*Appendix A.1. Two-Dimensional Mass and Momentum Equations for the Case of a Rotating Capillary*

We assume axis-symmetry and an incompressible flow, i.e., *ρ* = *const* and *<sup>∂</sup> ∂φ* = 0, with a finite and non-zero *v*. The mass conservation equation assumes the same form as Equation (6), and many terms of the components of the stress tensor are identical to those defined in Section 2.1.2 for the case of a non-rotating capillary. The only components of the stress tensor that differ are

and

$$\pi\_{r\phi} = \mu \left[ r \frac{\partial}{\partial r} \left( \frac{v}{r} \right] \right)$$

The momentum equations along the radial and the azimuthal direction assume the form

*τφ<sup>z</sup>* = *μ* -*∂v ∂z* 

$$\begin{split} \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} - \frac{v^2}{r} + w \frac{\partial u}{\partial z} \right) &= f\_r - \frac{\partial p}{\partial r} \\ + \frac{2}{r} \frac{\partial}{\partial r} \left( r \mu \left( \frac{\partial u}{\partial r} \right) \right) + \frac{\partial}{\partial z} \left( \mu \left( \frac{\partial w}{\partial r} + \frac{\partial u}{\partial z} \right) \right) - 2\mu \frac{u}{r^2} \\\\ \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + \frac{uv}{r} + w \frac{\partial v}{\partial z} \right) &= f\_\phi + \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \mu \frac{\partial}{\partial r} \left( \frac{v}{r} \right) \right] \\ &\quad + \frac{\partial}{\partial z} \left( \mu \frac{\partial v}{\partial z} \right) \end{split} \tag{A2}$$

while the one along the axial direction is identical to Equation (8). The kinematic and dynamic boundary conditions defined in the previous section for the case of a non-rotating capillary hold for the present case as well. In addition, we have two more tangential boundary conditions

$$\mathbf{s}\_1^T \cdot \mathbf{\pi} \cdot \mathbf{n}\_1 = 0 \tag{A3a}$$

$$\mathbf{s}\_2^T \cdot \mathbf{\tau} \cdot \mathbf{\pi}\_2 = 0 \tag{A3b}$$

in the tangential azimuthal direction. Now, the stress tensor *τ* reads

$$\mathbf{r} = \begin{bmatrix} 2\mu \left(\frac{\partial w}{\partial \overline{z}}\right) - p & \mu \left(\frac{\partial v}{\partial \overline{z}}\right) & \mu \left(\frac{\partial w}{\partial r} + \frac{\partial u}{\partial \overline{z}}\right) \\ \mu \left(\frac{\partial v}{\partial \overline{z}}\right) & 2\mu \left(\frac{\partial v}{\partial r}\right) - p & \mu \left[r \frac{\partial}{\partial r} \left(\frac{v}{r}\right)\right] \\ \mu \left(\frac{\partial w}{\partial r} + \frac{\partial u}{\partial \overline{z}}\right) & \mu \left[r \frac{\partial}{\partial r} \left(\frac{v}{r}\right)\right] & 2\mu \left(\frac{\partial u}{\partial r}\right) - p \end{bmatrix} \tag{A4}$$

and

$$\mathbf{s}\_1^T = \mathbf{s}\_2^T = 1\mathbf{e}\_\phi$$

while *n<sup>i</sup>* and *τ<sup>i</sup>* take the same form as Equations (11c) and (11b) for the case of a non-rotating capillary.

*Appendix A.2. Energy Equation for the Case of a Rotating Capillary*

In the case of a rotating capillary, the energy equation reads

$$\begin{split} \rho c\_{p} \left( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} \right) &= \frac{1}{r} \frac{\partial}{\partial r} \left( kr \frac{\partial T}{\partial r} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) \\ &+ 2\mu \left[ \left( \frac{\partial u}{\partial r} \right)^{2} + \left( \frac{u}{r} \right)^{2} + \left( \frac{\partial w}{\partial z} \right)^{2} \right] + \mu \left[ r \frac{\partial}{\partial r} \left( \frac{v}{r} \right) \right]^{2} \\ &+ \mu \left( \frac{\partial v}{\partial z} \right)^{2} + \mu \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \right]^{2} \end{split} \tag{A5}$$

Employing the Rosseland approximation and using Equation (15), we obtain

$$\begin{split} \rho c\_{p} \left( \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} \right) &= \frac{1}{r} \frac{\partial}{\partial r} \left( k\_{\varepsilon} r \frac{\partial T}{\partial r} \right) \\ + \frac{1}{r} \frac{\partial}{\partial r} \left( \bar{k}\_{r} r \frac{\partial T^{4}}{\partial r} \right) &+ \frac{\partial}{\partial z} \left( k\_{\varepsilon} \frac{\partial T}{\partial z} \right) + \frac{\partial}{\partial z} \left( \bar{k}\_{r} r \frac{\partial T^{4}}{\partial z} \right) \\ + 2\mu \left[ \left( \frac{\partial u}{\partial r} \right)^{2} + \left( \frac{u}{r} \right)^{2} + \left( \frac{\partial w}{\partial z} \right)^{2} \right] &+ \mu \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial r} \right]^{2} \\ &\quad \quad \quad \quad \quad \quad \mu \left[ r \frac{\partial}{\partial r} \left( \frac{v}{r} \right) \right]^{2} + \mu \left( \frac{\partial v}{\partial z} \right)^{2} \end{split} \tag{A6}$$

*Appendix A.3. Non-Dimensionalization of the Governing Equations in the Case of a Rotating Capillary*

We non-dimensionalize Equations (6), (8), (A1), (A2) and (A6) by setting

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

where overbars indicate non-dimensional quantities, *φ* is the azimuthal angle, and Ω is the angular frequency. The continuity and momentum equations in the *z* direction are equal to Equation (19) and Equation (20), respectively. In turn, the equations in the *r* and in the *φ* direction assume the form

$$
\epsilon^2 Re \left( \frac{\partial \overline{u}}{\partial \overline{t}} + \overline{u} \frac{\partial \overline{u}}{\partial \overline{r}} + \overline{w} \frac{\partial \overline{u}}{\partial \overline{z}} - \frac{S^2 \overline{v}^2}{\epsilon^2 \overline{r}} \right) = \epsilon \frac{Re}{Fr^2} - \frac{\partial \overline{p}}{\partial \overline{r}} \tag{A7}
$$

$$
\epsilon + \frac{2}{\overline{r}} \frac{\partial}{\partial \overline{r}} \left[ \overline{r} \overline{\mu} \left( \frac{\partial \overline{u}}{\partial \overline{r}} \right) \right] + \epsilon^2 \frac{\partial}{\partial \overline{z}} \left[ \overline{\mu} \left( \frac{\partial \overline{u}}{\partial \overline{z}} \right) \right] + \frac{\partial}{\partial \overline{z}} \left[ \overline{\mu} \left( \frac{\partial \overline{w}}{\partial \overline{r}} \right) \right] - 2\overline{\mu} \left( \frac{\overline{u}}{\overline{r}^2} \right)
$$

$$
\text{Re} \left( \frac{\partial \overline{v}}{\partial \overline{t}} + \overline{u} \frac{\partial \overline{v}}{\partial \overline{r}} + \overline{w} \frac{\partial \overline{v}}{\partial \overline{z}} + \frac{\overline{u} \ \overline{v}}{\overline{r}} \right) = \tag{A8}
$$

$$
\frac{1}{\varepsilon^2 \overline{r}^2} \frac{\partial}{\partial \overline{r}} \left[ \overline{r}^2 \overline{\mu} \left( \overline{r} \frac{\partial}{\partial \overline{r}} \left( \frac{\overline{v}}{\overline{r}} \right) \right) \right] + \frac{\partial}{\partial \overline{z}} \left[ \overline{\mu} \left( \frac{\partial \overline{v}}{\partial \overline{z}} \right) \right]
$$

The energy equation now reads

$$\begin{split} \epsilon^{2} \text{Pe} \left( \frac{\partial \overline{T}}{\partial \overline{t}} + \overline{u} \frac{\partial \overline{T}}{\partial \overline{r}} + \overline{w} \frac{\partial \overline{T}}{\partial \overline{z}} \right) &= \frac{1}{\overline{r}} \frac{\partial}{\partial \overline{r}} \left( \overline{r} \frac{\partial \overline{T}}{\partial \overline{r}} \right) \\ + \frac{1}{\overline{r}} \frac{\partial}{\partial \overline{r}} \left( \gamma\_{R} \overline{r} \frac{\partial \overline{T}^{4}}{\partial \overline{r}} \right) + \epsilon^{2} \frac{\partial}{\partial \overline{z}} \left( \overline{\frac{\partial \overline{T}}{\partial \overline{z}}} \right) + \epsilon^{2} \frac{\partial}{\partial \overline{z}} \left( \gamma\_{R} \frac{\partial \overline{T}^{4}}{\partial \overline{z}} \right) \\ + 2 \epsilon^{2} Br \overline{\mu} \left[ \left( \frac{\partial \overline{u}}{\partial \overline{r}} \right)^{2} + \left( \frac{\overline{u}}{\overline{r}} \right)^{2} + \left( \frac{\partial \overline{w}}{\partial \overline{z}} \right)^{2} \right] + \epsilon^{2} Br \overline{\mu} \left[ \frac{\partial \epsilon \overline{u}}{\partial z} + \frac{\partial \overline{w}}{\partial \epsilon \overline{r}} \right] \\ + \epsilon^{2} Br\_{R} \overline{\mu} \left( \frac{\partial \overline{v}}{\partial \overline{z}} \right)^{2} + \epsilon^{2} Br\_{R} \overline{\mu} \left[ \overline{r} \frac{\partial}{\partial \overline{r}} \left( \frac{\overline{v}}{\overline{c} \overline{r}} \right) \right]^{2} \end{split} \tag{A9}$$

where

$$Br\_R = \frac{\mu\_s \Omega^2 L^2}{T\_s k\_c}$$

is the Brinkman number associated with the rotational speed and *S* = <sup>Ω</sup>*<sup>L</sup> <sup>W</sup>*<sup>1</sup> . The dimensionless kinematic, normal, and axial tangential boundary conditions are the same as in Equations (23) and (24). The tangential boundary conditions in the azimuthal direction for the inner and outer surface read

$$\frac{\partial \overline{\boldsymbol{\sigma}}}{\partial \overline{\boldsymbol{\tau}}}\Big|\_{\overline{\boldsymbol{\tau}} = \overline{\boldsymbol{h}}\_1} - \frac{\overline{\boldsymbol{v}}}{\overline{\boldsymbol{\tau}}}\Big|\_{\overline{\boldsymbol{\tau}} = \overline{\boldsymbol{h}}\_1} - \epsilon^2 \frac{\partial \overline{\boldsymbol{h}}\_1}{\partial \overline{\boldsymbol{z}}} \frac{\partial \overline{\boldsymbol{w}}}{\partial \overline{\boldsymbol{z}}} = 0 \tag{A10a}$$

$$\frac{\partial \overline{\boldsymbol{\sigma}}}{\partial \overline{\boldsymbol{\tau}}} \Big|\_{\overline{\boldsymbol{\tau}} = \overline{\boldsymbol{h}}\_2} - \frac{\overline{\boldsymbol{v}}}{\overline{\boldsymbol{\tau}}} \Big|\_{\overline{\boldsymbol{\tau}} = \overline{\boldsymbol{h}}\_2} - \epsilon^2 \frac{\partial \overline{\boldsymbol{h}}\_2}{\partial \overline{\boldsymbol{\varepsilon}}} \frac{\partial \overline{\boldsymbol{v}}}{\partial \overline{\boldsymbol{\varepsilon}}} = 0 \tag{A10b}$$

Finally, the dimensionless thermal boundary conditions for the outer and the inner surface are equal to Equations (25).

#### *Appendix A.4. Final Asymptotic Equations for the Case of a Rotating Capillary*

In this case, we expand the unknowns as follows:

$$\begin{aligned} \overline{w} &= \overline{w}\_0(\overline{t}, \overline{z}) + \epsilon^2 \overline{w}\_1(\overline{t}, \overline{z}, \overline{r}) \\ \overline{u} &= \overline{u}\_0(\overline{t}, \overline{z}, \overline{r}) + \epsilon^2 \overline{u}\_1(\overline{t}, \overline{z}, \overline{r}) \\ \overline{v} &= \overline{v}\_0(\overline{t}, \overline{z}, \overline{r}) + \epsilon^2 \overline{v}\_1(\overline{t}, \overline{z}, \overline{r}) \\ \overline{p} &= \overline{p}\_a + \epsilon^2 \overline{P}(\overline{t}, \overline{z}, \overline{r}) \\ \overline{T} &= \overline{T}\_0(\overline{t}, \overline{z}, \overline{r}) + \epsilon^2 \overline{T}\_1(\overline{t}, \overline{z}, \overline{r}) \end{aligned}$$

including the azimuthal component of the velocity field *v*, then use them in the nondimensional continuity (Equation (19)), momentum (Equations (20), (A7) and (A8) ), and energy (Equation (A9)) equations and boundary conditions (Equations (23), (24) and (A10)). We obtain the equations of Voyce et al. [21] by expanding the unknowns up to order 2, as follows:

$$\frac{\partial(\overline{h}\_1^2 \overline{w}\_0)}{\partial \overline{z}} = \frac{2\overline{p}\_0 \overline{h}\_2^2 \overline{h}\_1^2 - 2\overline{\gamma}\overline{h}\_2 \overline{h}\_1(\overline{h}\_2 + \overline{h}\_1) + \operatorname{Re} S^2 \overline{B}^2 \overline{h}\_1^2 \overline{h}\_2^2 (\overline{h}\_2^2 - \overline{h}\_1^2)}{2\overline{\mu}(\overline{h}\_2^2 - \overline{h}\_1^2)}\tag{A11}$$

$$\frac{\partial(\overline{h}\_2^2 \overline{w}\_0)}{\partial \overline{z}} = \frac{2\overline{p}\_0 \overline{h}\_2^2 \overline{h}\_1^2 - 2\overline{\gamma}\overline{h}\_2 \overline{h}\_1(\overline{h}\_2 + \overline{h}\_1) + \operatorname{Re} S^2 \overline{B}^2 \overline{h}\_1^2 \overline{h}\_2^2 (\overline{h}\_2^2 - \overline{h}\_1^2)}{2\overline{\mu}(\overline{h}\_2^2 - \overline{h}\_1^2)}\tag{A12}$$

$$\frac{\partial}{\partial \overline{z}} \left( 3\overline{\mu} \frac{\partial \overline{w}\_0}{\partial \overline{z}} (\overline{h}\_2^2 - \overline{h}\_1^2) + \overline{\gamma} (\overline{h}\_2 + \overline{h}\_1) + \frac{1}{4} Re S^2 \overline{B}^2 (\overline{h}\_2^4 - \overline{h}\_1^4) \right) = 0 \tag{A13}$$

$$\begin{split} \frac{\partial}{\partial \overline{\varepsilon}} \Big( \overline{\mu} \frac{\partial \overline{B}}{\partial \overline{\varepsilon}} (\overline{h}\_{2}^{4} - \overline{h}\_{1}^{4}) \Big) &= \operatorname{Re} \overline{w}\_{0} \Big[ \overline{h}\_{2}^{2} \frac{\partial}{\partial \overline{\varepsilon}} \Big( \overline{h}\_{2}^{2} \overline{B} \Big) - \overline{h}\_{1}^{2} \frac{\partial}{\partial \overline{\varepsilon}} \Big( \overline{h}\_{1}^{2} \overline{B} \Big) \Big] + \overline{h}\_{1}^{2} \overline{h}\_{2}^{2} \overline{B} \frac{\operatorname{Re} \overline{p}\_{0}}{\overline{\mu}} \\ &+ \frac{\operatorname{Re}^{2} S^{2} \overline{h}\_{1}^{2} \overline{h}\_{2}^{2} \overline{B}^{3} \Big( \overline{h}\_{2}^{2} - \overline{h}\_{1}^{2} \Big)}{2 \overline{\mu}} - \left( \overline{h}\_{1} + \overline{h}\_{2} \right) \frac{\operatorname{Re} \overline{\gamma} \overline{B} \overline{h}\_{1} \overline{h}\_{2}}{\overline{\mu}} \end{split} \tag{A14}$$

Equations (A11)–(A14) are the equations for the inner and outer fibre radii, axial momentum, and non-dimensional angular frequency equation, respectively; for more details, see [21]. Additionally, in this case we consider the steady-state situation and neglect the convective inertial terms in Equation (A13). As far as the energy equation is concerned, we rescale the Péclet and Brinkman numbers as follows:

$$\text{Per} = \frac{\overleftarrow{\mathcal{P}}}{\mathfrak{E}^2}, \qquad \qquad \qquad Br = \frac{\overleftarrow{Br}}{\mathfrak{E}^2} \tag{A15} \\ \qquad \qquad \qquad Br\_R = \frac{\overleftarrow{Br}\_R}{\mathfrak{E}^2} \tag{A15}$$

obtaining

$$\begin{split} \tilde{P}\left(\frac{\partial \overline{T}}{\partial \overline{t}} + \overline{u}\frac{\partial \overline{T}}{\partial \overline{r}} + \overline{w}\frac{\partial \overline{T}}{\partial \overline{z}}\right) &= \frac{1}{\overline{r}}\frac{\partial}{\partial \overline{r}}\left(\overline{r}\frac{\partial \overline{T}}{\partial \overline{r}}\right) + \frac{1}{\overline{r}}\frac{\partial}{\partial \overline{r}}\left(\gamma\_R \overline{r}\frac{\partial \overline{T}^4}{\partial \overline{r}}\right) \\ + 2\overline{B}\overline{r}\overline{\mu}\left[\left(\frac{\partial \overline{u}}{\partial \overline{r}}\right)^2 + \left(\frac{\overline{u}}{\overline{r}}\right)^2 + \left(\frac{\partial \overline{w}}{\partial \overline{z}}\right)^2\right] &+ \overline{B}\overline{r}\overline{\mu}\left[\frac{\partial \epsilon\overline{u}}{\partial z} + \frac{\partial \overline{w}}{\partial \epsilon \overline{r}}\right]^2 \\ &+ \overline{B}\overline{r}\_R\overline{\mu}\left(\frac{\partial \overline{w}}{\partial \overline{z}}\right)^2 + \overline{B}\overline{r}\_R\overline{\mu}\left[\overline{r}\frac{\partial}{\partial \overline{r}}\left(\frac{\overline{v}}{\epsilon\overline{r}}\right)\right]^2 \end{split} \tag{A16}$$

Using the coordinate system defined in Equation (40), the axial momentum equation becomes

$$\frac{\partial \overline{w}\_0}{\partial \overline{\zeta}} = \left( \frac{\overline{F} - \overline{\gamma} (\overline{h}\_2 + \overline{h}\_1) - \frac{1}{4} R \epsilon S^2 \overline{B}^2 (\overline{h}\_2^4 - \overline{h}\_1^4)}{3 \overline{\mu} (\overline{h}\_{20}^2 - \overline{h}\_{10}^2) \overline{W}\_0} \right) \overline{w}\_0 \tag{A17}$$

where again the constant *F* is the tension required to pull the fibre. Utilizing the asymptotic expansions defined in this section and changing the variables in Equation (A16) according to Equation (40), we obtain

$$\begin{split} \frac{\partial \overline{T}}{\partial \overline{\zeta}} &= \overline{\Psi} \frac{\partial \overline{T}\_{0}}{\partial \overline{\mathbf{x}}} + \overline{\Omega} \left( \frac{\partial^{2} \overline{T}\_{0}}{\partial \overline{\mathbf{x}}^{2}} \right) + \overline{\Lambda} \left( \frac{\partial \overline{T}\_{0}}{\partial \overline{\mathbf{x}}} \right)^{2} \\ + \frac{\overline{Br} \overline{\mu}}{\overline{w}\_{0}} \left[ 3 \left( \frac{\partial \overline{w}\_{0}}{\partial \overline{\zeta}} \right)^{2} + \left( \frac{2 \overline{A}\_{R}}{\overline{\Theta}^{2}} \right)^{2} \right] + \frac{\overline{Br}\_{R} \overline{\mu}}{\overline{w}\_{0}} \left( \frac{\partial \overline{B}}{\partial \overline{\zeta}} \overline{\Theta} \right)^{2} \end{split} \tag{A18}$$

where

$$\begin{aligned} \overline{\Pi} &= [\overline{\pi}(\overline{h}\_{2} - \overline{h}\_{1}) + \overline{h}\_{1}](\overline{h}\_{2} - \overline{h}\_{1}) \\ \overline{\Theta} &= [\overline{\pi}(\overline{h}\_{2} - \overline{h}\_{1}) + \overline{h}\_{1}] \\ \overline{\Lambda} &= \frac{12\gamma\_{R}\overline{\Gamma}\_{0}^{2}}{\overline{w}\_{0}(\overline{h}\_{2} - \overline{h}\_{1})^{2}} \\ \overline{\Omega} &= \frac{1 + 4\gamma\_{R}\overline{\Gamma}\_{0}^{3}}{\overline{w}\_{0}(\overline{h}\_{2} - \overline{h}\_{1})^{2}} \\ \overline{\Psi} &= \frac{1}{\overline{w}\_{0}} \left[ \frac{1 + 4\gamma\_{R}\overline{\Gamma}\_{0}^{3}}{\overline{\Pi}} - \frac{\overline{A}\_{R}}{\overline{\Pi}} + \overline{P}\frac{\partial\overline{w}\_{0}}{\partial\overline{\zeta}} \frac{\overline{\Theta}}{2(\overline{h}\_{2} - \overline{h}\_{1})} \right] \\ \overline{A}\_{R} &= \frac{2\overline{p}\_{0}\overline{h}\_{2}^{2}\overline{h}\_{1}^{2} - 2\overline{\gamma}\overline{h}\_{2}\overline{h}\_{1}(\overline{h}\_{2} + \overline{h}\_{1}) - \operatorname{Re}S^{2}\overline{B}^{2}(\overline{h}\_{2}^{2}\overline{h}\_{1} - \overline{h}\_{1}^{2}\overline{h}\_{2})}{4\overline{\mu}(\overline{h}\_{2}^{2} - \overline{h}\_{1}^{2})} \end{aligned}$$

Finally, the thermal boundary conditions assume the same form as in Equation (46).
