2.1.3. Energy Equation

The three-dimensional energy equation in cylindrical coordinates assumes the form

$$\begin{aligned} \rho c\_p \left( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} + \frac{v}{r} \frac{\partial T}{\partial \phi} + w \frac{\partial T}{\partial z} \right) &= \frac{1}{r} \frac{\partial}{\partial r} \left( kr \frac{\partial T}{\partial r} \right) \\\ + \frac{1}{r} \frac{\partial}{\partial \phi} \left( \frac{k}{r} \frac{\partial T}{\partial \phi} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) + \beta T \left( \frac{\partial p}{\partial t} + u \frac{\partial p}{\partial r} + \frac{v}{r} \frac{\partial p}{\partial \phi} + w \frac{\partial p}{\partial z} \right) + \Phi \end{aligned} \tag{12}$$

In Equation (12), *k* and *cp* denote the thermal conductivity and the specific heat capacity. In addition, *T*, *β*, and Φ are the temperature, the coefficient of thermal expansion at constant pressure, and the viscous dissipation, respectively. The viscous dissipation Φ reads

$$\begin{split} \frac{\Phi}{\mu} = 2\left[ \left( \frac{\partial u}{\partial r} \right)^2 + \left( \frac{1}{r} \frac{\partial v}{\partial \phi} + \frac{u}{r} \right)^2 + \left( \frac{\partial w}{\partial z} \right)^2 \right] + \left[ \frac{1}{r} \frac{\partial w}{\partial \phi} + \frac{\partial v}{\partial z} \right]^2 \\ + \left[ r \frac{\partial}{\partial r} \left( \frac{v}{r} \right) + \frac{1}{r} \frac{\partial u}{\partial \phi} \right]^2 + \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \right]^2 - \frac{2}{3} (\nabla \cdot q) \end{split} \tag{13}$$

Assuming axis symmetry and constant density, Equation (12) reduces to

$$\begin{split} \rho c\_p \left( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} \right) &= \frac{1}{r} \frac{\partial}{\partial r} \left( kr \frac{\partial T}{\partial r} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) \\ &+ 2\mu \left[ \left( \frac{\partial u}{\partial r} \right)^2 + \left( \frac{u}{r} \right)^2 + \left( \frac{\partial w}{\partial z} \right)^2 \right] + \mu \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \right]^2 \end{split} \tag{14}$$

The left-hand side of the energy equation denotes the change of temperature with time and thermal convection. The right-hand side encompasses thermal conduction and viscous dissipation. Because glass is a transparent material, radiative heat transfer within the fibre is relevant, especially at high temperatures. Nevertheless, we avoid the full description of the problem in detail and opt for the common approximation in which the capillary is optically thick. Thereby, we assume that the capillary thickness is much greater than the absorption length scale. Following Taroni et al. [24], we utilize the Rosseland approximation and add a radiative contribution to the total thermal conductivity, say, *k*(*T*) = *kc* + *kr*(*T*)

$$k\_r(T) = \frac{16n\_0^2 \sigma T^3}{3\chi} \tag{15}$$

where *n*<sup>0</sup> and *χ* are the refractive index and the absorption coefficient, respectively. Therefore, Equation (14) assumes the form

$$\begin{split} \rho c\_p \left( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} \right) &= \frac{1}{r} \frac{\partial}{\partial r} \left( k\_\varepsilon r \frac{\partial T}{\partial r} \right) \\ + \frac{1}{r} \frac{\partial}{\partial r} \left( \bar{k}\_\varepsilon r \frac{\partial T^4}{\partial r} \right) &+ \frac{\partial}{\partial z} \left( k\_\varepsilon \frac{\partial T}{\partial z} \right) + \frac{\partial}{\partial z} \left( \bar{k}\_\varepsilon \frac{\partial T^4}{\partial z} \right) \\ + 2\mu \left[ \left( \frac{\partial u}{\partial r} \right)^2 + \left( \frac{u}{r} \right)^2 + \left( \frac{\partial w}{\partial z} \right)^2 \right] &+ \mu \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \right]^2 \end{split} \tag{16}$$

where ˜ *kr* = *kr*(*T*) <sup>4</sup>*T*<sup>3</sup> . The thermal boundary conditions at the outer glass-air interface read

$$-k\frac{\partial T}{\partial r} = \sigma \epsilon\_r \left( T^4 - T\_f^4 \right) + k\_h (T - T\_a) \tag{17}$$

where *σ* and *<sup>r</sup>* are the Stefan-Boltzmann constant and the specific emissivity of the furnace. In addition, *kh*, *Tf* , and *Ta* are the convective heat exchange coefficient, the furnace, and the ambient temperature, respectively. The boundary condition at the inner surface assumes the form

$$-k\frac{\partial T}{\partial r} = k\_h(T - T\_d) \tag{18}$$

Finally, we impose the temperature at the beginning of the drawing by numerically solving Equation (17) with the left-hand side set equal to zero and specified functional forms of the furnace and ambient temperature.
