2.1. Controlling parameters of the CS process

In the CS process, under a centrifugal force, a polymer solution or melt emerges from a nozzle of a rapidly rotating reservoir, also known as the spinneret, forming a nanosized curved jet as schematically sketched in figure 1. As the jet emerges from the nozzle (regime 1), it starts to stretch and it develops a time and position dependent trajectory and radius. However, at long times when the fibre touches the collectors, its behaviour becomes steady (regime 2) in a way that the jet trajectory does not change with time anymore, provided that there are no perturbations or jet breakups. To better visualize how the jet is initially developed and reaches the steady state, figure 2 shows the formation of a growing polymer solution jet over time, in our laboratory CS set-up, a detailed description of which can be found in Noroozi et al. (Reference Noroozi, Arne, Larson and Taghavi2020). As observed, at short times, the polymer solution is extruded into a straight jet and starts to bend in the anti-rotation direction. With increasing time, the fibre goes through a range of transient shapes, from a pedant drop, to an anti-S shape jet (Li et al. Reference Li, Lu, Hou, Zhou, Wang, Zhang and Yang2020), to a necking point, etc. Next, the fibre continues to grow until it finally reaches the steady condition where its trajectory no longer changes with time. According to the figure, one can realize that the transient part of the fibre formation in the CS process, i.e. regime 1, is very fast and it merely takes one or two spinning cycles for a growing fibre to turn into a steady jet (regime 2). Although the major part of the process occurs in regime 2, studying regime 1 is also of importance, since it will eventually make the analysis of the jet stability possible. Thus, in this work, we derive a complete set of model equations including the transient forms, to set the stage for future analysis of jet stability.

Figure 2. Experimental observation of a growing fibre over time in a typical CS experiment, using a poly(ethylene oxide) (PEO) solution. The field of view in all images is $140\,{\rm mm}\times 250\,{\rm mm}$. The solution is $7\,\%$wt PEO (with the molecular weight of $M_{v}=9 \times 10^{5}$ (g mol$^{-1}$)), and the experimental parameters are $\varOmega =370$ (rad s$^{-1}$), $s_0=8$ (cm) and $a=1$ (mm). The times with respect to the beginning of the experiment are given in seconds in the corner of each image.

During jet flight time, many parameters affect the jet dynamic behaviour and, consequently, its size (radius) and shape; these parameters may include rotational (comprising centrifugal and Coriolis effects), viscoelastic, inertial, gravitational, aerodynamic (air drag), surface tension and jet temperature variation effects. In this study, the effects of these parameters will be taken into account in our model, through several dimensionless numbers as the input parameters of the model, listed in Table 1. These dimensionless numbers quantify the aforementioned phenomena and forces. In particular, $Rb$, $Fr$, $Re$, $We$, $Pe^*$ and $Re^*$ can be interpreted as the inverse dimensionless rotation rate, gravitational acceleration, polymer solution viscosity, surface tension, air thermal diffusivity and air viscosity, respectively, provided that the flow rate, density and geometrical parameters are constant. Also, $Wi$ is similarly a representative of the polymer solution/melt relaxation time, $\lambda$.

Table 1. Definitions of the dimensionless groups along with their typical ranges in a typical CS process. The subscript ‘$noz$’ marks the polymer solution/melt jet parameters at the nozzle exit and the dimensional and dimensionless parameters describing properties in the air are marked with an asterisk ($*$). We also use the subscript ‘$p$’ to mark the polymer properties and ‘$s$’ to mark the solvent properties. Here, $\rho$ stands for the density, $U$ the velocity, $\bar {\sigma }$ the surface tension, $\mu =\mu _{s}+\mu _{p}$ the zero-shear viscosity (with $\mu _s$ and $\mu _p$ being the solvent and polymer contributions to the zero-shear viscosity, respectively), $\lambda$ the relaxation time, $\theta$ the temperature, ${c_{p}}$ the specific heat capacity, $k$ the conductivity and $\mathrm {g}$ the gravitational acceleration. In addition to the parameters presented, our steady state and transient models will respectively use $\ell =L/s_0$ and $\tau _{end}$ (the end time, made dimensionless using the characteristic time $s_0/U_{noz}$), as additional input parameters.

To preserve the generality of the results, we present all the variables and equations in their dimensionless forms throughout this study. To do so, we use the nozzle diameter ($a$) to scale the fibre radius and we use $s_0$ as a length scale, $s_0/U_{noz}$ as the time scale, $U_{noz}$ as the velocity scale, $U_{noz}/s_0$ as the angular velocity scale, $A_{ref}={\rm \pi} a^2/4$ as the reference area, $\mu _{noz} U_{noz}/s_0$ as the stress scale and, finally, $\theta _{noz}$ as the temperature scale.

2.2. Governing equations

The CS of a viscoelastic fibre can be described using the transient three-dimensional conservation equations, along with viscoelastic constitutive equations of the polymer solution/melt. To write out the governing equations, we treat the jet as a single incompressible phase and ignore the effects of the solvent evaporation on the fibre behaviour. Therefore, the dimensionless continuity, momentum, angular momentum, energy and Giesekus viscoelastic constitutive equations can be represented respectively as

(2.1)$$\begin{gather} {\partial _t}\bar \rho + \boldsymbol{\nabla} \boldsymbol{\cdot} \left({\bar \rho {\boldsymbol{v}}}\right) = 0, \end{gather}$$

(2.2)$$\begin{gather}{\partial _t}{\boldsymbol{v}} + {\boldsymbol{v}} \boldsymbol{\cdot}\boldsymbol{\nabla} {\boldsymbol{v}} = \boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{\varPi }}+{\boldsymbol{F}}, \end{gather}$$

(2.3)$$\begin{gather}{\partial _t}\left( {{\boldsymbol{d}} \times {\boldsymbol{v}}} \right) + {\boldsymbol{v}}\boldsymbol{\cdot}\boldsymbol{\nabla} \left({{\boldsymbol{d}} \times {\boldsymbol{v}}} \right) = \boldsymbol{\nabla}\boldsymbol{\cdot}\left( {{\boldsymbol{d}} \times {\boldsymbol{\varPi }}} \right) + {\boldsymbol{d}} \times {\boldsymbol{F}}, \end{gather}$$

(2.4)$$\begin{gather}{\partial _t}\theta + {\boldsymbol{v}}\boldsymbol{\cdot}\boldsymbol{\nabla} \theta = \frac{1}{{Pe}}{\nabla^2}\theta, \end{gather}$$

(2.5)$$\begin{gather}\bar{\lambda}Wi\left( {{\partial _t}{\boldsymbol{\mathsf{S}}} + {\boldsymbol{v}}\boldsymbol{\cdot}\boldsymbol{\nabla}{\boldsymbol{\mathsf{S}}} - \left({{\boldsymbol{\mathsf{S}}}\boldsymbol{\cdot}\boldsymbol{\nabla} {\boldsymbol{v}} + {{\left( {\boldsymbol{\nabla} {\boldsymbol{v}}} \right)}^{\rm T}}\boldsymbol{\cdot}{\boldsymbol{\mathsf{S}}}} \right)}\right) + \left({\frac{{\bar{\lambda} Wi\chi }}{{(1 - {\delta _s})}}{\boldsymbol{\mathsf{S}}} + {\boldsymbol{I}}} \right)\boldsymbol{\cdot}{\boldsymbol{\mathsf{S}}} = 2(1 - {\delta _s}){\boldsymbol{\gamma }}, \end{gather}$$

in which $\boldsymbol {v}$ denotes the velocity vector, $\bar {\rho }=\rho (\theta )/\rho _{noz}$ the relative density, $\boldsymbol {d}$ the position vector, $\boldsymbol {\varPi }$ the stress tensor (yet to be determined), $\boldsymbol{\mathsf{S}}$ the extra stress tensor related to viscoelasticity, $\theta$ the dimensionless jet temperature, $\boldsymbol {\gamma }$ the strain rate tensor, $\chi$ the mobility factor, $\bar {\lambda }=\lambda (\theta )/\lambda _{noz}$ the relative relaxation time, $\delta _s$ the viscosity ratio; also, $\bar {\mu }=\mu _p(\theta )/\mu _{pnoz}$ stands for the relative polymer zero-shear viscosity; see table 1 for the definitions of the dimensionless numbers. We also define $Wi=\lambda _{noz} U_{noz}/s_0$ in which $\lambda _{noz}$ is the polymer relaxation time at the nozzle. Finally, $\boldsymbol {F}$ stands for the external force vectors consisting of the Coriolis, centrifugal, gravitational and drag forces, defined as

(2.6)\begin{equation} {\boldsymbol{F}} ={-} \frac{2}{{Rb}}{\boldsymbol{\varOmega }} \times {\boldsymbol{v}} - \frac{1}{{R{b^2}}}{\boldsymbol{\varOmega }} \times \left( {{\boldsymbol{\varOmega }} \times {\boldsymbol{d}}} \right)+\frac{{\boldsymbol{g}}}{{F{r^2}}} + {{\boldsymbol{F}}_{{drag}}}, \end{equation}

where $\boldsymbol {\varOmega }=(1,0,0)$ denotes the angular velocity vector, $\boldsymbol {g}=(-1,0,0)$ is the gravity acceleration vector and $\boldsymbol {F}_{{drag}}$ stands for the aerodynamic drag force vector, yet to be defined. Solving the full 3-D set of equations coupled with the viscoelastic constitutive relations and the boundary conditions at the jet free surface (to be defined) is not feasible, in a transient or even stationary frame. Therefore, in this work, we define the variables based on the cross-averaged values in the axial direction and then develop a set of quasi-1-D equations using the asymptotic estimations of the key parameters. Prior to deriving the governing equations, however, we need to define some geometrical and kinematic relations to frame an unsteady viscoelastic curved jet, presented in the following.

2.3. Coordinate systems and basis vectors

In this section, we introduce the coordinate systems and their corresponding bases to frame our unsteady curved jet. To this aim, we use 3-tuple $(n_1, n_2, s)$ as our curvilinear coordinate system so that $s$ defines the arc length and $(n_1, n_2)$ defines the jet cross-section as rectangular coordinates normal to the jet baseline. When defining the jet cross-section, we can also find it helpful to use polar coordinates $(r, \varphi )$ whose relations with $(n_1, n_2)$ are

(2.7)\begin{equation} \left.\begin{gathered} r = \sqrt {n_1^2 + n_2^2},\quad \varphi = {\tan ^{ - 1}} \left( {\frac{{{n_1}}}{{{n_2}}}} \right),\quad \text{or} \\ {n_1} = r\sin (\varphi ),\quad {n_2} = r\cos (\varphi). \end{gathered}\right\} \end{equation}

In this study, whenever more appropriate, we use the polar coordinates to describe the jet cross-section using back-and-forth transformation relations expressed in (2.7).

Next, we present the basis vector expressions to define the baseline of the jet. We choose the Cartesian coordinate system as an outer frame to define the jet baseline, so that the nozzle, from which the jet emerges, is fixed. Given this, we can introduce a time-dependent three-dimensional curve function

(2.8)\begin{equation} \boldsymbol{D}(s,t)= X(s,t)\boldsymbol{\hat x } + Y(s,t)\boldsymbol{\hat y} + Z(s,t)\boldsymbol{\hat z}, \end{equation}

as the jet baseline position in which $t$ is time and $(\boldsymbol {\hat x}, \boldsymbol {\hat y}, \boldsymbol {\hat z})$ are the Cartesian basis vectors. We also use the Bishop basis vectors $(\boldsymbol {N_1}, \boldsymbol {N_2}, \boldsymbol {T})$, which are orthonormal, as the moving frame of reference for our curved jet centreline and derive the final uniaxial equations. In this work, we use the Bishop basis vectors instead of the commonly used Frenet basis vectors to frame our curved baseline for two reasons: first, we would like to exclude the baseline torsion from our calculation when deriving our set of equations and, second, we would like to ease the cumbersome derivation procedure of the governing equations in the previous works in this area; see for instance Noroozi et al. (Reference Noroozi, Arne, Larson and Taghavi2020) or Shikhmurzaev & Sisoev (Reference Shikhmurzaev and Sisoev2017). Furthermore, to simplify the basis vector derivatives with respect to time and space and also the kinematic expressions (yet to be introduced), it may be a good idea to use the fibre angles ($\alpha, \beta$; see figure 1), to compute the baseline function derivatives as

(2.9)\begin{equation} \left.\begin{gathered} {X_{,s}} = \cos (\beta),\\ {Y_{,s}} = \cos (\alpha )\sin (\beta),\\ {Z_{,s}} = \sin (\alpha )\sin (\beta), \end{gathered}\right\} \end{equation}

which automatically satisfy the arc length condition expressed as

(2.10)\begin{equation} X_{,s}^2 + Y_{,s}^2 + Z_{,s}^2 = 1. \end{equation}

It is noted that, herein, for convenience we use $H_{,s}$ or $H_{,t}$ as the partial derivative of an arbitrary function $H(s, t)$ with respect to $s$ or $t$. We also define ${\langle {H} \rangle _T}$ as the projection of a given vector $\boldsymbol {H}$ onto an arbitrary base vector such as $\boldsymbol {T}$. We further introduce the baseline curvature components, i.e. $\kappa _i,\ i=1, 2, 3$, determined as

(2.11a–c)\begin{equation} {\kappa _1} = {\beta _{,s}},\quad {\kappa _2} = {\alpha _{,s}}\sin (\beta ),\quad {\kappa _3} = {\alpha _{,s}}\cos (\beta ). \end{equation}

The two former components are also known as the Bishop curvatures. Using (2.11a–c) the baseline curvature $\kappa$ and torsion $\mathcal {T}$ can be obtained as

(2.12)$$\begin{gather} \kappa= \sqrt {\kappa _1^2 + \kappa _2^2}, \end{gather}$$

(2.13)$$\begin{gather}\mathcal{T} = {\kappa _3} + {\left| \kappa \right|^{ - 2}} \left( {{\kappa _1}{\kappa _{2,s}} - {\kappa _2}{\kappa _{1,s}}} \right). \end{gather}$$

Now, using the aforementioned expressions, we can define the Bishop basis vectors $({\boldsymbol {N}}_1, {\boldsymbol {N}}_2, \boldsymbol {T})$ as (Bishop Reference Bishop1975):

(2.14)$$\begin{gather} {{\boldsymbol{N}}_1} = {\boldsymbol{B}}\frac{{{\kappa _2}}}{\kappa } - {\boldsymbol{N}} \frac{{{\kappa _1}}}{\kappa }, \end{gather}$$

(2.15)$$\begin{gather}{{\boldsymbol{N}}_2} ={-}{\boldsymbol{B}}\frac{{{\kappa _1}}}{\kappa } - {\boldsymbol{N}} \frac{{{\kappa _2}}}{\kappa }, \end{gather}$$

(2.16)$$\begin{gather}{\boldsymbol{T}} = {\boldsymbol{T}}, \end{gather}$$

in which $(\boldsymbol {T}, \boldsymbol {N}, \boldsymbol {B})$ are the Frenet basis vectors, i.e. the tangent, normal and binormal unit vectors, respectively, defined as

(2.17)$$\begin{gather} \boldsymbol{T} = \frac{{{\rm d}\boldsymbol{D}}}{{{\rm d} s}} = {X_{,s}}\boldsymbol{\hat x }+ {Y_{,s}}\boldsymbol{\hat y} + {Z_{,s}}\boldsymbol{\hat z}, \end{gather}$$

(2.18)$$\begin{gather}\boldsymbol{N} = \frac{{{\rm d}\boldsymbol{T}}}{{{\rm d} s}}{\left|{\frac{{{\rm d}\boldsymbol{T}}}{{{\rm d} s}}} \right|^{ - 1}} = \frac{{{X_{,ss}}\boldsymbol{\hat x} + {Y_{,ss}}\boldsymbol{\hat y } + {Z_{,ss}}\boldsymbol{\hat z }}}{{\sqrt {{{\left( {{X_{,ss}}} \right)}^2} + {{\left( {{Y_{,ss}}}\right)}^2} + {{\left( {{Z_{,ss}}} \right)}^2}} }}, \end{gather}$$

(2.19)$$\begin{gather}\boldsymbol{B} = \boldsymbol{T} \times \boldsymbol{N} = \frac{{\left({{Y_{,s}}{Z_{,ss}} - {Y_{,ss}}{Z_{,s}}} \right)\boldsymbol{\hat x } + \left( {{Z_{,s}}{X_{,ss}} - {Z_{,ss}}{X_{,s}}} \right)\boldsymbol{\hat y } + \left( {{X_{,s}}{Y_{,ss}} - {X_{,ss}}{Y_{,s}}} \right) \boldsymbol{\hat z }}}{{\sqrt {{{\left({{X_{,ss}}} \right)}^2} + {{\left( {{Y_{,ss}}} \right)}^2} + {{\left({{Z_{,ss}}} \right)}^2}} }}. \end{gather}$$

Using (2.9), (2.11a–c) and (2.14)–(2.19), we can find

(2.20)\begin{equation} \left.\begin{gathered} {{\boldsymbol{N}}_1} = \left( {\sin (\beta )} \right){\boldsymbol{\hat x}} - (\cos (\alpha )\cos (\beta )){\boldsymbol{\hat y}}- \left( {\sin (\alpha )\cos (\beta )} \right){\boldsymbol{\hat z}}, \\ {{\boldsymbol{N}}_2} = (\sin (\alpha )){\boldsymbol{\hat y}} - \left( { \cos (\alpha )} \right){\boldsymbol{\hat z}}, \\ {\boldsymbol{T}} = \left( {\cos (\beta )} \right){\boldsymbol{\hat x}}+(\cos (\alpha )\sin (\beta )){\boldsymbol{\hat y}} + \left( {\sin (\alpha )\sin (\beta )} \right){\boldsymbol{\hat z}}. \end{gathered}\right\} \end{equation}

Next, using (2.11a–c) and (2.20), the derivatives of the Bishop basis vectors can be obtained with respect to the arc length $s$ as

(2.21)\begin{equation} \left.\begin{gathered} \frac{{\partial {{\boldsymbol{N}}_1}(s,t)}}{{\partial s}} = {\kappa _1}{\boldsymbol{T}} + {\kappa _3}{{\boldsymbol{N}}_2}, \\ \frac{{\partial {{\boldsymbol{N}}_2}(s,t)}}{{\partial s}} = {\kappa _2}{\boldsymbol{T}} - {\kappa _3}{{\boldsymbol{N}}_1}, \\ \frac{{\partial {\boldsymbol{T}}(s,t)}}{{\partial s}} ={-} {\kappa _1}{{\boldsymbol{N}}_1} - {\kappa _2}{{\boldsymbol{N}}_2}, \end{gathered}\right\} \end{equation}

and with respect to time $t$ as

(2.22)\begin{equation} \left.\begin{gathered} \frac{{\partial {{\boldsymbol{N}}_1}(s,t)}}{{\partial t}} = \left( {{\beta _{,t}}} \right){\boldsymbol{T}} + \left( {{\alpha _{,t}}\cos (\beta )} \right){{\boldsymbol{N}}_2}, \\ \frac{{\partial {{\boldsymbol{N}}_2}(s,t)}}{{\partial t}} = \left( {{\alpha _{,t}}\sin (\beta )} \right){\boldsymbol{T}} - \left( {{\alpha _{,t}}\cos (\beta )} \right){{\boldsymbol{N}}_1}, \\ \frac{{\partial {\boldsymbol{T}}(s,t)}}{{\partial t}} ={-} \left( {{\alpha _{,t}}\sin (\beta )} \right){{\boldsymbol{N}}_2} - \left( {{\beta _{,t}}} \right){{\boldsymbol{N}}_1}. \end{gathered}\right\} \end{equation}

Now, we turn from the baseline geometry to that of the fibre as a whole, for the definition of which we use the covariant basis vectors, ${\boldsymbol {g}_{1}}$, ${\boldsymbol {g}_{2}}$ and ${\boldsymbol {g}_{3}}$ corresponding to $n_1$, $n_2$ and $s$ directions, respectively; see figure 1. Based on our definition here, the time-dependent position vector of an arbitrary point $\boldsymbol {d}$ inside the fibre at a given time $t$ can be expressed as

(2.23)\begin{equation} {\boldsymbol{d}}(n_1,n_2,s,t) = {\boldsymbol{D}}(s,t) + \varepsilon{n_1}{{\boldsymbol{N}}_1} + \varepsilon{n_2}{{\boldsymbol{N}}_2} = {\boldsymbol{D}}(s,t) + \varepsilon{\boldsymbol{r}}, \end{equation}

in which $\boldsymbol {r}$ is $(n_1, n_2, 0)$, so as $| {\boldsymbol {r}} |=r$, and $\varepsilon$ is the aspect ratio. Given the position vector of an arbitrary point $\boldsymbol {d}$, i.e. (2.23), we can define the covariant basis vectors as

(2.24)\begin{equation} {{\boldsymbol{g}}_i} = \frac{{\partial {\boldsymbol{d}}}}{{\partial {s^i}}}\quad (i = 1, 2, 3)\quad {\rm{and}}\quad ({s^1} = {n_1},\ {s^2} = {n_2},\ {s^3} = s), \end{equation}

and therefore we have

(2.25)\begin{equation} \left.\begin{gathered} {{\boldsymbol{g}}_1} = \frac{{\partial {\boldsymbol{d}}}}{{\partial {n_1}}} = {\varepsilon{\boldsymbol{N}}_1},\quad {{\boldsymbol{g}}_2} = \frac{{\partial {\boldsymbol{d}}}}{{\partial {n_2}}} = {\varepsilon{\boldsymbol{N}}_2}, \\ {{\boldsymbol{g}}_3} = \frac{{\partial {\boldsymbol{d}}}}{{\partial s}} ={-} \left( {\varepsilon {n_2}{\kappa _3}} \right){{\boldsymbol{N}}_1} + \left( {\varepsilon {n_1}{\kappa _3}} \right){{\boldsymbol{N}}_2} + \left( {1 + \varepsilon {n_1}{\kappa _1} + \varepsilon {n_2}{\kappa _2}} \right){\boldsymbol{T}}. \end{gathered}\right\} \end{equation}

Thus, we can define the metric tensor as

(2.26) \begin{equation} \left.\begin{gathered} ({g_{ij}})= {{\boldsymbol{g}}_i}\boldsymbol{\cdot}{{\boldsymbol{g}}_j} = \begin{pmatrix} {{\varepsilon ^2}} & 0 & { - {\varepsilon ^2}{n_2}{\kappa _3}}\\ 0 & {{\varepsilon ^2}} & {{\varepsilon ^2}{n_1}{\kappa _3}}\\ { - {\varepsilon ^2}{n_2}{\kappa _3}} & {{\varepsilon ^2}{n_1}{\kappa _3}} & {{h^2} + {\varepsilon ^2}{r^2}\kappa _3^2}\end{pmatrix}, \\ \left| {{g_{ij}}} \right| = \varepsilon^4 {h^2},\quad h = \left( {1 + \varepsilon{n_1}{\kappa _1} + \varepsilon{n_2}{\kappa _2}} \right), \end{gathered}\right\} \end{equation}

in which $| {{g_{ij}}} |$ denotes the determinant of the metric tensor. As vividly seen, our local bases are neither orthogonal nor normalized due to non-zero off-diagonal elements and non-unity of the diagonal ones. Using (2.26), we can attain the conjugate metric tensor ${{g}}^{ij}$ such that

(2.27)\begin{equation} {g}^{ik}\cdot{g}_{kj}=\delta^{i}_{j}, \end{equation}

where $\delta ^{i}_{j}$ stands for the Kronecker delta; therefore, we arrive at

(2.28) \begin{equation} ({g^{ij}}) = \frac{{{\varepsilon ^2}}}{{\left| {{g_{ij}}} \right|}} \begin{pmatrix} {{h^2} + {\varepsilon ^2}n_2^2\kappa _3^2} & { - {\varepsilon ^2}{n_1}{n_2}\kappa _3^2} & {{\varepsilon ^2}{n_2}{\kappa _3}}\\ { - {\varepsilon ^2}{n_1}{n_2}\kappa _3^2} & {{h^2} + {\varepsilon ^2}n_1^2\kappa _3^2} & { - {\varepsilon ^2}{n_1}{\kappa _3}}\\ {{\varepsilon ^2}{n_2}{\kappa _3}} & { - {\varepsilon ^2}{n_1}{\kappa _3}} & {{\varepsilon ^2}} \end{pmatrix}. \end{equation}

Using (2.25), (2.26) and (2.28), we can obtain the gradient operator in our curvilinear coordinate system as

(2.29)\begin{equation} \boldsymbol{\nabla} = {g^{ik}}{{\boldsymbol{g}}_k}\frac{\partial }{{\partial {s^i}}} = \left(\frac{1}{\varepsilon }{{\boldsymbol{N}}_1} + \frac{{{n_2}{\kappa _3}}}{h}{\boldsymbol{T}}\right) \frac{\partial }{{\partial {n_1}}} + \left(\frac{1}{\varepsilon }{{\boldsymbol{N}}_2} - \frac{{{n_1}{\kappa _3}}}{h}{\boldsymbol{T}}\right)\frac{\partial }{{\partial {n_2}}} + \left(\frac{1}{h}{\boldsymbol{T}}\right)\frac{\partial }{{\partial s}}, \end{equation}

from which we can later determine the strain rate tensor. In the next step, we derive the corresponding kinematic relations, the stress tensor and the dynamic boundary condition equations using the aforementioned expressions.

2.4. Kinematic relations

To study an unsteady growing curved jet, two kinds of velocities can be realized at each point inside the liquid jet. The first one is the fluid velocity and it can be expressed as the material derivative of the position vector $\boldsymbol {d}$ as

(2.30)\begin{equation} {\boldsymbol{v}} = \frac{{{\rm d}{\boldsymbol{d}}(n_1, n_2, s, t)}}{{{\rm d} t}} = \frac{{\partial {\boldsymbol{d}}}}{{\partial t}} + \frac{{\partial {\boldsymbol{d}}}}{{\partial {n_1}}} \frac{{\partial {n_1}}}{{\partial t}} + \frac{{\partial {\boldsymbol{d}}}}{{\partial {n_2}}} \frac{{\partial {n_2}}}{{\partial t}} + \frac{{\partial {\boldsymbol{d}}}}{{\partial s}} \frac{{\partial s}}{{\partial t}} = \frac{{\partial {\boldsymbol{d}}}}{{\partial t}} + {u_i} \frac{{\partial {\boldsymbol{d}}}}{{\partial {s^i}}}, \end{equation}

in which $u_i$ stands for the convectional velocity vector, i.e. $u_i = ({{{\partial {n_1}}}/{{\partial t}}, {{\partial {n_2}}}/{{\partial t}}, {{\partial s}}/{{\partial t}}})=(u_1, u_2, u_3)$. The second velocity field is the coordinate velocity $\boldsymbol {w}$ expressed as

(2.31)\begin{equation} {\boldsymbol{w}} = \frac{{\partial {\boldsymbol{d}}}}{{\partial t}} = \frac{{\partial {\boldsymbol{D}}(s,t)}}{{\partial t}} + \varepsilon {n_1} \frac{{\partial {{\boldsymbol{N}}_1}}}{{\partial t}} +\varepsilon {n_2} \frac{{\partial {{\boldsymbol{N}}_2}}}{{\partial t}}. \end{equation}

Now, the fluid velocity can be calculated using the coordinate velocity $\boldsymbol {w}$ and the convectional velocity $\boldsymbol {u}$ vectors, with the help of (2.30); in the steady state condition, however, the coordinate velocity is zero. In this work, to avoid an arduous procedure to derive the dynamic equations and deviatoric stress terms, we use the velocity projections onto the Bishop basis vectors so as ${\boldsymbol {v}} = ( {{v_{{N_1}}},{v_{{N_2}}},{v_T}} )$. However, one can also easily obtain the equations based on the velocity vector projections onto the covariant basis vectors using (2.25).

2.5. Stress tensor

The stress tensor $\boldsymbol {\varPi }$ in our curvilinear coordinate system is defined as

(2.32a,b)\begin{equation} {\boldsymbol{\varPi}} ={-} P{\boldsymbol{I}} + \frac{1}{{{{Re}}}} \left( {2{\delta _s}\bar{\mu} {\boldsymbol{\gamma }} + \boldsymbol{\mathsf{S}}} \right),\quad {\boldsymbol{\gamma }} = \frac{1}{2}\left( {\boldsymbol{\nabla} {\boldsymbol{v}} + \boldsymbol{\nabla} {{\boldsymbol{v}}^{\rm T}}} \right),\end{equation}

in which $P$ is the pressure, $\boldsymbol {I}$ the identity matrix and $\delta _s$ the viscosity ratio. Also, $\boldsymbol {\gamma }$ stands for the strain rate tensor whose components can be obtained with the help of (2.29) as

(2.33)\begin{equation} \left.\begin{gathered} {\gamma _{TT}} = \frac{1}{h}\left( {{v_{T,s}} + {\kappa _2}{v_{{N_2}}} + {\kappa _1}{v_{{N_1}}}} \right) + \frac{{{\kappa _3}}}{h} \left( {{n_2}{v_{T,{n_1}}} - {n_1}{v_{T,{n_2}}}} \right), \\ {\gamma _{{N_1}{N_1}}} = \frac{1}{\varepsilon }{v_{{N_1},{n_1}}},\quad {\gamma _{{N_2}{N_2}}} = \frac{1}{\varepsilon }{v_{{N_2},{n_2}}}, \\ {\gamma _{T{N_1}}} = {\gamma _{{N_1}T}} = \frac{1}{{2h}}\left( { - {\kappa _1}{v_T} + {v_{{N_1},s}} + \frac{h}{\varepsilon }{v_{T,{n_1}}}} \right) + \frac{{{\kappa _3}}}{{2h}} \left( {{n_2}{v_{{N_1},{n_1}}} - {n_1}{v_{{N_1},{n_2}}} - {v_{{N_2}}}} \right), \\ {\gamma _{{N_2}T}} = {\gamma _{T{N_2}}} = \frac{1}{{2h}}\left( { - {\kappa _2}{v_T} + {v_{{N_2},s}} + \frac{h}{\varepsilon }{v_{T,{n_2}}}} \right) + \frac{{{\kappa _3}}}{{2h}} \left( {{n_2}{v_{{N_2},{n_1}}} - {n_1}{v_{{N_2},{n_2}}} + {v_{{N_1}}}} \right), \\ {\gamma _{{N_1}{N_2}}} = {\gamma _{{N_2}{N_1}}} = \frac{1}{{2\varepsilon }} \left( {{v_{{N_1},{n_2}}} + {v_{{N_2},{n_1}}}} \right). \end{gathered}\right\} \end{equation}

2.6. Dynamic boundary conditions

Due to the existence of the surface tension, at the jet free surface, we have a stress balance equation known as dynamic boundary conditions. The interface causes a jump in the normal component of the stress, leading to altering the mean interface curvature, expressed as

(2.34)\begin{equation} {{\boldsymbol{\varPi}}^*}\boldsymbol{\cdot}{\boldsymbol{n}} - {\boldsymbol{\varPi }} \boldsymbol{\cdot}{\boldsymbol{n}} = \frac{1}{{We}}{\boldsymbol{n}}{\mathcal{H}},\quad {\rm at}\ r = R(s,t), \end{equation}

where $R(s,t)$ is a function characterizing the free surface of the jet and $\boldsymbol {\varPi }^*$ indicates the stresses exerted by air on the jet interface; we also have $\boldsymbol {f}_{drag}={{\boldsymbol {\varPi }}^*}\boldsymbol {\cdot }{\boldsymbol {n}}$, which is the air drag force per unit area. Furthermore, $\boldsymbol {n}$ is the unit normal vector to the interface pointing outwards from the jet surface, and $\mathcal {H}$ is the mean local curvature of the interface which can be obtained as

(2.35)\begin{equation} \mathcal{H} = \frac{{{\mathbb{I}_1}{\mathbb{J}_3} + {\mathbb{I}_3}{\mathbb{J}_1} - 2{\mathbb{I}_2}{\mathbb{J}_2}}}{{{\mathbb{I}_1}{\mathbb{I}_3} - \mathbb{I}_2^2}}, \end{equation}

where

(2.36)\begin{equation} \left.\begin{gathered} \mathbb{I}_1 = \frac{{\partial \boldsymbol{d}}}{{\partial s}}\boldsymbol{\cdot} \frac{{\partial \boldsymbol{d}}}{{\partial s}},\quad \mathbb{I}_2 = \frac{{\partial \boldsymbol{d}}}{{\partial s}}\boldsymbol{\cdot} \frac{{\partial \boldsymbol{d}}}{{\partial \varphi }},\quad \mathbb{I}_3 = \frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}\boldsymbol{\cdot} \frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}\quad {\rm{at}}\ r = R(s,t), \\ \mathbb{J}_1 = \frac{{{\partial ^2}\boldsymbol{d}}}{{\partial {s^2}}}\boldsymbol{\cdot}\boldsymbol{n},\quad \mathbb{J}_2 = \frac{{{\partial ^2}\boldsymbol{d}}}{{\partial s\partial \varphi }}\boldsymbol{\cdot} \boldsymbol{n},\quad \mathbb{J}_3 = \frac{{{\partial ^2}\boldsymbol{d}}}{{\partial {\varphi ^2}}}\boldsymbol{\cdot} \boldsymbol{n}\quad {\rm{at}}\ r = R(s,t), \\ \boldsymbol{n} = \left( {\frac{{\partial \boldsymbol{d}}}{{\partial s}} \times \frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}} \right){\left| {\frac{{\partial \boldsymbol{d}}}{{\partial s}} \times \frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}} \right|^{ - 1}} \quad {\rm{at}}\ r = R(s,t), \end{gathered}\right\} \end{equation}

wherein $\{{\mathbb {I}_1,\ \mathbb {I}_2,\ \mathbb {I}_3}\}$ and $\{{\mathbb {J}_1,\ \mathbb {J}_2,\ \mathbb {J}_3}\}$ stand for the first and second fundamental forms. It is noted that, when setting the dynamic boundary conditions to describe the jet cross-section, one finds the polar coordinates ($r, \varphi$) to be more appropriate than the rectangular coordinate ($n_1, n_2$). In this sense, one can simply use (2.7) and (2.23) to attain corresponding expressions for $\boldsymbol {d}$ and then compute the related terms in mean curvature and normal vector definitions (see, for example, Marheineke & Wegener (Reference Marheineke and Wegener2007) and Shikhmurzaev & Sisoev (Reference Shikhmurzaev and Sisoev2017)). To write out the stress jumps in the normal and tangential directions, one can simply take the inner product of (2.34) with the unit normal $\boldsymbol {n}$ and tangent $\boldsymbol {t}_i$ vectors to the interface. The latter has two components in the $s$ and $\varphi$ directions defined as

(2.37a,b)\begin{equation} \boldsymbol{{t}}_s = \left( {\frac{{\partial \boldsymbol{d}}}{{\partial s}}} \right){\left| {\frac{{\partial \boldsymbol{d}}}{{\partial s}}} \right|^{ - 1}},\quad \boldsymbol{{t_\varphi }} = \left({\frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}} \right) {\left|{\frac{{\partial \boldsymbol{d}}}{{\partial \varphi }}} \right|^{ - 1}}\quad {\rm{at}}\ r = R(s,\varphi). \end{equation}

Therefore, the jump conditions in the normal and tangential directions at the jet surface can be expressed as

(2.38)$$\begin{gather} {\boldsymbol{n}}\boldsymbol{\cdot}\boldsymbol{\varPi}\boldsymbol{\cdot}{\boldsymbol{n}} ={-}\frac{1}{{We}}\mathcal{H} + {\boldsymbol{n}}\boldsymbol{\cdot}{{\boldsymbol{f}}_{{{drag}}}}, \end{gather}$$

(2.39)$$\begin{gather}{{\boldsymbol{t}}_i}\boldsymbol{\cdot}{\boldsymbol{\varPi }}\boldsymbol{\cdot}{\boldsymbol{n}} = {{\boldsymbol{t}}_i}\boldsymbol{\cdot}{{\boldsymbol{f}}_{{{drag}}}},\quad {\text{at}}\ {\text{r}} = R(s,\varphi ),\quad i = s,\varphi. \end{gather}$$

In what follows, we will express the relations needed to simplify our equations presented so far by integrating the equations over the jet cross-section and then using asymptotic series to estimate the key variables.

2.7. Asymptotic analysis

The uniaxial asymptotic model to be developed here is first based on the cross-sectional averaging of the conservation equations, followed by use of asymptotic expansions of cross-averaged variables in the leading-, first- and second-order terms; this allows one to more easily evaluate the key variables throughout the computational domain. Here, the cross-sectional averaging is performed to dimensionally reduce the conservation equations; to do so, we use the Reynolds transport theorem as our averaging rule (see also Marheineke et al. Reference Marheineke, Liljegren-Sailer, Lorenz and Wegener2016) expressed as

(2.40)$$\begin{gather} \int_A {\left( \boldsymbol{\nabla}\boldsymbol{\cdot} f \right)\,{\rm d} A} = {\partial _s} \int_A {\left( {f\boldsymbol{\cdot} \boldsymbol{T}} \right)} \,{\rm d} A + \int_{l} {\left( {f\boldsymbol{\cdot} \boldsymbol{n}} \right)}\,{\rm d} l, \end{gather}$$

(2.41)$$\begin{gather}{\partial_t}\int_A {\left( f \right)\,{\rm d} A} = \int_A {{\partial _t}f} \,{\rm d} A + \int_l {\left( {\left( {\boldsymbol{n}\boldsymbol{\cdot} \boldsymbol{u}} \right)f} \right)}\,{\rm d} l, \end{gather}$$

in which $A=R^2$ is the given dimensionless cross-section of the jet and $l$ is its perimeter and $\boldsymbol {u}=(u_1, u_2, u_3)$ is the convectional velocity vector. Additionally, $f$ stands for any differentiable and integrable scalar, vector or tensor valued function. Applying these averaging rules on our three-dimensional set of equations, under the assumption of radial symmetry, leads to a set of uniaxial equations.

2.7.1. Asymptotic expansion

Considering the jet as a long thin object with the aspect ratio $\varepsilon$, we can expand the governing equations based on the asymptotic series of the cross-averaged key parameters and use the dominant terms as a reasonable approximation to the jet behaviour. To do so, we assume that the velocity components can be expanded as (see Yarin Reference Yarin1993; Marheineke et al. Reference Marheineke, Liljegren-Sailer, Lorenz and Wegener2016)

(2.42)\begin{equation} {\boldsymbol{v}} = {{\boldsymbol{v}}_0} + {v_1}\left( {\varepsilon {\boldsymbol{r}}} \right) + {\boldsymbol{\omega }} \times \left( {\varepsilon {\boldsymbol{r}}} \right) + {\boldsymbol{\varPhi }}, \end{equation}

where $v_1$ is the first-order velocity expansion term in the $\boldsymbol {N}_1$ and $\boldsymbol {N}_2$ directions and $\boldsymbol {\varPhi }$ is the second-order velocity expansion in $\boldsymbol {r}$. In addition, $\boldsymbol {\omega }$ denotes the angular velocity of the cross-section whose components, i.e. $(\omega _{N_1}, \omega _{N_2}, \omega _{T})$, can be defined using ${{\partial {{\boldsymbol {N}}_1}}}/{{\partial t}}$ and ${{\partial {{\boldsymbol {N}}_2}}}/{{\partial t}}$; see (2.31) and figure 1. Next, we expand stress and pressure terms in powers of $\varepsilon \boldsymbol {r}$ and $R, X, Z, Y, \theta$ in powers of $\varepsilon$ (see Eggers Reference Eggers1997; Hohman et al. Reference Hohman, Shin, Rutledge and Brenner2001); therefore,

(2.43) \begin{equation} \left.\begin{gathered} {v_{{N_1}}} = {v_{0{N_1}}} + \varepsilon \left( {{n_1}{v_1} - {n_2}{\omega _T}} \right) + {\varepsilon ^2}{\varPhi _{{N_1}}} + O\left( {{{({{\varepsilon \boldsymbol{r} }})}^3}} \right), \\ {v_{{N_2}}} = {v_{0{N_2}}} + \varepsilon \left( {{n_2}{v_1} + {n_1}{\omega _T}} \right) + {\varepsilon ^2}{\varPhi _{{N_2}}} + O\left( {{{({{\varepsilon \boldsymbol{r} }})}^3}} \right), \\ {v_T} = {v_{0T}} + \varepsilon \left( {{n_2}{\omega _{{N_1}}} - {n_1}{\omega _{{N_2}}}} \right) + {\varepsilon ^2}{\varPhi _T} + O\left( {{{({{\varepsilon \boldsymbol{r} }})}^3}} \right), \\ {S_{TT}} = S_{TT}^{(0)} + \varepsilon {n_1}S_{TT1}^{(1)} + \varepsilon {n_2}S_{TT2}^{(1)} + O\left( {{{\left( {\varepsilon {\boldsymbol{r}}} \right)}^2}} \right), \\ {S_{NN}} = S_{NN}^{(0)} + \varepsilon {n_1}S_{NN1}^{(1)} + \varepsilon {n_2}S_{NN2}^{(1)} + O\left( {{{\left( {\varepsilon {\boldsymbol{r}}} \right)}^2}} \right), \\ {S_{ij}} = \left( {\varepsilon {\boldsymbol{r}}} \right)S_{ij}^{(1)} + O\left( {{{\left( {\varepsilon {\boldsymbol{r}}} \right)}^2}} \right),\quad i \ne j, \\ P = {P_0} + (\varepsilon {\boldsymbol{r}}){P_1} + O\left( {{{\left( {\varepsilon {\boldsymbol{r}}} \right)}^2}} \right), \\ C = {C_0} + \varepsilon {C_1} + O\left( {{\varepsilon ^2}} \right),\quad C = \{ \theta,A, X, Z, Y\}. \end{gathered}\right\} \end{equation}

Now, we compute the governing equations using the velocity expansion relation in our coordinate system. Starting from the strain rate tensor, we obtain the projection of its components onto the Bishop basis vectors as

(2.44) \begin{gather} \left.\begin{gathered} {\gamma _{{N_1}T}} = {\gamma _{T{N_1}}} = \tfrac{1}{2}\left( {{v_{0{N_1},s}} - {\kappa _1}{v_{0T}} - {\kappa _3}{v_{0{N_2}}} - {\omega _{{N_2}}}} \right) \\ \quad + \tfrac{1}{2}\left( {{\varPhi _{T,{n_1}}} + {n_2}\left( {{\kappa _1}{\kappa _2}{v_{0T}} - {\kappa _1}{\omega _{{N_1}}} - {\omega _{T,s}} - {\kappa _2}{v_{0{N_1},s}} + {\kappa _3}{\kappa _2}{v_{0{N_2}}}} \right)} \right. \\ \quad + {n_1}\left. {\left( {\kappa _1^2{v_{0T}} + {\kappa _1}{\omega _{{N_2}}} + {v_{1,s}} - {\kappa _1}{v_{0{N_1},s}} + {\kappa _3}{\kappa _1}{v_{0{N_2}}}} \right)} \right) \varepsilon + O\left( {{\varepsilon ^2}} \right), \\ {\gamma _{{N_2}T}} = {\gamma _{T{N_2}}} = \tfrac{1}{2}\left( {{v_{0{N_2},s}} - {\kappa _2}{v_{0T}}+ {\kappa _3}{v_{0{N_1}}} + {\omega _{{N_1}}}} \right) \\ \quad + \tfrac{1}{2}\left( {{n_1}\left( {{\kappa _2}{\omega _{{N_2}}} + {\omega _{T,s}} - {\kappa _1}{v_{0{N_2},s}} + {\kappa _1}{\kappa _2}{v_{0T}}}-{\kappa _3}{\kappa _1}{v_{0{N_1}}} \right)} \right. \\ \quad + {n_2}\left( {\kappa _2^2{v_{0T}} - {\kappa _2}{\omega _{{N_1}}} + {v_{1,s}} - {\kappa _2}{v_{0{N_2},s}}-{\kappa _3}{\kappa _2}{v_{0{N_1}}}} \right) + \left. {{\varPhi _{T,{n_2}}}} \right)\varepsilon + O\left( {{\varepsilon ^2}} \right), \\ {\gamma _{{N_2}{N_1}}} = {\gamma _{{N_1}{N_2}}} = \tfrac{1}{2}\left( {{\varPhi _{{N_2},{n_1}}} + {\varPhi _{{N_1},{n_2}}}} \right)\varepsilon+ O\left( {{\varepsilon ^2}} \right), \\ {\gamma _{TT}} = {v_{0T,s}} + {\kappa _1}{v_{0{N_1}}} + {\kappa _2}{v_{0{N_2}}} \\ \quad + \left( {{n_2}\left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T}-{\kappa _3}{\omega _{N_2}} + {\kappa _2}{v_1} - {\kappa _2}{v_{0T,s}} - \kappa _2^2{v_{0{N_2}}} - {\kappa _1}{\kappa _2}{v_{0{N_1}}}} \right)} \right. \\ \quad + \left. {\left. { {n_1}\left( { {\kappa _2}{\omega _T}- {\omega _{{N_2},s}} -{\kappa _3}{\omega _{N_1}} + {\kappa _1}{v_1} - {\kappa _1}{v_{0T,s}} - \kappa _1^2{v_{0{N_1}}} - {\kappa _1}{\kappa _2}{v_{0{N_2}}}} \right)} \right)\varepsilon + O\left( {{\varepsilon ^2}} \right)} \right), \\ {\gamma _{{N_1}{N_1}}} = {v_1} + ({\varPhi _{{N_1},{n_1}}})\varepsilon+ O\left( {{\varepsilon ^2}} \right), \\ {\gamma _{{N_2}{N_2}}} = {v_1} + ({\varPhi _{{N_2},{n_2}}})\varepsilon+ O\left( {{\varepsilon ^2}} \right). \end{gathered}\right\} \end{gather}

Next, we proceed to evaluate the leading-order terms to simplify our equations. First, the incompressibility condition, i.e. $\textrm {tr}(\boldsymbol \gamma )=0$, at the leading and first order reads

(2.45)\begin{equation} \left.\begin{gathered} {v_1} ={-} \tfrac{1}{2}\left( {{v_{0T,s}} + {\kappa _1}{v_{0{N_1}}} + {\kappa _2}{v_{0{N_2}}}} \right), \\ {\varPhi _{{N_1},{n_1}}} + {\varPhi _{{N_2},{n_2}}} = {n_2}\left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} + 3{\kappa _2}{v_1}} \right) \\ + {n_1}\left( { - {\omega _{{N_2},s}} + {\kappa _2}{\omega _T} - {\kappa _3}{\omega _{{N_1}}} + 3{\kappa _1}{v_1}} \right). \end{gathered}\right\} \end{equation}

Afterwards, from the normal stress jump condition at the jet surface, we have

(2.46)\begin{align} P &={-} \frac{1}{{{{Re}}}}\left( {{\delta _s}\bar \mu \left( {{v_{0T,s}} + {\kappa _1}{v_{0{N_1}}} + {\kappa _2}{v_{0{N_2}}}} \right) - S_{NN}^{(0)}} \right) + \frac{1}{{We\sqrt A_0 }} - {\boldsymbol{f}_{{drag}}}\boldsymbol{\cdot}\boldsymbol{n} \nonumber\\ &\quad + \varepsilon {n_1}\left( {\frac{{{\delta _s}\bar \mu }}{{Re}} \left( { - {\omega _{{N_2},s}} + {\kappa _2}{\omega _T} - {\kappa _3}{\omega _{{N_1}}} + 3{\kappa _1}{v_1}} \right) - \frac{1}{{Re}}S_{NN1}^{(1)} + \frac{{{\kappa _1}}}{{We{{\sqrt A }_0}}}} \right) \nonumber\\ &\quad - \varepsilon {n_2}\left( {\frac{{{\delta _s}\bar \mu }}{{Re}}\left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} + 3{\kappa _2}{v_1}} \right) - \frac{1}{{Re}}S_{NN2}^{(1)} + \frac{{{\kappa _2}}}{{We{{\sqrt A }_0}}}} \right). \end{align}

From the tangential stress balances, we can obtain

(2.47)\begin{gather} \left.\begin{gathered} {\omega _{{N_1}}} ={-} {v_{0{N_2},s}} + {\kappa _2}{v_{0T}} - {\kappa _3}{v_{0{N_1}}}, \\ {\omega _{{N_2}}} = {v_{0{N_1},s}} - {\kappa _1}{v_{0T}} - {\kappa _3}{v_{0{N_2}}}, \end{gathered}\right\} \end{gather}

(2.48)\begin{gather} \left( {{\varPhi _{{N_2},{n_1}}} + {\varPhi _{{N_1},{n_2}}}} \right) = 0. \end{gather}

Using (2.45) and (2.48), we can simply obtain

(2.49)\begin{equation} \left.\begin{gathered} {\varPhi _{{N_1}}} ={-} \tfrac{1}{2}{n_1}{n_2}\left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} + 3{\kappa _2}{v_1}} \right) \\ \quad + \tfrac{1}{4}\left( {n_1^2 - n_2^2} \right)\left( { - {\omega _{{N_2},s}} + {\kappa _2}{\omega _T} - {\kappa _3}{\omega _{{N_1}}} + 3{\kappa _1}{v_1}} \right), \\ {\varPhi _{{N_2}}} = \tfrac{1}{2}{n_1}{n_2}\left( { - {\omega _{{N_2},s}} + {\kappa _2}{\omega _T} - {\kappa _3}{\omega _{{N_1}}} + 3{\kappa _1}{v_1}} \right) \\ \quad + \tfrac{1}{4}\left( {n_2^2 - n_1^2} \right)\left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} + 3{\kappa _2}{v_1}} \right). \end{gathered}\right\} \end{equation}

Now, using the relations presented, we derive the kinematic and dynamic equations.

2.8. Kinematic expressions

As a next step, we derive the kinematic equations in the leading order. To do so, starting from the projections of (2.30) onto the Cartesian basis vectors, we have at the leading order

(2.50)\begin{equation} \left.\begin{gathered} {X_{0,t}} + u{X_{0,s}} = {v_{0{N_1}}}\sin (\beta ) + {v_{0T}}\cos (\beta ), \\ {Y_{0,t}} + u{Y_{0,s}} ={-} {v_{0{N_1}}}\cos (\alpha )\cos (\beta ) + {v_{0{N_2}}}\sin (\alpha ) + {v_{0T}}\cos (\alpha )\sin (\beta ), \\ {Z_{0,t}} + u{Z_{0,s}} ={-} {v_{0{N_1}}}\sin (\alpha )\cos (\beta ) - {v_{0{N_2}}}\cos (\alpha ) + {v_{0T}}\sin (\alpha )\sin (\beta ). \end{gathered}\right\} \end{equation}

Differentiating the expressions in (2.50) with respect to $s$ and after some manipulation, we end up with

(2.51a–c)\begin{equation} {\beta _{,t}} + u{\beta _{,s}}={-} {\omega _{{N_2}}},\quad {\alpha _{,t}} + u{\alpha _{,s}} = \frac{{{\omega _{{N_1}}}}}{{\sin (\beta )}},\quad {u_{,s}} = {v_{0T,s}} + {\kappa _1}{v_{0{N_1}}} + {\kappa _2}{v_{0{N_2}}}, \end{equation}

and consequently from (2.11a–c) we arrive at

(2.52a,b)\begin{equation} {\kappa _1}_{,t} + {\left( {u{\kappa _1}} \right)_{,s}} = {\left( { - {\omega _{{N_2}}}} \right)_{,s}},\quad {\left( {\frac{{{\kappa _2}}}{{\sin (\beta )}}} \right)_{,t}} + {\left( {\frac{{u{\kappa _2}}}{{\sin (\beta )}}} \right)_{,s}} = {\left( {\frac{{{\omega _{{N_1}}}}}{{\sin (\beta )}}} \right)_{,s}}. \end{equation}

In the following, we derive the dynamic equations that govern the behaviours of a slender curved jet and, afterwards, we present the corresponding constitutive relations to capture the rheological features of a viscoelastic jet through the CS process.

2.9. Dynamic equations

In this section, using the asymptotic expressions for pressure (2.46) and velocity (2.43) we deliver the dynamic equations of a viscoelastic curved jet consisting of mass, momentum and angular momentum equations. Considering a control volume $V$ bounded by the jet lateral surface and two cross-sections, $A_1$ and $A_2$ corresponding to $s_1$ and $s_2$ in figure 1, and ignoring the density change by temperature variations, i.e. $\bar {\rho }=1$, the mass conservation equation reads

(2.53)\begin{equation} \frac{\partial }{{\partial t}}\int_V {{\rm d} V + } \frac{\partial }{{\partial s}}\int_{{A_1}}^{{A_2}} {\left( {\left( {{\boldsymbol{v}} - \boldsymbol{w}} \right)\boldsymbol{\cdot}{\boldsymbol{T}}} \right)} {\rm d} A = 0. \end{equation}

It is worth mentioning that mass transfer velocity through the boundary of our infinitesimal element is equal to $({\boldsymbol {v}} - \boldsymbol {w})\boldsymbol {\cdot }{\boldsymbol {T}}$, which, in the leading order, is the convectional velocity at the baseline $u_3$; hereafter, for simplicity of the presentation, we use $u$ instead of $u_3$, representing the projection of the convectional velocity onto the Bishop frame in the tangential direction. Using the cross-sectional averaging rules, i.e. (2.40) and (2.41), and the asymptotic expansion expressions (2.43), we have

(2.54)\begin{equation} \frac{{\partial A_0}}{{\partial t}} + \frac{{\partial \left( {uA_0} \right)}}{{\partial s}} = 0. \end{equation}

Now, we turn to derive the momentum balance equations. After integrating (2.2) over our control volume and using the averaging rule (2.40) and (2.41), we arrive at

(2.55)\begin{align} \frac{\partial }{{\partial t}}\left[ {\int_V {{\boldsymbol{v}}\,{\rm d} V}} \right] + \frac{\partial }{{\partial s}}\left[ {\int_{{A_1}}^{{A_2}} {\left( {({\boldsymbol{v}} - {\boldsymbol{w}})\boldsymbol{\cdot}{\boldsymbol{T}}} \right){\boldsymbol{v}}\,{\rm d} A} } \right] = \frac{\partial }{{\partial s}}\left[ {\int_{{A_1}}^{{A_2}} {\left( {{\boldsymbol{\varPi}}\boldsymbol{\cdot}{\boldsymbol{T}}} \right)\,{\rm d} A}} \right] + \int_V {{\boldsymbol{F}}\,{\rm d} V}.\end{align}

After applying the asymptotic expressions and some manipulations, we end up having

(2.56)\begin{equation} \frac{{\partial \left( {A_0{{\boldsymbol{v}}_0}} \right)}}{{\partial t}} + \frac{{\partial \left( {uA_0{{\boldsymbol{v}}_0}} \right)}}{{\partial s}} = \frac{1}{{{{Re}}}} \frac{{\partial {\boldsymbol{\eta }}}}{{\partial s}} + A_0{\boldsymbol{F}}, \end{equation}

in which ${\boldsymbol {\eta }} = \int _A {{{\langle \boldsymbol {\varPi } \rangle }_T}\,\textrm {d} A} = ( {{\eta _{{N_1}}},\ {\eta _{{N_2}}},\ {\eta _T}})$ denotes the tensile force vector, in which $\eta _{N_1}$ and $\eta _{N_2}$ stand for the shearing forces and ${\eta _T}$ stands for the longitudinal force. From the definition of $\boldsymbol {\eta }$, we have

(2.57)\begin{equation} {\eta _T} = {A_0}\left( {3{\delta _s}\bar \mu {u_{,s}} + \left({S_{TT}^{(0)} - S_{NN}^{(0)}} \right) + \frac{{Re}}{{We{{\sqrt A }_0}}}}\right). \end{equation}

It is noted that the shearing force components, i.e. ${\eta _{{N_1}}}$ and ${\eta _{{N_2}}}$, are rather complicated and thus cannot be obtained explicitly. Next, to attain the external forces $\boldsymbol {F}$, we project the related terms in the outer bases (here Cartesian bases) onto the Bishop basis vectors; in doing so, we arrive at

(2.58)\begin{gather} \left.\begin{gathered} {F_{{N_1}}} ={-} \cos (\beta )\frac{{\left( {Y_0\cos (\alpha ) + Z_0\sin (\alpha )} \right)}}{{R{b^2}}} + \frac{{2{v_{0{N_2}}}\cos (\beta )}}{{Rb}} - \frac{{\sin (\beta )}}{{F{r^2}}} + {\left\langle {{F_{drag}}} \right\rangle _{{N_1}}}, \\ {F_{{N_2}}} = \frac{{\left( {Y_0\sin (\alpha ) - Z_0\cos (\alpha )} \right)}}{{R{b^2}}} + \frac{2}{{Rb}}\left( {{v_{0T}}\sin (\beta ) - {v_{0{N_1}}}\cos (\beta )} \right) + {\left\langle {{F_{drag}}} \right\rangle _{{N_2}}}, \\ {F_T} = \sin (\beta )\frac{{\left( {Y_0\cos (\alpha ) + Z_0\sin (\alpha )} \right)}}{{R{b^2}}} - \frac{{2{v_{0{N_2}}}\sin (\beta )}}{{Rb}} - \frac{{\cos (\beta )}}{{F{r^2}}} + {\left\langle {{F_{drag}}} \right\rangle _T}. \end{gathered}\right\} \end{gather}

To calculate the cross-averaged aerodynamic drag force $\boldsymbol {F}_{{drag}}$, we rely on the drag model of Marheineke & Wegener (Reference Marheineke and Wegener2011), in which we assume a one-way coupling. In this model, we calculate the drag force as a function of the dimensionless air-jet relative velocity, $\boldsymbol {V^*_{rel}}$, the local tangent vector to the fibre baseline, $\boldsymbol {\hat T}$, and the air and jet physical properties; thus, we have

(2.59)\begin{equation} {{\boldsymbol{F}}_{{{drag}}}} = {\boldsymbol{F}}\left({\boldsymbol{\hat T}},\quad Re_w^*\frac{{{\boldsymbol{V}}_{{\boldsymbol{rel}}}^*}}{{\left\| {{\boldsymbol{V}}_{{\boldsymbol{rel}}}^*} \right\|}}\right)= {F_n}({{W}_n}){\boldsymbol{\hat n}} + {F_T}({{W}_T},{{W}_n}){\boldsymbol{\hat T}}, \end{equation}

with $R{e^*_w}={R Re^{*}}\| {{{\boldsymbol {V}}^*_{{\boldsymbol {rel}}}}} \|$, ${W_T} = Re_w^*{\langle {V_{rel}^*} \rangle _{\hat T}}{\| {{\boldsymbol {V}}_{{\boldsymbol {rel}}}^*} \|^{ - 1}}$ and ${W_n} = Re_w^*{\langle {V_{rel}^*} \rangle _{\hat n}}{\| {{\boldsymbol {V}}_{{\boldsymbol {rel}}}^*} \|^{ - 1}}$; we also have

(2.60a–c)\begin{equation} {\boldsymbol{\hat n}} = \frac{{{{\boldsymbol{V}}^*_{{\boldsymbol{rel}}}} - {{\left\langle {{V^*_{rel}}} \right\rangle }_{\hat T}}{\boldsymbol{\hat T}}}}{{{{\left\langle {{V^*_{rel}}} \right\rangle }_{\hat n}}}},\quad {\left\langle {{V^*_{rel}}} \right\rangle _{\hat T}} = {{\boldsymbol{V}}^*_{{\boldsymbol{rel}}}}\boldsymbol{\cdot}{\boldsymbol{\hat T}},\quad {\left\langle {{V^*_{rel}}} \right\rangle _{\hat n}} = \sqrt {{{\left( {{{\boldsymbol{V}}^*_{{\boldsymbol{rel}}}}} \right)}^2} - {{\left( {{{\left\langle {{V^*_{rel}}} \right\rangle }_{\hat T}}} \right)}^2}}. \end{equation}

Here, the double bars $\| {.} \|$ represents the norm of the quantity. To calculate the relative velocity, we use

(2.61)\begin{equation} {\boldsymbol{V}}_{{\boldsymbol{rel}}}^* ={-} \frac{1}{{Rb}}\left( {\boldsymbol{\varOmega} \times {\boldsymbol{D}}} \right) - {\boldsymbol{v}}\boldsymbol{\cdot}{\boldsymbol{\hat T}}, \end{equation}

in which we assume that the surrounding air is stagnant and, thus, its velocity is equal to the velocity of our rotating frame. It is worth mentioning that, in practical situations or experiments, as the CS spinneret rotates around its axis, it induces airflow (known as free vortex flow) affecting the velocity of the ambient air and accordingly the air-to-fibre drag force. To include the effect of the free vortex flow due to the spinneret head motion, i.e. the non-stagnant air, one can modify the ${\boldsymbol {V}}_{{\boldsymbol {rel}}}^*$ as

(2.62)\begin{equation} {\boldsymbol{V}}_{{\boldsymbol{rel}}}^* ={-} \frac{1}{{Rb}}\left( {{\boldsymbol{\varOmega }} \times \left( {{\boldsymbol{D}} - \frac{{\boldsymbol{D}}}{{{{\left\| {\boldsymbol{D}} \right\|}^2}}}} \right)} \right) - {\boldsymbol{v}}\boldsymbol{\cdot}{\boldsymbol{\hat T}}. \end{equation}

Most results presented in this work will make the stagnant air assumption, although one figure near the end will consider the effect of the vortex flow. Eventually, we have

(2.63)$$\begin{gather} {F_n}({{W}_n}) = {\left( {{{W}_n}} \right)^2}{c_n}, \end{gather}$$

(2.64)$$\begin{gather}{F_T}({{W}_n},{{W}_T}) = {{W}_n}{{W}_T}{c_T}, \end{gather}$$

with $c_{n}$ and $c_{T}$ are the drag coefficients in the normal and tangential directions as functions of the normal velocity $W_{n}$, presented in Appendix A. Now, having the drag force as

(2.65)\begin{equation} \boldsymbol{F}_{{drag}} = {F_1}\boldsymbol{\hat x }+{F_2} \boldsymbol{\hat y }+{F_3}\boldsymbol{\hat z}, \end{equation}

we merely need to have their projections onto the Bishop basis vectors, giving

(2.66)$$\begin{gather} {\left\langle {{F_{{{drag}}}}} \right\rangle _{{N_1}}} = \left( {\sin (\beta )} \right){F_1}- (\cos (\alpha )\cos (\beta )){F_2} - (\sin (\alpha)\cos (\beta )){F_3}, \end{gather}$$

(2.67)$$\begin{gather}{\left\langle {{F_{{{drag}}}}} \right\rangle _{{N_2}}} =(\sin (\alpha )){F_2} - (\cos (\alpha )){F_3}, \end{gather}$$

(2.68)$$\begin{gather}{\left\langle {{F_{{{drag}}}}} \right\rangle _T} = (\cos (\beta )){F_1}+(\cos (\alpha )\sin (\beta )){F_2} + (\sin (\alpha )\sin (\beta )){F_3}. \end{gather}$$

We refer the reader to Marheineke & Wegener (Reference Marheineke and Wegener2011) for more details on the drag model extended herein.

Now, following the same procedure as for the momentum equation, we derive the angular momentum equation (2.3). After integrating over the jet cross-section, we obtain

(2.69)\begin{align} & \frac{\partial }{{\partial t}}\left[ {\int_V {\left( {{\boldsymbol{d}} \times {\boldsymbol{v}}} \right) {\rm d} V}}\right] + \frac{\partial }{{\partial s}}\left[ {\int_{{A_1}}^{{A_2}} {\left( {({\boldsymbol{v}} - {\boldsymbol{w}}).{\boldsymbol{T}}} \right) \left( {{\boldsymbol{d}} \times {\boldsymbol{v}}} \right){\rm d} A}}\right] \nonumber\\ &\quad = \frac{\partial }{{\partial s}}\left[ {\int_{{A_1}}^{{A_2}} {{\boldsymbol{d}} \times \left( {{\boldsymbol{\varPi }}\boldsymbol{\cdot}{\boldsymbol{T}}} \right){\rm d} A} } \right] + \int_V {{\boldsymbol{d}} \times {\boldsymbol{F}}{\rm d} V}. \end{align}

Afterwards, taking a cross-product of the momentum equations (2.55) with $\boldsymbol {D}$, and then subtracting them from the angular momentum equations (2.69) followed by using the asymptotic expressions and keeping only the terms of order $\varepsilon ^2$, we arrive at

(2.70)\begin{align} & \frac{\partial }{{\partial t}}\left[ {\int_V {\left( {{\boldsymbol{r}} \times {\boldsymbol{\omega }} \times {\boldsymbol{r}}} \right){\rm d} V}} \right] + \frac{\partial }{{\partial s}} \left[ {\int_A {u\left( {{\boldsymbol{r}} \times {\boldsymbol{\omega }} \times {\boldsymbol{r}}} \right){\rm d} A} } \right] \nonumber\\ &\quad = \frac{{\partial {\boldsymbol{M}}}}{{\partial s}} + \frac{4}{{{{Re}} }}\frac{1}{{{\varepsilon ^2}}}{\boldsymbol{T}} \times {\boldsymbol{\eta }} +\int_V {{\boldsymbol{r}} \times {\boldsymbol{F}}\,{\rm d} V}, \end{align}

where ${\boldsymbol {M}} = \int _A {{\boldsymbol {r}} \times {{\langle \boldsymbol {\varPi } \rangle }_T}\,\textrm {d} A}$ is the moment of stresses vector whose components can be decomposed into

(2.71)\begin{equation} \left.\begin{gathered} {M_{{N_1}}} = \frac{{{I_{{1}}}}}{{Re}}\left[ {3{\delta _s}\bar \mu \left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} - \frac{3}{2}{\kappa _2}{u_{,s}}} \right) + \left( {S_{TT2}^{\left( 1 \right)} - S_{NN2}^{\left( 1 \right)}} \right) - \frac{{Re{\kappa _2}}}{{We{{\sqrt A }_0}}}} \right] \\ \quad + \frac{{{I_{12}}}}{{Re}}\left[ {3{\delta _s}\bar \mu \left( {{\kappa _2}{\omega _T} - {\omega _{{N_2},s}} - {\kappa _3}{\omega _{{N_1}}} - \frac{3}{2}{\kappa _1}{u_{,s}}} \right) + \left( {S_{TT1}^{\left( 1 \right)} - S_{NN1}^{\left( 1 \right)}} \right) - \frac{{Re{\kappa _1}}}{{We{{\sqrt A }_0}}}} \right], \\ {M_{{N_2}}} ={-} \frac{{{I_{12}}}}{{Re}}\left[ {3{\delta _s}\bar \mu \left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} - \frac{3}{2}{\kappa _2}{u_{,s}}} \right) + \left( {S_{TT2}^{\left( 1 \right)} - S_{NN2}^{\left( 1 \right)}} \right) - \frac{{Re{\kappa _2}}}{{We{{\sqrt A }_0}}}} \right] \\ \quad - \frac{{{I_2}}}{{Re}}\left[ {3{\delta _s}\bar \mu \left( {{\kappa _2}{\omega _T} - {\omega _{{N_2},s}} - {\kappa _3}{\omega _{{N_1}}} - \frac{3}{2}{\kappa _1}{u_{,s}}} \right) + \left( {S_{TT1}^{\left( 1 \right)} - S_{NN1}^{\left( 1 \right)}} \right) - \frac{{Re{\kappa _1}}}{{We{{\sqrt A }_0}}}} \right], \\ {M_T} = \frac{{{\delta _s}\bar \mu }}{{{{Re}}}}\left( {{I_{{1}}} \left( {{\kappa _1}{\omega _{{N_1}}} + {\kappa _2}{\omega _{{N_2}}} + {\omega _{T,s}}} \right) + {I_{{2}}}\left( {{\kappa _2}{\omega _{{N_2}}} + {\kappa _1}{\omega _{{N_1}}} + {\omega _{T,s}}} \right)} \right), \end{gathered}\right\} \end{equation}

in which $I_{1}$, $I_{2}$, $I_{12}$ are the second moments of area of the cross-section defined as

(2.72a–c)\begin{equation} {I_{{1}}} = \int_A {n_2^2\,{\rm d} A},\quad {I_{{2}}} = \int_A {n_1^2dA},\quad {I_{{1}{2}}} = \int_A {{n_2}{n_1}\,{\rm d} A}. \end{equation}

Since the cross-section is assumed to be fully circular, we have in dimensionless form $I=I_{{2}}=I_{{1}}=A^2/4{\rm \pi}$ and $I_{{12}}=0$ and, therefore, we arrive at

(2.73)\begin{equation} \left.\begin{gathered} {M_{{N_1}}} = \frac{{A_0^2}}{{4Re}}\left[ {3{\delta _s}\bar \mu \left( {{\omega _{{N_1},s}} - {\kappa _1}{\omega _T} - {\kappa _3}{\omega _{{N_2}}} - \frac{3}{2}{\kappa _2}{u_{,s}}} \right) + \left( {S_{TT2}^{\left( 1 \right)} - S_{NN2}^{\left( 1 \right)}} \right) -\frac{{Re{\kappa _2}}}{{We{{\sqrt A }_0}}}} \right], \\ {M_{{N_2}}} = \frac{{A_0^2}}{{4Re}}\left[ {3{\delta _s}\bar \mu \left( {{\omega _{{N_2},s}} - {\kappa _2}{\omega _T} + {\kappa _3}{\omega _{{N_1}}} + \frac{3}{2}{\kappa _1}{u_{,s}}} \right) - \left( {S_{TT1}^{\left( 1 \right)} - S_{NN1}^{\left( 1 \right)}} \right) + \frac{{Re{\kappa _1}}}{{We{{\sqrt A }_0}}}} \right],\\ {M_T} = \frac{{{A_0^2}}}{4}\left( {\frac{{2{\delta _s}\bar \mu }}{{{{Re}}}} \left( {{\kappa _1}{\omega _{{N_1}}} + {\kappa _2}{\omega _{{N_2}}} + {\omega _{T,s}}} \right)} \right). \end{gathered}\right\} \end{equation}

Now, using the related expressions and after some manipulation, we have

(2.74)\begin{equation} \frac{{\partial \left( {{A_0^2}{\boldsymbol{\omega }}} \right)}}{{\partial t}} + \frac{{\partial \left( {u{A_0^2}{\boldsymbol{\omega }}} \right)}}{{\partial s}}= 4\left( {\frac{{\partial {\boldsymbol{M}}}}{{\partial s}} + \frac{4}{{{{Re}}}} \frac{1}{{{\varepsilon ^2}}}{\boldsymbol{T}} \times {\boldsymbol{\eta }}} \right) + {\boldsymbol{F'}}, \end{equation}

with $\boldsymbol {F'}$ being the moment of external forces defined as

(2.75)\begin{equation} \left.\begin{gathered} {{F'}_{{N_1}}} = \frac{1}{{Rb}}{A_0^2}{u_{,s}}\sin (\beta ),\quad {{F'}_{{N_2}}} =\frac{2}{{Rb}}{A_0^2}{\omega _T}\sin (\beta ), \\ {{F'}_T} ={-} \frac{2}{{Rb}}{A_0^2}\left( {{\omega _{{N_1}}}\sin (\beta ) - {u_{,s}}\cos (\beta )} \right). \end{gathered}\right\} \end{equation}

Of note, when deriving the moment of the forces here, we use $\int _A {\boldsymbol {r}\,\textrm {d} A = 0}$. Throughout this study we also assume that $| {\kappa \times \varepsilon \boldsymbol {r}} | \ll 1$ always holds.

Now, it is time to derive the constitutive relations and the energy equation to take the viscoelastic extra stresses and temperature effects into consideration, as carried out in the upcoming subsections.

2.10. Constitutive modelling

To take the viscoelastic effects into account, here we use the Giesekus constitutive model, (2.5), which is a nonlinear viscoelastic model. In this study, to be consistent with experiments, we use the material properties of PEO dissolved in deionized water and set $\chi =0.15$ (see Gauri & Koelling Reference Gauri and Koelling1997), unless otherwise stated. After applying the corresponding asymptotic relations into the constitutive equations, in the leading order, we end up having

(2.76)\begin{equation} \left.\begin{gathered} \bar \lambda Wi\left( {S_{TT,t}^{(0)} + uS_{TT,s}^{(0)} - 2{u_{,s}}S_{TT}^{(0)} + \frac{\chi }{{\left( {1 - {\delta _s}} \right)}}{{\left( {S_{TT}^{(0)}} \right)}^2}} \right) + S_{TT}^{(0)} = 2\bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \\ -\bar \lambda Wi\left( {S_{NN,t}^{(0)} + uS_{NN,s}^{(0)} + {u_{,s}}S_{NN}^{(0)} + \frac{\chi }{{\left( {1 - {\delta _s}} \right)}}{{\left( {S_{NN}^{(0)}} \right)}^2}} \right) - S_{NN}^{(0)} =\bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \end{gathered}\right\} \end{equation}

and in the first order, we arrive at

(2.77)\begin{equation} \left.\begin{gathered} \bar \lambda Wi \left( {S_{TT1,t}^{(1)} + uS_{TT1,s}^{(1)} - 2{u_{,s}}S_{TT1}^{(1)} - 2{K_2}S_{TT}^{(0)}} \right) + S_{TT1}^{(1)} = 2\bar \mu \left( {1 - {\delta _s}} \right){K_2},\\ \bar \lambda Wi \left( {S_{TT2,t}^{(1)} + uS_{TT2,s}^{(1)} - 2{u_{,s}}S_{TT2}^{(1)} - 2{K_1}S_{TT}^{(0)}} \right) + S_{TT2}^{(1)} = 2\bar \mu \left( {1 - {\delta _s}} \right){K_1}, \\ -\bar \lambda Wi \left( {S_{NN1,t}^{(1)} + uS_{NN1,s}^{(1)} + {u_{,s}}S_{NN1}^{(1)} + {K_2}S_{NN}^{(0)}} \right) - S_{NN1}^{(1)} =\bar \mu \left( {1 - {\delta _s}} \right){K_2}, \\ -\bar \lambda Wi \left( {S_{NN2,t}^{(1)} + uS_{NN2,s}^{(1)} + {u_{,s}}S_{NN2}^{(1)} + {K_1}S_{NN}^{(0)}} \right) - S_{NN2}^{(1)} =\bar \mu \left( {1 - {\delta _s}} \right){K_1}, \end{gathered}\right\} \end{equation}

with ${K_1}$ and ${K_2}$ as

(2.78a,b)\begin{equation} {K_1} = {\omega _{{N_1},s}} - {\kappa _1}{\omega _T} -\tfrac{3}{2}{\kappa _2}{u_{,s}},\quad {K_2} ={-} {\omega _{{N_2},s}} - {\kappa _2}{\omega _T} - \tfrac{3}{2}{\kappa _1}{u_{,s}}. \end{equation}

It is of note that the leading-order equations of the normal stresses in the $\boldsymbol {N}_1$ and $\boldsymbol {N}_2$ directions are exactly the same and, therefore, we simply mark the corresponding normal stresses in both directions by the subscript $NN$.

2.11. Energy equation

Here, we derive the asymptotic energy conservation (2.4); using the cross-averaging technique and having the diffusion flux at the jet surface equal to the air convectional heat flux, we arrive at

(2.79)\begin{equation} {\partial _t}{\theta _0} + u{\partial _s}{\theta _0} ={-} \frac{{2N{u^*}({\hbar ^*},Re_w^*,P{r^*}){{\tilde c}_p}\tilde \rho }}{{\varepsilon P{e^*}{{\sqrt A }_0}}} \left( {{\theta _0} - {\theta _\infty }} \right), \end{equation}

in which ${\theta _\infty }$ and ${\theta _0}$ denote the dimensionless ambient temperature and the zeroth-order temperature in $\varepsilon$, respectively; $Nu^*$ stands for the air Nusselt number, $\hbar ^*$ the air attack angle, $Pr^*$ the air Prandtl number, $R{e^*_w}$ the air local Reynolds number, $Pe^*$ the air Péclet number and finally ${{\tilde c}_p}$ and $\tilde \rho$ denote the ratios of the air specific heat and air density to that of the polymer melt/solution, respectively. To compute $Nu^*$, we extend the model of Wieland et al. (Reference Wieland, Arne, Feßler, Marheineke and Wegener2019) to our curved fibre in the CS process expressed as

(2.80)\begin{gather} Nu^*(\hbar^*,R{e^*_w},Pr^*) = \left( {1 - 0.5\mathbb{G}(\hbar^*,R{e^*_w})} \right) \left\{\begin{array}{l} {\mathfrak{n}_1}(R{e^*_w},Pr^*),\quad R{e^*_w}Pr^* \ge 7.3 \times {10^{ - 5}}\\ {\mathfrak{n}_2}(R{e^*_w},Pr^*),\quad R{e^*_w}Pr^* < 7.3 \times {10^{ - 5}} \end{array}\right., \end{gather}

(2.81)\begin{gather} \mathbb{G}(\hbar^*,R{e^*_w}) = \left\{ \begin{array}{@{}l} {\left( {\hbar^*{Re} _w^{ - 1}} \right)^2},\quad {{Re} _w} \ge \xi = {10^{ - 7}}\\ {\left( {1 - {{\left( {{{{Re} }_w}{\xi ^{ - 1}}} \right)}^2}} \right)^2} + \left( {3 - 2{{\left( {{{{Re} }_w}{\xi ^{ - 1}}} \right)}^2}} \right) {\left( {\hbar^*{{{Re} }_w}{\xi ^{ - 2}}} \right)^2},\quad {{Re} _w} < \xi. \end{array}\right. \end{gather}

(2.82)\begin{gather} \left.\begin{gathered} {\mathfrak{n}_1}(R{e^*_w},Pr^*) = 0.462{\left( {R{e^*_w}Pr^*} \right)^{0.1}} + f(Pr^*) \frac{{{{\left({R{e^*_w}Pr^*} \right)}^{0.7}}}}{{1 + 2.79{{\left({R{e^*_w}Pr^*} \right)}^{0.2}}}}, \\ {\mathfrak{n}_2}(R{e^*_w},Pr^*) = {\mathfrak{m}_1}(Pr^*){\left( {R{e^*_w}Pr^*} \right)^3} + {\mathfrak{m}_2}(Pr^*){\left( {R{e^*_w}Pr^*} \right)^2} + 0.1, \\ {\mathfrak{m}_1}(Pr^*) ={-} 3.5636 \times {10^{11}} - 3.138 \times {10^9} \times f(Pr^*), \\ {\mathfrak{m}_2}(Pr^*) = 4.0694 \times {10^7} - 3.9768 \times {10^5} \times f(Pr^*), \\ f(Pr^*) = \frac{{2.5}}{{{{\left( {1 + {{(1.25{Pr^*}{^{1/6}})}^{2.5}}} \right)}^{0.4}}}}, \end{gathered}\right\} \end{gather}

where $\xi$ is a regulating parameter. To compute $\hbar ^*$ and $R{e^*_w}$, we use

(2.83)\begin{equation} \hbar^*=\left( {\sqrt A_0{Re^{*}}{\boldsymbol{V^*_{rel}}}} \right)\boldsymbol{\cdot}\hat{\boldsymbol{T}}, \end{equation}

and

(2.84)\begin{equation} R{e^*_w}={\sqrt A_0 Re^{*}}\left\| {{{\boldsymbol{V}}^*_{{\boldsymbol{rel}}}}} \right\|. \end{equation}

It is worth mentioning that since the air attack angle and the relative velocity vary along the fibre length, the value of $Nu^*$ is computed locally, as an output of the model. Moreover, the variations of the jet and air specific heat capacities and convection heat transfer coefficient as well as the surface tension coefficient with temperature are neglected. However, the temperature variation during the CS process causes the rheological behaviour of the fibre to change rapidly, which, in turn, remarkably alters the fibre dynamic behaviour. Thus, to take this into account, we calculate the relative viscosity and relaxation time based on the Arrhenius expressions (see Yarin, Sinha-Ray & Pourdeyhimi Reference Yarin, Sinha-Ray and Pourdeyhimi2010; Yarin, Pourdeyhimi & Ramakrishna Reference Yarin, Pourdeyhimi and Ramakrishna2014), represented respectively as

(2.85a–c)\begin{equation} \bar \mu = \exp \left( {B\left( {\frac{1}{\theta_0 } - 1} \right)} \right),\quad \bar \lambda = \frac{1}{\theta_0 }\exp \left( {B\left( {\frac{1}{\theta_0 } - 1} \right)} \right),\quad B = \frac{E}{{Re {\theta _{noz}}}}, \end{equation}

wherein $E$ is the activation energy and $Re$ is the absolute gas constant; in this work, we set $B=10$ (Yarin et al. Reference Yarin, Sinha-Ray and Pourdeyhimi2010).

Thus far, we have derived the asymptotic cross-averaged kinematic, dynamic and energy equations along with the boundary condition equations and viscoelastic constitutive relations in the 3-D space. In the following section, using our assumptions and conditions, we tailor the set of equations presented so far to the one appropriate to analyse the CS process.

2.12. Transient set of equations for the CS process

Now, we derive a set of transient equations customized for CS applications. Since in the CS process, the gravitational force is very small in comparison with the centrifugal one ($Rb \ll Fr$), the jet primarily flies in a horizontal $Y-Z$ plane until it sits on the collectors. Therefore, we ignore the gravitational effects (quantified via $Fr$) and solve the set of equations in the $Y-Z$ plane to analyse the fibre behaviour. In such a case, $\beta ={\rm \pi} /2$ and $X(s,t)=0$ always hold true and, hence, all the equations concerning the components of $\boldsymbol {\omega }$, $\boldsymbol {M}$ and $\boldsymbol {\kappa }$ are ruled out from our set of equations; this is with the exception of $\omega _{N_1}$, $M_{{N_1}}$ and $\kappa _2$ which for convenience are termed hereinafter as $(\omega, M, \kappa )$, respectively. To avoid a continual repetition of indices, we put $m_0={S_{TT}^{(0)} - S_{NN}^{(0)}}$, $m_1={S_{TT2}^{(1)} - S_{NN2}^{(1)}}$, $p_0=S_{NN}^{(0)}$, $p_1=S_{NN2}^{(1)}$ and, for simplicity, we drop the zeros from the leading-order parameter subscripts. Given these, our transient coupled system of partial differential equations in the 2-D space can be obtained as

(2.86)\begin{equation} \left.\begin{gathered} {Y_{,t}} + u{Y_{,s}} = {v_T}\cos (\alpha ) + {v_{{N_2}}}\sin (\alpha ),\quad {Z_{,t}} + u{Z_{,s}} = {v_T}\sin (\alpha ) - {v_{{N_2}}}\cos (\alpha ), \\ {u_{,s}} = {v_{T,s}} + \kappa {v_{{N_2}}},\quad {\omega } = \kappa {v_T} - {v_{{N_2},s}},\quad {\alpha _{,t}}+ u{\alpha _{,s}} = {\omega},\quad {\kappa _{,t}} + {\left( {u\kappa } \right)_{,s}} = {\omega}_{,s}, \\ {A_{,t}} + {(uA)_{,s}} = 0,\quad {\partial _t}{\theta} + u{\partial _s}{\theta} ={-} \frac{{2N{u^*}{{\tilde c}_p}\tilde \rho }}{{\varepsilon P{e^*}{\sqrt A }}} \left( {{\theta} - {\theta _\infty }} \right), \\ {v_{{N_2},t}} + u{v_{{N_2},s}} - {v_T}{\omega} = \frac{1}{{A{{Re}}}}\left( {{\eta _{{N_2},s}} - \kappa {\eta _T}} \right) + {F_{{N_2}}}, \\ {v_{T,t}} + u{v_{T,s}} + {v_{{N_2}}}{\omega} = \frac{1}{{A{{Re}}}}\left( {{\eta _{T,s}} + \kappa {\eta _{{N_2}}}} \right) + {F_T}, \\ {\left( {{A^2}{\omega}} \right)_{,t}} + {\left( {u{A^2}{\omega}} \right)_{,s}} = \frac{{4}}{{{{Re}}}}\frac{{\partial {M}}}{{\partial s}} - \frac{{16 }}{{{\varepsilon ^2}{{Re}}}}{\eta _{{N_2}}} + \frac{1}{{Rb}}{A^2}{u_{,s}}, \\ {m _0} + \bar \lambda Wi\left( {{m _{0,t}} + u{m _{0,s}} - \left( {2{m _0} + 3{p_0}} \right){u_{,s}} + \frac{{\chi \left( {m _0^2 + {p_0}{m _0}} \right)}}{{\left( {1 - {\delta _s}} \right)}}} \right) = 3 \bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \\ {p_0} + \bar \lambda Wi\left( {{p_{0,t}} + u{p_{0,s}} + {p_0}{u_{,s}} + \frac{{\chi p_0^2}}{{\left( {1 - {\delta _s}} \right)}}} \right) ={-}\bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \\ {m _1} + \bar \lambda Wi\left( {{m _{1,t}} + u{m _{1,s}} - \left( {2{m _1} + 3{p_1}} \right){u_{,s}} - \left( {2{m _0} + 3{p_0}} \right){K_1}} \right) = 3\bar \mu \left( {1 - {\delta _s}} \right){K_1}, \\ {p_1} + \bar \lambda Wi\left( {{p_{1,t}} + u{p_{1,s}} + {p_1}{u_{,s}} +{p_0}{K_1}} \right) ={-}\bar \mu \left( {1 - {\delta _s}} \right){K_1}, \\ {\eta _T} = A\left( {\left( {3{\delta _s}\bar \mu{u_{,s}} + {m _0}} \right) + \frac{{{{Re}}}}{{We\sqrt A }}} \right), \\ {M} = \frac{{{A^2}}}{4}\left( { \left( {3{\delta _s}\bar \mu{K_1} + {m _1}} \right) - \frac{{{{Re}}\kappa }}{{We\sqrt A }}} \right), \\ {K_1} = {\omega _{,s}} - \frac{3}{2}\kappa {u_{,s}}, \\ {F_{{N_2}}} = \frac{{\left( {Y\sin (\alpha ) - Z\cos (\alpha )} \right)}}{{R{b^2}}} + \frac{{2{v_T}}}{{Rb}} + {\left\langle {{F_{{drag}}}} \right\rangle _{{N_2}}},\\ {F_T} = \frac{{\left( {Y\cos (\alpha ) + Z\sin (\alpha )} \right)}}{{R{b^2}}} - \frac{{2{v_{{N_2}}}}}{{Rb}} + {\left\langle {{F_{{drag}}}} \right\rangle _T}. \end{gathered}\right\} \end{equation}

The set of transient equations outlined is developed for a growing curved jet in an Eulerian frame, for the solution of which one requires the fibre end velocity, which is not known a priori; this makes the problem at hand very hard to solve; namely, in the Eulerian description, the computational nodes are fixed in space and the jet can only grow at the end boundary. To deal with the problem, however, we can use a simple procedure to map our equations of interest from the Eulerian into the Lagrangian configuration; in this way, we would not need the end velocity in order to solve the equations numerically, thanks to a moving domain, i.e. the computational nodes are moving and, thus, the new nodes are added at the nozzle location whose conditions are known beforehand. To derive the equations in the Lagrangian frame, we first determine the Jacobian matrix of the transformation from $(s, t)$ in the Eulerian setting to $(\sigma,\tau )$ in the Lagrangian one. Introducing the elongation parameter $e$ in a way that $\partial s=e \partial \sigma$ and having $\partial t= \partial \tau$, we can compute the mapping Jacobian matrix $J$ as

(2.87) \begin{equation} J = \frac{{\partial (t,s)}}{{\partial \left( {\tau,\sigma } \right)}} = \begin{pmatrix} 1 & 0\\ u & e \end{pmatrix}\Rightarrow \left[ {{J^{ - 1}}} \right]^{{\rm T}} = \begin{pmatrix} 1 & { - u{e^{ - 1}}}\\ 0 & {{{\rm e}^{ - 1}}}\end{pmatrix}, \end{equation}

in which ${{\rm T}}$ represents the matrix transpose operator; therefore, we have

(2.88)\begin{equation} \left.\begin{gathered} \frac{\partial }{{\partial t}} = \frac{\partial }{{\partial \tau }} - \frac{u}{e} \frac{\partial }{{\partial \sigma}}, \\ \frac{\partial }{{\partial s}} = \frac{1}{e}\frac{\partial }{{\partial \sigma }}, \end{gathered}\right\} \end{equation}

using which we can simply derive the corresponding differential terms in time and space in the Lagrangian setting. For the expression of $u_{,s}$, we also have

(2.89)\begin{equation} {u_{,s}} = \frac{\partial }{{\partial s}}\left( {\frac{{\partial s}}{{\partial t}}} \right) = \frac{1}{e}\frac{\partial }{{\partial \sigma }}\left( {\frac{{\partial s}}{{\partial \tau }}} \right) = \frac{1}{e}\frac{\partial }{{\partial \tau }}\left( {\frac{{\partial s}}{{\partial \sigma }}} \right) = \frac{1}{e}\frac{{\partial e}}{{\partial \tau }}, \end{equation}

and for the continuity we end up with

(2.90)\begin{equation} {A_{,t}}+{(uA)_{,s}} = 0 \Rightarrow {\left( {Ae} \right)_{,\tau }} = 0, \end{equation}

and then all the equations can be transformed into the Lagrangian setting using (2.88), (2.89) and (2.90), giving

(2.91)\begin{equation} \left.\begin{gathered} {Y_{,\tau }} = {v_T}\cos (\alpha ) + {v_{{N_2}}}\sin (\alpha ),\quad {Z_{,\tau }} = {v_T}\sin (\alpha ) - {v_{{N_2}}}\cos (\alpha ), \\ {e_{,\tau }} = {v_{T,\sigma }} + \tilde \kappa {v_{{N_2}}},\quad {e^2}\varpi ={-} {v_{{N_2},\sigma}} + \tilde \kappa {v_T}, \\ {\alpha _{,\tau }} = e\varpi,\quad {{\tilde \kappa }_{,\tau }} = {\left( {e\varpi } \right)_{,\sigma }},\quad {\theta _{,\tau }} ={-} \frac{{2N{u^*}{{\tilde c}_p}\tilde \rho {\sqrt e } }}{{\varepsilon P{e^*}}} \left( {{\theta} - {\theta _\infty }} \right), \\ {v_{{N_2},\tau }} - e\varpi {v_T} = \frac{1}{{{{Re}}}} \left( {{\eta _{{N_2},\sigma }} - \tilde \kappa {\eta _T}} \right) + \frac{{\left( {Y\sin (\alpha ) - Z\cos (\alpha )} \right)}}{{R{b^2}}} + \frac{{2{v_T}}}{{Rb}} + {\left\langle {{F_{{drag}}}} \right\rangle _{{N_2}}}, \\ {v_{T,\tau }} + e\varpi {v_{{N_2}}} = \frac{1}{{{{Re}}}} \left( {{\eta _{T,\sigma }} + \tilde \kappa {\eta _{{N_2}}}} \right) + \frac{{\left( {Y\cos (\alpha ) + Z\sin (\alpha )} \right)}}{{R{b^2}}} - \frac{{2{v_{{N_2}}}}}{{Rb}} + {\left\langle {{F_{{drag}}}} \right\rangle _T}, \\ {\varpi _{,\tau }} = \frac{4}{{{{Re}}}}{M_{,\sigma }} - \frac{{16}}{{{\varepsilon ^2}{{Re}}}}e{\eta _{{N_2}}} + \frac{{{v_{T,\sigma }} + \tilde \kappa {v_{{N_2}}}}}{{{e^2}Rb}}, \\ {{\tilde m}_0} + \bar \lambda Wi\left( {{{\tilde m}_{0,\tau }} - 3({{\tilde m}_0} + {{\tilde p}_0}) \frac{{{e_{,\tau }}}}{e} + \frac{{\chi \left( {\tilde m_0^2 + 2{{\tilde p}_0}{{\tilde m}_0}} \right)}}{{e\left( {1 - {\delta _s}} \right)}}} \right) = 3\bar \mu \left( {1 - {\delta _s}} \right){e_{,\tau }}, \\ {{\tilde p}_0} + \bar \lambda Wi\left( {{{\tilde p}_{0,\tau }} + \frac{{\chi \tilde p_0^2}}{{e\left( {1 - {\delta _s}} \right)}}} \right) ={-} \bar \mu \left( {1 - {\delta _s}} \right){e_{,\tau }}, \\ {{\tilde m}_1} + \bar \lambda Wi\left( {{{\tilde m}_{1,\tau }} - 3\left( {{{\tilde m}_1} + {{\tilde p}_1}} \right)\frac{{{e_{,\tau }}}}{e} - \left( {2{{\tilde m}_0} + 3{{\tilde p}_0}} \right)\frac{{{{\tilde K}_1}}}{e}} \right) = 3\bar \mu \left( {1 - {\delta _s}} \right){{\tilde K}_1}, \\ {{\tilde p}_1} + \bar \lambda Wi\left( {{{\tilde p}_{1,\tau }} + {{\tilde p}_0} \frac{{{{\tilde K}_1}}}{e}} \right) ={-} \bar \mu \left( {1 - {\delta _s}} \right){{\tilde K}_1}, \end{gathered}\right\} \end{equation}

in which

(2.92)\begin{equation} \left.\begin{gathered} {\eta _T} = \frac{1}{{{e^2}}}\left( {3{\delta _s}\bar \mu {e_{,\tau }} + {{\tilde m}_0} + \frac{{{{Re}}}}{{We}}{{\rm e}^{{3}/{2}}}} \right), \\ M = \frac{1}{{4{e^3}}}\left( {3{\delta _s}\bar \mu {{\tilde K}_1} + {{\tilde m}_1} - \frac{{Re}}{{We}}\tilde \kappa \,{{\rm e}^{{1}/{2}}}}\right), \\ {{\tilde K}_1} = {{\tilde \kappa }_{,\tau }} - \frac{3}{2}\tilde \kappa \frac{{{e_{,\tau }}}}{e}, \end{gathered}\right\} \end{equation}

with $\varpi =\omega /e$, $\tilde {m}_0=e m_0$, $\tilde {p}_0=e p_0$, $\tilde {m}_1=e m_1$, $\tilde {p}_1=e p_1$ and $\tilde {\kappa }=e \kappa$. To numerically solve the set of partial differential equations presented here, we need several boundary and initial conditions, coupled to a rigorous solution algorithm, presented in the following sections.

2.12.2. Solution procedure

To solve our set of equations in an unsteady frame, we implement the finite volume method to estimate the spatial derivatives and source terms. To this aim, first we classify the governing equations into flux function terms and source terms. In the next step, we use different schemes to discretize the flux terms based on their directions: we use the upwind scheme for the convective terms $f^u$, the downwind scheme for the normal stresses $f^d$ and the central differences for the viscous terms $f^c$. Following this procedure, we can represent our set of equations in a general form as

(2.93)\begin{equation} {\partial _\tau }G(\phi ) = {\partial _\sigma }{f^d}(\phi ) + {\partial _\sigma }{f^u}(\phi ) + {\partial _\sigma }{f^c}(\phi,{\partial _\sigma }H(\phi )) + {\mathcal{S}_1}\left( {\phi,{\partial _\sigma }H(\phi )} \right) + {\mathcal{S}_2}(\phi), \end{equation}

with $\mathcal {S}_1$ and $\mathcal {S}_2$ denoting the source terms with and without the spatial derivative terms, respectively. Here, $G$ and $H$ are arbitrary functions and $\phi$ is a given variable. Now, the idea is to integrate the set of equations in the form of (2.93) over a control volume, leading to a system of integro-differential equations, expressed in a generalized form as

(2.94)\begin{align} \frac{{{\rm d}{G_i}}}{{{\rm d}\tau }} &= \frac{1}{{\Delta \sigma }}\left[ {\left( {f_{i + 1/2}^d - f_{i - 1/2}^d} \right) + \left( {f_{i + 1/2}^u - f_{i - 1/2}^u} \right) + \left( {f_{i + 1/2}^c - f_{i - 1/2}^c} \right)} \right] \nonumber\\ &\quad + \frac{1}{{\Delta \sigma }}\int_{i - 1/2}^{i + 1/2} {{{\mathcal{S}}_1} \left({\phi,{\partial _\sigma }H(\phi )} \right){\rm d}\sigma } + \frac{1}{{\Delta \sigma }}\int_{i - 1/2}^{i + 1/2} {{{\mathcal{S}}_2}(\phi )\,{\rm d}\sigma}, \end{align}

in which the cell centres are indexed as $i$ and, thus, $i \pm 1/2$ gives the finite volume up- and down-wind boundaries. Moreover, to implement the Lagrangian setting and handle the growing jet, we determine the dynamic cells, such that at the end of each iteration the new distance of the first cell centre from the nozzle is calculated, and the new cells are added at the nozzle with the aid of the floor function. The new cells are initialized with boundary conditions imposed at the nozzle. Henceforth, we use the implicit Runge–Kutta scheme, i.e. the Radau IIa technique, for time integration and solve the resultant set of equations using Newton's method. We refer the reader to Arne et al. (Reference Arne, Marheineke, Meister, Schiessl and Wegener2015) for further details of the numerical approach implemented here.

2.13. Steady state set of equations for the CS process

In a typical CS process, at short times, the fibre flow passes through an unsteady phase before reaching a steady state at long times (see e.g. the experimental results of Noroozi et al. Reference Noroozi, Arne, Larson and Taghavi2020). Since the production of fibres at long times may be a steady process in the CS method, it is reasonable to also present and analyse the steady set of equations in this study; this allows us to more easily investigate the effects of different flow parameters on the jet dynamic behaviour.

At steady state, in our moving frame of reference, the coordinate velocity is zero and, thus, the leading-order fluid velocity at the jet axis direction is equal to the convectional velocity of the fluid, i.e. $(v_{N_2}, v_T)=(0, u)$. On the other hand, the angular velocity $\omega$ is reduced to $u\kappa$ and, due to the constant mass flux, we have $A=1/u$. Giving the new relations for the velocities and angular velocity, removing the unsteady terms from our transient set of equations in the Eulerian setting (2.86), and using some rearrangement, we eventually end up with our set of equations in the steady frame as

(2.95)\begin{equation} \left.\begin{gathered} {Y_{,s}} = \cos (\alpha ),\quad {Z_{,s}} = \sin (\alpha ),\quad {\alpha _{,s}} = \kappa,\quad {\partial _s}\theta ={-} \frac{{2N{u^*}{{\tilde c}_p}\tilde \rho }}{{\varepsilon P{e^*} \sqrt u}} \left( {{\theta} - {\theta _\infty }} \right), \\ {\eta _{{N_2},s}} = \left( {\eta _T} - {Re}u \right)\kappa - \frac{{Re\left( {Y\sin (\alpha ) - Z\cos (\alpha )} \right)}}{{uR{b^2}}} - \frac{{2Re}}{{Rb}} - \frac{{{{Re}}}}{u}{\left\langle {{F_{{drag}}}} \right\rangle _{{N_2}}}, \\ {\eta _{T,s}} = Re{u_{,s}} - \kappa {\eta _{{N_2}}} - \frac{{Re\left( {Y\cos (\alpha ) + Z\sin (\alpha )} \right)}}{{uR{b^2}}} - \frac{{{{Re}}}}{u}{\left\langle {{F_{{drag}}}} \right\rangle _T}, \\ {M_{,s}} = \frac{{Re}}{4}{\kappa _{,s}} + \frac{4}{{{\varepsilon^2}}}{\eta _{{N_2}}} - \frac{{Re}}{{4{u^2}Rb}}{u_{,s}}, \\ {u_{,s}} = \frac{1}{{3{\delta _s}\bar \mu }}\left( {u{\eta _T} -\frac{{{{Re}} \sqrt u }}{{We}} - {m_0}} \right), \\ {\kappa _{,s}} = \frac{1}{{3{\delta _s}\bar \mu u}}\left( {4{u^2}M + \frac{{Re\kappa \sqrt u }}{{We }} - {m_1}} \right) + \frac{1}{{2u}}\kappa {u_{,s}}, \\ {m_0} + \bar \lambda Wi\left( {u{m_{0,s}} - \left( {2{m_0} + 3{p_0}} \right){u_{,s}} + \frac{{\chi \left( {m_0^2 + {p_0}{m_0}} \right)}}{{\left( {1 - {\delta _s}} \right)}}} \right) = 3\bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \\ {p_0} + \bar \lambda Wi\left( {u{p_{0,s}} + {p_0}{u_{,s}} + \frac{{\chi p_0^2}}{{\left( {1 - {\delta _s}} \right)}}} \right) ={-} \bar \mu \left( {1 - {\delta _s}} \right){u_{,s}}, \\ {m_1} + \bar \lambda Wi\left( {u{m_{1,s}} - \left( {2{m_1} + 3{p_1}} \right){u_{,s}} - \left( {2{m_0} + 3{p_0}} \right){K_1}} \right) = 3\bar \mu \left( {1 - {\delta _s}} \right){K_1}, \\ {p_1} + \bar \lambda Wi\left( {u{p_{1,s}} + {p_1}{u_{,s}} + {K_1}{p_0}} \right) ={-} \bar \mu \left( {1 - {\delta _s}} \right){K_1}, \end{gathered}\right\} \end{equation}

in which ${K_1}= u{\kappa _{,s}} - \frac {1}{2}\kappa {u_{,s}}$. In the ensuing sections, we present the boundary conditions and solution algorithm to solve our set of steady equations.

2.13.3. Polymer melt jet modelling constraint

As seen explicitly in (2.95), the stationary (steady state) equations for $u_{,s}$ and $\kappa _{,s}$ become singular when $\delta _s=0$; this is the case when the fluid is a polymer melt. For such a condition, the viscous effects of solvent play no role on the fibre dynamics. To solve the problem, one can derive the set of stationary equations anew applying $\delta _s=0$, as carried out in Appendix B. However, as seen in Appendix B, the terms $q_2$ and $q_3$ appear in the equations for $u_{,s}$ and $\kappa _{,s}$, implying further limitations: when $q_2$ and $q_3$ approach zero, the equations for $u_{,s}$ and $\kappa _{,s}$ become singular; in addition, when $q_2$ and $q_3$ are negative, the aforementioned equations have no physically relevant stationary solutions and they would need a regularization. Therefore, to avoid any singularity issues that may arise in the solution of the set of (B1), for $\delta _s \to 0$, we instead use the set of (2.95) but consider a small value of $\delta _s=10^{-4}$, to keep this set from becoming singular (see also Lorenz, Marheineke & Wegener Reference Lorenz, Marheineke and Wegener2014).

Before we proceed, let us note a limitation on the assumptions used in our study. Our results for polymer melts reveal that the fibre temperature can decrease significantly during the CS process; in reality, this considerable temperature drop may bring the fibre temperature below its solidification or crystallization temperature, which we ignore in this study. While outside of the scope of this paper, these phenomena and their effects on the fibre dynamics could be accounted for within our framework by introducing appropriate crystallization rate equations and the constitutive equations for the amorphous melt and semi-crystalline phases, into the string model equations (see for instance Ettmüller et al. Reference Ettmüller, Arne, Marheineke and Wegener2021).

In the upcoming section, we analyse the CS process using the input parameters given in table 1. The solutions of our transient and stationary mathematical models with certain input parameters provide us with several output parameters, presented in table 2, with the help of which we can quantify the effects of the flow parameters in the CS process.

Table 2. Main model output parameters. These parameters are at the leading order and they are functions of the arc length $s$ and time $t$.