**2. Analysis**

The WOW-type coating process is illustrated in Figure 2. The glass fiber is pulled with constant velocity *U* through the primary coating die, which is filled with a primary coating resin. Afterwards, the uncured coated fiber optics enters the secondary coating die, which is filled with a secondary resin. After the secondary die the fiber leaves the system with two-coated layers, as displayed in Figure 2. At the end these coated-layers, they are cured by ultraviolet lamps. Where *Rw*, *R*, and *R*<sup>d</sup> are the radius of the fiber optics, interface radius location, and radius of the die, *L* is the length of the die. The present study is investigated under the assumption that the flow is incompressible, laminar, length of the die is sufficient large, the fiber optics moves along the centerline with constant speed, negligible small radial flow, as compared to the axial flow, because of high viscosity of the polymer-melt, the viscous impacts are dominant, as compared to the inertial effects, axial heat conduction is negligible, and the thermal conductivity, specific heat, melt density do not depend on the temperature and neglect the gravitational effect. To analyze the flow, the cylindrical coordinate system (*r*, θ, *z*) is used in which *r* is the radial coordinate and *z* is the axial coordinate of the wire means centerline of the die.

**Figure 2.** Geometry of double-layer optical fiber coating in wet-on-wet coating process [17].

The basic equations governing the flow of incompressible fluids are:

$$
\nabla \cdot \boldsymbol{\mu} = 0 \tag{1}
$$

$$
\rho \frac{\mathbf{d}u}{\mathbf{d}t} = \nabla.\mathbf{T} \tag{2}
$$

$$
\rho c\_p \frac{d\Theta}{dt} = k\nabla^2 \Theta + \Phi \tag{3}
$$

$$f(tr\mathbf{S})\mathbf{S} + \lambda\mathbf{S} = \eta\mathbf{A} \tag{4}$$

where ρ is the density of the fluid, *T* is the shear stress tensor, *c*<sup>p</sup> is the the specific heat, *D/Dt* denotes the material derivative, *k* is the thermal conductivity, Θ is the fluid temperature, Φ is the dissipation function, *trS* is the trace of extra stress tensor, S˙ is the upper contra-variant convicted tensor, μ is the viscosity of the fluid, and A is the deformation rate tensor.

The shear stress tensor is given in Equation (2) and the deformation rate tensor is given in Equation (4), defined as:

$$\mathbf{T} = -p\mathbf{I} + \mathbf{S} \tag{5}$$

$$\mathbf{A} = \mathbf{L}^T + \mathbf{L} \tag{6}$$

where I is the identity tensor and the superscript, *T* stands for the transpose of a matrix, and *L* = ∇*u*.

The upper contra-variant convicted tensor S in Equation (4) is given by ˙

$$\mathbf{S} = \frac{d\mathbf{S}}{dt} - \left[ (\nabla \boldsymbol{\mu})^T \mathbf{S} + \mathbf{S} (\nabla \boldsymbol{\mu}) \right] \tag{7}$$

The function *f*(*tr*S) is given by Tanner [19–21],

$$f(tr\text{S}) = 1 + \frac{\varepsilon\lambda}{\eta}(tr\text{S})\tag{8}$$

In Equation (8), *f*(*tr*S) is the stress function in which ε is related to the elongation behavior of the fluid. For ε = 0, the model reduces to the well-known Maxwell model and for λ = 0, the model reduces to a Newtonian one.

With the above frame of reference and assumptions the fluid velocity, extra stress tensor and temperature filed are considered as

$$\mu = (0, 0, w(r)), \mathbf{S} = \mathbf{S}(r), \Theta = \Theta(r) \tag{9}$$

Using assumptions and Equation (9), the continuity Equation (1) satisfied identically and from Equations (2–8), we arrive at:

$$\frac{\partial p}{\partial r} = 0\tag{10}$$

$$\frac{\partial p}{\partial \theta} = 0\tag{11}$$

$$\frac{\partial p}{\partial z} = \frac{1}{r} \frac{\mathbf{d}}{\mathbf{d}r} (r S\_{\text{rz}}) \tag{12}$$

$$k\left(\frac{\text{d}^2}{\text{d}r^2} + \frac{1}{r}\frac{\text{d}}{\text{d}r}\right)\Theta + S\_{\text{rx}}\frac{\text{d}w}{\text{d}r} = 0\tag{13}$$

$$(f(trS)S\_{\rm xz} = 2\lambda S\_{\rm rx} \frac{\rm dw}{\rm dr})\tag{14}$$

$$f(trS)S\_{\rm rz} = \eta \frac{\rm dw}{\rm dr} \tag{15}$$

$$
\Phi = \mathcal{S}\_{\text{rx}} \frac{\text{d}w}{\text{d}r} \tag{16}
$$

From Equations (10) and (11), it is concluded that *p* is a function of *z* only. Assuming that the pressure gradient along the axial direction is constant. Thus, we have <sup>d</sup>*p*/d*<sup>z</sup>* = Ω.

Integrating Equation (12) with respect to *r*, we get

$$S\_{\rm rz} = \frac{\Omega}{2}r + \frac{\mathbb{C}}{r} \tag{17}$$

where *C* is an arbitrary constant of integration.

By substituting Equation (17) in Equation (15), we have

$$f(trS) = \frac{\eta \frac{dw}{dr}}{\left(\frac{\Omega}{2}r + \frac{C}{r}\right)}\tag{18}$$

Combining Equations (14), (15) and (17), we obtain the explicit expression for a normal stress component *S*zz as:

$$S\_{\rm zx} = 2\frac{\lambda}{\eta} \left( \frac{\Omega}{2} r + \frac{C}{r} \right)^2 \tag{19}$$

From Equations (8) and (18), we have

$$
\eta \frac{\mathrm{d}w}{\mathrm{d}r} = \left(1 + \varepsilon \frac{\lambda}{\eta} \mathrm{S}\_{zz}\right) \left(\frac{\Omega}{2}r + \frac{\mathsf{C}}{r}\right) \tag{20}
$$

Inserting Equation (19) in Equation (20), we obtain an analytical expression for axial velocity as:

$$\frac{\mathrm{d}w\_{(j)}}{\mathrm{d}r} = \frac{1}{\eta\_{(j)}} \left( \frac{\Omega}{2}r + \frac{C\_{(j)}}{r} \right) + 2\varepsilon \frac{\lambda^2}{\eta\_{(j)}^3} \left( \frac{\Omega}{2}r + \frac{C\_{(j)}}{r} \right)^3 \tag{21}$$

Additionally, the temperature distribution is

$$k\_{(j)} \left(\frac{\text{d}^2}{\text{d}r^2} + \frac{1}{r}\frac{\text{d}}{\text{d}r}\right) \theta\_{(j)} + S\_{r z\_{(j)}} \frac{\text{d}w\_{(j)}}{\text{d}r} = 0 \tag{22}$$

Here, *j* = 1, 2 represents the primary layer and secondary layer flow, respectively.

The boundary condition on θ(*j*) is θ<sup>w</sup> at the fiber optics and θ<sup>d</sup> at the die wall. For the problem displayed in Figure 1, at the fluid interface, we utilize the assumptions that the velocity, the shear stress, and the pressure gradient along the flow direction and the temperature and the heat flux are continuous, which are given as follows.

The relevant boundary and interface conditions [17–22] on the velocity are

$$w\_1 = \mathcal{U} \text{ at } r = R\_\text{w and } w\_2 = 0 \text{ at } r = R\_\text{d} \tag{23}$$

$$\mathbf{w}\_1 = w\_2 \text{ and } \mathbf{S}\_{\mathbf{r}\mathbf{z}1} = \mathbf{S}\_{\mathbf{r}\mathbf{z}2} \text{ at } \mathbf{r} = \mathbb{R} \tag{24}$$

The relevant boundary and interface conditions [17–22] on the temperature are

$$
\theta\_1 = \theta\_\text{w} \text{ at } r = R\_\text{w} \text{ and } \theta\_2 = \theta\_\text{d} \text{ at } r = R\_\text{d} \tag{25}
$$

$$\theta\_1 = \theta\_2 \text{ and } k\_1 \frac{\mathbf{d}\theta\_1}{\mathbf{d}r} = k\_2 \frac{\mathbf{d}\theta\_2}{\mathbf{d}r} \text{ at } r = R \tag{26}$$

We introduce the non-dimensional flow variables as

$$\begin{array}{c} \Gamma^\* = \frac{r}{\mathbb{R}\_{\mathbf{w}}}, w^\*\_{\langle j\rangle} = \frac{\mathbb{w}\_{\langle j\rangle}}{\mathbb{L}}, \mathfrak{G}^\*\_{\langle j\rangle} = \frac{\theta\_{\langle j\rangle} - \theta\_{\mathbf{d}}}{\theta\_{\mathbf{d}} - \theta\_{\mathbf{w}}}, C^\*\_{\langle j\rangle} = \frac{2\mathbb{C}\_{\langle j\rangle}}{\mathbb{R}\_{\mathbf{w}}^2 \mathbb{L} \Omega'} Br\_{\langle j\rangle} = \frac{\eta\_{\langle j\rangle} \mathbb{L}^2}{k\_{\langle j\rangle} (\theta\_{\mathbf{d}} - \theta\_{\mathbf{w}})}, \varepsilon D^2\_{\langle j\rangle} = \frac{\lambda \mathbb{L}\_{\mathbf{c}}}{\mathbb{R}\_{\mathbf{w}}}, \mathsf{X}\_{\langle j\rangle} = \frac{\mathsf{U}}{\mathsf{L}},\\ \Gamma^\* = \frac{\mathsf{R}}{\mathbb{R}\_{\mathbf{w}}^\*}, \frac{\mathsf{R}\_{\mathbf{d}}}{\mathsf{L}\_{\mathbf{w}}} = \delta > 1, K = \frac{k\_2}{\mathsf{k}\_1}, j = 1, 2. \end{array} \tag{27}$$

$$\frac{d w\_{(j)}}{d \tau} = -4 \tau X\_{(j)} - 4 \mathcal{C}\_{(j)} X\_{(j)} \frac{1}{r} - 128 X\_{(j)} \varepsilon D\_{(j)}^2 r^3 - 384 X\_{(j)} \varepsilon D\_{(j)}^2 \mathcal{C}\_{(j)} r - 384 X\_{(j)} \mathcal{C}^2\_{(j)} \varepsilon D\_{(j)}^2 \frac{1}{r} - \tag{28}$$

$$\frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d}\theta\_{(j)}}{\mathrm{d}r} \right) - 4Br\_{(j)}X\_{(j)} \left( r^2 + \mathcal{C}\_{(j)} \right) \frac{\mathrm{d}w\_{(j)}}{\mathrm{d}r} = 0 \tag{29}$$

$$w\_1(1) = 1, w\_2(\delta) = 0\tag{30}$$

$$w\_1(\Gamma) = w\_2(\Gamma), \mathcal{S}\_{\text{rz1}}(\Gamma) = \mathcal{S}\_{\text{rz2}}(\Gamma) \tag{31}$$

$$\theta\_1(1) = 0, \theta\_2(\delta) = 1, \theta\_1(\Gamma) = \theta\_2(\Gamma), \frac{\mathbf{d}\theta\_1(\Gamma)}{\mathbf{d}r} = K \frac{\mathbf{d}\theta\_2(\Gamma)}{\mathbf{d}r}.\tag{32}$$

where *U*<sup>c</sup> = <sup>−</sup> *<sup>R</sup>*<sup>2</sup> <sup>w</sup>Γ/8η(*j*) is the characteristic velocity scale, and ε*D*<sup>2</sup> (*j*) is the characteristic Deborah number based on velocity scale *U*c, *X*(*j*) has physical meaning of a non-dimensional pressure gradient and *Br*(*j*) is the Brinkman number. Here, Γ is the dimensionless parameter that is the ratio of the radius of the liquid-liquid interface to the radius of the optical fiber and *j* = 1, 2 stands for primary and secondary coating layer flows, respectively.

#### **3. Analytical Solution (Exact Solution)**

Analytical solution is given in the Appendix A.

#### **4. Numerical Solution**

We shall solve the above equations numerically. For this purpose, the Runge–Kutta–Fehlberg method is employed. The computations are carried out for δ = 2. Before proceeding to the results and their discussion, we first validate our results of numerical solution for comparing them with the corresponding results based on exact solution (given in Appendix A). To this end, Figure 3 is prepared, which shows the velocity curve obtained through both numerical and exact solutions. This figure clearly demonstrates an excellent correlation between both the solutions. This establishes the confidence on both exact and numerical solutions and also on the results predicted by these solutions.

**Figure 3.** Comparison of analytical and numerical solutions when ε*D*<sup>2</sup> <sup>1</sup> = 5, <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> = 10, *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, δ = 2.

#### **5. Results of Analysis and Discussion**

This section shows the impact of different emerging parameters of interest including the Deborah numbers (viscoelastic parameter) ε*D*<sup>2</sup> <sup>1</sup> and <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> ,pressure gradient parameters *X*<sup>1</sup> and *X*2, Brinkman numbers *Br*<sup>1</sup> and *Br*<sup>2</sup> and the radii ration δ on the velocity and temperature profiles, volume flow rate, thickness of the coated fiber optics, shear stress, and force required to pulling the fiber optics (later referred as force only). This purpose is achieved graphically in 4–11. Figure 4 shows the effect of dimensionless pressure gradient *X*<sup>1</sup> and *X*<sup>2</sup> on the velocity profile when ε*D*<sup>2</sup> <sup>1</sup> = 0.5, <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> = 1, δ = 2. This figure shows that, as the pressure gradient parameter increases, the velocity profile increases. The effect of Deborah number ε*D*<sup>2</sup> <sup>1</sup> on velocity profile is shown in Figure 5. Since Deborah number is the measure of the ratio of the rate of the pressure drop in the flow to the viscosity, i.e., ε*D*<sup>2</sup> (*j*) <sup>=</sup> <sup>λ</sup>*U*<sup>c</sup> *R*w where *U*<sup>c</sup> = <sup>−</sup> *<sup>R</sup>*<sup>2</sup> <sup>w</sup>Ω/8η(*j*) is the characteristic velocity and Ω is constant pressure gradient in the axial direction. That is why the velocity follows as an increasing trend with increasing Deborah number. From Figures 4 and 5, it is clear that nonlinear behavior is occurred in the velocity profiles. Since the velocity of fluid first increase up to a certain value and then decreases, which shows the shear thickening effect. For low elasticity means for low Deborah number, the velocity disparity diverges a little from the Newtonian one, however, when the Deborah number is increased, these profiles turn into a more flattened one, showing the shear-thinning effect. It can be seen that, as ε is reduced, the profiles turn to the Newtonian one and the result is therefore independent of *D*<sup>2</sup> <sup>1</sup> and *<sup>D</sup>*<sup>2</sup> <sup>2</sup>. As *<sup>X</sup>*(*j*) <sup>=</sup> *<sup>U</sup>*<sup>c</sup> *<sup>U</sup>* is the pressure gradient in which *U*<sup>c</sup> = <sup>−</sup> *<sup>R</sup>*<sup>2</sup> <sup>w</sup>Ω/8η(*j*) is the characteristic velocity where *U* is the optical fiber velocity. That is why the velocity inside the die exceeds from the fiber optics velocity due to large values of the pressure gradient parameter.

**Figure 4.** Effect of *X*<sup>1</sup> and *X*<sup>2</sup> on velocity when ε*D*<sup>2</sup> <sup>1</sup> = 5, <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> = 10, δ = 2.

**Figure 5.** Effect of ε*D*<sup>2</sup> <sup>1</sup> and <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> on velocity profile when *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, δ = 2.

Figure 6 reveals that the volume flow rate increases with the increasing values of Deborah number along with increasing radii ratio δ. The dimensionless temperature profile inside the die for various values of emerging parameters is shown in Figures 7–9. Figure 7 depicts the effect of Brinkman number on temperature profile. A rise in temperature is observed with increasing the Brinkman number. Additionally, the temperature increases with an increasing Deborah number and pressure gradient parameters, as shown in Figures 8 and 9, respectively.

**Figure 6.** Effect of ε*D*<sup>2</sup> <sup>1</sup> and <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> on volume flow rate when *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0.

**Figure 7.** Effect of *Br*<sup>1</sup> and *Br*<sup>2</sup> on temperature *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, ε*D*<sup>2</sup> <sup>2</sup> = 10, δ = 2.

The thickness of the coated fiber optics or coating thickness (*h*c) is shown in Figures 10 and 11. It is observed that the thickness of the coated fiber optics increases with the increasing values of Deborah number and radii ratio δ, as shown in Figures 10 and 11, respectively. For the sake of validity, the present work is also compared with the published work in Reference [17] and good agreement is found by taking the non-Newtonian parameter, which tends to zero, i.e., λ → 0.

**Figure 8.** Effect of ε*D*<sup>2</sup> <sup>1</sup> and <sup>ε</sup>*D*<sup>2</sup> <sup>2</sup> on temperature *Br*<sup>2</sup> = 0.5 *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, δ = 2.

**Figure 9.** Effect of *X*<sup>1</sup> and *X*<sup>2</sup> on temperature *Br*<sup>2</sup> = 0.5, ε*D*<sup>2</sup> <sup>2</sup> = 10, δ = 2.

**Figure 10.** Effect of ε*D*<sup>2</sup> <sup>1</sup> on thickness of coated fiber optics when *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, δ = 2.

**Figure 11.** Effect of δ on thickness of coated fiber optics when *X*<sup>1</sup> = 0.5, *X*<sup>1</sup> = 1.0, ε*D*<sup>2</sup> <sup>2</sup> = 10.
