2.1.4. Non-Dimensionalization

We non-dimensionalize the previous equations by exploiting the slenderness of the geometry, that is, utilizing the ratio = *<sup>h</sup>*20−*h*<sup>10</sup> *<sup>L</sup>* << 1, where *L* is a "hot-zone" length and *h*<sup>20</sup> − *h*<sup>10</sup> is the initial size of a capillary. We set

$$\begin{aligned} r &= \epsilon L \overline{r} & z &= L \overline{\varepsilon} & \overline{u} &= \varepsilon W\_1 \overline{u} \\ w &= W\_1 \overline{w} & h\_1 &= \varepsilon L \overline{h}\_1 & h\_2 &= \varepsilon L \overline{h}\_2 \\ p &= \frac{\mu\_s W\_1}{\varepsilon^2 L} \overline{p} & T &= T\_s \overline{T} & t &= \frac{L}{W\_1} \overline{t} \end{aligned}$$

where overbars indicate non-dimensional quantities, *W*<sup>1</sup> is a typical draw speed, and *Ts* and *μ<sup>s</sup>* are typical glass softening temperature and viscosity at the glass softening temperature, respectively. The continuity equation becomes

$$\frac{1}{\overline{r}}\frac{\partial(\overline{r}\,\overline{u})}{\partial\overline{r}} + \frac{\partial\overline{w}}{\partial\overline{z}} = 0\tag{19}$$

The momentum equation in the *z* direction reads

$$\begin{split} \epsilon^{2} \operatorname{Re} \left( \frac{\partial \overline{w}}{\partial \overline{t}} + \overline{u} \frac{\partial \overline{w}}{\partial \overline{r}} + \overline{w} \frac{\partial \overline{w}}{\partial \overline{z}} \right) &= \epsilon^{2} \frac{\operatorname{Re}}{\operatorname{Fr}^{2}} - \frac{\partial \overline{p}}{\partial \overline{z}} + \frac{1}{\overline{r}} \frac{\partial}{\partial \overline{r}} \left( \overline{r} \overline{\mu} \left( \frac{\partial \overline{w}}{\partial \overline{r}} \right) \right) \\ &+ \epsilon^{2} \frac{1}{\overline{r}} \frac{\partial}{\partial \overline{r}} \left( \overline{r} \overline{\mu} \left( \frac{\partial \overline{u}}{\partial \overline{z}} \right) \right) + 2 \epsilon^{2} \frac{\partial}{\partial \overline{z}} \left( \overline{\mu} \frac{\partial \overline{w}}{\partial \overline{z}} \right) \end{split} \tag{20}$$

while the one in the *r* direction becomes

$$\begin{split} \epsilon^{4} \text{Re}\left(\frac{\partial\overline{u}}{\partial\overline{t}} + \overline{u}\frac{\partial\overline{u}}{\partial\overline{r}} + \overline{w}\frac{\partial\overline{u}}{\partial\overline{z}}\right) &= \epsilon^{3}\frac{\text{Re}}{Fr^{2}} - \frac{\overline{\partial\overline{p}}}{\partial\overline{r}}\\ +2\epsilon\frac{\epsilon^{2}}{\overline{r}}\frac{\partial}{\partial\overline{r}}\left(\overline{r}\overline{\mu}\left(\frac{\partial\overline{u}}{\partial\overline{r}}\right)\right) + \epsilon^{2}\frac{\partial}{\partial\overline{z}}\left(\overline{\mu}\left(\frac{\partial\overline{w}}{\partial\overline{r}}\right)\right) + \epsilon^{4}\frac{\partial}{\partial\overline{z}}\left(\overline{\mu}\left(\frac{\partial\overline{u}}{\partial\overline{z}}\right)\right) - 2\epsilon^{2}\overline{\mu}\left(\frac{\overline{u}}{\overline{r}^{2}}\right) \end{split} \tag{21}$$

where we have set *μ* = *μsμ* and *f* = *ρg*. Moreover,

$$Re = \frac{L\rho W\_1}{\mu\_s}$$

$$Fr = \frac{W\_1^2}{\sqrt{L\mathcal{g}}}$$

where *Re* and *Fr* are the Reynolds and the Froude numbers, respectively. The energy equation assumes the form

$$\epsilon^{2}Pe\left(\frac{\partial\overline{T}}{\partial\overline{t}} + \overline{u}\frac{\partial\overline{T}}{\partial\overline{t}} + \overline{w}\frac{\partial\overline{T}}{\partial\overline{z}}\right) = \frac{1}{\overline{r}}\frac{\partial}{\partial\overline{r}}\left(\overline{r}\frac{\partial\overline{T}}{\partial\overline{r}}\right) + \frac{1}{\overline{r}}\frac{\partial}{\partial\overline{r}}\left(\gamma\_{R}\overline{r}\frac{\partial\overline{T}^{4}}{\partial\overline{r}}\right) + \epsilon^{2}\frac{\partial}{\partial\overline{z}}\left(\frac{\partial\overline{T}}{\partial\overline{z}}\right)$$

$$+ \epsilon^{2}\frac{\partial}{\partial\overline{z}}\left(\gamma\_{R}\frac{\partial\overline{T}^{4}}{\partial\overline{z}}\right) + 2\epsilon^{2}Br\overline{\mu}\left[\left(\frac{\partial\overline{u}}{\partial\overline{r}}\right)^{2} + \left(\frac{\overline{u}}{\overline{r}}\right)^{2} + \left(\frac{\partial\overline{w}}{\partial\overline{z}}\right)^{2}\right] + \epsilon^{2}Br\overline{\mu}\left[\frac{\partial\epsilon\overline{w}}{\partialz} + \frac{\partial\overline{w}}{\partial\epsilon\overline{r}}\right]^{2}$$

where

$$\begin{aligned} Pe &= \frac{L\rho W\_1 c\_P}{k\_\varsigma} \\ Br &= \frac{\mu\_s W\_1^2}{T\_s k\_\varsigma} \\ \gamma\_R &= \frac{4n\_0^2 \sigma T\_s^3}{3\chi k\_\varsigma} \end{aligned}$$

In the previous equation, *Pe* and *Br* are the Peclet and the Brinkman number, respectively. The former denotes the ratio between the convective transport of thermal energy to the fluid to the conduction of thermal energy within the fluid, and the latter is a ratio between the heat originated by mechanical dissipation to the heat transferred by conduction. Furthermore, *γ<sup>R</sup>* is a parameter that indicates the strength of bulk diffusion [24]. The dimensionless kinematic boundary conditions for the inner and the outer surfaces read

$$\frac{\partial \overline{h}\_1}{\partial \overline{t}} + \overline{w} \frac{\partial \overline{h}\_1}{\partial \overline{z}} = \overline{u} \tag{23a}$$

$$\frac{\partial \overline{h}\_2}{\partial \overline{t}} + \overline{w} \frac{\partial \overline{h}\_2}{\partial \overline{z}} = \overline{u} \tag{23b}$$

while the dimensionless dynamic boundary conditions in the normal and tangential direction assume the form

<sup>−</sup>2*μ*<sup>2</sup> *<sup>∂</sup><sup>w</sup> ∂z* - *∂h*<sup>1</sup> *∂z* .2 + 2*μ ∂h*<sup>1</sup> *∂z ∂w ∂r r*=*h*<sup>1</sup> + 22*μ ∂h*<sup>1</sup> *∂z ∂u ∂z* −2*μ ∂u ∂r r*=*h*<sup>1</sup> <sup>+</sup> *<sup>γ</sup> h*1 ⎛ <sup>⎝</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> - *∂h*<sup>1</sup> *∂z* .2 ⎞ ⎠ <sup>=</sup> <sup>1</sup> <sup>2</sup> (*pH* <sup>−</sup> *<sup>p</sup>*) ⎛ <sup>⎝</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> - *∂h*<sup>1</sup> *∂z* .2 ⎞ ⎠ (24a) <sup>−</sup>2*μ*<sup>2</sup> *<sup>∂</sup><sup>w</sup> ∂z* - *∂h*<sup>2</sup> *∂z* .2 + 2*μ ∂h*<sup>2</sup> *∂z ∂w ∂r r*=*h*<sup>2</sup> + 22*μ ∂h*<sup>2</sup> *∂z ∂u ∂z* −2*μ ∂u ∂r r*=*h*<sup>2</sup> − *γ h*2 ⎛ <sup>⎝</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> - *∂h*<sup>2</sup> *∂z* .2 ⎞ ⎠ <sup>=</sup> <sup>1</sup> <sup>2</sup> (*pa* <sup>−</sup> *<sup>p</sup>*) ⎛ <sup>⎝</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> - *∂h*<sup>2</sup> *∂z* .2 ⎞ ⎠ (24b) 2*μ ∂w ∂z ∂h*<sup>1</sup> *<sup>∂</sup><sup>z</sup>* <sup>−</sup> <sup>2</sup>*μ ∂h*<sup>1</sup> *∂z ∂u ∂r r*=*h*<sup>1</sup> +*μ* - *∂h*<sup>1</sup> *∂z* .2- *∂w ∂<sup>r</sup>* <sup>+</sup> *∂u ∂z* − *μ* - *∂w ∂<sup>r</sup>* <sup>+</sup> *∂u ∂z* = 0 (24c)

$$\begin{aligned} 2\overline{\mu}\epsilon \frac{\partial \overline{w}}{\partial \overline{z}} \frac{\partial \overline{h}\_2}{\partial \overline{z}} - 2\overline{\mu}\epsilon \frac{\partial \overline{h}\_2}{\partial \overline{z}} \frac{\partial \overline{u}}{\partial \overline{r}} \Big|\_{\overline{r} = \overline{l}\_{\overline{l}\_2}} \\ + \overline{\mu} \left( \frac{\partial \epsilon \overline{h}\_2}{\partial \overline{z}} \right)^2 \left( \frac{\partial \overline{w}}{\partial \epsilon \overline{r}} + \frac{\partial \epsilon \overline{u}}{\partial \overline{z}} \right) - \overline{\mu} \left( \frac{\partial \overline{w}}{\partial \epsilon \overline{r}} + \frac{\partial \epsilon \overline{u}}{\partial \overline{z}} \right) = 0 \end{aligned} \tag{24d}$$

Finally, the dimensionless thermal boundary conditions for the outer and the inner surface read

$$-\left(1+4\gamma\_R \overline{T}^3 \frac{\partial \overline{T}}{\partial \overline{r}}\right) = \epsilon a(\overline{T}^4 - \overline{T}\_f^4) + \epsilon \beta(\overline{T} - \overline{T}\_a) \tag{25a}$$

$$-\left(1+4\gamma\_R \overline{T}^3\right)\frac{\partial \overline{T}}{\partial \overline{r}} = \epsilon \beta (\overline{T} - \overline{T}\_a) \tag{25b}$$

In Equation (25), *<sup>α</sup>* <sup>=</sup> *σrT*<sup>3</sup> *s L kc* represents the ratio between the radiative and the conductive heat exchange, while *β* = *khL kc* is the Nusselt number, that is, the ratio between convective and conductive heat transfer [24].
