We are interested in the spinning of ten thousands of slender glass jets by fast air streams, MN=26\text{,}950. The glass jets form a kind of curtain that interact and crucially affect the surrounding air. The determination of the fluid-fiber-interactions requires in principle the simulation of the three-dimensional multiphase problem with appropriate interface conditions. However, regarding the complexity and enormous computational effort, this is practically impossible. Therefore, we propose a coupling concept for slender bodies and fluid flows that is based on drag force and heat exchange models. In this section we first present the two-way coupling of a single viscous thermal Cosserat rod and the compressible Navier-Stokes equations and then generalize the concept to many rods. Thereby, we choose an invariant formulation in the three-dimensional Euclidian space {\mathbb{E}}^{3}.

Note that we mark all quantities associated to the air flow by the subscript _{⋆} throughout the paper. Moreover, to facilitate the readability of the coupling concept, we introduce the abbreviations **Ψ** and {\mathbf{\Psi}}_{\star} that represent all quantities of the glass jets and the air flow, respectively.

### 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}

(1)

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},\phantom{\rule{1em}{0ex}}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.

#### 2.1.2 Navier-Stokes equations

A compressible air flow in the space-time domain {\Omega}_{t}=\{(\mathbf{x},t)|\mathbf{x}\in \Omega \subset {\mathbb{E}}^{3},t>0\} is described by density {\rho}_{\star}, velocity {\mathbf{v}}_{\star}, temperature {T}_{\star}. Its model equations consist of the balance laws for mass, momentum and energy,

\begin{array}{rl}{\partial}_{t}{\rho}_{\star}+\nabla \cdot ({\mathbf{v}}_{\star}{\rho}_{\star})& =0,\\ {\partial}_{t}({\rho}_{\star}{\mathbf{v}}_{\star})+\nabla \cdot ({\mathbf{v}}_{\star}\otimes {\rho}_{\star}{\mathbf{v}}_{\star})& =\nabla \cdot {\mathbf{S}}_{\star}^{T}+{\rho}_{\star}g{\mathbf{e}}_{\mathbf{g}}+{\mathbf{f}}_{\mathit{jets}},\end{array}

(2)

{\partial}_{t}({\rho}_{\star}{e}_{\star})+\nabla \cdot ({\mathbf{v}}_{\star}{\rho}_{\star}{e}_{\star})={\mathbf{S}}_{\star}:\nabla {\mathbf{v}}_{\star}-\nabla \cdot {\mathbf{q}}_{\star}+{q}_{\mathit{jets}}

supplemented with the Newtonian stress tensor {\mathbf{S}}_{\star}, the Fourier law for heat conduction {\mathbf{q}}_{\star}

\begin{array}{rl}{\mathbf{S}}_{\star}& =-{p}_{\star}\mathbf{I}+{\mu}_{\star}(\nabla {\mathbf{v}}_{\star}+\nabla {\mathbf{v}}_{\star}^{T})+{\lambda}_{\star}\nabla \cdot {\mathbf{v}}_{\star}\mathbf{I},\\ {\mathbf{q}}_{\star}& =-{k}_{\star}\nabla {T}_{\star},\end{array}

as well as thermal and caloric equations of state of a ideal gas

{p}_{\star}={\rho}_{\star}{R}_{\star}{T}_{\star},\phantom{\rule{1em}{0ex}}{e}_{\star}={\int}_{0}^{{T}_{\star}}{c}_{p\star}(T)\phantom{\rule{0.2em}{0ex}}\mathrm{d}T-\frac{{p}_{\star}}{{\rho}_{\star}}

with pressure {p}_{\star} and inner energy {e}_{\star}. The specific gas constant for air is denoted by {R}_{\star}. The temperature-dependent viscosities {\mu}_{\star}{\lambda}_{\star}, heat capacity {c}_{p\star} and heat conductivity {k}_{\star} can be modeled by standard polynomial laws, see, for example, [33, 35]. External loads rise from gravity {\rho}_{\star}g{\mathbf{e}}_{\mathbf{g}} and forces due to the immersed fiber jets {\mathbf{f}}_{\mathit{jets}}. These fiber jets also imply a heat source {q}_{\mathit{jets}} in the energy equation. Appropriate initial and boundary conditions close the 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].

#### 2.2.1 Drag forces - {\mathbf{f}}_{\mathit{air}} vs {\mathbf{f}}_{\mathit{jets}}

Let **Ψ** and {\mathbf{\Psi}}_{\star} represent all glass jet and air flow quantities, respectively. Thereby, {\mathbf{\Psi}}_{\star} is the spatially averaged solution of (2). This delocation is necessary to avoid singularities in the two-way coupling. Then, the drag forces are given by

\begin{array}{rl}{\mathbf{f}}_{\mathit{air}}(s,t)& =\mathcal{F}(\mathbf{\Psi}(s,t),{\mathbf{\Psi}}_{\star}(\mathbf{r}(s,t),t\left)\right),\\ {\mathbf{f}}_{\mathit{jets}}(\mathbf{x},t)& =-{\int}_{I(t)}\delta (\mathbf{x}-\mathbf{r}(s,t))\mathcal{F}(\mathbf{\Psi}(s,t),{\mathbf{\Psi}}_{\star}(\mathbf{x},t))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\\ \mathcal{F}(\mathbf{\Psi},{\mathbf{\Psi}}_{\star})& =\frac{{\mu}_{\star}^{2}}{d{\rho}_{\star}}\mathbf{F}({\mathbf{d}}_{\mathbf{3}},\frac{d{\rho}_{\star}}{{\mu}_{\star}}({\mathbf{v}}_{\star}-\mathbf{v})),\end{array}

where *δ* is the Dirac distribution. By construction, they fulfill the principle that action equals reaction and hence the momentum is conserved, that is,

{\int}_{{I}_{V}(t)}{\mathbf{f}}_{\mathit{air}}(s,t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=-{\int}_{V}{\mathbf{f}}_{\mathit{jets}}(\mathbf{x},t)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\mathbf{x}

for an arbitrary domain *V* and {I}_{V}(t)=\{s\in I(t)|\mathbf{r}(s,t)\in V\}. The (line) force \mathcal{F} acting on a slender body is caused by friction and inertia. It depends on material and geometrical properties as well as on the specific inflow situation. The number of dependencies can be reduced to two by help of non-dimensionalizing which yields the dimensionless drag force **F** in dependence on the jet orientation (tangent) and the dimensionless relative velocity between air flow and glass jet. Due to the rotational invariance of the force, the function

\mathbf{F}:{S}^{2}\times {\mathbb{E}}^{3}\to {\mathbb{E}}^{3}

can be associated with its component tuple F for every representation in an orthonormal basis, that is,

\begin{array}{r}\mathrm{F}:{S}_{{\mathbb{R}}^{3}}^{2}\times {\mathbb{R}}^{3}\to {\mathbb{R}}^{3},\\ \mathrm{F}=({F}_{1},{F}_{2},{F}_{3})\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}\sum _{i=1}^{3}{F}_{i}(\tau ,\mathsf{w}){\mathbf{e}}_{\mathbf{i}}=\mathbf{F}(\sum _{i=1}^{3}{\tau}_{i}{\mathbf{e}}_{\mathbf{i}},\sum _{i=1}^{3}{w}_{i}{\mathbf{e}}_{\mathbf{i}})\end{array}

for every orthonormal basis \{{\mathbf{e}}_{\mathbf{i}}\}.

For F we use the drag model [24] that was developed on top of Oseen and Stokes theory [25–27], Taylor heuristic [28] and numerical simulations and validated with measurements [29–32]. It shows to be applicable for all air flow regimes and incident flow directions. Let \{\mathbf{n},\mathbf{b},\mathit{\tau}\} be the orthonormal basis induced by the specific inflow situation (\mathit{\tau},\mathbf{w}) with orientation *τ* and velocity **w**, assuming \mathbf{w}\nparallel \mathit{\tau}

\begin{array}{rl}\mathbf{n}& =\frac{\mathbf{w}-{w}_{\tau}\mathit{\tau}}{{w}_{n}},\phantom{\rule{1em}{0ex}}\mathbf{b}=\mathit{\tau}\times \mathbf{n},\\ {w}_{\tau}& =\mathbf{w}\cdot \mathit{\tau},\phantom{\rule{1em}{0ex}}{w}_{n}=\sqrt{{\mathbf{w}}^{2}-{w}_{\tau}^{2}}.\end{array}

Then, the force is given by

\begin{array}{rl}\mathbf{F}(\mathit{\tau},\mathbf{w})& ={F}_{n}({w}_{n})\mathbf{n}+{F}_{\tau}({w}_{n},{w}_{\tau})\mathit{\tau},\\ {F}_{n}({w}_{n})& ={w}_{n}^{2}{c}_{n}({w}_{n})={w}_{n}{r}_{n}({w}_{n}),\end{array}

(3)

{F}_{\tau}({w}_{n},{w}_{\tau})={w}_{\tau}{w}_{n}{c}_{\tau}({w}_{n})={w}_{\tau}{r}_{\tau}({w}_{n})

according to the Independence Principle [38]. The differentiable normal and tangential drag functions {c}_{n}{c}_{\tau} are

\begin{array}{rl}{c}_{n}({w}_{n})& =\{\begin{array}{ll}\frac{4\pi}{S{w}_{n}}[1-{w}_{n}^{2}\frac{{S}^{2}-S/2+5/16}{32S}],& {w}_{n}<{w}_{1},\\ exp\left(\sum _{j=0}^{3}{p}_{n,j}{ln}^{j}{w}_{n}\right),& {w}_{1}\le {w}_{n}\le {w}_{2},\\ \frac{2}{\sqrt{{w}_{n}}}+0.5,& {w}_{2}<{w}_{n},\end{array}\\ {c}_{\tau}({w}_{n})& =\{\begin{array}{ll}\frac{4\pi}{(2S-1){w}_{n}}[1-{w}_{n}^{2}\frac{2{S}^{2}-2S+1}{16(2S-1)}],& {w}_{n}<{w}_{1},\\ exp\left(\sum _{j=0}^{3}{p}_{\tau ,j}{ln}^{j}{w}_{n}\right),& {w}_{1}\le {w}_{n}\le {w}_{2},\\ \frac{\gamma}{\sqrt{{w}_{n}}},& {w}_{2}<{w}_{n},\end{array}\end{array}

with S({w}_{n})=2.0022-ln{w}_{n}, transition points {w}_{1}=0.1{w}_{2}=100, amplitude \gamma =2. The regularity involves the parameters {p}_{n,0}=1.6911{p}_{n,1}=-6.7222\cdot {10}^{-1}{p}_{n,2}=3.3287\cdot {10}^{-2}{p}_{n,3}=3.5015\cdot {10}^{-3} and {p}_{\tau ,0}=1.1552{p}_{\tau ,1}=-6.8479\cdot {10}^{-1}{p}_{\tau ,2}=1.4884\cdot {10}^{-2}{p}_{\tau ,3}=7.4966\cdot {10}^{-4}. To be also applicable in the special case of a transversal incident flow \mathbf{w}\parallel \mathit{\tau} and to ensure a realistic smooth force **F**, the drag is modified for {w}_{n}\to 0. A regularization based on the slenderness parameter *δ* matches the associated resistance functions {r}_{n}{r}_{\tau} (3) to Stokes resistance coefficients of higher order for {w}_{n}\ll 1, for details see [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))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\\ \mathcal{Q}(\mathbf{\Psi},{\mathbf{\Psi}}_{\star})& =2{k}_{\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

\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

In case of *k* slender bodies in the air flow, we have {\mathbf{\Psi}}_{\mathbf{i}}, i=1,\dots ,k, representing the quantities of each Cosserat rod, here k=MN. Assuming no contact between neighboring fiber jets, every single jet can be described by the stated rod system (1). Their multiple effect on the air flow is reflected in {\mathbf{f}}_{\mathit{jets}} and {q}_{\mathit{jets}}. The source terms in the momentum and energy equations of the air flow (2) become

\begin{array}{rl}{\mathbf{f}}_{\mathit{jets}}(\mathbf{x},t)& =-\sum _{i=1}^{k}{\int}_{{I}_{i}(t)}\delta (\mathbf{x}-{\mathbf{r}}_{\mathbf{i}}(s,t))\mathcal{F}({\mathbf{\Psi}}_{\mathbf{i}}(s,t),{\mathbf{\Psi}}_{\star}(\mathbf{x},t))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\\ {q}_{\mathit{jets}}(\mathbf{x},t)& =-\sum _{i=1}^{k}{\int}_{{I}_{i}(t)}\delta (\mathbf{x}-{\mathbf{r}}_{\mathbf{i}}(s,t))\mathcal{Q}({\mathbf{\Psi}}_{\mathbf{i}}(s,t),{\mathbf{\Psi}}_{\star}(\mathbf{x},t))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\end{array}