xi = \xi^{\prime}$. Let $h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ ($\tau ,\;\tau^{\prime} > 0$, $0 \le \xi ,\;\xi^{\prime} \le \pi$) be such that

$$ \frac{\partial }{\partial \tau }\left( {\sinh \tau \frac{\partial h}{{\partial \tau }}} \right) + \sinh \tau \frac{{\partial^{2} h}}{{\partial \xi^{2} }} - \sinh \tau \left( {\frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)h = \delta \left( {\tau - \tau^{\prime}} \right)\delta \left( {\xi - \xi^{\prime}} \right) $$

(D.1)

$$ \frac{\partial h}{{\partial \xi }} = 0\quad \xi = 0,\;\pi $$

(D.2)

where $e^{\tau /2} h$ is bounded as $\tau \to 0$ and $\tau \to \infty$. Thus, $\left( {\cosh \tau + \cos \xi } \right)^{1/2} h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)\cos k\phi$ is a solution of the Laplace equation everywhere in $z \ge 0$, apart from the circle $\tau = \tau^{\prime}$, $\xi = \xi^{\prime}$. This solution satisfies the boundary conditions of zero normal derivative on $z = 0$ and decays to zero as $r \to \infty$ ($\tau \to 0$ and $\xi \to \pi$). $h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ is a smooth function of its arguments except when $\tau = \tau^{\prime}$, $\xi = \xi^{\prime}$.

Consider the behaviour of $h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ as $\tau \to \tau^{\prime}$ and $\xi \to \xi^{\prime}$ for given $\tau^{\prime}$ and $\xi^{\prime} \ne \pi$. When $\xi^{\prime} \ne 0$, $\tau$ is close to $\tau^{\prime}$ and $\xi$ is close to $\xi^{\prime}$, the terms involving derivatives dominate the left-hand side of (D.1) and we can approximate $\sinh \tau$ by $\sinh \tau^{\prime}$. Hence,

$$ \frac{{\partial^{2} h}}{{\partial \tau^{2} }} + \frac{{\partial^{2} h}}{{\partial \xi^{2} }}\sim \frac{{\delta \left( {\tau - \tau^{\prime}} \right)\delta \left( {\xi - \xi^{\prime}} \right)}}{{\sinh \tau^{\prime}}} $$

(D.3)

The solution of (D.3) follows from the usual two-dimensional Green’s function of the Laplace equation as

$$ h\sim \frac{1}{{2\pi \sinh \tau^{\prime}}}\ln \left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \left( {\xi - \xi^{\prime}} \right)^{2} } \right)^{1/2} $$

(D.4)

This shows the logarithmic singularity at $\tau = \tau^{\prime}$, $\xi = \xi^{\prime}$ which reflects the ring source at this location. If $\xi^{\prime} = 0$, the situation is a bit more complicated. This case can be treated by examining the limit in which $\tau$ is close to $\tau^{\prime}$ and both $\xi$ and $\xi^{\prime}$ are small. Thus, we have (D.3) and $\partial h/\partial \xi = 0$ at $\xi = 0$. The solution is

$$ h\sim \frac{1}{{2\pi \sinh \tau^{\prime}}}\left( {\ln \left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \left( {\xi - \xi^{\prime}} \right)^{2} } \right)^{1/2} + \ln \left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \left( {\xi + \xi^{\prime}} \right)^{2} } \right)^{1/2} } \right) $$

(D.5)

which shows an image source as the effect of the wall. Setting $\xi^{\prime} = 0$,

$$ h\left( {\tau ,\xi ;\tau^{\prime},0} \right)\sim \frac{1}{{\pi \sinh \tau^{\prime}}}\ln \left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \xi^{2} } \right)^{1/2} $$

(D.6)

as $\tau \to \tau^{\prime}$ and $\xi \to 0$. Note that (D.6) implies a logarithmic singularity of $h\left( {\tau ,0;\tau^{\prime},0} \right)$ at $\tau = \tau^{\prime}$.

Next define $H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ ($\tau ,\;\tau^{\prime} > 0$, $0 \le \xi \le \overline{\theta }$, $0 \le \xi^{\prime} < \overline{\theta }$) by

$$ \frac{\partial }{\partial \tau }\left( {\sinh \tau \frac{\partial H}{{\partial \tau }}} \right) + \sinh \tau \frac{{\partial^{2} H}}{{\partial \xi^{2} }} - \sinh \tau \left( {\frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)H = 0 $$

(D.7)

$$ \frac{\partial H}{{\partial \xi }} = 0\quad \xi = 0 $$

(D.8)

$$ \frac{\partial H}{{\partial \xi }} = - \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},\xi^{\prime}} \right)\quad \xi = \overline{\theta } $$

(D.9)

and bounded $e^{\tau /2} H$ as $\tau \to 0$ and $\tau \to \infty$. $\left( {\cosh \tau + \cos \xi } \right)^{1/2} H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)\cos k\phi$ is a solution of the Laplace equation inside the drop. Given the smoothness properties of $h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$, $H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ is a smooth function of its arguments.

Defining

$$ H_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right) = \left( { - 1} \right)^{n} \frac{2}{{\overline{\theta }}}\int_{0}^{{\overline{\theta }}} {H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}d\xi } $$

(D.10)

we have

$$ H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right) = \frac{1}{2}H_{0} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right) + \sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} H_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}} $$

(D.11)

Multiplying (D.7) by $\cos \frac{n\pi \xi }{{\overline{\theta }}}$ and integrating over $0 \le \xi \le \overline{\theta }$, integration by parts and (D.8), (D.9) give

$$ \frac{d}{d\tau }\left( {\sinh \tau \frac{{dH_{n} }}{d\tau }} \right) - \sinh \tau \left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} + \frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)H_{n} = \frac{2}{{\overline{\theta }}}\sinh \tau \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},\xi^{\prime}} \right) $$

(D.12)

If the right-hand side is known, (D.12) and bounded $e^{\tau /2} H_{n}$ as $\tau \to 0$ and $\tau \to \infty$ determine $H_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right)$.

Finally, define $g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right) = h\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right) + H\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$, where $\tau ,\;\tau^{\prime} > 0$, $0 \le \xi \le \overline{\theta }$, $0 \le \xi^{\prime} < \overline{\theta }$. (D.1), (D.2) and (D.7)-(D.9) give

$$ \frac{\partial }{\partial \tau }\left( {\sinh \tau \frac{\partial g}{{\partial \tau }}} \right) + \sinh \tau \frac{{\partial^{2} g}}{{\partial \xi^{2} }} - \sinh \tau \left( {\frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)g = \delta \left( {\tau - \tau^{\prime}} \right)\delta \left( {\xi - \xi^{\prime}} \right) $$

(D.13)

$$ \frac{\partial g}{{\partial \xi }} = 0\quad \xi = 0,\;\overline{\theta } $$

(D.14)

(D.13) implies that $\left( {\cosh \tau + \cos \xi } \right)^{1/2} g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)\cos k\phi$ satisfies the Laplace equation inside the drop, apart from the circle $\tau = \tau^{\prime}$, $\xi = \xi^{\prime}$, where it has a ring-source singularity. Elsewhere, $g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ is a smooth function of $\tau$, $\tau^{\prime}$, $\xi$ and $\xi^{\prime}$. Furthermore, given boundedness of $e^{\tau /2} h$ and $e^{\tau /2} H$, $e^{\tau /2} g$ is bounded as $\tau \to 0$ and $\tau \to \infty$.

4.2 D.2 Expressions for $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$

Let

$$ g_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right) = \left( { - 1} \right)^{n} \frac{2}{{\overline{\theta }}}\int_{0}^{{\overline{\theta }}} {g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}d\xi } $$

(D.15)

so that

$$ g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right) = \frac{1}{2}g_{0} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right) + \sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} g_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}} $$

(D.16)

Multiplying (D.13) by $\cos \frac{n\pi \xi }{{\overline{\theta }}}$ and integrating over $0 \le \xi \le \overline{\theta }$, integration by parts and use of (D.14) give

$$ \frac{\partial }{\partial \tau }\left( {\sinh \tau \frac{{\partial g_{n} }}{\partial \tau }} \right) - \sinh \tau \left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} + \frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)g_{n} = \left( { - 1} \right)^{n} \frac{2}{{\overline{\theta }}}\cos \frac{{n\pi \xi^{\prime}}}{{\overline{\theta }}}\delta \left( {\tau - \tau^{\prime}} \right) $$

(D.17)

Boundedness of $e^{\tau /2} g$ as $\tau \to 0$ and $\tau \to \infty$ means that the same is true of $g_{n}$. Comparison with the defining problem of $G_{n} \left( {\tau ;\tau^{\prime}} \right)$ implies

$$ g_{n} \left( {\tau ;\tau^{\prime},\xi^{\prime}} \right) = \left( { - 1} \right)^{n} \frac{2}{{\overline{\theta }}}G_{n} \left( {\tau ;\tau^{\prime}} \right)\cos \frac{{n\pi \xi^{\prime}}}{{\overline{\theta }}} $$

(D.18)

Using this result in (D.16),

$$ g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right) = \frac{2}{{\overline{\theta }}}\left( {\frac{1}{2}G_{0} \left( {\tau ;\tau^{\prime}} \right) + \sum\limits_{n = 1}^{\infty } {G_{n} \left( {\tau ;\tau^{\prime}} \right)\cos \frac{n\pi \xi }{{\overline{\theta }}}\cos \frac{{n\pi \xi^{\prime}}}{{\overline{\theta }}}} } \right) $$

(D.19)

which, given reciprocity of $G_{n} \left( {\tau ;\tau^{\prime}} \right)$, shows that $g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$ is unchanged by permutation of either $\tau$ and $\tau^{\prime}$ or $\xi$ and $\xi^{\prime}$. Using (A.17), (A.23), (A.24) and (D.19),

$$ G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right) = \frac{1}{2}\overline{\theta }\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} g\left( {\tau ,0;\tau^{\prime},0} \right) $$

(D.20)

$$ G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right) = \frac{1}{2}\overline{\theta }\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} g\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) $$

(D.21)

$$ G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right) = \frac{1}{2}\overline{\theta }\left( {\cosh \tau + 1} \right)^{1/2} \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} g\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) $$

(D.22)

Given the smoothness properties of $g\left( {\tau ,\xi ;\tau^{\prime},\xi^{\prime}} \right)$, as stated in the main text, $G^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ are smooth functions of $\tau$ and $\tau^{\prime}$ and $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ is also a smooth function of $\tau$ and $\tau^{\prime}$, apart from at $\tau = \tau^{\prime}$.

According to (D.20), (D.22) and the definition of $g$,

$$ \begin{aligned} G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right) = & \frac{1}{2}\overline{\theta }\left( {\cosh \tau + \cos \overline{\theta }} \right)^{1/2} \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} \\ & \quad \left( {h\left( {\tau ,0;\tau^{\prime},0} \right) + H\left( {\tau ,0;\tau^{\prime},0} \right)} \right) \\ \end{aligned} $$

(D.23)

$$ \begin{aligned} G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right) = & \frac{1}{2}\overline{\theta }\left( {\cosh \tau + 1} \right)^{1/2} \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} \\ & \quad \left( {h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right) \\ \end{aligned} $$

(D.24)

The quantity $h\left( {\tau ,0;\tau^{\prime},0} \right)$ is smooth, apart from a logarithmic singularity at $\tau = \tau^{\prime}$, while $h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)$, $H\left( {\tau ,0;\tau^{\prime},0} \right)$ and $H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)$ are everywhere smooth. Thus, (D.23) expresses $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ as the sum of singular and nonsingular components, whereas (D.24) reflects smoothness of $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$. (A.26) and (D.24) give

$$ \begin{aligned} G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right) = & \frac{1}{2}\overline{\theta }\left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)^{1/2} \\ & \quad \frac{\partial }{\partial \tau }\left( {\left( {\cosh \tau + 1} \right)^{1/2} \left( {h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right)} \right) \\ \end{aligned} $$

(D.25)

Equations (D.23)-(D.25) are used to calculate $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$, but this requires $h\left( {\tau ,\xi ;\tau^{\prime},0} \right)$, which is the subject of the final two sections of this appendix.

4.3 D.3 Determination of $h\left( {\tau ,\xi ;\tau^{\prime},0} \right)$

For given $\tau^{\prime} > 0$, define a second toroidal coordinate system, $\tau_{*} > 0$, $0 \le \xi_{*} \le \pi$, $\phi$ by

$$ \frac{{\sinh \tau_{*} }}{{\cosh \tau_{*} + \cos \xi_{*} }} = \frac{{\left( {\cosh \tau^{\prime} + 1} \right)\sinh \tau }}{{\sinh \tau^{\prime}\left( {\cosh \tau + \cos \xi } \right)}} $$

(D.26)

$$ \frac{{\sin \xi_{*} }}{{\cosh \tau_{*} + \cos \xi_{*} }} = \frac{{\left( {\cosh \tau^{\prime} + 1} \right)\sin \xi }}{{\sinh \tau^{\prime}\left( {\cosh \tau + \cos \xi } \right)}} $$

(D.27)

This system has $\tau_{*} = 0$ on the symmetry axis, $\tau = 0$, while $\tau_{*} = \infty$ corresponds to $\tau = \tau^{\prime}$, $\xi = 0$. $\xi_{*} = 0,\;\pi$ represent the wall, $z = 0$. It can be shown that

$$ \tanh \tau_{*} = \frac{{\sinh \tau \sinh \tau^{\prime}}}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }} $$

(D.28)

$$ \cosh \tau_{*} + \cos \xi_{*} = \frac{{\left( {\cosh \tau^{\prime} - 1} \right)\left( {\cosh \tau + \cos \xi } \right)}}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }}\cosh \tau_{*} $$

(D.29)

The right-hand side of (D.28) is positive and, unless $\tau = \tau^{\prime}$ and $\xi = 0$, is less than $1$, hence (D.28) can be solved for $\tau_{*}$ as a function of $\tau$, $\tau^{\prime}$ and $\xi$, while (D.29) provides $\cosh \tau_{*} + \cos \xi_{*}$.

(D.1) expresses the fact that $\cos k\phi \left( {\cosh \tau + \cos \xi } \right)^{1/2} h\left( {\tau ,\xi ;\tau^{\prime},0} \right)$ is a solution of the Laplace equation away from $\tau = \tau^{\prime}$, $\xi = 0$, implying that

$$ h_{*} \left( {\tau_{*} ,\xi_{*} } \right) = \left( {\frac{\cosh \tau + \cos \xi }{{\cosh \tau_{*} + \cos \xi_{*} }}} \right)^{1/2} h\left( {\tau ,\xi ;\tau^{\prime},0} \right) $$

(D.30)

satisfies

$$ \frac{1}{{\sinh \tau_{*} }}\frac{\partial }{{\partial \tau_{*} }}\left( {\sinh \tau_{*} \frac{{\partial h_{*} }}{{\partial \tau_{*} }}} \right) + \frac{{\partial^{2} h_{*} }}{{\partial \xi_{*}^{2} }} - \left( {\frac{{k^{2} }}{{\sinh^{2} \tau_{*} }} - \frac{1}{4}} \right)h_{*} = 0 $$

(D.31)

We want to show that $h_{*} = Cf_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)$ for a suitable choice of the constant $C$. Firstly, it satisfies (D.31), so, using (D.29) and (D.30),

$$ h\left( {\tau ,\xi ;\tau^{\prime},0} \right) = C\left( {\frac{{\left( {\cosh \tau^{\prime} - 1} \right)\cosh \tau_{*} }}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }}} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right) $$

(D.32)

respects (D.1) with $\xi^{\prime} = 0$, apart from at $\tau = \tau^{\prime}$, $\xi = 0$ ($\tau_{*} = \infty$). It follows from (D.28) that $\partial \tau_{*} /\partial \xi = 0$ when $\xi = 0,\;\pi$, thus (D.32) satisfies (D.2). $\tau_{*}$, and hence $f_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)\cosh^{1/2} \tau_{*}$, is finite unless $\tau = \tau^{\prime}$, $\xi = 0$. In particular, (D.32) is bounded as $\tau \to 0$, as it should be. As $\tau \to \infty$, $\tau_{*} \to \tau^{\prime}$ and (D.32) gives

$$ h\left( {\tau ,\xi ;\tau^{\prime},0} \right)\sim Ce^{ - \tau /2} \left( {2\left( {\cosh \tau^{\prime} - 1} \right)} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right) $$

(D.33)

Thus, $e^{\tau /2} h$ is bounded for $\tau \to \infty$, again as it should be.

It remains to consider the behaviour as $\tau \to \tau^{\prime}$, $\xi \to 0$, corresponding to $\tau_{*} \to \infty$. In that limit, $f_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)\sim \tau_{*} e^{{ - \tau_{*} /2}}$, hence (D.32) implies

$$ h\left( {\tau ,\xi ;\tau^{\prime},0} \right)\sim C\left( {2\left( {\cosh \tau^{\prime} + 1} \right)} \right)^{ - 1/2} \tau_{*} $$

(D.34)

(D.28) gives

$$ 1 - \tanh \tau_{*} = \frac{{\cosh \left( {\tau - \tau^{\prime}} \right) - \cos \xi }}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }} $$

(D.35)

which yields

$$ 2e^{{ - 2\tau_{*} }} \sim \frac{1}{{2\sinh^{2} \tau^{\prime}}}\left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \xi^{2} } \right) $$

(D.36)

when $\tau \to \tau^{\prime}$, $\xi \to 0$, hence

$$ \tau_{*} \sim - \ln \left( {\left( {\tau - \tau^{\prime}} \right)^{2} + \xi^{2} } \right)^{1/2} $$

(D.37)

in the given limit. Agreement of (D.34) with (D.6) requires

$$ C = - \frac{1}{\pi }\left( {\frac{2}{{\cosh \tau^{\prime} - 1}}} \right)^{1/2} $$

(D.38)

Thus, (D.32) gives

$$ h\left( {\tau ,\xi ;\tau^{\prime},0} \right) = - \frac{1}{\pi }\left( {\frac{{2\cosh \tau_{*} }}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }}} \right)^{1/2} f_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right) $$

(D.39)

as the final expression for $h\left( {\tau ,\xi ;\tau^{\prime},0} \right)$, where $\tau_{*}$ is the solution of (D.28).

4.4 D.4 An alternative expression for $h\left( {\tau ,0;\tau^{\prime},0} \right)$

Although (D.39) can be used to calculate $h\left( {\tau ,0;\tau^{\prime},0} \right)$ for any $\tau \ne \tau^{\prime}$, its logarithmic singularity at $\tau = \tau^{\prime}$ can be numerically troublesome. $f_{0}^{\left( 1 \right)} \left( \tau \right)$ satisfies (A.20) with $n = 0$, i.e.

$$ \frac{{d^{2} f_{0}^{\left( 1 \right)} }}{{d\tau^{2} }} + \coth \tau \frac{{df_{0}^{\left( 1 \right)} }}{d\tau } - \left( {\frac{{k^{2} }}{{\sinh^{2} \tau }} - \frac{1}{4}} \right)f_{0}^{\left( 1 \right)} = 0 $$

(D.40)

Using the transformation $X = e^{ - \tau }$, $f_{0}^{\left( 1 \right)} \left( X \right)$ satisfies a differential equation having regular singular points at $X = 0$ and $X = \pm 1$. Focusing on $X = 0$, the usual Frobenius approach leads to solutions of (D.40) of the form

$$ f_{0}^{\left( 1 \right)} \left( \tau \right) = e^{ - \tau /2} \sum\limits_{l = 0}^{\infty } {\left( {c_{l} \tau + d_{l} } \right)e^{ - 2l\tau } } $$

(D.41)

General theory says that the power series in $X$ which result in (D.41) are convergent for $\left| X \right| < 1$ and so (D.41) converges for all $\tau > 0$. We have

$$ \coth \tau = 1 + 2\sum\limits_{m = 1}^{\infty } {e^{ - 2m\tau } } \quad \frac{1}{{\sinh^{2} \tau }} = 4\sum\limits_{m = 1}^{\infty } {me^{ - 2m\tau } } $$

(D.42)

hence (D.40) gives

$$ \begin{aligned} & \frac{{d^{2} f_{0}^{{\left( 1 \right)}} }}{{d\tau ^{2} }} + \frac{{df_{0}^{{\left( 1 \right)}} }}{{d\tau }} + \frac{1}{4}f_{0}^{{\left( 1 \right)}} \\ & = 2\sum\limits_{{m = 1}}^{\infty } {\left( {2mk^{2} f_{0}^{{\left( 1 \right)}} - \frac{{df_{0}^{{\left( 1 \right)}} }}{{d\tau }}} \right)e^{{ - 2m\tau }} } \\ \end{aligned} $$

(D.43)

Introducing (D.41) in (D.43),

$$ c_{l} = l^{ - 2} \sum\limits_{m = 1}^{l} {\left( {m\left( {k^{2} - 1} \right) + l + \frac{1}{4}} \right)c_{l - m} } $$

(D.44)

$$ d_{l} = l^{ - 1} c_{l} + l^{ - 2} \sum\limits_{m = 1}^{l} {\left( {\left( {m\left( {k^{2} - 1} \right) + l + \frac{1}{4}} \right)d_{l - m} - \frac{1}{2}c_{l - m} } \right)} $$

(D.45)

for $l \ge 1$. Given $f_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau e^{ - \tau /2}$ as $\tau \to \infty$, $c_{0} = 1$ and (D.44) allows calculation of the remaining $c_{l}$. (D.44), (D.45) and $c_{0} = 1$ imply

$$ d_{l} = d_{0} c_{l} + e_{l} $$

(D.46)

where $e_{0} = 0$ and

$$ e_{l} = l^{ - 1} c_{l} + l^{ - 2} \sum\limits_{m = 1}^{l} {\left( {\left( {m\left( {k^{2} - 1} \right) + l + \frac{1}{4}} \right)e_{l - m} - \frac{1}{2}c_{l - m} } \right)} $$

(D.47)

determine $e_{l}$. (D.41) and (D.46) give

$$ f_{0}^{\left( 1 \right)} \left( \tau \right) = e^{ - \tau /2} \left( {\left( {\tau + d_{0} } \right)\sum\limits_{l = 0}^{\infty } {c_{l} e^{ - 2l\tau } } + \sum\limits_{l = 0}^{\infty } {e_{l} e^{ - 2l\tau } } } \right) $$

(D.48)

If $f_{0}^{\left( 1 \right)} \left( \tau \right)$ is known at any single value of $\tau$, (D.48) allows the calculation of $d_{0}$.

Using (D.48), (D.39) yields

$$ h\left( {\tau ,0;\tau^{\prime},0} \right) = - \frac{1}{\pi }\left( {\frac{{1 + e^{{ - 2\tau_{*} }} }}{{\cosh \tau \cosh \tau^{\prime} - 1}}} \right)^{1/2} \left( {\left( {\tau_{*} + d_{0} } \right)\sum\limits_{l = 0}^{\infty } {c_{l} e^{{ - 2l\tau_{*} }} } + \sum\limits_{l = 0}^{\infty } {e_{l} e^{{ - 2l\tau_{*} }} } } \right) $$

(D.49)

According to (D.35) with $\xi = 0$,

$$ e^{{ - 2\tau_{*} }} = \frac{{F\left( {\tau ,\tau^{\prime}} \right)}}{{2 - F\left( {\tau ,\tau^{\prime}} \right)}} $$

(D.50)

where

$$ F\left( {\tau ,\tau^{\prime}} \right) = \frac{{\cosh \left( {\tau - \tau^{\prime}} \right) - 1}}{{\cosh \tau \cosh \tau^{\prime} - 1}} $$

(D.51)

is a smooth function of $\tau$ and $\tau^{\prime}$ which lies in the range $0 \le F \le 1$. Thus, $e^{{ - 2\tau_{*} }}$ is also a smooth function of $\tau$ and $\tau^{\prime}$, including when $\tau = \tau^{\prime}$. It follows that the only term in (D.49) which is singular for $\tau = \tau^{\prime}$ is $\tau_{*}$. (D.50) gives

$$ \tau_{*} = - \ln \left| {\mu - \mu^{\prime}} \right| - \frac{1}{2}\ln \left( {\frac{{F\left( {\tau ,\tau^{\prime}} \right)}}{{\left( {\mu - \mu^{\prime}} \right)^{2} \left( {2 - F\left( {\tau ,\tau^{\prime}} \right)} \right)}}} \right) $$

(D.52)

in which $\mu = \ln \left( {\sinh \tau } \right)$, $\mu^{\prime} = \ln \left( {\sinh \tau^{\prime}} \right)$ and the term containing $F$ is smooth. Using (D.52) in (D.49),

$$ h\left( {\tau ,0;\tau^{\prime},0} \right) = h^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)\ln \left| {\mu - \mu^{\prime}} \right| + h^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right) $$

(D.53)

where

$$ h^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right) = \frac{1}{\pi }\left( {\frac{{1 + e^{{ - 2\tau_{*} }} }}{{\cosh \tau \cosh \tau^{\prime} - 1}}} \right)^{1/2} \sum\limits_{l = 0}^{\infty } {c_{l} e^{{ - 2l\tau_{*} }} } $$

(D.54)

$$ \begin{aligned} h^{\left( 2 \right)} \left( {\tau ;\tau^{\prime}} \right) = & \frac{1}{\pi }\left( {\frac{{1 + e^{{ - 2\tau_{*} }} }}{{\cosh \tau \cosh \tau^{\prime} - 1}}} \right)^{1/2} \\ & \quad \left( {\left( {\frac{1}{2}\ln \left( {\frac{{F\left( {\tau ,\tau^{\prime}} \right)}}{{\left( {\mu - \mu^{\prime}} \right)^{2} \left( {2 - F\left( {\tau ,\tau^{\prime}} \right)} \right)}}} \right) - d_{0} } \right)\sum\limits_{l = 0}^{\infty } {c_{l} e^{{ - 2l\tau_{*} }} } - \sum\limits_{l = 0}^{\infty } {e_{l} e^{{ - 2l\tau_{*} }} } } \right) \\ \end{aligned} $$

(D.55)

are smooth functions. In the limit $\tau \to \tau^{\prime}$, $\tau_{*} \to \infty$ and $c_{0} = 1$, $e_{0} = 0$, (D.51), (D.54), (D.55) and $\mu - \mu^{\prime}\sim \left( {\tau - \tau^{\prime}} \right)\coth \tau$ yield

$$ h^{\left( 1 \right)} \left( {\tau ;\tau } \right) = \frac{1}{\pi \sinh \tau } $$

(D.56)

$$ h^{\left( 2 \right)} \left( {\tau ;\tau } \right) = - \frac{1}{\pi \sinh \tau }\left( {\ln \left( {2\cosh \tau } \right) + d_{0} } \right) $$

(D.57)

(D.53) has the advantage over (D.39) that it makes the logarithmic singularity explicit. On the other hand, the infinite series in (D.54) and (D.55) are slowly convergent at small $\tau_{*}$. Mathematically, either (D.39) or (D.53) can be used to express $h\left( {\tau ,0;\tau^{\prime},0} \right)$. According to the discussion of integrals involving $G^{\left( 1 \right)}$ in appendix B, the integration range is divided into intervals which are of two types. (D.39) is used for the first type and (D.53) for the second. (D.53) leads to (B.5). When they are used numerically, the infinite sums in (D.48), (D.54) and (D.55) are treated by truncation.

E. Numerical calculations

5.1 E.1 Calculation of $f_{n}^{\left( 1 \right)} \left( \tau \right)$

Let us suppress the exponential behaviour apparent in (C.1) and (C.2) by defining

$$ \tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right) = \exp \left( {\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\tau } \right)f_{n}^{\left( 1 \right)} \left( \tau \right) $$

(E.1)

As $\tau \to \infty$, it follows from (C.1) and (C.2) that $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau$ and $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right) \to 1$ for $n \ne 0$. It can be shown that the large-$\tau$ expansion of $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)$ continues as $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau + D$, where $D$ is a constant.

Recall that $f_{n}^{\left( 1 \right)}$ satisfies (A.20). Thus,

$$ \begin{gathered} \frac{{d^{2} \tilde{f}_{n}^{\left( 1 \right)} }}{{d\tau^{2} }} + \left( {\coth \tau - 1 + \frac{2n\pi }{{\overline{\theta }}}} \right)\frac{{d\tilde{f}_{n}^{\left( 1 \right)} }}{d\tau } + \\ \left( {\left( {\frac{n\pi }{{\overline{\theta }}} - \frac{1}{2}} \right)\left( {\coth \tau - 1} \right) - \frac{{k^{2} }}{{\sinh^{2} \tau }}} \right)\tilde{f}_{n}^{\left( 1 \right)} = 0 \\ \end{gathered} $$

(E.2)

The idea behind the numerical calculation of $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$ is to integrate (E.2) upwards in $\tau$, starting at small enough $\tau$. Small-$\tau$ analysis of (A.20) gives

$$ f_{n}^{\left( 1 \right)} \left( \tau \right)\sim C_{n} \left( {1 + \frac{1}{4}\left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} - \frac{1}{4}} \right)\tau^{2} + O\left( {\tau^{4} } \right)} \right) $$

(E.3)

for $k = 0$ and

$$ f_{n}^{\left( 1 \right)} \left( \tau \right)\sim C_{n} \left( {\tau + \frac{1}{8}\left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} - \frac{11}{{12}}} \right)\tau^{3} + O\left( {\tau^{5} } \right)} \right) $$

(E.4)

when $k = 1$, where $C_{n}$ are constants determined later. Accuracy of these results for large $n$ requires $\tau < < \overline{\theta }/n\pi$. This can be achieved by taking $\tau < < \overline{\theta }/N\pi$, where $N$ is the largest value of $n$ which is treated. The starting values for integration are obtained using (E.1) and the above expansions, but without the factors $C_{n}$. Hence, the approximations

$$ \tilde{f}_{n}^{\left( 1 \right)} = 1 + \left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\left( {\tau + \frac{1}{4}\left( {\frac{1}{2} - \frac{3n\pi }{{\overline{\theta }}}} \right)\tau^{2} - \frac{1}{12}\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\left( {\frac{1}{2} + \frac{5n\pi }{{\overline{\theta }}}} \right)\tau^{3} } \right) $$

(E.5)

$$ \frac{{d\tilde{f}_{n}^{\left( 1 \right)} }}{d\tau } = \left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\left( {1 + \frac{1}{2}\left( {\frac{1}{2} - \frac{3n\pi }{{\overline{\theta }}}} \right)\tau - \frac{1}{4}\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\left( {\frac{1}{2} + \frac{5n\pi }{{\overline{\theta }}}} \right)\tau^{2} } \right) $$

(E.6)

for $k = 0$ and

$$ \tilde{f}_{n}^{\left( 1 \right)} = \tau + \left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\tau^{2} + \frac{1}{2}\left( {\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)^{2} + \frac{1}{4}\left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} - \frac{11}{{12}}} \right)} \right)\tau^{3} $$

(E.7)

$$ \frac{{d\tilde{f}_{n}^{\left( 1 \right)} }}{d\tau } = 1 + 2\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)\tau + \frac{3}{2}\left( {\left( {\frac{1}{2} - \frac{n\pi }{{\overline{\theta }}}} \right)^{2} + \frac{1}{4}\left( {\frac{{n^{2} \pi^{2} }}{{\overline{\theta }^{2} }} - \frac{11}{{12}}} \right)} \right)\tau^{2} $$

(E.8)

when $k = 1$ are used. Of course, the resulting solution of (E.2) will not be normalised as it should be. This will be remedied later by multiplication by $C_{n}$.

The difficulty is that (E.2) is stiff when $n\pi /\overline{\theta }$ is large. It is therefore important to choose the numerical integration scheme appropriately, otherwise numerical instability may occur. We used the Radau IIA scheme of order 5. The total range of integration is $\tau_{{ - \hat{M}}} \le \tau \le \tau_{{\tilde{M}}}$, where $\hat{M}$ and $\tilde{M}$ are as in appendix B. This range is split into intervals $\tau_{m} \le \tau \le \tau_{m + 1}$ ($- \hat{M} \le m < \tilde{M}$), where $\ln \left( {\sinh \tau_{m} } \right) = m\Delta$ as in appendix B. Each interval is divided into four subintervals of length $\left( {\tau_{m + 1} - \tau_{m} } \right)/4$. Starting at $\tau = \tau_{{ - \hat{M}}}$ with (E.5), (E.6) or (E.7), (E.8), Radau integration of (E.2) is applied to each subinterval in turn. This gives $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$ and its derivative at the $4\left( {\hat{M} + \tilde{M}} \right)$ points which are the upper boundaries of subintervals. As $\tau \to \infty$, (C.1), (C.2) and (E.1) imply that $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$ should respect the normalisation conditions $d\tilde{f}_{0}^{\left( 1 \right)} /d\tau \to 1$ and $\tilde{f}_{n}^{\left( 1 \right)} \to 1$ for $n \ne 0$. As noted above, the numerically determined values require normalisation, which is achieved by multiplying the unnormalised $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$ and its derivative by $C_{n}$, where $C_{0}^{ - 1} = \frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau_{{\tilde{M}}} } \right)$ and $C_{n}^{ - 1} = \tilde{f}_{n}^{\left( 1 \right)} \left( {\tau_{{\tilde{M}}} } \right)$ ($n \ne 0$). Henceforth, $\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$ and its derivative have these normalised values. Because $f_{0}^{\left( 1 \right)} \left( {\tau_{0} } \right)$ is now known from (E.1), the value of the coefficient $d_{0}$ in (D.55) and (D.57) can be calculated using (D.48) with $\tau = \tau_{0}$. Note that, for $k = 0$, $f_{0}^{\left( 1 \right)} \left( 0 \right)$, which is needed in (A.36), has the value $C_{0}$.

We will later require values of $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)$ and its derivative for $\tau$ other than those for which they have been determined by the above procedure. If $\tau \le \tau_{{ - \hat{M}}}$, the small-$\tau$ approximations (E.5)-(E.8) are normalised using multiplication by $C_{0}$. When $\tau \ge \tau_{{\tilde{M}}}$, the large-$\tau$ approximations $\tilde{f}_{0}^{\left( 1 \right)} = \tau + \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{{\tilde{M}}} } \right) - \tau_{{\tilde{M}}}$, $d\tilde{f}_{0}^{\left( 1 \right)} /d\tau = 1$ are applied. Finally, if $\tau_{{ - \hat{M}}} < \tau < \tau_{{\tilde{M}}}$, one-step Radau integration of (E.2) with $n = 0$ is performed, starting with the values of $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)$ and its derivative at the point nearest $\tau$ at which they are known.

5.2 E.2 Determination of $H_{n} \left( {\tau ;\tau^{\prime},0} \right)$

(D.12) with $\xi^{\prime} = 0$ and bounded $e^{\tau /2} H_{n}$ as $\tau \to 0$ and $\tau \to \infty$ determine $H_{n} \left( {\tau ;\tau^{\prime},0} \right)$. This requires $\partial h/\partial \xi \left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)$. (D.28) gives

$$ \frac{{\partial \tau_{*} }}{\partial \xi } = - \frac{{\sin \xi \sinh \tau_{*} \cosh \tau_{*} }}{{\cosh \tau \cosh \tau^{\prime} - \cos \xi }} $$

(E.9)

hence

$$ \begin{aligned} \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = & \frac{{\sin \overline{\theta }\left( {1 + e^{{ - 2\tau_{*} }} } \right)^{1/2} \cosh \tau_{*} }}{{\pi \left( {\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{3/2} }} \\ & \quad \left( {\sinh \tau_{*} \frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau_{*} } \right) + \frac{1}{2}e^{{ - \tau_{*} }} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)} \right) \\ \end{aligned} $$

(E.10)

according to (D.39) and (E.1), where $\tau_{*}$ is the solution of (D.28) with $\xi = \overline{\theta }$, i.e.

$$ \tanh \tau_{*} = \frac{{\sinh \tau \sinh \tau^{\prime}}}{{\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }}} $$

(E.11)

For given $\tau^{\prime}$, define

$$ I_{n} \left( {\tau ;\tau^{\prime}} \right) = \exp \left( { - \frac{n\pi }{{\overline{\theta }}}\tau } \right)\sinh \tau \left( {\frac{{\partial H_{n} }}{\partial \tau }\left( {\tau ;\tau^{\prime},0} \right)f_{n}^{\left( 1 \right)} \left( \tau \right) - H_{n} \left( {\tau ;\tau^{\prime},0} \right)\frac{{df_{n}^{\left( 1 \right)} }}{d\tau }} \right) $$

(E.12)

(A.20) and (D.12) imply

$$ \frac{d}{d\tau }\left( {\exp \left( {\frac{n\pi }{{\overline{\theta }}}\tau } \right)I_{n} } \right) = \frac{2}{{\overline{\theta }}}\sinh \tau \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)f_{n}^{\left( 1 \right)} \left( \tau \right) $$

(E.13)

Integrating (E.13) and using (E.1) and $I_{n} \left( {0;\tau^{\prime}} \right) = 0$,

$$ \begin{aligned} I_{n} \left( {\tau ;\tau^{\prime}} \right) = & \frac{2}{{\overline{\theta }}}\exp \left( { - \frac{n\pi }{{\overline{\theta }}}\tau } \right) \\ & \quad \int_{0}^{\tau } {\sinh \tau^{\prime\prime}\frac{\partial h}{{\partial \xi }}\left( {\tau^{\prime\prime},\overline{\theta };\tau^{\prime},0} \right)\tilde{f}_{n}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right)\exp \left( {\left( {\frac{n\pi }{{\overline{\theta }}} - \frac{1}{2}} \right)\tau^{\prime\prime}} \right)d\tau^{\prime\prime}} \\ \end{aligned} $$

(E.14)

Let $\tau_{m + 1/2} = \left( {\tau_{m} + \tau_{m + 1} } \right)/2$, $- \hat{M} \le m < \tilde{M}$. The numerical evaluation of $I_{n} \left( {\tau ;\tau^{\prime}} \right)$ for $\tau = \tau_{l/2}$, $- 2\hat{M} \le l \le 2\tilde{M}$ is carried out as follows.

As $\tau \to 0$, $\tau_{*} = \tau \sinh \tau^{\prime}/\left( {\cosh \tau^{\prime} - \cos \overline{\theta }} \right) + O\left( {\tau^{3} } \right)$ according to (E.11). Using this result, it can be shown that

$$ \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = \frac{{C_{0} \sin \overline{\theta }}}{{2^{1/2} \pi \left( {\cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{3/2} }} + O\left( {\tau^{2} } \right) $$

(E.15)

for $k = 0$ and

$$ \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = \frac{{3C_{0} \sinh \tau^{\prime}\sin \overline{\theta }}}{{2^{1/2} \pi \left( {\cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{5/2} }}\tau + O\left( {\tau^{3} } \right) $$

(E.16)

for $k = 1$. Employing (E.1), $f_{n}^{\left( 1 \right)} \left( \tau \right) = C_{n} + O\left( {\tau^{2} } \right)$ for $k = 0$ and $f_{n}^{\left( 1 \right)} \left( \tau \right) = C_{n} \tau + O\left( {\tau^{3} } \right)$ for $k = 1$, (E.14) gives

$$ {I_n}\left( {\tau ;{\tau ^\prime }} \right) = \frac{{{C_0}{C_n}\sin \bar \theta }}{{{2^{1/2}}\pi \bar \theta {{\left( {\cosh {\tau ^\prime } - \cos \bar \theta } \right)}^{3/2}}}}{\tau ^2}\left( {1 - \frac{{n\pi }}{{\bar \theta }}\tau } \right) + O\left( {{\tau ^4}} \right) $$

(E.17)

when $k = 0$ and

$$ I_{n} \left( {\tau ;\tau^{\prime}} \right) = \frac{{3C_{0} C_{n} \sinh \tau^{\prime}\sin \overline{\theta }}}{{2^{3/2} \pi \overline{\theta }\left( {\cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{5/2} }}\tau^{4} \left( {1 - \frac{n\pi }{{\overline{\theta }}}\tau } \right) + O\left( {\tau^{6} } \right) $$

(E.18)

when $k = 1$. (E.17) for $k = 0$ and (E.18) for $k = 1$ are used to approximate $I_{n} \left( {\tau_{{ - \hat{M}}} ;\tau^{\prime}} \right)$.

(E.14) implies

$$ \begin{aligned} I_{n} \left( {\tau_{l/2} ;\tau^{\prime}} \right) = & \exp \left( { - \frac{n\pi }{{\overline{\theta }}}\left( {\tau_{l/2} - \tau_{{\left( {l - 1} \right)/2}} } \right)} \right)I_{n} \left( {\tau_{{\left( {l - 1} \right)/2}} ;\tau^{\prime}} \right) + \\ & \quad \frac{2}{{\overline{\theta }}}\int_{{\tau_{{\left( {l - 1} \right)/2}} }}^{{\tau_{l/2} }} {e^{{ - \tau^{\prime\prime}/2}} \sinh \tau^{\prime\prime}\frac{\partial h}{{\partial \xi }}\left( {\tau^{\prime\prime},\overline{\theta };\tau^{\prime},0} \right)\tilde{f}_{n}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right)\exp \left( { - \frac{n\pi }{{\overline{\theta }}}\left( {\tau_{l/2} - \tau^{\prime\prime}} \right)} \right)d\tau^{\prime\prime}} \\ \end{aligned} $$

(E.19)

in which the integral is treated as follows. Quadratic approximation of

$$ e^{{ - \tau^{\prime\prime}/2}} \sinh \tau^{\prime\prime}\frac{\partial h}{{\partial \xi }}\left( {\tau^{\prime\prime},\overline{\theta };\tau^{\prime},0} \right)\tilde{f}_{n}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right) $$

(E.20)

is carried out for the interval $\tau_{{\left( {l - 1} \right)/2}} \le \tau^{\prime\prime} \le \tau_{l/2}$ using its values at the mid- and end-points. The values of $\tilde{f}_{n}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right)$ are known from earlier calculation. Those of $\partial h/\partial \xi$ follow from (E.10), but this requires $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)$ and its derivative at $\tau = \tau_{*}$, where $\tau_{*}$ is the solution of (E.11) with $\tau$ replaced by $\tau^{\prime\prime}$. In general, $\tau_{*}$ is not one of the points for which $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)$ and its derivative are as yet known so the procedure described earlier is used. Once the coefficients of the quadratic approximation of (E.20) are determined, the integral in (E.19) is calculated analytically. Starting with the value of $I_{n} \left( {\tau_{{ - \hat{M}}} ;\tau^{\prime}} \right)$ from (E.17) or (E.18), (E.19) determines $I_{n} \left( {\tau ;\tau^{\prime}} \right)$ for $\tau = \tau_{l/2}$, $- 2\hat{M} < l \le 2\tilde{M}$.

(E.12) implies

$$ \frac{d}{d\tau }\left( {\exp \left( { - \frac{n\pi }{{\overline{\theta }}}\tau } \right)K_{n} } \right) = \exp \left( {\frac{n\pi }{{\overline{\theta }}}\tau } \right)\frac{{I_{n} \left( {\tau ;\tau^{\prime}} \right)}}{{f_{n}^{\left( 1 \right)2} \left( \tau \right)\sinh \tau }} $$

(E.21)

where $K_{n} \left( {\tau ;\tau^{\prime}} \right) = e^{\tau /2} H_{n} \left( {\tau ;\tau^{\prime},0} \right)/\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)$. Given (C.1), (C.2) and boundedness of $e^{\tau /2} H_{n}$ as $\tau \to \infty$, $e^{{ - n\pi \tau /\overline{\theta }}} K_{n} \to 0$ in that limit. Thus, (E.1) and integration of (E.21) gives

$$ K_{n} \left( {\tau ;\tau^{\prime}} \right) = - \exp \left( {\frac{n\pi }{{\overline{\theta }}}\tau } \right)\int_{\tau }^{\infty } {\frac{{e^{{\tau^{\prime\prime}}} I_{n} \left( {\tau^{\prime\prime};\tau^{\prime}} \right)}}{{\tilde{f}_{n}^{\left( 1 \right)2} \left( {\tau^{\prime\prime}} \right)\sinh \tau^{\prime\prime}}}\exp \left( { - \frac{n\pi }{{\overline{\theta }}}\tau^{\prime\prime}} \right)d\tau^{\prime\prime}} . $$

(E.22)

which is used to calculate $K_{n} \left( {\tau ;\tau^{\prime}} \right)$ numerically at $\tau = \tau_{m}$, $- \hat{M} \le m \le \tilde{M}$, as follows. We first treat $K_{n} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right)$.

Let us consider the case of large, positive $\tau - \tau^{\prime}$. (E.11) implies $\tau_{*} \sim \tau^{\prime}$ and (E.10) gives

$$ \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)\sim \frac{{4\sin \overline{\theta }}}{\pi }e^{ - 3\tau /2} e^{{ - \tau^{\prime}/2}} \left( {\sinh \tau^{\prime}\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) + \frac{1}{2}e^{{ - \tau^{\prime}}} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)} \right) $$

(E.23)

Furthermore, since $\tau$ is large, $\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)\sim \tau + D$, where the constant $D$ can be approximated by $\tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{{\tilde{M}}} } \right) - \tau_{{\tilde{M}}}$. Using these results in (E.14) with $n = 0$,

$$ \begin{aligned} & I_{0} \left( {\tau ;\tau^{\prime}} \right)\sim I_{0} \left( {\infty ;\tau^{\prime}} \right) - \\ & \quad \frac{{4\sin \overline{\theta }}}{{\pi \overline{\theta }}}\left( {\tau + D + 1} \right)e^{ - \tau } e^{{ - \tau^{\prime}/2}} \left( {\sinh \tau^{\prime}\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) + \frac{1}{2}e^{{ - \tau^{\prime}}} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)} \right) \\ \end{aligned} $$

(E.24)

where $I_{0} \left( {\infty ;\tau^{\prime}} \right)$ is the $\tau \to \infty$ limit of $I_{0} \left( {\tau ;\tau^{\prime}} \right)$ and is approximated using (E.24) with $\tau = \tau_{{\tilde{M}}}$, i.e.

$$ \begin{aligned} I_{0} \left( {\infty ;\tau^{\prime}} \right) = & I_{0} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right) + \\ & \quad \frac{{4\sin \overline{\theta }}}{{\pi \overline{\theta }}}\left( {\tau_{{\tilde{M}}} + D + 1} \right)e^{{ - \tau_{{\tilde{M}}} }} e^{{ - \tau^{\prime}/2}} \left( {\sinh \tau^{\prime}\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) + \frac{1}{2}e^{{ - \tau^{\prime}}} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)} \right) \\ \end{aligned} $$

(E.25)

Using (E.24), (E.25) and $\tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right)\sim \tau^{\prime\prime} + D$, (E.22) applied at $\tau = \tau_{{\tilde{M}}}$ gives

$$ \begin{aligned} & K_{0} \left( {\tau_{{\tilde{M}}} ;\tau^{{\prime }} } \right)\sim - \frac{{2I_{0} \left( {\tau_{{\tilde{M}}} ;\tau^{{\prime }} } \right)}}{{\tau_{{\tilde{M}}} + D}} - \\ & \quad \frac{{8\sin \overline{\theta }}}{{\pi \overline{\theta }}}e^{{ - \tau_{{\tilde{M}}} }} e^{{ - \tau^{{\prime }} /2}} \left( {\sinh \tau^{{\prime }} \frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{{\prime }} } \right) + \frac{1}{2}e^{{ - \tau^{{\prime }} }} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{{\prime }} } \right)} \right) \\ \end{aligned} $$

(E.26)

When $n \ne 0$, using (E.23) and $f_{n}^{\left( 1 \right)} \left( {\tau^{\prime\prime}} \right)\sim 1$, (E.14) yields

$$ I_{n} \left( {\tau ;\tau^{\prime}} \right)\sim \frac{{4\sin \overline{\theta }}}{{\pi \left( {n\pi - \overline{\theta }} \right)}}e^{ - \tau } e^{{ - \tau^{\prime}/2}} \left( {\sinh \tau^{\prime}\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) + \frac{1}{2}e^{{ - \tau^{\prime}}} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)} \right) $$

(E.27)

for large, positive $\tau - \tau^{\prime}$. Thus, (E.22) implies

$$ K_{n} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right)\sim - \frac{{8\overline{\theta }\sin \overline{\theta }}}{{\pi \left( {n^{2} \pi^{2} - \overline{\theta }^{2} } \right)}}e^{{ - \tau_{{\tilde{M}}} }} e^{{ - \tau^{\prime}/2}} \left( {\sinh \tau^{\prime}\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau^{\prime}} \right) + \frac{1}{2}e^{{ - \tau^{\prime}}} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau^{\prime}} \right)} \right) $$

(E.28)

Although the approximations (E.26) and (E.28) are only strictly valid for large, positive $\tau_{{\tilde{M}}} - \tau^{\prime}$, they are used to express $K_{n} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right)$ numerically for all $\tau^{\prime}$. This does not mean that the resulting $K_{n} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right)$ are accurate when $\tau^{\prime}$ approaches $\tau_{{\tilde{M}}}$, but rather that we expect such inaccuracies to be unimportant as regards the precision of the overall numerical scheme.

(E.22) implies

$$ \begin{aligned} K_{n} \left( {\tau_{m} ;\tau^{\prime}} \right) = & \exp \left( { - \frac{n\pi }{{\overline{\theta }}}\left( {\tau_{m + 1} - \tau_{m} } \right)} \right)K_{n} \left( {\tau_{m + 1} ;\tau^{\prime}} \right) - \\ & \quad \int_{{\tau_{m} }}^{{\tau_{m + 1} }} {\frac{{e^{{\tau^{\prime\prime}}} I_{n} \left( {\tau^{\prime\prime};\tau^{\prime}} \right)}}{{\tilde{f}_{n}^{\left( 1 \right)2} \left( {\tau^{\prime\prime}} \right)\sinh \tau^{\prime\prime}}}\exp \left( { - \frac{n\pi }{{\overline{\theta }}}\left( {\tau^{\prime\prime} - \tau_{m} } \right)} \right)d\tau^{\prime\prime}} \\ \end{aligned} $$

(E.29)

The integral is treated as follows. Quadratic approximation of

$$ \frac{{e^{{\tau^{\prime\prime}}} I_{n} \left( {\tau^{\prime\prime};\tau^{\prime}} \right)}}{{\tilde{f}_{n}^{\left( 1 \right)2} \left( {\tau^{\prime\prime}} \right)\sinh \tau^{\prime\prime}}} $$

(E.30)

is carried out for the interval $\tau_{m} \le \tau^{\prime\prime} \le \tau_{m + 1}$ using its values at the mid- and end-points. Knowing the coefficients of the quadratic approximation of (E.30), the integral in (E.29) is calculated analytically. Given $K_{n} \left( {\tau_{{\tilde{M}}} ;\tau^{\prime}} \right)$ from (E.26) or (E.28), (E.29) determines $K_{n} \left( {\tau_{m} ;\tau^{\prime}} \right)$, $- \hat{M} \le m < \tilde{M}$. $H_{n} \left( {\tau ;\tau^{\prime},0} \right)$ follows from the definition of $K_{n}$.

5.3 E.3 Final expressions for $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$

$G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ are calculated using (D.11), (D.23) and (D.24). Convergence of the infinite sums can be accelerated using the large-$n$ result

$$ H_{n} \left( {\tau ;\tau^{\prime},0} \right)\sim - \frac{{2\overline{\theta }}}{{n^{2} \pi^{2} }}\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + O\left( {n^{ - 4} } \right) $$

(E.31)

which follows from (D.12). According to (D.11),

$$ \begin{aligned} H\left( {\tau ,0;\tau^{\prime},0} \right) = & \frac{1}{2}H_{0} \left( {\tau ;\tau^{\prime},0} \right) + \\ & \quad \frac{1}{6}\overline{\theta }\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + \sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} \left( {H_{n} \left( {\tau ;\tau^{\prime},0} \right) + \frac{{2\overline{\theta }}}{{n^{2} \pi^{2} }}\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right)} \\ \end{aligned} $$

(E.32)

$$ \begin{aligned} H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = & \frac{1}{2}H_{0} \left( {\tau ;\tau^{\prime},0} \right) - \\ & \quad \frac{1}{3}\overline{\theta }\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + \sum\limits_{n = 1}^{\infty } {\left( {H_{n} \left( {\tau ;\tau^{\prime},0} \right) + \frac{{2\overline{\theta }}}{{n^{2} \pi^{2} }}\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right)} \\ \end{aligned} $$

(E.33)

where $\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} n^{ - 2} } = - \pi^{2} /12$ and $\sum\limits_{n = 1}^{\infty } {n^{ - 2} } = \pi^{2} /6$ have been used. Truncating the sums,

$$ \begin{aligned} H\left( {\tau ,0;\tau^{\prime},0} \right) = & \frac{1}{6}\overline{\theta }\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)\left( {\frac{12}{{\pi^{2} }}\sum\limits_{n = 1}^{N} {\left( { - 1} \right)^{n} n^{ - 2} } + 1} \right) + \\ & \quad \frac{1}{2}H_{0} \left( {\tau ;\tau^{\prime},0} \right) + \sum\limits_{n = 1}^{N} {\left( { - 1} \right)^{n} H_{n} \left( {\tau ;\tau^{\prime},0} \right)} \\ \end{aligned} $$

(E.34)

$$ \begin{aligned} H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = & \frac{1}{3}\overline{\theta }\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)\left( {\frac{6}{{\pi^{2} }}\sum\limits_{n = 1}^{N} {n^{ - 2} } - 1} \right) + \\ & \quad \frac{1}{2}H_{0} \left( {\tau ;\tau^{\prime},0} \right) + \sum\limits_{n = 1}^{N} {H_{n} \left( {\tau ;\tau^{\prime},0} \right)} \\ \end{aligned} $$

(E.35)

$H_{n} \left( {\tau ;\tau^{\prime},0} \right)$, $\partial h/\partial \xi \left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)$ are calculated as described in the previous section. For $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$, (D.24) requires $h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)$, which is determined using (D.39) with $\tau_{*}$ the solution of (E.11). On the other hand, (D.23) brings in $h\left( {\tau ,0;\tau^{\prime},0} \right)$, for which, as discussed earlier, there is a choice of representations, one being (D.39), the other (D.53). The method for choosing between the two was described at the end of appendix D.

$G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$ is given by (D.25). According to (D.28),

$$ \frac{{\partial \tau_{*} }}{\partial \tau } = \frac{{\sinh \tau^{\prime}\left( {\cosh \tau^{\prime} - \cos \xi \cosh \tau } \right)}}{{\left( {\cosh \tau \cosh \tau^{\prime} - \cos \xi } \right)^{2} }}\cosh^{2} \tau_{*} $$

(E.36)

(D.39) and (E.1) give

$$ h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) = - \frac{1}{\pi }\left( {\frac{{1 + e^{{ - 2\tau_{*} }} }}{{\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }}}} \right)^{1/2} \tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right) $$

(E.37)

(E.36) and (E.37) imply

$$ \begin{aligned} & \frac{\partial }{\partial \tau }\left( {\left( {\cosh \tau + 1} \right)^{1/2} h\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right) \\ & \quad = \frac{{\left( {\cosh \tau + 1} \right)^{1/2} \left( {1 + e^{{ - 2\tau_{*} }} } \right)^{1/2} }}{{\pi \left( {\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{3/2} }}\left( {\frac{{\sinh \tau \left( {\cosh \tau^{\prime} + \cos \overline{\theta }} \right)\tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)}}{{2\left( {\cosh \tau + 1} \right)}} + \begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \right. \\ \quad & \left. {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} \frac{{\sinh \tau^{\prime}\left( {\cos \overline{\theta }\cosh \tau - \cosh \tau^{\prime}} \right)}}{{\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }}}\cosh^{2} \tau_{*} \left( {\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau_{*} } \right) - \frac{1}{{1 + e^{{2\tau_{*} }} }}\tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)} \right)} \right) \\ \end{aligned} $$

(E.38)

where $\tau_{*}$ is the solution of (E.11). Using the identity $\left( {\cosh \tau + 1} \right)^{1/2} = 2^{ - 1/2} e^{\tau /2} \left( {1 + e^{ - \tau } } \right)$, (E.35) and the definition of $K_{n}$,

$$ \begin{aligned} & \frac{\partial }{\partial \tau }\left( {\left( {\cosh \tau + 1} \right)^{1/2} H\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right) \\ \quad & = \frac{1}{3}\overline{\theta }\frac{\partial }{\partial \tau }\left( {\left( {\cosh \tau + 1} \right)^{1/2} \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right)\left( {\frac{6}{{\pi^{2} }}\sum\limits_{n = 1}^{N} {n^{ - 2} } - 1} \right) + \\ & \quad 2^{ - 3/2} \left( {1 + e^{ - \tau } } \right)\left( {\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)\frac{{\partial K_{0} }}{\partial \tau }\left( {\tau ;\tau^{\prime}} \right) + \left( {\frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau } - \frac{{\tilde{f}_{0}^{\left( 1 \right)} \left( \tau \right)}}{{1 + e^{\tau } }}} \right)K_{0} \left( {\tau ;\tau^{\prime}} \right)} \right) + \\ \quad & 2^{ - 1/2} \left( {1 + e^{ - \tau } } \right)\sum\limits_{n = 1}^{N} {\left( {\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)\frac{{\partial K_{n} }}{\partial \tau }\left( {\tau ;\tau^{\prime}} \right) + \left( {\frac{{d\tilde{f}_{n}^{\left( 1 \right)} }}{d\tau } - \frac{{\tilde{f}_{n}^{\left( 1 \right)} \left( \tau \right)}}{{1 + e^{\tau } }}} \right)K_{n} \left( {\tau ;\tau^{\prime}} \right)} \right)} \\ \end{aligned} $$

(E.39)

(E.22) implies

$$ \frac{{\partial K_{n} }}{\partial \tau }\left( {\tau ;\tau^{\prime}} \right) = \frac{n\pi }{{\overline{\theta }}}K_{n} \left( {\tau ;\tau^{\prime},0} \right) + \frac{{e^{\tau } I_{n} \left( {\tau ;\tau^{\prime}} \right)}}{{\tilde{f}_{n}^{\left( 1 \right)2} \left( \tau \right)\sinh \tau }} $$

(E.40)

Multiplying (E.10) by $\left( {\cosh \tau + 1} \right)^{1/2}$ and taking the $\tau$-derivative, use of (E.2) with $n = 0$ and (E.36) leads to

$$ \begin{aligned} & \frac{\partial }{\partial \tau }\left( {\left( {\cosh \tau + 1} \right)^{1/2} \frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right)} \right) \\ \quad & = \frac{{\left( {\cosh \tau + 1} \right)^{1/2} \sinh \tau^{\prime}\left( {\cosh \tau^{\prime} - \cos \overline{\theta }\cosh \tau } \right)}}{{\left( {\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{2} }}\cosh^{2} \tau_{*} \\ \quad & \left\{ {\frac{{1 - 2e^{{ - 2\tau_{*} }} }}{{1 + e^{{ - 2\tau_{*} }} }}\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) + \frac{{\sin \overline{\theta }\left( {1 + e^{{ - 2\tau_{*} }} } \right)^{1/2} \cosh \tau_{*} }}{{\pi \left( {\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }} \right)^{3/2} }}\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \right. \\ \quad & \left. {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} \left( {\frac{1}{2}e^{{\tau_{*} }} \frac{{d\tilde{f}_{0}^{\left( 1 \right)} }}{d\tau }\left( {\tau_{*} } \right) + \frac{{k^{2} }}{{\sinh \tau_{*} }}\tilde{f}_{0}^{\left( 1 \right)} \left( {\tau_{*} } \right)} \right)} \right\} - \\ \quad & \frac{{\sinh \tau \left( {\cosh \tau^{\prime}\left( {2\cosh \tau + 3} \right) + \cos \overline{\theta }} \right)}}{{2\left( {\cosh \tau + 1} \right)^{1/2} \left( {\cosh \tau \cosh \tau^{\prime} - \cos \overline{\theta }} \right)}}\frac{\partial h}{{\partial \xi }}\left( {\tau ,\overline{\theta };\tau^{\prime},0} \right) \\ \end{aligned} $$

(E.41)

where $\tau_{*}$ is the solution of (E.11). Equations (D.25), (E.10) and (E.38)-(E.41) determine $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$.

The implementation of the above numerical procedure was checked by comparing the values of $G^{\left( 1 \right)} \left( {\tau ;\tau^{\prime}} \right)$, $G^{\left( 3 \right)} \left( {\tau ;\tau^{\prime}} \right)$ and $G^{\left( 4 \right)} \left( {\tau ;\tau^{\prime}} \right)$ with those obtained using (A.17), (A.23), (A.24), (A.26), (C.10) and (C.11) for values of $\tau$, $\tau^{\prime}$ for which the infinite series in (A.17) and (A.23) converge rapidly, i.e. $\left| {\tau - \tau^{\prime}} \right|$ of order one or larger. This involves the numerical determination of $f_{n}^{\left( 2 \right)}$ and its derivative. Defining

$$ \tilde{f}_{n}^{\left( 2 \right)} \left( \tau \right) = \exp \left( {\left( {\frac{1}{2} + \frac{n\pi }{{\overline{\theta }}}} \right)\tau } \right)f_{n}^{\left( 2 \right)} \left( \tau \right) $$

(E.42)

backwards Radau integration of the differential equation corresponding to (E.2) for $\tilde{f}_{n}^{\left( 2 \right)}$ and the initial conditions $\tilde{f}_{n}^{\left( 2 \right)} = 1$ and $d\tilde{f}_{n}^{\left( 2 \right)} /d\tau = 0$ at $\tau = \tau_{{\tilde{M}}}$ are used. The calculations of $\tilde{f}_{n}^{\left( 1 \right)}$, $\tilde{f}_{n}^{\left( 2 \right)}$ and their derivatives were checked using (C.8) and the known values of $W_{n}$.