## Abstract

We report the first analytical description of the drawing of fibres with holes that does not require knowledge of the viscosity (or temperature) of the glass in the furnace. The model yields expressions for the size of a hole that is isolated from other holes and small compared to the outer diameter of the fibre, but includes the effects of surface tension, pressurisation and arbitrary viscosity profiles. The effect of viscosity is represented by the fibre draw tension which, unlike viscosity, can readily be measured in practice by the fibre fabricator. The model matches experiments without recourse to any adjustable fitting parameters.

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### Equations (29)

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(1)
$$u(x)=\frac{dx}{dt}=\frac{{r}_{1}^{2}}{{r}^{2}(x)}{u}_{f}\equiv \frac{{u}_{f}}{{\rho}^{2}(x)},$$
(2)
$$\rho (x)=\frac{r(x)}{{r}_{1}}\le 1.$$
(3)
$${\rho}_{0}=\rho (\infty )=\frac{{r}_{2}}{{r}_{1}}=\sqrt{\frac{{u}_{f}}{{u}_{d}}}.$$
(4)
$$F=3\pi {r}^{2}(x)\mu (x)\frac{du}{dx}.$$
(5)
$$\frac{dr}{dx}=\frac{dr}{dt}/\frac{dx}{dt}=-\frac{Fr(x)}{6\pi {u}_{f}{r}_{1}^{2}\mu (x)}.$$
(6)
$$\frac{d\rho}{dx}=-\frac{F\rho (x)}{6\pi {u}_{f}{r}_{1}^{2}\mu (x)},$$
(7)
$$r(x)={r}_{1}\rho (x)=\sqrt{\frac{A(x)}{\pi}},$$
(8)
$$\frac{dR}{dx}=-\frac{FR(x)}{6\pi {u}_{f}{r}_{1}^{2}\mu (x)}.$$
(9)
$${p}_{st}=\frac{\gamma}{R}.$$
(10)
$${p}_{v}=\frac{2\mu}{R}\frac{dR}{dt},$$
(11)
$${p}_{o}={p}_{H}-{p}_{a}.$$
(12)
$$\frac{dR}{dx}=\frac{{\rho}^{2}}{2{u}_{f}\mu}\left(R{p}_{o}-\gamma \right).$$
(13)
$$\frac{dR}{dx}=-\frac{F}{6\pi {u}_{f}{r}_{1}^{2}\mu (x)}\left\{\left[1-\frac{3\pi {r}_{1}^{2}{p}_{o}}{F}{\rho}^{2}(x)\right]R(x)+\frac{3\pi {r}_{1}^{2}\gamma}{F}{\rho}^{2}(x)\right\}.$$
(14)
$$\frac{dR}{d\rho}+\left[\frac{3\pi {r}_{1}^{2}{p}_{o}}{F}\rho -\frac{1}{\rho}\right]R(\rho )=\frac{3\pi {r}_{1}^{2}\gamma}{F}\rho .$$
(15)
$$R(\rho )=\rho {e}^{-E{\rho}^{2}}\left[{e}^{E}{R}_{0}-\frac{3\gamma}{s{\rho}_{0}^{2}}{\displaystyle {\int}_{\rho}^{1}{e}^{E{z}^{2}}dz}\right],$$
(16)
$$s=\frac{F}{\pi {r}_{2}^{2}},$$
(17)
$$E=\frac{3{p}_{o}}{2s{\rho}_{0}^{2}}.$$
(18)
$${\int}_{\rho}^{1}{e}^{E{z}^{2}}dz}=\{\begin{array}{cc}\frac{\sqrt{\pi}}{2\sqrt{E}}\left[erfi\left(\sqrt{E}\right)-erfi\left(\sqrt{E}\rho \right)\right]& E>0\\ \frac{\sqrt{\pi}}{2\sqrt{-E}}\left[erf\left(\sqrt{-E}\right)-erf\left(\sqrt{-E}\rho \right)\right]& E<0\\ 1-\rho & E=0\end{array}.$$
(19)
$${R}_{2}-{R}_{1}=\rho {e}^{E(1-{\rho}^{2})}\left[{R}_{0,2}-{R}_{0,1}\right].$$
(20)
$$\mathrm{ln}\rho (x)=-\frac{F}{6\pi {u}_{f}{r}_{1}^{2}}{\displaystyle {\int}_{-\infty}^{x}\frac{dz}{\mu (z)}}.$$
(21)
$$\mathrm{ln}{\rho}_{0}=-\frac{F}{6\pi {u}_{f}{r}_{1}^{2}}{\displaystyle {\int}_{-\infty}^{\infty}\frac{dz}{\mu (z)}}$$
(22)
$$\mathrm{ln}\rho (x)=\mathrm{ln}{\rho}_{0}\times \frac{{\displaystyle {\int}_{-\infty}^{x}\frac{dz}{\mu (z)}}}{{\displaystyle {\int}_{-\infty}^{\infty}\frac{dz}{\mu (z)}}},$$
(23)
$$F=-\frac{6\pi {u}_{f}{r}_{1}^{2}\mu \mathrm{ln}{\rho}_{0}}{L},$$
(24)
$$\rho (x)={\rho}_{0}{}^{x/L}={e}^{(\mathrm{ln}{\rho}_{0})x/L},$$
(25)
$$a=\frac{R}{\rho {R}_{0}}.$$
(26)
$$S=\frac{{R}_{0}s{\rho}_{0}^{2}}{3\gamma}=\frac{{R}_{0}{\rho}_{0}^{2}F}{3\pi \gamma {r}_{2}^{2}}=\frac{{R}_{0}F}{3\pi \gamma {r}_{1}^{2}}$$
(27)
$$P=ES=\frac{{p}_{0}{R}_{0}}{2\gamma}$$
(28)
$$a(\rho )={e}^{E(1-{\rho}^{2})}-\frac{{e}^{-E{\rho}^{2}}}{S}{\displaystyle {\int}_{\rho}^{1}{e}^{E{z}^{2}}dz},$$
(29)
$$S=\frac{{e}^{-E{\rho}_{0}^{2}}{\displaystyle {\int}_{{\rho}_{0}^{2}}^{1}{e}^{E{z}^{2}}dz}}{{e}^{E(1-{\rho}_{0}^{2})}-{a}_{0}}.$$