Fluid-fiber-interactions in rotational spinning process of glass wool production

2.1 Models for glass jets and air flows

2.1.1 Cosserat rod

A glass jet is a slender body, that is, a rod in the context of three-dimensional continuum mechanics. Because of its slender geometry, its dynamics might be reduced to a one-dimensional description by averaging the underlying balance laws over its cross-sections. This procedure is based on the assumption that the displacement field in each cross-section can be expressed in terms of a finite number of vector- and tensor-valued quantities. In the special Cosserat rod theory, there are only two constitutive elements: a curve specifying the position $\mathbf{r}:Q\to {\mathbb{E}}^3$ and an orthonormal director triad $\{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\}:Q\to {\mathbb{E}}^3$ characterizing the orientation of the cross-sections, where $Q=\{(s,t)\in {\mathbb{R}}^2|s\in I(t)=[0,l(t)],t>0\}$ with arclength parameter s and time t. For a schematic sketch of a Cosserat rod see Figure 3, for more details on the Cosserat theory we refer to [6]. In the following we use an incompressible viscous Cosserat rod model that was derived on basis of the work [20, 34] on viscous rope coiling and investigated for isothermal curved inertial jets in rotational spinning processes [16, 19]. We extend the model by incorporating temperature effects and aerodynamic forces. The rod system describes the variables of jet curve r, orthonormal triad $\{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\}$, generalized curvature κ, convective speed u, cross-section A, linear velocity v, angular velocity ω, temperature T and normal contact forces $\mathbf{n}\cdot {\mathbf{d}}_{\mathit{\alpha }}$$\alpha =1,2$. It consists of four kinematic and four dynamic equations, that is, balance laws for mass (cross-section), linear and angular momentum and temperature,

$\begin{array}{cc}\hfill {\partial }_t\mathbf{r}& =\mathbf{v}-u{\mathbf{d}}_{\mathbf{3}},\hfill \\ \hfill {\partial }_t{\mathbf{d}}_{\mathbf{i}}& =(\mathit{\omega }-u\mathit{\kappa })\times {\mathbf{d}}_{\mathbf{i}},\hfill \\ \hfill {\partial }_s\mathbf{r}& ={\mathbf{d}}_{\mathbf{3}},\hfill \\ \hfill {\partial }_s{\mathbf{d}}_{\mathbf{i}}& =\mathit{\kappa }\times {\mathbf{d}}_{\mathbf{i}},\hfill \\ \hfill {\partial }_{t}A+{\partial }_s(uA)& =0,\hfill \\ \hfill \rho ({\partial }_t(A\mathbf{v})+{\partial }_s(uA\mathbf{v}))& ={\partial }_s\mathbf{n}+\rho Ag{\mathbf{e}}_{\mathbf{g}}+{\mathbf{f}}_{\mathit{air}},\hfill \\ \hfill \rho ({\partial }_t(\mathbf{J}\cdot \mathit{\omega })+{\partial }_s(u\mathbf{J}\cdot \mathit{\omega }))& ={\partial }_s\mathbf{m}+{\mathbf{d}}_{\mathbf{3}}\times \mathbf{n},\hfill \\ \hfill \rho c_p({\partial }_t(AT)+{\partial }_s(uAT))& =q_{\mathit{rad}}+q_{\mathit{air}}\hfill \end{array}$

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supplemented with an incompressible geometrical model of circular cross-sections with diameter d

$\begin{array}{rl}\mathbf{J}& =I({\mathbf{d}}_{\mathbf{1}}\otimes {\mathbf{d}}_{\mathbf{1}}+{\mathbf{d}}_{\mathbf{2}}\otimes {\mathbf{d}}_{\mathbf{2}}+2{\mathbf{d}}_{\mathbf{3}}\otimes {\mathbf{d}}_{\mathbf{3}}),\\ I& =\frac{\pi }{64}{d}^4,A=\frac{\pi }{4}{d}^2\end{array}$

as well as viscous material laws for the tangential contact force $\mathbf{n}\cdot {\mathbf{d}}_{\mathbf{3}}$ and contact couple m

$\begin{array}{rl}\mathbf{n}\cdot {\mathbf{d}}_{\mathbf{3}}& =3\mu A{\partial }_{s}u\\ \mathbf{m}& =3\mu I({\mathbf{d}}_{\mathbf{1}}\otimes {\mathbf{d}}_{\mathbf{1}}+{\mathbf{d}}_{\mathbf{2}}\otimes {\mathbf{d}}_{\mathbf{2}}+\frac{2}{3}{\mathbf{d}}_{\mathbf{3}}\otimes {\mathbf{d}}_{\mathbf{3}})\cdot {\partial }_s\mathit{\omega }.\end{array}$

Rod density ρ and heat capacity $c_p$ are assumed to be constant. The temperature-dependent dynamic viscosity μ is modeled according to the Vogel-Fulcher-Tamman relation, that is, $\mu (T)={10}^{p_1+p_2/(T-p_3)}$ Pa s where we use the parameters $p_1=-2.56$$p_2=4\text{,}289.18$ K and $p_3=(150.74+273.15)$ K, [33]. The external loads rise from gravity $\rho Ag{\mathbf{e}}_{\mathbf{g}}$ with gravitational acceleration g and aerodynamic forces ${\mathbf{f}}_{\mathit{air}}$. In the temperature equation we neglect inner friction and heat conduction and focus exclusively on radiation $q_{\mathit{rad}}$ and aerodynamic heat sources $q_{\mathit{air}}$. The radiation effect depends on the geometry of the plant and is incorporated in the system by help of the simple model

$q_{\mathit{rad}}={\epsilon }_R\sigma \pi d(T_{\mathit{ref}}^4-T^4)$

with emissivity ${\epsilon }_R$, Stefan-Boltzmann constant σ and reference temperature $T_{\mathit{ref}}$. Appropriate initial and boundary conditions close the rod system.

2.2 Models for momentum and energy exchange

The coupling of the Cosserat rod and the Navier-Stokes equations is performed by help of drag forces and heat sources. Taking into account the conservation of momentum and energy, ${\mathbf{f}}_{\mathit{air}}$ and ${\mathbf{f}}_{\mathit{jets}}$ as well as $q_{\mathit{air}}$ and $q_{\mathit{jets}}$ satisfy the principle that action equals reaction and obey common underlying relations. Hence, we can handle the delicate fluid-fiber-interactions by help of two surrogate models, so-called exchange functions, that is, a dimensionless drag force F (inducing ${\mathbf{f}}_{\mathit{air}}$${\mathbf{f}}_{\mathit{jets}}$) and Nusselt number Nu (inducing $q_{\mathit{air}}$$q_{\mathit{jets}}$). For a flow around a slender long cylinder with circular cross-sections there exist plenty of theoretical, numerical and experimental investigations to these relations in literature, for an overview see [24] as well as, for example, [29, 30, 33, 36] and references within. We use this knowledge locally and globalize the models by superposition to apply them to our curved moving Cosserat rod. This strategy follows a Global-from-Local concept [37] that turned out to be very satisfying in the derivation and validation of a stochastic drag force in a one-way coupling of fibers in turbulent flows [24].

2.2.2 Heat sources - $q_{\mathit{air}}$ vs $q_{\mathit{jets}}$

Analogously to the drag forces, the heat sources are given by

$\begin{array}{rl}{q}_{\mathit{air}}(s,t)& =\mathcal{Q}(\mathbf{\Psi }(s,t),{\mathbf{\Psi }}_{\star }(\mathbf{r}(s,t),t\left)\right),\\ q_{\mathit{jets}}(\mathbf{x},t)& =-{\int }_{I(t)}\delta (\mathbf{x}-\mathbf{r}(s,t))\mathcal{Q}(\mathbf{\Psi }(s,t),{\mathbf{\Psi }}_{\star }(\mathbf{x},t))\mathrm{d}s,\\ \mathcal{Q}(\mathbf{\Psi },{\mathbf{\Psi }}_{\star })& =2k_{\star }(T_{\star }-T)\mathrm{Nu}(\frac{{\mathbf{v}}_{\star }-\mathbf{v}}{\parallel {\mathbf{v}}_{\star }-\mathbf{v}\parallel }\cdot {\mathbf{d}}_{\mathbf{3}},\frac{\pi }{2}\frac{d{\rho }_{\star }}{{\mu }_{\star }}\parallel {\mathbf{v}}_{\star }-\mathbf{v}\parallel ,\frac{{\mu }_{\star }{c}_{p\star }}{k_{\star }}).\end{array}$

The (line) heat source $\mathcal{Q}$ acting on a slender body also depends on several material and geometrical properties as well as on the specific inflow situation. The number of dependencies can be reduced to three by help of non-dimensionalizing which yields the dimensionless Nusselt number Nu in dependence of the cosine of the angle of attack, Reynolds and Prandtl numbers. The Reynolds number corresponds to the relative velocity between air flow and glass jet, the typical length is the half jet circumference.

For Nu we use a heuristic model. It originates in the studies of a perpendicular flow around a cylinder [33] and is modified for different inflow directions (angles of attack) with regard to experimental data. A regularization ensures the smooth limit for a transversal incident flow in analogon to the drag model for F in (3). We apply

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$\begin{array}{rcl}{\mathrm{Nu}}_{tu}(\mathrm{Re},\mathrm{Pr})& =& \frac{0.037{\mathrm{Re}}^{0.9}\mathrm{Pr}}{{\mathrm{Re}}^{0.1}+2.443({\mathrm{Pr}}^{2/3}-1)},\\ h(c,\mathrm{Re})& =& \{\begin{array}{ll}c\mathrm{Re}/{\delta }_h,& \mathrm{Re}<{\delta }_h,\\ c,& \mathrm{Re}\ge {\delta }_h.\end{array}\end{array}$

2.3 Generalization to many rods

3.1 Transition to stationarity

3.1.1 Representative spun jet of certain length

For the viscous Cosserat rods (1), the mass flux Q is constant in the stationarity, that is, $uA=Q/\rho =\mathit{const}$. We deal with Ω-adapted linear and angular velocities, ${\mathbf{v}}_{\Omega }=\mathbf{v}-\mathbf{\Omega }\times \mathbf{r}$ and ${\mathit{\omega }}_{\Omega }=\mathit{\omega }-\mathbf{\Omega }$, which fulfill the explicit stationarity relations

${\mathbf{v}}_{\Omega }=u{\mathbf{d}}_{\mathbf{3}},{\mathit{\omega }}_{\Omega }=u\mathit{\kappa }$

resulting from the first two equations of (1). Moreover, fictitious Coriolis and centrifugal forces and associated couples enter the linear and angular momentum equations. Using the material laws we can formulate the stationary rod model in terms of a boundary value problem of first order differential equations. Thereby, we present it in the director basis $\{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}}\}$ for convenience (see (5) and compare to [19] except for the temperature equation). Note that to an arbitrary vector field $\mathbf{z}={\sum }_{i=1}^3{\breve{z}}_i{\mathbf{a}}_{\mathbf{i}}={\sum }_{i=1}^3{z}_i{\mathbf{d}}_{\mathbf{i}}\in {\mathbb{E}}^3$, we indicate the component tuples corresponding to the rotating outer basis and the director basis by $\breve{\mathsf{z}}=({\breve{z}}_1,{\breve{z}}_2,{\breve{z}}_3)\in {\mathbb{R}}^3$ and $\mathsf{z}=(z_1,z_2,z_3)\in {\mathbb{R}}^3$, respectively. The director basis can be transformed into the rotating outer basis by the tensor-valued rotation R, that is, $\mathbf{R}={\mathbf{a}}_{\mathbf{i}}\otimes {\mathbf{d}}_{\mathbf{i}}=R_{ij}{\mathbf{a}}_{\mathbf{i}}\otimes {\mathbf{a}}_{\mathbf{j}}\in {\mathbb{E}}^3\otimes {\mathbb{E}}^3$ with associated orthogonal matrix $\mathsf{R}=(R_{ij})=({\mathbf{d}}_{\mathbf{i}}\cdot {\mathbf{a}}_{\mathbf{j}})\in SO(3)$. Its transpose and inverse matrix is denoted by ${\mathsf{R}}^{\mathrm{T}}$. For the components, $\mathsf{z}=\mathsf{R}\cdot \breve{\mathsf{z}}$ holds. The cross-product $\mathsf{z}\times \mathsf{R}$ is defined as mapping $(\mathsf{z}\times \mathsf{R}):{\mathbb{R}}^3\to {\mathbb{R}}^3$$\mathsf{y}\mapsto \mathsf{z}\times (\mathsf{R}\cdot \mathsf{y})$. Moreover, canonical basis vectors in ${\mathbb{R}}^3$ are denoted by ${\mathsf{e}}_i$$i=1,2,3$, for example, ${\mathsf{e}}_{\mathsf{1}}=(1,0,0)$. Then, the stationary Cosserat rod model stated in the director basis for a spun glass jet reads

$\begin{array}{cc}{\partial }_s\breve{\mathsf{r}}=& {\mathsf{R}}^{\mathrm{T}}\cdot {\mathsf{e}}_{\mathsf{3}},\hfill \\ {\partial }_s\mathsf{R}=& -\kappa \times \mathsf{R},\hfill \\ {\partial }_s\kappa =& -\frac{\rho }{3Q}\frac{\kappa n_3}{\mu }+\frac{4\pi {\rho }^2}{3Q^2}\frac{u}{\mu }{\mathsf{P}}_{3/2}\cdot \mathsf{m},\hfill \\ {\partial }_{s}u=& \frac{\rho }{3Q}\frac{u{n}_3}{\mu },\hfill \\ {\partial }_s\mathsf{n}=& -\kappa \times \mathsf{n}+Qu\kappa \times {\mathsf{e}}_{\mathsf{3}}+\frac{\rho }{3}\frac{u{n}_3}{\mu }{\mathsf{e}}_{\mathsf{3}}+2Q\Omega (\mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}})\times {\mathsf{e}}_{\mathsf{3}}\hfill \\ +Q{\Omega }^2\frac{1}{u}\mathsf{R}\cdot ({\mathsf{e}}_{\mathsf{1}}\times ({\mathsf{e}}_{\mathsf{1}}\times \breve{\mathsf{r}}))+Qg\frac{1}{u}\mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}}-\mathsf{R}\cdot {\breve{\mathsf{f}}}_{\mathit{air}},\hfill \\ {\partial }_s\mathsf{m}=& -\kappa \times \mathsf{m}+\mathsf{n}\times {\mathsf{e}}_{\mathsf{3}}+\frac{\rho }{3}\frac{u}{\mu }{\mathsf{P}}_{\mathsf{3}}\cdot \mathsf{m}-\frac{Q}{12\pi }\frac{n_3}{\mu }{\mathsf{P}}_{\mathsf{2}}\cdot \kappa \hfill \\ -\frac{Q\Omega }{12\pi }\frac{n_3}{\mu u}{\mathsf{P}}_{\mathsf{2}}\cdot (\mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}})-\frac{Q^2\Omega }{4\pi \rho }\frac{1}{u}{\mathsf{P}}_{\mathsf{2}}\cdot (\kappa \times \mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}})\hfill \\ -\frac{Q^2}{4\pi \rho }\frac{1}{u^2}{\mathsf{P}}_{\mathsf{2}}\cdot (u\kappa +\Omega \mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}})\times (u\kappa +\Omega \mathsf{R}\cdot {\mathsf{e}}_{\mathsf{1}}),\hfill \\ {\partial }_{s}T=& \frac{1}{c_{p}Q}(q_{\mathit{rad}}+q_{\mathit{air}})\hfill \end{array}$

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with $q_{\mathit{rad}}=2\sqrt{\pi }{\epsilon }_R\sigma \sqrt{Q/\rho }(T_{\mathit{ref}}^4-T^4)/\sqrt{u}$ and diagonal matrix ${\mathsf{P}}_k=diag(1,1,k)$$k\in \mathbb{R}$. For a spun jet emerging from the centrifugal disk at $s=0$ with stress-free end at $s=L$, the equations are supplemented with

(cf. Table 1). Considering the jet as representative of one spinning row, we choose the nozzle position to be $(H,R,0)$ with respective height H R is here the disk radius. The initialization $\mathsf{R}(0)$ prescribes the jet direction at the nozzle as $({\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}},{\mathbf{d}}_{\mathbf{3}})(0)=({\mathbf{a}}_{\mathbf{1}},-{\mathbf{a}}_{\mathbf{3}},{\mathbf{a}}_{\mathbf{2}})$.

Remark 1 The rotations$\mathsf{R}\in SO(3)$can be parameterized for example in Euler angles or unit quaternions[39]. The last variant offers a very elegant way of rewriting the second equation of (5). Define

$\mathsf{R}(\mathsf{q})=\left(\begin{array}{ccc}{q}_1^2-q_2^2-q_3^2+q_0^2& 2(q_1{q}_2-q_0{q}_3)& 2(q_1{q}_3+q_0{q}_2)\\ 2(q_1{q}_2+q_0{q}_3)& -q_1^2+q_2^2-q_3^2+q_0^2& 2(q_2{q}_3-q_0{q}_1)\\ 2(q_1{q}_3-q_0{q}_2)& 2(q_2{q}_3+q_0{q}_1)& -q_1^2-q_2^2+q_3^2+q_0^2\end{array}\right),$

$\mathsf{q}=(q_0,q_1,q_2,q_3)$

with$\parallel \mathsf{q}\parallel =1$then we have${\partial }_s\mathsf{q}=\mathcal{A}(\kappa )\cdot \mathsf{q}$with skew-symmetric matrix

$\mathcal{A}(\kappa )=\frac{1}{2}\left(\begin{array}{cccc}0& {\kappa }_1& {\kappa }_2& {\kappa }_3\\ -{\kappa }_1& 0& {\kappa }_3& -{\kappa }_2\\ -{\kappa }_2& -{\kappa }_3& 0& {\kappa }_1\\ -{\kappa }_3& {\kappa }_2& -{\kappa }_1& 0\end{array}\right).$

3.1.2 Rotationally invariant air flow

Due to the spinning set-up the jets emerging from the rotating device form row-wise dense curtains. As a consequence of a row-wise homogenization, the air flow (2) can be treated as stationary not only in the rotating outer basis $\{{\mathbf{a}}_{\mathbf{1}}(t),{\mathbf{a}}_{\mathbf{2}}(t),{\mathbf{a}}_{\mathbf{3}}(t)\}$, but also in a fixed outer one. Because of the symmetry with respect to the rotation axis, it is convenient to introduce cylindrical coordinates $(x,r,\varphi )\in \mathbb{R}\times {\mathbb{R}}^{+}\times [0,2\pi )$ for the space and to attach a cylindrical basis $\{{\mathbf{e}}_{\mathbf{x}},{\mathbf{e}}_{\mathbf{r}},{\mathbf{e}}_{\mathit{\varphi }}\}$ with ${\mathbf{e}}_{\mathbf{x}}={\mathbf{a}}_{\mathbf{1}}$ to each space point. The components to an arbitrary vector field $\mathbf{z}\in {\mathbb{E}}^3$ are indicated by $\hat{\mathsf{z}}=(z_x,z_r,z_{\varphi })\in {\mathbb{R}}^3$. Then, taking advantage of the rotational invariance, the stationary Navier-Stokes equations in $(x,r)$ simplify to

$\begin{array}{r}{\partial }_x({\rho }_{\star }{v}_{x\star })+\frac{1}{r}{\partial }_r(r{\rho }_{\star }{v}_{r\star })=0,\\ {\partial }_x({\rho }_{\star }{v}_{x\star }^2)+\frac{1}{r}{\partial }_r(r{\rho }_{\star }{v}_{r\star }{v}_{x\star })\\ =-{\partial }_x{p}_{\star }+{\partial }_x(2{\mu }_{\star }{\partial }_x{v}_{x\star }+{\lambda }_{\star }\nabla \cdot {\hat{\mathsf{v}}}_{\star })\\ +\frac{1}{r}{\partial }_r(r{\mu }_{\star }({\partial }_x{v}_{r\star }+{\partial }_r{v}_{x\star }))-{\rho }_{\star }g+{(f_x)}_{\mathit{jets}},\\ {\partial }_x({\rho }_{\star }{v}_{x\star }{v}_{r\star })+\frac{1}{r}{\partial }_r(r{\rho }_{\star }{v}_{r\star }^2)-\frac{1}{r}{\rho }_{\star }{v}_{\varphi \star }^2\\ =-{\partial }_r{p}_{\star }+{\partial }_x({\mu }_{\star }({\partial }_x{v}_{r\star }+{\partial }_r{v}_{x\star }))\\ +\frac{2}{r}{\partial }_r(r{\mu }_{\star }{\partial }_r{v}_{r\star })+{\partial }_r({\lambda }_{\star }\nabla \cdot {\hat{\mathsf{v}}}_{\star })-\frac{2}{r^2}{\mu }_{\star }{v}_{r\star }+{(f_r)}_{\mathit{jets}},\\ {\partial }_x({\rho }_{\star }{v}_{x\star }{v}_{\varphi \star })+\frac{1}{r}{\partial }_r(r{\rho }_{\star }{v}_{r\star }{v}_{\varphi \star })+\frac{1}{r}{\rho }_{\star }{v}_{r\star }{v}_{\varphi \star }\\ ={\partial }_x({\mu }_{\star }{\partial }_x{v}_{\varphi \star })+\frac{1}{r^2}{\partial }_r\left(r^3{\mu }_{\star }{\partial }_r\right(\frac{1}{r}{v}_{\varphi \star }\left)\right)+{(f_{\varphi })}_{\mathit{jets}},\end{array}$

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with $\nabla \cdot {\hat{\mathsf{v}}}_{\star }={\partial }_x{v}_{x\star }+({\partial }_r(r{v}_{r\star }))/r$ and equipped with appropriate inflow, outflow and wall boundary conditions, cf. Figures 1 and 2.

3.2 Exchange functions

Here, ${\hat{\mathsf{f}}}_{\mathit{jets}}$ and $q_{\mathit{jets}}$ represent the homogenized effect of the N glass jets emerging from the equidistantly placed holes in an arbitrary spinning row. Correspondingly, system (5) with ${\breve{\mathsf{f}}}_{\mathit{air}}$ and $q_{\mathit{air}}$ describes one representative glass jet for this row. To simulate the full problem with all MN glass jets in the air, jet representatives ${\Psi }_i$, $i=1,\dots ,M$ for all M spinning rows with the respective boundary and air flow conditions have to be determined. Their common effect on the air flow is

4.1 Collocation-continuation method for dimensionless rod boundary value problem

Here, $T_{\mathit{ref}}$ and the air flow associated $T_{\star }$ and ${\breve{\mathsf{v}}}_{\mathit{rel}}$ are scaled with θ and U, respectively. System (7) contains the slenderness parameter ϵ ($ϵ\ll 1$) explicity in the equation for the couple $\mathsf{m}$ and is hence no asymptotic model of zeroth order. In the slenderness limit $ϵ\to 0$, the rod model reduces to a string system and their solutions for $(\breve{\mathsf{r}},\breve{\tau },u,N=n_3,T)$ coincide. Only these jet quantities are relevant for the two-way coupling, as they enter in the exchange functions. However, the simpler string system is not well-posed for all parameter ranges, [15, 16]. Thus, it makes sense to consider (7) as ϵ-regularized string system, [19]. We treat ϵ as moderate fixed regularization parameter in the following to stabilize the numerics, in particular we set $ϵ=0.1$.

for $i=0,\dots ,N-1$. The convergence and hence the computational performance of the Newton method depends crucially on the initial guess. Thus, we adapt the initial guess iteratively by help of a continuation strategy. We scale the drag function F with the factor ${\mathrm{C}}_{\mathrm{F}}^{-2}$ and the right-hand side of the temperature equation with ${\mathrm{C}}_T$ and treat Re, Rb, Fr, ℓ${\mathrm{C}}_{\mathrm{F}}$ and ${\mathrm{C}}_T$ as continuation parameters. We start from the solution for $(\mathrm{Re},\mathrm{Rb},\mathrm{Fr},\ell ,{\mathrm{C}}_{\mathrm{F}},{\mathrm{C}}_T)=(1,1,1,0.15,\infty ,0)$ which corresponds to an isothermal rod without aerodynamic forces that has been intensively numerically investigated in [19]. Its determination is straight forward using the related string model as initial guess. Note that we choose ℓ so small to ensure that the glass jet lies in the air flow domain. The actual continuation is then divided into three parts. First, $(\mathrm{Re},\mathrm{Rb},\mathrm{Fr},\mathrm{CF})$ are adjusted, then ${\mathrm{C}}_T$ and finally ℓ. In the continuation we use an adaptive step size control. Thereby, we always compute the interim solutions by help of one step and two half steps and decide with regard to certain quality criteria whether the step size should be increased or decreased.

4.2 Weak iterative coupling algorithm

The ratio $|I_V|/|V|$ can be considered as the jet length density for the cell V. In case of M jet representatives, we deal with $I_{\Delta ,V,i}$ and $|I_{V,i}|$ for $i=1,\dots ,M$. Consequently, we have $I_{\Delta ,V}={\bigcup }_{i=1}^M{I}_{\Delta ,V,i}$ and $|I_V|={\sum }_{i=1}^M|I_{V,i}|$. Note, that the interpolation and averaging approximation strategies have the disadvantage that they are qualitatively different. Thus, momentum and energy conservation are only ensured for very fine resolutions.

Summing up, the algorithm that we use to couple glass jet ${\mathcal{S}}_{\mathit{jets}}$ and air flow ${\mathcal{S}}_{\mathit{air}}$ computations has the form:

Algorithm 1 Generate flow mesh ${\Omega }_h$

Perform flow simulation ${\mathcal{S}}_{\mathit{air}}$ without jets to obtain ${\Psi }_{\star }^{(0)}$

Initialize $k=0$

while $\parallel {\Psi }^{(k)}-{\Psi }^{(k-1)}\parallel >\mathit{tol}$