3.1 A two-viscida junction

To gain an understanding of the general behaviour at a junction, we first analyse a join between two viscidas of uniform and equal thickness, as shown schematically in figure 5(a). Our aim is to determine the time evolution of the interior corner angle $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}+\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{-},\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{+},\unicode[STIX]{x1D70F})$ made between the two connected viscidas. Our initial approach is to smooth the resulting corner, modelling the junction as a single viscida in which $\unicode[STIX]{x1D703}$ varies rapidly as a function of $\unicode[STIX]{x1D709}$ , over a small neighbourhood whose size is of order $\unicode[STIX]{x1D6FF}\ll 1$ , as in figure 5(b). The thin-viscida approximation remains valid locally provided $\unicode[STIX]{x1D716}\ll \unicode[STIX]{x1D6FF}$ ; however, Griffiths & Howell (Reference Griffiths and Howell2007) showed that the problem may be solved for arbitrarily small $\unicode[STIX]{x1D6FF}$ without introducing any artefacts from the corner, and the solution approaches a well-defined limit as $\unicode[STIX]{x1D6FF}\rightarrow 0$ .

Since the initial thickness is assumed to be uniform, we have $h_{0}(\unicode[STIX]{x1D709})=1$ , so the thickness evolves as $h=1+\unicode[STIX]{x1D70F}/2$ from (2.7), and the evolution equation (2.9) is then simply

(3.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[(1+\unicode[STIX]{x1D70F}/2)\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{2}}\right]=\frac{3A(\unicode[STIX]{x1D70F})}{(1+\unicode[STIX]{x1D70F}/2)^{4}}\sin \unicode[STIX]{x1D703}+\frac{3B(\unicode[STIX]{x1D70F})}{(1+\unicode[STIX]{x1D70F}/2)^{4}}\cos \unicode[STIX]{x1D703}.\end{eqnarray}$$

We rescale into the corner region via $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{0}^{+}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}$ and define $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}):=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})$ within this region. Substituting this local scaling into (3.1) and retaining only leading-order terms in $\unicode[STIX]{x1D6FF}$ gives

(3.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[(1+\unicode[STIX]{x1D70F}/2)\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}\right]=0,\end{eqnarray}$$

which is subject to the matching conditions

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})\rightarrow \unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{\pm },\unicode[STIX]{x1D70F})\quad \text{as }\unicode[STIX]{x1D702}\rightarrow \pm \infty .\end{eqnarray}$$

We integrate (3.2) and apply the far-field conditions (3.3) to obtain

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}-\frac{\unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}(0)}{1+\unicode[STIX]{x1D70F}/2}.\end{eqnarray}$$

Therefore the junction angle straightens out with increasing  $\unicode[STIX]{x1D70F}$ , approaching $\unicode[STIX]{x03C0}$ as $\unicode[STIX]{x1D70F}\rightarrow \infty$ . The result (3.4) also makes sense for inverse time provided $\unicode[STIX]{x1D70F}>-2$ , and we see that the internal angle sharpens for decreasing $\unicode[STIX]{x1D70F}$ , becoming zero when $\unicode[STIX]{x1D70F}=-2\unicode[STIX]{x1D711}(0)/\unicode[STIX]{x03C0}$ , corresponding to eventual self-intersection.

3.2 A junction of $n$ viscidas

We now attempt to extend our theory to describe the evolution of a junction where several viscidas meet. Let us first consider an alternative derivation of (3.4) governing the interior angle at a junction between two viscidas with uniform and equal thickness meeting with interior angle $\unicode[STIX]{x1D711}$ and exterior angle $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}$ , as shown in figure 5(a). As above, we can treat the two viscidas as a single continuous viscida containing a local jump in the inclination angle $\unicode[STIX]{x1D703}$ , at $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{0}$ , say. From (2.8), we know that the bending moment is given by

(3.5) $$\begin{eqnarray}M=-\frac{(1+\unicode[STIX]{x1D70F}/2)^{3}}{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right].\end{eqnarray}$$

Since $\unicode[STIX]{x1D703}\sim \unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{-},\unicode[STIX]{x1D70F})+\unicode[STIX]{x0394}\unicode[STIX]{x1D703}{\mathcal{H}}(\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}_{0})$ , near the corner, where ${\mathcal{H}}$ is the Heaviside function, the local bending moment behaves as

(3.6) $$\begin{eqnarray}M\sim -\frac{(1+\unicode[STIX]{x1D70F}/2)^{3}}{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}_{0}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}_{0})$ denotes the Dirac delta function. To avoid an unbalanced point singularity in $M$ , we must therefore have

(3.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x0394}\unicode[STIX]{x1D703}\right]=0,\end{eqnarray}$$

which upon integration with respect to $\unicode[STIX]{x1D70F}$ and use of $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}-\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ , yields precisely the same angle evolution law (3.4) as found above by smoothing the corner.

Figure 6.

A junction between three viscidas of equal thicknesses. The behaviour is characterized by the evolution of the internal angles $\unicode[STIX]{x1D711}_{i}$ ( $i=1,2,3$ ).

We now generalize the above argument to the case of a junction where three viscidas meet, as shown schematically in figure 6. Here the viscidas are labelled 1–3, with $\unicode[STIX]{x1D711}_{i}$ denoting the interior angle between viscida $i$ and viscida $i+1$ (mod 3). The bend through an exterior angle $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{3}$ between viscidas 3 and 1 produces a point anticlockwise moment of magnitude

(3.8) $$\begin{eqnarray}M_{3}=-\frac{(1+\unicode[STIX]{x1D70F}/2)^{3}}{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{3}\right].\end{eqnarray}$$

Similarly, the bend between viscidas 3 and 2 produces a point moment of magnitude

(3.9) $$\begin{eqnarray}M_{2}=\frac{(1+\unicode[STIX]{x1D70F}/2)^{3}}{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{2}\right].\end{eqnarray}$$

To ensure that no unbalanced point moment acts on viscida 3, we must have $M_{3}+M_{2}=0$ , which may be simplified to

(3.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)(\unicode[STIX]{x1D711}_{3}-\unicode[STIX]{x1D711}_{2})\right]=0.\end{eqnarray}$$

Similarly, a balance of moments acting on each of viscidas 1 and 2 gives

(3.11a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)(\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}_{3})\right]=0\quad \text{and}\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)(\unicode[STIX]{x1D711}_{2}-\unicode[STIX]{x1D711}_{1})\right]=0.\end{eqnarray}$$

We also note that the interior angles must satisfy

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{1}+\unicode[STIX]{x1D711}_{2}+\unicode[STIX]{x1D711}_{3}=2\unicode[STIX]{x03C0}.\end{eqnarray}$$

Using (3.12) and combinations of (3.10) and (3.11), we find that

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{i}(\unicode[STIX]{x1D70F})=\frac{2\unicode[STIX]{x03C0}}{3}+\frac{\unicode[STIX]{x1D711}_{i}(0)-2\unicode[STIX]{x03C0}/3}{1+\unicode[STIX]{x1D70F}/2},\end{eqnarray}$$

where $i\in \{1,2,3\}$ . Therefore, the junction of three viscidas evolves in a way that ensures that, as $\unicode[STIX]{x1D70F}\rightarrow \infty$ , the interior angles $\unicode[STIX]{x1D711}_{1},\unicode[STIX]{x1D711}_{2},\unicode[STIX]{x1D711}_{3}$ all tend to $2\unicode[STIX]{x03C0}/3$ .

We hypothesize, based on the examples we have shown, that this method is generalizable for a junction of $n$ viscidas, for any $n\geqslant 2$ . Balancing moments at the $n$ -viscida junction yields, after some algebra,

(3.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)(\unicode[STIX]{x1D711}_{i}-\unicode[STIX]{x1D711}_{j})\right]=0,\end{eqnarray}$$

for all $1\leqslant i<j\leqslant n$ , where $\unicode[STIX]{x1D711}_{i}$ again labels the interior angle between viscidas $i$ and $i+1$ (mod  $n$ ). Also, in the general case, the relation (3.12) becomes

(3.15) $$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D711}_{i}=2\unicode[STIX]{x03C0},\end{eqnarray}$$

which in combination with (3.14), gives

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{i}(\unicode[STIX]{x1D70F})=\frac{2\unicode[STIX]{x03C0}}{n}+\frac{\unicode[STIX]{x1D711}_{i}(0)-2\unicode[STIX]{x03C0}/n}{1+\unicode[STIX]{x1D70F}/2}.\end{eqnarray}$$

Again, all the interior angles tend to the same value, $2\unicode[STIX]{x03C0}/n$ , as $\unicode[STIX]{x1D70F}\rightarrow \infty$ .

4.1 A two-viscida network

Figure 7.

Schematic diagram of the two-viscida problem. The top viscida has a centreline described by $\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ and the bottom viscida has a centreline described by $\unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ . The interior angle at $\unicode[STIX]{x1D709}=0$ for any time $\unicode[STIX]{x1D70F}$ is given by $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})$ and the angle at $\unicode[STIX]{x1D709}=1$ by $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70F})$ . The two nodes represent the viscida junctions.

In this section, we analyse a set-up consisting of two interacting viscidas of equal and uniform thickness, connected by junctions at their ends, as illustrated in figure 7. By combining the governing equation (3.1) for each viscida with the angle evolution equation (3.4) for each node, we obtain a closed model for the coupled system of viscidas and junctions. The ideas developed in this section will be generalized below to a more general network of viscidas connected by junctions.

The inclination angle $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ in viscida $i\in \{1,2\}$ satisfies (3.1), which we write in the form

(4.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{2}}\right]=A_{i}(\unicode[STIX]{x1D70F})\sin \unicode[STIX]{x1D703}_{i}+B_{i}(\unicode[STIX]{x1D70F})\cos \unicode[STIX]{x1D703}_{i},\end{eqnarray}$$

by absorbing factors $\pm 3/(1+\unicode[STIX]{x1D70F}/2)^{4}$ into the functions $A_{i}(\unicode[STIX]{x1D70F})$ and $B_{i}(\unicode[STIX]{x1D70F})$ . Since each equation (4.1) is second order in space and has two additional unknowns, eight boundary conditions are needed in total to close the problem.

To ensure that the ends join up, we impose

(4.2a,b ) $$\begin{eqnarray}\int _{0}^{1}\cos \unicode[STIX]{x1D703}_{1}\,\text{d}\unicode[STIX]{x1D709}=\int _{0}^{1}\cos \unicode[STIX]{x1D703}_{2}\,\text{d}\unicode[STIX]{x1D709}\quad \text{and}\quad \int _{0}^{1}\sin \unicode[STIX]{x1D703}_{1}\,\text{d}\unicode[STIX]{x1D709}=\int _{0}^{1}\sin \unicode[STIX]{x1D703}_{2}\,\text{d}\unicode[STIX]{x1D709}.\end{eqnarray}$$

It is further required that, at each junction, there is zero net bending moment, i.e. $M_{1}+M_{2}=0$ where the bending moment in each viscida is given by (3.5). Therefore we impose the boundary condition

(4.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[(1+\unicode[STIX]{x1D70F}/2)\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right)\right]=0\quad \text{at }\unicode[STIX]{x1D709}=0,1.\end{eqnarray}$$

Equation (3.4) provides the evolution of the corners at $\unicode[STIX]{x1D709}=0,1$ ,

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}+\frac{\unicode[STIX]{x1D6FC}(0)-\unicode[STIX]{x03C0}}{1+\unicode[STIX]{x1D70F}/2}=\unicode[STIX]{x1D703}_{1}(0,\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}_{2}(0,\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}+\frac{\unicode[STIX]{x1D6FD}(0)-\unicode[STIX]{x03C0}}{1+\unicode[STIX]{x1D70F}/2}=\unicode[STIX]{x1D703}_{2}(1,\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}_{1}(1,\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

respectively, where $\unicode[STIX]{x1D6FC}(0)$ is the initial internal angle at $\unicode[STIX]{x1D709}=0$ and $\unicode[STIX]{x1D6FD}(0)$ is the initial internal angle at  $\unicode[STIX]{x1D709}=1$ . Force balances at $\unicode[STIX]{x1D709}=0,1$ imply that there must be zero resultant forces in the $x$ - and $y$ -direction, for all $\unicode[STIX]{x1D70F}$ , i.e.

(4.6a,b ) $$\begin{eqnarray}A_{1}(\unicode[STIX]{x1D70F})+A_{2}(\unicode[STIX]{x1D70F})=0\quad \text{and}\quad B_{1}(\unicode[STIX]{x1D70F})+B_{2}(\unicode[STIX]{x1D70F})=0.\end{eqnarray}$$

Equations (4.2)–(4.6) provide eight boundary conditions in total, and so appear to close the problem. However, there remains an indeterminate rigid-body motion: for example, one can easily see that the entire problem (4.1)–(4.6) is invariant under the transformation $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\mapsto \unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})+\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70F})$ for any function $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D70F})$ . Therefore, we will find that three further conditions are needed to fix the arbitrary translation and rotation.

Upon integration of (4.1) with respect to $\unicode[STIX]{x1D709}$ , we obtain

(4.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x1D703}_{i}\right]=A_{i}(\unicode[STIX]{x1D70F})Y_{i}+B_{i}(\unicode[STIX]{x1D70F})X_{i}+C_{i}(\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where

(4.8) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}X_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\\ Y_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\end{array}\right)=(1+\unicode[STIX]{x1D70F}/2)\left(\begin{array}{@{}c@{}}x_{i}(s,t)\\ y_{i}(s,t)\end{array}\right)=\left(\begin{array}{@{}c@{}}X_{0}(\unicode[STIX]{x1D70F})\\ Y_{0}(\unicode[STIX]{x1D70F})\end{array}\right)+\int _{0}^{\unicode[STIX]{x1D709}}\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{i}(\hat{\unicode[STIX]{x1D709}},\unicode[STIX]{x1D70F})\\ \sin \unicode[STIX]{x1D703}_{i}(\hat{\unicode[STIX]{x1D709}},\unicode[STIX]{x1D70F})\end{array}\right)\text{d}\hat{\unicode[STIX]{x1D709}}\end{eqnarray}$$

gives the centreline of viscida $i\in \{1,2\}$ , expressed in Lagrangian variables. At $\unicode[STIX]{x1D709}=0$ , the two viscidas meet at a junction whose position $(X_{0}(\unicode[STIX]{x1D70F}),Y_{0}(\unicode[STIX]{x1D70F}))$ is thus far arbitrary. By applying (4.3) and (4.6) at $\unicode[STIX]{x1D709}=0$ , we deduce that

(4.9) $$\begin{eqnarray}C_{1}(\unicode[STIX]{x1D70F})+C_{2}(\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

and then the corresponding moment balance (4.3) at $\unicode[STIX]{x1D709}=1$ is satisfied identically: a manifestation of the rotational invariance alluded to above.

Integration once more with respect to  $\unicode[STIX]{x1D709}$ gives

(4.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x1D703}_{i}\right]=A_{i}(\unicode[STIX]{x1D70F})\int _{0}^{\unicode[STIX]{x1D709}}Y_{i}\,\text{d}\hat{\unicode[STIX]{x1D709}}+B_{i}(\unicode[STIX]{x1D70F})\int _{0}^{\unicode[STIX]{x1D709}}X_{i}\,\text{d}\hat{\unicode[STIX]{x1D709}}+C_{i}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D709}+D_{i}(\unicode[STIX]{x1D70F}),\end{eqnarray}$$

for $i\in \{1,2\}$ . If we subtract (4.10) with $i=1$ from (4.10) with $i=2$ and use the angle conditions (4.4) and (4.5) at $\unicode[STIX]{x1D709}=0$ and $\unicode[STIX]{x1D709}=1$ , we obtain the relations

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle D_{1}(\unicode[STIX]{x1D70F})-D_{2}(\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x03C0}}{2}, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle C_{1}(\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}[A_{2}(\unicode[STIX]{x1D70F})\overline{Y_{2}}(\unicode[STIX]{x1D70F})+B_{2}(\unicode[STIX]{x1D70F})\overline{X_{2}}(\unicode[STIX]{x1D70F})-A_{1}(\unicode[STIX]{x1D70F})\overline{Y_{1}}(\unicode[STIX]{x1D70F})-B_{1}(\unicode[STIX]{x1D70F})\overline{X_{1}}(\unicode[STIX]{x1D70F})-\unicode[STIX]{x03C0}], & \displaystyle\end{eqnarray}$$

where the bar is used to denote the averaged value

(4.13) $$\begin{eqnarray}\overline{F}(\unicode[STIX]{x1D70F}):=\int _{0}^{1}F(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D709}.\end{eqnarray}$$

At this point it is helpful to make some specific choices that eliminate the arbitrary rigid-body motion and simplify the problem somewhat. We fix the rotation by taking $D_{1}(\unicode[STIX]{x1D70F})+D_{2}(\unicode[STIX]{x1D70F})\equiv 0$ . We also assume that the net translation is chosen to fix the net centroid of the system, so that $\overline{X_{1}}(\unicode[STIX]{x1D70F})+\overline{X_{2}}(\unicode[STIX]{x1D70F})\equiv 0$ and $\overline{Y_{1}}(\unicode[STIX]{x1D70F})+\overline{Y_{2}}(\unicode[STIX]{x1D70F})\equiv 0$ . Given these specific choices, we can evaluate

(4.14a-d ) $$\begin{eqnarray}C_{1}=-\frac{\unicode[STIX]{x03C0}}{2},\quad C_{2}=\frac{\unicode[STIX]{x03C0}}{2},\quad D_{1}=\frac{\unicode[STIX]{x03C0}}{4},\quad D_{2}=-\frac{\unicode[STIX]{x03C0}}{4},\end{eqnarray}$$

as well as

(4.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\begin{array}{@{}c@{}}X_{0}(\unicode[STIX]{x1D70F})\\ Y_{0}(\unicode[STIX]{x1D70F})\end{array}\right)=-\frac{1}{2}\int _{0}^{1}(1-\unicode[STIX]{x1D709})\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})+\cos \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\\ \sin \unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})+\sin \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\end{array}\right)\,\text{d}\unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

(4.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\begin{array}{@{}c@{}}X_{i}(1,\unicode[STIX]{x1D70F})\\ Y_{i}(1,\unicode[STIX]{x1D70F})\end{array}\right)=\frac{1}{2}\int _{0}^{1}\unicode[STIX]{x1D709}\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})+\cos \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\\ \sin \unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})+\sin \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\end{array}\right)\,\text{d}\unicode[STIX]{x1D709}. & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\unicode[STIX]{x1D703}_{i}\right]=(-1)^{i}\left[A_{1}(\unicode[STIX]{x1D70F})\int _{0}^{\unicode[STIX]{x1D709}}Y_{i}\,\text{d}\hat{\unicode[STIX]{x1D709}}+B_{1}(\unicode[STIX]{x1D70F})\int _{0}^{\unicode[STIX]{x1D709}}X_{i}\,\text{d}\hat{\unicode[STIX]{x1D709}}+\frac{\unicode[STIX]{x03C0}}{4}(1-2\unicode[STIX]{x1D709})\right],\end{eqnarray}$$

which can be solved numerically for $\unicode[STIX]{x1D703}_{i}$ once we have determined $A_{1}(\unicode[STIX]{x1D70F})$ and $B_{1}(\unicode[STIX]{x1D70F})$ .

To this end, we multiply (4.7) by $X_{i}$ for $i\in \{1,2\}$ and integrate over the viscida length to obtain

(4.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}[(1+\unicode[STIX]{x1D70F}/2)\unicode[STIX]{x1D703}_{i}]\right|_{\unicode[STIX]{x1D709}=1}X_{i}(1,\unicode[STIX]{x1D70F})-\left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}[(1+\unicode[STIX]{x1D70F}/2)\unicode[STIX]{x1D703}_{i}]\right|_{\unicode[STIX]{x1D709}=1}X_{0}(\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad -\,(1+\unicode[STIX]{x1D70F}/2)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\int _{0}^{1}\sin \unicode[STIX]{x1D703}_{i}\,\text{d}\unicode[STIX]{x1D709}-\frac{1}{2}\int _{0}^{1}\unicode[STIX]{x1D703}_{i}\cos \unicode[STIX]{x1D703}_{i}\,\text{d}\unicode[STIX]{x1D709}=A_{i}\overline{X_{i}Y_{i}}+B_{i}\overline{X_{i}^{2}}.\end{eqnarray}$$

In a similar fashion, by multiplying (4.7) by $Y_{i}$ , integrating over the viscida length and using (4.2b ), we obtain

(4.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}[(1+\unicode[STIX]{x1D70F}/2)\unicode[STIX]{x1D703}_{i}]\right|_{\unicode[STIX]{x1D709}=1}Y_{i}(1,\unicode[STIX]{x1D70F})-\left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}[(1+\unicode[STIX]{x1D70F}/2)\unicode[STIX]{x1D703}_{i}]\right|_{\unicode[STIX]{x1D709}=1}Y_{0}(\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & & \displaystyle \quad +\,(1+\unicode[STIX]{x1D70F}/2)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\int _{0}^{1}\cos \unicode[STIX]{x1D703}_{i}\,\text{d}\unicode[STIX]{x1D709}-\frac{1}{2}\int _{0}^{1}\unicode[STIX]{x1D703}_{i}\sin \unicode[STIX]{x1D703}_{i}\,\text{d}\unicode[STIX]{x1D709}=A_{i}\overline{Y_{i}^{2}}+B_{i}\overline{X_{i}Y_{i}},\end{eqnarray}$$

with $i\in \{1,2\}$ . We subtract the cases $i=1$ and $i=2$ of (4.17) and (4.18) and rearrange to obtain a linear system for $A_{1}$ and $B_{1}$ as follows:

(4.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(\begin{array}{@{}cc@{}}\overline{X_{1}Y_{1}}+\overline{X_{2}Y_{2}} & \overline{X_{1}^{2}}+\overline{X_{2}^{2}}\\ \overline{Y_{1}^{2}}+\overline{Y_{2}^{2}} & \overline{X_{1}Y_{1}}+\overline{X_{2}Y_{2}}\end{array}\right)\left(\begin{array}{@{}c@{}}\vphantom{\overline{X_{1}^{2}}}A_{1}\\ \vphantom{\overline{X_{1}^{2}}}B_{1}\end{array}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{2}\int _{0}^{1}\left[\left(\begin{array}{@{}c@{}}\vphantom{\overline{X_{1}^{2}}}\cos \unicode[STIX]{x1D703}_{2}\\ \vphantom{\overline{X_{1}^{2}}}\sin \unicode[STIX]{x1D703}_{2}\end{array}\right)[\unicode[STIX]{x1D703}_{2}-\unicode[STIX]{x03C0}(\unicode[STIX]{x1D709}-1/2)]-\left(\begin{array}{@{}c@{}}\vphantom{\overline{X_{1}^{2}}}\cos \unicode[STIX]{x1D703}_{1}\\ \vphantom{\overline{X_{1}^{2}}}\sin \unicode[STIX]{x1D703}_{1}\end{array}\right)[\unicode[STIX]{x1D703}_{1}-\unicode[STIX]{x03C0}(1/2-\unicode[STIX]{x1D709})]\right]\text{d}\unicode[STIX]{x1D709}.\qquad\end{eqnarray}$$

With $A_{1}(\unicode[STIX]{x1D70F})$ and $B_{1}(\unicode[STIX]{x1D70F})$ determined at every time $\unicode[STIX]{x1D70F}$ by (4.19), the evolution equation (4.16) for $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ is easily solved numerically, for example by using the method of lines. Before proceeding, we observe that the coupled problem (4.16), (4.19) admits an exact steady solution in which

(4.20a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}\left({\textstyle \frac{1}{2}}-\unicode[STIX]{x1D709}\right),\quad \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}\left(\unicode[STIX]{x1D709}-{\textstyle \frac{1}{2}}\right).\end{eqnarray}$$

Clearly these forms render the right-hand side of (4.19) identically zero and thus give $A_{1}=B_{1}=0$ (since the matrix on the left-hand side of (4.19) is negative definite). Then the left- and right-hand sides of (4.16) are also identically equal. The solution (4.20) corresponds to a pair of semi-circular viscidas whose ends meet smoothly with an interior angle of $\unicode[STIX]{x03C0}$ , which is tantamount to a single complete circular viscida. We expect the system to converge to such a configuration for large positive values of  $\unicode[STIX]{x1D70F}$ .

We illustrate the strategy developed above to solve for the evolution of a pair of symmetric coupled viscidas whose initial centreline geometries are defined by

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},0)=-\unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},0)=\tanh (20(1/2-\unicode[STIX]{x1D709})).\end{eqnarray}$$

As demonstrated in figure 8(a,b), the reflectional symmetry is preserved, so that $\unicode[STIX]{x1D703}_{2}\equiv -\unicode[STIX]{x1D703}_{1}$ for all time. As $\unicode[STIX]{x1D70F}$ increases, the two-viscida system retracts under the action of surface tension and becomes increasingly circular, approaching the limiting solution (4.20) as expected. When the solution is calculated for negative time, the reverse happens, with each viscida becoming increasingly elongated and deformed, and a non-local self-intersection occurs when $\unicode[STIX]{x1D70F}\approx -1.16$ .

We note that the set-up shown in figure 7 may equivalently be viewed as a single closed-loop viscida containing two corners, rather than as a system of two viscidas whose ends are joined. Moreover, the initial condition (4.21) corresponds to a viscida centreline that is symmetric under rotation through $\unicode[STIX]{x03C0}$ . As shown by Griffiths & Howell (Reference Griffiths and Howell2007), this rotational invariance implies that the two tensions $A_{1}(\unicode[STIX]{x1D70F})$ and $B_{1}(\unicode[STIX]{x1D70F})$ must be identically zero, and we can therefore obtain the exact solution of the evolution equation (4.16) in this case, namely

(4.22) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=-\unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},0)-\unicode[STIX]{x03C0}(1/2-\unicode[STIX]{x1D709})}{1+\unicode[STIX]{x1D70F}/2}+\unicode[STIX]{x03C0}(1/2-\unicode[STIX]{x1D709}).\end{eqnarray}$$

It is evident in figure 8(a,b) that our two-viscida solutions computed following the general approach described above agree essentially perfectly with the analytical solution (4.22), which is plotted using dotted curves.

Figure 8.

Solutions of the system (4.16), (4.19) for two connected viscidas, plotted using Eulerian coordinates $(x,y)$ , with initial conditions given by (4.21) (a,b) and (4.23) (c,d). The times shown are (a) $\unicode[STIX]{x1D70F}\in \{0,1,2,4,8\}$ , (b) $\unicode[STIX]{x1D70F}\in \{-1.16,-1.1,-1.0,-0.8,-0.4,0\}$ , (c) $\unicode[STIX]{x1D70F}\in \{0,1,2,4,8\}$ , (d) $\unicode[STIX]{x1D70F}\in \{-2/3,-1/2,-1/3,-1/6,0\}$ , with the direction of increasing $\unicode[STIX]{x1D70F}$ indicated by an arrow in each panel. In (a,b), the dotted lines show the analytical result (4.22), which perfectly overlay the solid numerical curves.

Next we demonstrate our method on another two-viscida example, which has reflectional symmetry again, but not rotational symmetry. We impose the initial conditions

(4.23) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709},0)=-\unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709},0)=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\unicode[STIX]{x03C0}}{3}-\left(\sqrt{3}+\frac{\unicode[STIX]{x03C0}}{2}-1\right)\unicode[STIX]{x1D709},\quad & \unicode[STIX]{x1D709}<{\displaystyle \frac{\unicode[STIX]{x03C0}}{2\sqrt{3}+\unicode[STIX]{x03C0}-2}},\\ \displaystyle -\frac{\unicode[STIX]{x03C0}}{6},\quad & \unicode[STIX]{x1D709}\geqslant {\displaystyle \frac{\unicode[STIX]{x03C0}}{2\sqrt{3}+\unicode[STIX]{x03C0}-2}},\end{array}\right.\end{eqnarray}$$

which describe a circular arc merged smoothly with a straight line, as can be seen in the ‘ $t=0$ ’ curves in figure 8(c,d). In this case, one can show that $A_{1}(\unicode[STIX]{x1D70F})$ is identically zero but $B_{1}(\unicode[STIX]{x1D70F})$ is strictly negative, so an analytical solution akin to (4.22) does not exist, and it is necessary to solve the system (4.16), (4.19) numerically.

The resulting solutions plotted in figure 8(c,d) again show the anticipated behaviour. As $\unicode[STIX]{x1D70F}$ increases, the corner angles increase towards $\unicode[STIX]{x03C0}$ and the system approaches a circular arc. As $\unicode[STIX]{x1D70F}$ decreases from zero, the corner angles sharpen, with a cusp forming at $\unicode[STIX]{x1D709}=1$ when $\unicode[STIX]{x1D70F}=-2/3$ , so that self-intersection would occur for $\unicode[STIX]{x1D70F}<-2/3$ .

4.2 A three-viscida network

Next we analyse a network of three viscidas connected in the configuration illustrated in figure 9, with a three-way junction at either end. To describe such a situation, we must solve an evolution equation for each viscida, i.e.

(4.24) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{2}}\right]=A_{i}(\unicode[STIX]{x1D70F})\sin \unicode[STIX]{x1D703}_{i}+B_{i}(\unicode[STIX]{x1D70F})\cos \unicode[STIX]{x1D703}_{i},\end{eqnarray}$$

for $i\in \{1,2,3\}$ . We identify each spatial coordinate $\unicode[STIX]{x1D709}_{i}$ separately for clarity, since each viscida may have a different initial length, say $\ell _{i}$ . Twelve independent boundary conditions are required to solve for the three deflection angles $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709}_{i},\unicode[STIX]{x1D70F})$ and the six tensions $A_{i}(\unicode[STIX]{x1D70F})$ , $B_{i}(\unicode[STIX]{x1D70F})$ .

Figure 9.

A system of three connected viscidas. The top, bottom and middle viscidas have centrelines described by $\unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D70F})$ , $\unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D703}_{3}(\unicode[STIX]{x1D709}_{3},\unicode[STIX]{x1D70F})$ , respectively. For $i\in \{1,2,3\}$ , the interior angles at $\unicode[STIX]{x1D709}_{i}=0$ are given by $\unicode[STIX]{x1D6FC}_{1,2}(\unicode[STIX]{x1D70F})$ , and the angles at $\unicode[STIX]{x1D709}_{i}=\ell _{i}$ are given by $\unicode[STIX]{x1D6FD}_{1,2}(\unicode[STIX]{x1D70F})$ . The two nodes represent the viscida junctions.

Since the ends must meet at the two junctions, we enforce

(4.25) $$\begin{eqnarray}\int _{0}^{\ell _{1}}\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{1}\\ \sin \unicode[STIX]{x1D703}_{1}\end{array}\right)\,\text{d}\unicode[STIX]{x1D709}_{1}=\int _{0}^{\ell _{2}}\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{2}\\ \sin \unicode[STIX]{x1D703}_{2}\end{array}\right)\,\text{d}\unicode[STIX]{x1D709}_{2}=\int _{0}^{\ell _{3}}\left(\begin{array}{@{}c@{}}\cos \unicode[STIX]{x1D703}_{3}\\ \sin \unicode[STIX]{x1D703}_{3}\end{array}\right)\text{d}\unicode[STIX]{x1D709}_{3}.\end{eqnarray}$$

The condition of zero net bending moment at each junction becomes, in this case,

(4.26) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{2}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{3}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{3}}\right)\right]=0\quad \text{at }\unicode[STIX]{x1D709}_{j}=0,\ell _{j}\text{ for }j=1,2,3.\end{eqnarray}$$

Using (3.16) with $n=3$ , we write the evolution of the angles at the corners as

(4.27a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{i}(\unicode[STIX]{x1D70F})=\frac{2\unicode[STIX]{x03C0}}{3}+\frac{\unicode[STIX]{x1D6FC}_{i}(0)-2\unicode[STIX]{x03C0}/3}{1+\unicode[STIX]{x1D70F}/2}=(-1)^{i+1}[\unicode[STIX]{x1D703}_{i}(0,\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}_{3}(0,\unicode[STIX]{x1D70F})], & \displaystyle\end{eqnarray}$$

(4.27b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}_{i}(\unicode[STIX]{x1D70F})=\frac{2\unicode[STIX]{x03C0}}{3}+\frac{\unicode[STIX]{x1D6FD}_{i}(0)-2\unicode[STIX]{x03C0}/3}{1+\unicode[STIX]{x1D70F}/2}=(-1)^{i+1}[\unicode[STIX]{x1D703}_{3}(\ell _{3},\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}_{i}(\ell _{i},\unicode[STIX]{x1D70F})], & \displaystyle\end{eqnarray}$$

$i\in \{1,2\}$

$\unicode[STIX]{x1D6FC}_{1,2}(0)$

$\unicode[STIX]{x1D6FD}_{1,2}(0)$

$\unicode[STIX]{x1D709}_{i}=0,\ell _{i}$

$i\in \{1,2,3\}$

$x$

$y$

$\unicode[STIX]{x1D70F}$

(4.28a,b ) $$\begin{eqnarray}\mathop{\sum }_{i=1}^{3}A_{i}(\unicode[STIX]{x1D70F})=0\quad \text{and}\quad \mathop{\sum }_{i=1}^{3}B_{i}(\unicode[STIX]{x1D70F})=0.\end{eqnarray}$$

Equations (4.25)–(4.28) provide twelve boundary conditions for the system (4.24). However, as in § 4.1, the problem suffers from a rotational indeterminacy. By integrating the evolution equation (4.24) along the length of each viscida from $\unicode[STIX]{x1D709}_{i}=0$ to $\unicode[STIX]{x1D709}_{i}=\ell _{i}$ and applying the force balances (4.28), we see that the moment balance (4.26) is satisfied identically at $\unicode[STIX]{x1D709}_{i}=\ell _{i}$ if it is enforced at $\unicode[STIX]{x1D709}_{i}=0$ . Therefore one more condition must be imposed to determine all of the angles $\unicode[STIX]{x1D703}_{i}(\unicode[STIX]{x1D709}_{i},\unicode[STIX]{x1D70F})$ uniquely, and then a further two conditions are needed to fix the translation of the entire network. The details of the calculation to determine the evolution of the three-viscida network may be found in appendix A.

We illustrate the evolution of a three-viscida network with an initial configuration given by

(4.29a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D703}_{1}(\unicode[STIX]{x1D709}_{1},0)=\unicode[STIX]{x03C0}(1/2-\unicode[STIX]{x1D709}_{1}),\quad \text{for }0\leqslant \unicode[STIX]{x1D709}_{1}\leqslant 1, & \displaystyle\end{eqnarray}$$

(4.29b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D703}_{2}(\unicode[STIX]{x1D709}_{2},0)=\unicode[STIX]{x03C0}(\unicode[STIX]{x1D709}_{2}-1/2),\quad \text{for }0\leqslant \unicode[STIX]{x1D709}_{2}\leqslant 1, & \displaystyle\end{eqnarray}$$

(4.29c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D703}_{3}(\unicode[STIX]{x1D709}_{3},0)=0,\quad \text{for }0\leqslant \unicode[STIX]{x1D709}_{3}\leqslant 2/\unicode[STIX]{x03C0}, & \displaystyle\end{eqnarray}$$

For forward time, the internal angles formed at the junctions evolve towards $2\unicode[STIX]{x03C0}/3$ , while the outer viscidas want to approach circular arcs, but the specified (Lagrangian) lengths are incompatible with such a configuration. The system resolves this frustration by approaching two semi-circular arcs (similar to the initial condition), with the angle adjusting rapidly to satisfy the required angle conditions in boundary layers near the two junctions. The profile for negative time shows the cross-sectional profile that we should begin with to achieve a final circular tube cross-section with a central partitioning viscida.

Figure 10.

Solution of the problem (4.24), (4.25)–(4.28) for three connected viscidas with initial profiles given by (4.29), plotted in Eulerian coordinates for (a) $\unicode[STIX]{x1D70F}\in \{0,1.03,2.53,4.03,6\}$ and (b) $\unicode[STIX]{x1D70F}\in \{-1,-0.75,-0.5,-0.25,0\}$ .

4.3 A general viscida network model

We now combine the ideas presented so far to construct a general framework that describes the evolution of an arbitrary system of $N$ thin viscidas connected at $J$ junctions. In principle the resulting model can ultimately be used to describe the fabrication of a MOF.

As shown schematically in figure 11(a), we describe the shape of a viscida that goes from junction $i$ to junction $j$ (with $1\leqslant i\neq j\leqslant J$ ) by the inclination angle $\unicode[STIX]{x1D703}_{ij}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ . For each of the $N$ such viscidas, we have to solve a version of the evolution equation (4.1), i.e.

(4.30) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\left(1+\frac{\unicode[STIX]{x1D70F}}{2}\right)\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}_{ij}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{2}}\right]=A_{ij}(\unicode[STIX]{x1D70F})\sin \unicode[STIX]{x1D703}_{ij}+B_{ij}(\unicode[STIX]{x1D70F})\cos \unicode[STIX]{x1D703}_{ij},\end{eqnarray}$$

for $\unicode[STIX]{x1D709}\in (0,\ell _{ij})$ , where $\ell _{ij}$ is the viscida’s initial length and $(A_{ij},B_{ij})$ are the scaled tension components. Considering the same viscida instead as going from junction $j$ to junction  $i$ , we deduce the symmetry conditions

(4.31a-c ) $$\begin{eqnarray}A_{ji}(\unicode[STIX]{x1D70F})\equiv -A_{ij}(\unicode[STIX]{x1D70F}),\quad B_{ji}(\unicode[STIX]{x1D70F})\equiv -B_{ij}(\unicode[STIX]{x1D70F}),\quad \ell _{ji}\equiv \ell _{ij},\end{eqnarray}$$

and

(4.32) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{ji}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\equiv \pm \unicode[STIX]{x03C0}+\unicode[STIX]{x1D703}_{ij}(\ell _{ij}-\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F}).\end{eqnarray}$$

Figure 11.

(a) Schematic of a viscida connecting two junctions labelled $i$ and $j$ . (b) A central junction $i$ connected to five others $\{a,b,c,d,e\}$ .

We note in passing that the set-up here appears to assume that there is only one viscida connecting each pair of junctions $(ij)$ and also that there are no viscidas that start and end at the same junction. Such cases can be dealt with either by introducing additional passive junctions (connecting two viscidas with interior angle $\unicode[STIX]{x03C0}$ ) or by defining slightly more cumbersome notation, such as $\unicode[STIX]{x1D703}_{ij}^{k}$ , with the index $k$ enumerating all of the viscidas connecting junctions $i$ and $j$ . In either case, the arguments developed below follow through.

We denote the (anticlockwise) moment exerted on node $i$ by the viscida that connects it to node $j$ by $M_{ij}$ , which may be expressed in the form

(4.33) $$\begin{eqnarray}M_{ij}(\unicode[STIX]{x1D70F})=\left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[(1+\unicode[STIX]{x1D70F}/2)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{ij}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]\right|_{\unicode[STIX]{x1D709}=0}.\end{eqnarray}$$

By integrating (4.30) between $\unicode[STIX]{x1D709}=0$ and $\unicode[STIX]{x1D709}=\ell _{ij}$ , and using the identity (4.32), we find a symmetry condition satisfied by $M_{ij}$ , namely

(4.34) $$\begin{eqnarray}M_{ij}(\unicode[STIX]{x1D70F})+M_{ji}(\unicode[STIX]{x1D70F})\equiv [X_{i}(\unicode[STIX]{x1D70F})-X_{j}(\unicode[STIX]{x1D70F})]\,B_{ij}(\unicode[STIX]{x1D70F})+[Y_{i}(\unicode[STIX]{x1D70F})-Y_{j}(\unicode[STIX]{x1D70F})]\,A_{ij}(\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where $(X_{i},Y_{i})$ and $(X_{j},Y_{j})$ denote the positions of junction $i$ and junction $j$ , respectively.

Now, to impose the corner-evolution equation (3.16), we need to define the relevant interior angles between adjacent viscidas. To this end, let $v_{i}$ be an ordered list, enumerating the indices $j$ of the other junctions which are connected to junction  $i$ , sorted in order of increasing $\unicode[STIX]{x1D703}_{ij}(0,0)$ ; for example, in the scenario sketched in figure 11(b), we could take $v_{i}=\{a,b,c,d,e\}.$ Then the total number of viscidas connected at junction $i$ is $n_{i}=|v_{i}|$ , and the total number of viscidas in the network is

(4.35) $$\begin{eqnarray}N=\frac{1}{2}\mathop{\sum }_{i=1}^{J}n_{i}.\end{eqnarray}$$

It is also clear that $j\in v_{i}$ if and only if $i\in v_{j}$ .

We can now write the interior angles in terms of the differences between the angles corresponding to successive terms in the ordered list $v_{i}$ , i.e.

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{ik}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D703}_{iv_{i}(k+1)}(0,\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}_{iv_{i}(k)}(0,\unicode[STIX]{x1D70F}),\end{eqnarray}$$

where $v_{i}(k)$ denotes the $k$ th element in $v_{i}$ . Finally, applying the corner rule (3.16), we deduce that the angle with which each viscida leaves junction $i$ is given by

(4.37) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{iv_{i}(k)}(0,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E9}_{i}(\unicode[STIX]{x1D70F})+\frac{\unicode[STIX]{x1D703}_{iv_{i}(k)}(0,0)}{1+\unicode[STIX]{x1D70F}/2}+\frac{(k-1)\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}}{n_{i}(1+\unicode[STIX]{x1D70F}/2)},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}_{i}$ is an arbitrary function of $\unicode[STIX]{x1D70F}$ , which measures the net rotation of junction $i$ .

Let us suppose for a moment that we know the position $(X_{i}(\unicode[STIX]{x1D70F}),Y_{i}(\unicode[STIX]{x1D70F}))$ and rotation $\unicode[STIX]{x1D6E9}_{i}(\unicode[STIX]{x1D70F})$ of every junction $i=1,2,\ldots ,J$ . If so, then (4.37) determines the angle $\unicode[STIX]{x1D703}_{ij}$ at both ends of each viscida. In addition, the displacement between each pair of connected junctions is given by

(4.38a,b ) $$\begin{eqnarray}\int _{0}^{\ell _{ij}}\cos \unicode[STIX]{x1D703}_{ij}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D709}=X_{j}(\unicode[STIX]{x1D70F})-X_{i}(\unicode[STIX]{x1D70F}),\quad \int _{0}^{\ell _{ij}}\sin \unicode[STIX]{x1D703}_{ij}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D709}=Y_{j}(\unicode[STIX]{x1D70F})-Y_{i}(\unicode[STIX]{x1D70F}).\end{eqnarray}$$

In total, then, for each pair $(ij)$ with $j\in v_{i}$ , we have the four boundary conditions required to solve the evolution equation (4.30) uniquely for $A_{ij}(\unicode[STIX]{x1D70F})$ , $B_{ij}(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D703}_{ij}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ .

Assuming we have solved for $\unicode[STIX]{x1D703}_{ij}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ , we can then evaluate all of the moments acting at each junction using (4.33). Finally, we impose force and moment balances at each junction, namely

(4.39) $$\begin{eqnarray}\mathop{\sum }_{j\in v_{i}}A_{ij}=\mathop{\sum }_{j\in v_{i}}B_{ij}=\mathop{\sum }_{j\in v_{i}}M_{ij}=0,\end{eqnarray}$$

for every $i=1,2,\ldots ,J$ . On the face of it, equation (4.39) provides the $3J$ equations required to determine the remaining unknowns $X_{i}$ , $Y_{i}$ and $\unicode[STIX]{x1D6E9}_{i}$ which we had temporarily imagined to be given. However, the problem suffers from a by now familiar degeneracy. By virtue of the symmetry conditions (4.31) and (4.34), it is easily shown that

(4.40) $$\begin{eqnarray}\mathop{\sum }_{i,j}A_{ij}\equiv \mathop{\sum }_{i,j}B_{ij}\equiv \mathop{\sum }_{i,j}M_{ij}\equiv 0,\end{eqnarray}$$

and therefore (4.39) actually provides only $3J-3$ independent equations, and there remain three degrees of freedom in the problem. To determine the solution uniquely, it is necessary to eliminate the arbitrary rigid-body motion, for example by fixing the position and rotation of a specific junction, say $X_{1}(\unicode[STIX]{x1D70F})=Y_{1}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E9}_{1}(\unicode[STIX]{x1D70F})=0$ for all $\unicode[STIX]{x1D70F}$ .

We thus argue that the system of equations, boundary conditions and balance laws developed above results in a well-posed evolution problem for the shape of each viscida and the position and rotation of each junction. A general solution strategy is sketched out in appendix B.

We illustrate the general approach by solving for the evolution of a six-viscida tri-junction network, similar to the one shown in figure 3(b). We prescribe the initial configuration

(4.41a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{23}(\unicode[STIX]{x1D709},0)=\frac{2\unicode[STIX]{x03C0}}{3}\unicode[STIX]{x1D709},\quad \unicode[STIX]{x1D703}_{34}(\unicode[STIX]{x1D709},0)=\frac{2\unicode[STIX]{x03C0}}{3}(\unicode[STIX]{x1D709}+1),\quad \unicode[STIX]{x1D703}_{42}(\unicode[STIX]{x1D709},0)=\frac{2\unicode[STIX]{x03C0}}{3}(\unicode[STIX]{x1D709}+2),\end{eqnarray}$$

for $0\leqslant \unicode[STIX]{x1D709}\leqslant 1$ , and

(4.41d-f ) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{13}(\unicode[STIX]{x1D709},0)=\frac{\unicode[STIX]{x03C0}}{6},\quad \unicode[STIX]{x1D703}_{14}(\unicode[STIX]{x1D709},0)=\frac{5\unicode[STIX]{x03C0}}{6},\quad \unicode[STIX]{x1D703}_{15}(\unicode[STIX]{x1D709},0)=\frac{9\unicode[STIX]{x03C0}}{6},\end{eqnarray}$$

for $0\leqslant \unicode[STIX]{x1D709}\leqslant 3/2\unicode[STIX]{x03C0}$ , impose the angle-evolution equation (3.16) at each junction, and ensure that the ends join up by setting

(4.42a ) $$\begin{eqnarray}\displaystyle & (x_{13}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}),y_{13}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}))=(x_{23}(1,\unicode[STIX]{x1D70F}),y_{23}(1,\unicode[STIX]{x1D70F}))=(x_{34}(0,\unicode[STIX]{x1D70F}),y_{34}(0,\unicode[STIX]{x1D70F})),\qquad & \displaystyle\end{eqnarray}$$

(4.42b ) $$\begin{eqnarray}\displaystyle & (x_{14}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}),y_{14}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}))=(x_{34}(1,\unicode[STIX]{x1D70F}),y_{34}(1,\unicode[STIX]{x1D70F}))=(x_{42}(1,\unicode[STIX]{x1D70F}),y_{42}(1,\unicode[STIX]{x1D70F})),\qquad & \displaystyle\end{eqnarray}$$

(4.42c ) $$\begin{eqnarray}\displaystyle & (x_{12}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}),y_{12}(3/2\unicode[STIX]{x03C0},\unicode[STIX]{x1D70F}))=(x_{42}(1,\unicode[STIX]{x1D70F}),y_{42}(1,\unicode[STIX]{x1D70F}))=(x_{23}(0,\unicode[STIX]{x1D70F}),y_{23}(0,\unicode[STIX]{x1D70F})),\qquad & \displaystyle\end{eqnarray}$$

(4.42d ) $$\begin{eqnarray}\displaystyle & (x_{12}(0,\unicode[STIX]{x1D70F}),y_{12}(0,\unicode[STIX]{x1D70F}))=(x_{13}(0,\unicode[STIX]{x1D70F}),y_{13}(0,\unicode[STIX]{x1D70F}))=(x_{14}(0,\unicode[STIX]{x1D70F}),y_{14}(0,\unicode[STIX]{x1D70F})).\qquad & \displaystyle\end{eqnarray}$$

We show the evolution of such a profile for both forward and inverse time in figure 12. The rotational symmetry of the initial condition (4.41) is preserved by the dynamics, so the central struts remain straight and the interior angles at the centre remain equal to $2\unicode[STIX]{x03C0}/3$ for all time. The junctions at the edge become more symmetric as $\unicode[STIX]{x1D70F}$ increases, with all of the interior angles approaching $2\unicode[STIX]{x03C0}/3$ , but become increasingly asymmetric as $\unicode[STIX]{x1D70F}$ becomes increasingly negative. In either case, the evolution of the angles at the junctions imposes forces and moments on the curved viscidas and causes them to lose their initially circular shapes. At $\unicode[STIX]{x1D70F}=-3/2$ , the angles between the radial struts and the curved viscidas become zero, leading to self-intersection for $\unicode[STIX]{x1D70F}<-3/2$ .

Figure 12.

Solution of the problem (4.30), (4.33), (4.37)–(4.39) for six connected viscidas with initial profiles given by (4.41), plotted in Eulerian coordinates for (a) $\unicode[STIX]{x1D70F}\in \{0,0.90,2.15,3.40,5\}$ and (b) $\unicode[STIX]{x1D70F}\in \{-1.5,-1.12,-0.75,-0.37,0\}$ .

5.1 Viscida evolution

Thus far in this paper, we have studied systems of thin viscidas that are joined together in a variety of configurations, to capture the complex network structure of the cross-section of a microstructured optical fibre. However, so far we have assumed throughout that every viscida in our network structure has equal and uniform thickness. In this section we return to the full equation of motion (2.9) to study the dynamics of a viscida with non-uniform initial thickness profile $h_{0}(\unicode[STIX]{x1D709})$ . In many of the real-life examples shown in figure 3, the connecting struts do not have uniform thickness. In particular, the honeycomb structure of figure 3(d) has struts that thicken as they approach junctions.

To explore the possible effects of thickness variations on the dynamics, we first consider the example of a single viscida with free ends at which no force or moment is applied. In this case, the evolution is governed by (2.9) with the tensions $A=B=0$ and zero-moment boundary conditions at the ends,

(5.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left[\frac{h_{0}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D70F}/2}{h_{0}(\unicode[STIX]{x1D709})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]=0\quad \text{at }\unicode[STIX]{x1D709}=0,1.\end{eqnarray}$$

We can therefore integrate (2.9) directly with respect to $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D70F}$ , subject to a specified initial centreline profile $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709},0)=\unicode[STIX]{x1D703}_{0}(\unicode[STIX]{x1D709})$ to give

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=D(\unicode[STIX]{x1D70F})+\int _{0}^{\unicode[STIX]{x1D709}}\frac{h_{0}(\hat{\unicode[STIX]{x1D709}})}{h_{0}(\hat{\unicode[STIX]{x1D709}})+\unicode[STIX]{x1D70F}/2}\unicode[STIX]{x1D703}_{0}^{\prime }(\hat{\unicode[STIX]{x1D709}})\,\text{d}\hat{\unicode[STIX]{x1D709}},\end{eqnarray}$$

where $D(\unicode[STIX]{x1D70F})$ is an integration function. Two further integrations are required to evaluate the centreline profile, which is given by

(5.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle x(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=x(0,\unicode[STIX]{x1D70F})+\int _{0}^{\unicode[STIX]{x1D709}}\frac{h_{0}(\hat{\unicode[STIX]{x1D709}})\cos \unicode[STIX]{x1D703}(\hat{\unicode[STIX]{x1D709}},\unicode[STIX]{x1D70F})}{h_{0}(\hat{\unicode[STIX]{x1D709}})+\unicode[STIX]{x1D70F}/2}\,\text{d}\hat{\unicode[STIX]{x1D709}}, & \displaystyle\end{eqnarray}$$

(5.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle y(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=y(0,\unicode[STIX]{x1D70F})+\int _{0}^{\unicode[STIX]{x1D709}}\frac{h_{0}(\hat{\unicode[STIX]{x1D709}})\sin \unicode[STIX]{x1D703}(\hat{\unicode[STIX]{x1D709}},\unicode[STIX]{x1D70F})}{h_{0}(\hat{\unicode[STIX]{x1D709}})+\unicode[STIX]{x1D70F}/2}\,\text{d}\hat{\unicode[STIX]{x1D709}}. & \displaystyle\end{eqnarray}$$

$D(\unicode[STIX]{x1D70F})$

$x(0,\unicode[STIX]{x1D70F})$

$y(0,\unicode[STIX]{x1D70F})$

(5.4) $$\begin{eqnarray}x(0,\unicode[STIX]{x1D70F})+\int _{0}^{1}\frac{(1-\unicode[STIX]{x1D709})h_{0}(\unicode[STIX]{x1D709})\cos \unicode[STIX]{x1D703}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})}{h_{0}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D70F}/2}\,\text{d}\unicode[STIX]{x1D709}=y(0,\unicode[STIX]{x1D70F})=y(1,\unicode[STIX]{x1D70F})=0.\end{eqnarray}$$

This choice results in the viscida having both ends lying on the $x$ -axis and the centroid at $x=0$ .

We calculate the solutions given by (5.2) and (5.3) for a free viscida with specific initial centreline and thickness configurations, given respectively by

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{0}(\unicode[STIX]{x1D709})=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\unicode[STIX]{x03C0}}{4},\quad & 0\leqslant \unicode[STIX]{x1D709}\leqslant \unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D6FF},\\ \displaystyle \frac{\unicode[STIX]{x03C0}}{4}\left(\frac{\unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D709}}{\unicode[STIX]{x1D6FF}}\right),\quad & \unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D6FF},\\ -\displaystyle \frac{\unicode[STIX]{x03C0}}{4},\quad & \unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D6FF}\leqslant \unicode[STIX]{x1D709}\leqslant 1,\end{array}\right.\end{eqnarray}$$

and

(5.6) $$\begin{eqnarray}h_{0}(\unicode[STIX]{x1D709})=\left\{\begin{array}{@{}ll@{}}1-\unicode[STIX]{x1D6FD},\quad & 0\leqslant \unicode[STIX]{x1D709}\leqslant \unicode[STIX]{x1D709}_{2}-\unicode[STIX]{x1D708},\\ 1+\unicode[STIX]{x1D6FD}\displaystyle \left(\frac{\unicode[STIX]{x1D709}-\unicode[STIX]{x1D709}_{2}}{\unicode[STIX]{x1D708}}\right),\quad & \unicode[STIX]{x1D709}_{2}-\unicode[STIX]{x1D708}<\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{2}+\unicode[STIX]{x1D708},\\ 1+\unicode[STIX]{x1D6FD},\quad & \unicode[STIX]{x1D709}_{2}+\unicode[STIX]{x1D708}\leqslant \unicode[STIX]{x1D709}\leqslant 1.\end{array}\right.\end{eqnarray}$$

This formulation corresponds to an initial viscida whose angle transitions linearly from $\unicode[STIX]{x03C0}/4$ to $-\unicode[STIX]{x03C0}/4$ over a region of width $2\unicode[STIX]{x1D6FF}$ centred at $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{1}$ and whose thickness varies linearly from $1-\unicode[STIX]{x1D6FD}$ to $1+\unicode[STIX]{x1D6FD}$ over a region of width $2\unicode[STIX]{x1D708}$ centred at $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{2}$ . The functions defined by (5.5) and (5.6) are sketched in figure 13. We will take small values of $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D708}$ so that the transitions in both angle and thickness are localized.

Figure 13.

Sketch of the initial condition (a) for the centreline given by (5.5) and (b) for the thickness profile (5.6).

Figure 14.

Evolution of the centreline for a free viscida whose initial angle and thickness profiles are given by (5.5) and (5.6), with $\unicode[STIX]{x1D6FD}=0.3$ and $\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D709}_{2}=1/2$ . (a) We fix $\unicode[STIX]{x1D708}=0$ and vary $\unicode[STIX]{x1D6FF}\in \{0.01,0.08,0.15\}$ (dotted, dot-dashed and solid lines). (b) We fix $\unicode[STIX]{x1D6FF}=0$ and vary $\unicode[STIX]{x1D708}\in \{0.1,0.3,0.5\}$ .

In figure 14 we plot solutions for the evolution of a free viscida, as given by (5.2)–(5.4), for an initial angle and thickness profile given by (5.5) and (5.6). We choose $\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D709}_{2}=1/2$ so that the transitions in angle and thickness both occur close to the central point of the viscida.

In figure 14(a), we set $\unicode[STIX]{x1D708}=0$ , so that the thickness increases abruptly across $\unicode[STIX]{x1D709}=1/2$ , and then gradually decrease the angle smoothing parameter $\unicode[STIX]{x1D6FF}$ . As the viscida evolves, we observe that the effect of smoothing the angle remains localized near $\unicode[STIX]{x1D709}=1/2$ , and the solution appears to approach a well-defined limiting profile containing a corner as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . These observations echo the behaviour found for a uniformly thick viscida by Griffiths & Howell (Reference Griffiths and Howell2007). The thickness variations lead to a loss of symmetry in the centreline profile as $\unicode[STIX]{x1D70F}$ increases, caused by the more rapid retraction of the thinner region under the action of surface tension.

In figure 14(b), we perform the complementary experiment of setting $\unicode[STIX]{x1D6FF}=0$ , so the profile contains a sharp corner at $\unicode[STIX]{x1D709}=1/2$ , while varying the value of $\unicode[STIX]{x1D708}$ . We find that the corner is again pulled towards the left-hand end by capillary retraction, but that the way in which the transition in thickness is smoothed has very little effect on the global viscida behaviour.

Next we explore the influence on the dynamics of the relative positioning of the transition in angle relative to the transition in thickness. We find that the effect is most prominent when the viscida ends are fixed in place, rather than being free. We therefore solve the full evolution equation (2.9) subject to the boundary conditions

(5.7a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D703}(0,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D703}(0,0),\quad \unicode[STIX]{x1D703}(1,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D703}(1,0),\end{eqnarray}$$

(5.7c,d ) $$\begin{eqnarray}x(0,\unicode[STIX]{x1D70F})=0,\quad x(1,\unicode[STIX]{x1D70F})=x(1,0),\end{eqnarray}$$

(5.7e,f ) $$\begin{eqnarray}y(0,\unicode[STIX]{x1D70F})=0,\quad y(1,\unicode[STIX]{x1D70F})=0,\end{eqnarray}$$

corresponding to clamping the end points of the viscida at fixed angles at fixed locations on the $x$ -axis. For the details of the calculations to determine the resulting viscida evolution, see appendix C.

Figure 15.

Evolution in Eulerian coordinates of a profile with rapid local angle and thickness variations at different locations along the viscida. The initial angle and thickness are given by (5.5) and (5.6) with parameter values $\unicode[STIX]{x1D6FF}=0.02$ , $\unicode[STIX]{x1D708}=0.01$ , $\unicode[STIX]{x1D6FD}=0.6$ , $\unicode[STIX]{x1D709}_{1}=1/2$ and $\unicode[STIX]{x1D709}_{2}=0.4$ . The dot-dashed line indicates the centreline and the shaded region depicts the viscida thickness. (a) The forward time evolution with $\unicode[STIX]{x1D70F}=0,0.5$ . (b) The inverse time evolution with $\unicode[STIX]{x1D70F}=-0.5,0$ . The circled region in (a) highlights the deflection generated with time at the thickness transition point.

Figure 15 shows the evolution in both forward and inverse time of an initial profile with a smoothed corner at the central point $\unicode[STIX]{x1D709}=1/2$ and a sharp transition in thickness at a different location $\unicode[STIX]{x1D709}=0.4$ . As above we observe that the thinner section of the viscida retracts more rapidly as $\unicode[STIX]{x1D70F}$ increases, and it is also more susceptible to deformation. Since the effective bending stiffness is proportional to $h^{3}$ , the thicker portion of the viscida approximately preserves its shape, while the thinner portion undergoes significant bending. The effect is particularly marked in a neighbourhood of the thickness transition point, where the thinner section is effectively clamped by the thicker section. For inverse time, the thin part of the viscida rapidly becomes even thinner and longer while, again, being restrained by the thick part, which remains relatively unchanged. Consequently, the thin region adopts a profile akin to a buckled state (Buckmaster et al. Reference Buckmaster, Nachman and Ting1975; Buckmaster & Nachman Reference Buckmaster and Nachman1978). In the manufacture of microstructured optical fibres, however, the original preform is usually constructed by sintering identical glass tubes, so it is unlikely that one would observe large variations in thickness.

5.2 A junction of two viscidas of different thicknesses

Figure 14 shows the evolution of several viscida profiles in which sharp changes in angle and thickness both occur at the same location. Here we consider in detail the limit as the smoothing parameters $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D708}$ both tend to zero, so that the limiting configuration describes a junction between two viscidas of differing thicknesses. As in § 3.1, we scale into the region over which the angle and thickness both vary rapidly via $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{0}^{+}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}$ , and define $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F}):=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})$ and $H_{0}(\unicode[STIX]{x1D702}):=h_{0}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702})$ . We substitute these scalings into (2.9) and retain only leading-order terms in $\unicode[STIX]{x1D6FF}$ to obtain

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\left[(H_{0}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D70F}/2)^{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\left(\frac{H_{0}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D70F}/2}{H_{0}(\unicode[STIX]{x1D702})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\right)\right]=0,\end{eqnarray}$$

which is subject to the initial condition $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702},0)=\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})$ and the matching conditions $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702},\unicode[STIX]{x1D70F})\rightarrow \unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{\pm },\unicode[STIX]{x1D70F})$ as $\unicode[STIX]{x1D702}\rightarrow \pm \infty$ . By integrating (5.8), we obtain the following evolution law for the internal angle $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x03C0}+\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{-},\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709}_{0}^{+},\unicode[STIX]{x1D70F})$ :

(5.9a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F}) & = & \displaystyle \int _{-\infty }^{\infty }\frac{H_{0}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6E9}_{0}^{\prime }(\unicode[STIX]{x1D702})}{H_{0}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D70F}/2}\,\text{d}\unicode[STIX]{x1D702}\end{eqnarray}$$

(5.9b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{H_{+}\unicode[STIX]{x1D6E9}_{+}}{H_{+}+\unicode[STIX]{x1D70F}/2}-\frac{H_{-}\unicode[STIX]{x1D6E9}_{-}}{H_{-}+\unicode[STIX]{x1D70F}/2}-\frac{\unicode[STIX]{x1D70F}}{2}\int _{-\infty }^{\infty }\frac{H_{0}^{\prime }(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})}{(H_{0}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D70F}/2)^{2}}\,\text{d}\unicode[STIX]{x1D702},\end{eqnarray}$$

$H_{\pm }=h_{0}(\unicode[STIX]{x1D709}_{0}^{\pm })=H_{0}(\pm \infty )$

$\unicode[STIX]{x1D6E9}_{\pm }=\unicode[STIX]{x1D703}_{0}(\unicode[STIX]{x1D709}_{0}^{\pm })=\unicode[STIX]{x1D6E9}_{0}(\pm \infty )$

$H_{0}(\unicode[STIX]{x1D702})=1$

$\unicode[STIX]{x1D711}\rightarrow \unicode[STIX]{x03C0}$

$\unicode[STIX]{x1D70F}\rightarrow \infty$

$\unicode[STIX]{x1D70F}$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D70F}$

In the case of a uniformly thick viscida, the corner evolution given by (3.4) depends only on the initial angle $\unicode[STIX]{x1D711}(0)$ and not on the precise way in which the initial angle profile $\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})$ varies from $\unicode[STIX]{x1D6E9}_{-}$ to $\unicode[STIX]{x1D6E9}_{+}$ . In contrast, equation (5.9) shows that, for a non-uniformly thick profile, the evolution of $\unicode[STIX]{x1D711}$ depends on the specific local behaviour of both the thickness profile $H_{0}(\unicode[STIX]{x1D702})$ and the angle variation $\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})$ . We illustrate this dependence by examining the two limiting cases where either the thickness variation occurs much more rapidly than the angle variation or vice versa.

The first limit corresponds to a viscida with a relatively slow change in angle around a smoothed corner, in the midst of which there is an abrupt jump in thickness, which we may take to occur at $\unicode[STIX]{x1D702}=0$ without loss of generality. In this case, the inner thickness profile $H_{0}(\unicode[STIX]{x1D702})$ collapses to a Heaviside function, and the angle-evolution equation (5.9) thus reduces to

(5.10) $$\begin{eqnarray}\unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})=\frac{H_{+}}{H_{+}+\unicode[STIX]{x1D70F}/2}(\unicode[STIX]{x1D6E9}_{+}-\unicode[STIX]{x1D6E9}_{0}(0))+\frac{H_{-}}{H_{-}+\unicode[STIX]{x1D70F}/2}(\unicode[STIX]{x1D6E9}_{0}(0)-\unicode[STIX]{x1D6E9}_{-}).\end{eqnarray}$$

We can interpret (5.10) as the sum of two evolution laws of the form (3.4) for the portions of the inner region in $\unicode[STIX]{x1D702}>0$ and $\unicode[STIX]{x1D702}<0$ , with effective initial thicknesses $H_{+}$ and $H_{-}$ , respectively.

In the reverse limit, the thickness varies relatively smoothly while the angle suddenly switches at a sharp corner at $\unicode[STIX]{x1D702}=0$ , so that $\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})$ may be replaced by a Heaviside function and (5.9) reduces to

(5.11) $$\begin{eqnarray}\unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})=\frac{H_{0}(0)}{H_{0}(0)+\unicode[STIX]{x1D70F}/2}(\unicode[STIX]{x03C0}-\unicode[STIX]{x1D711}(0)).\end{eqnarray}$$

Thus the uniform-thickness evolution law (3.4) is recovered in this case, with initial thickness effectively constant in the inner region, and equal to its value $H_{0}(0)$ at the corner.

Let us discuss the above results in the context of our piecewise linear example, with the initial angle and thickness given by (5.5) and (5.6), respectively. If $\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D709}_{2}$ , then the rapid variations in $\unicode[STIX]{x1D703}_{0}(\unicode[STIX]{x1D709})$ and $h_{0}(\unicode[STIX]{x1D709})$ are centred at the same point, as in the solutions plotted in figure 14, and the configuration approaches a junction between two viscidas of differing thicknesses as $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D708}$ both tend to zero. We have shown that the limiting behaviour depends on the order in which the limits $\unicode[STIX]{x1D6FF}\rightarrow 0$ and $\unicode[STIX]{x1D708}\rightarrow 0$ are taken: the interior angle satisfies (5.10) if $\unicode[STIX]{x1D708}\ll \unicode[STIX]{x1D6FF}\ll 1$ or (5.11) if $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D708}\ll 1$ . In the distinguished limit where $\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D6FF}\ll 1$ , then the integrals in (5.9) must be computed, and the behaviour depends on the entire inner profiles of $\unicode[STIX]{x1D6E9}_{0}(\unicode[STIX]{x1D702})$ and $H_{0}(\unicode[STIX]{x1D702})$ .

As shown in § 3.1, a corner in a viscida with spatially uniform thickness may be analysed by regularizing the corner over a small length scale $\unicode[STIX]{x1D6FF}$ and then letting $\unicode[STIX]{x1D6FF}\rightarrow 0$ . The resulting limiting configuration also describes a junction between two viscidas of uniform and equal thickness, and the same approach also works when the thickness is non-uniform, provided it varies smoothly across the join. Of course, if $\unicode[STIX]{x1D6FF}$ becomes too small, then the asymptotic approximations used to derive the governing equations (2.1)–(2.3) cease to be valid. In principle one should then solve a two-dimensional inner Stokes-flow problem for the behaviour near the corner and match to the viscida model away from the corner. However, as argued by Griffiths & Howell (Reference Griffiths and Howell2007), the outer solution approaches a unique well-defined limit as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , which does not depend on the behaviour in the inner region.

In this section we have found that the same regularization approach does not work for a junction where both the thickness and the angle vary rapidly. The introduction of a smoothing parameter $\unicode[STIX]{x1D6FF}$ does not result in a unique solution being selected as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , and the evolution of the corner depends on the precise local behaviour of both $h$ and $\unicode[STIX]{x1D703}$ . A generalized moment-balance argument analogous to that in § 3.2 does not work either, since there is no well-defined interpretation of the product of distributions involved in the calculation of $M$ if both $h$ and $\unicode[STIX]{x1D703}$ are discontinuous. We conclude that the global behaviour of a junction between viscidas of different thicknesses ultimately depends on the solution in an inner region that is not described by the thin-viscida approximation, and there is no universal corner evolution law for such a configuration. To resolve the indeterminacy one would need to consider an inner Stokes-flow problem whose solution depends on precisely how the two viscidas are joined together.