**6. Conclusions**

To provide protection from signal attenuation and mechanical damage, optical fibers required a double-layer resin coating on the glass fiber. Wet-on-wet coating processes are considered for double-layer coating in optical fiber manufacturing. Expressions are presented for the radial variation of axial velocity and temperature distribution analytically and numerically. Analytical expressions of velocity, volume flow rate, final radius of the coated fiber optics and force required the full fiber optics, which are reported. The effect of physical parameters such as Deborah number, dimensionless parameter, radii ratio δ and Brinkman number has been obtained numerically. It was found that velocity increases with increasing values of these parameters. The volume flow rate increases with increasing Deborah number. The thickness of coated fiber optic increase with an increase in ε*D*<sup>2</sup> <sup>1</sup>, <sup>ε</sup>*D*<sup>2</sup> 2, and δ. The temperature depends upon *Br*1, *Br*2, ε*D*<sup>2</sup> <sup>1</sup>, <sup>ε</sup>*D*<sup>2</sup> <sup>1</sup>, *X*1, and *X*2, and it increases with increasing these parameters. For ε = 0 and λ = 0, our results respectively, reduce to Maxwell and linear viscous model. According to the best of our knowledge, there is no previous literature about the discussed problem, which is our first attempt to handle this problem with two-layer coating flows.

**Author Contributions:** Z.K. modelled the physical problem. H.U.R. solved it. I.K. and S.O.A. computed the results. T.A. and D.L.C.C. wrote the physical discussion of the results and conclusion. All the authors equally contributed in writing manuscript.

**Acknowledgments:** Authors would like to thanks YUTP 015LC0-078 for the financial support. and Deanship of Scientific Research, Majmaah University for supporting this work.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

Analytical solution

Solutions of Equations (28) and (29) corresponding to the boundary conditions Equations (30–32) become:

$$w\_1 = -2r^2\mathcal{X}\_1 - 4\mathcal{C}\_1X\_1\ln r - 32\varepsilon D\varepsilon\_1^2r^4 - 192X\_1\mathcal{C}\_1\varepsilon D\varepsilon\_1^2r^2 - 384X\_1\mathcal{C}\_1^2\varepsilon D\varepsilon\_1^2\ln r + \\\tag{A1}$$

$$\begin{split} \Delta w\_{2} &= -2r^{2}\mathbb{X}\_{2} - 4\mathbb{C}\_{2}\mathbb{X}\_{2}\ln r - 32\mathbb{X}\_{2}\varepsilon D\mathbf{e}\_{2}^{2}r^{4} - 192\mathbb{X}\_{2}\mathbb{C}\_{2}\varepsilon D\mathbf{e}\_{2}^{2}r^{2} - 384\mathbb{X}\_{2}\mathbb{C}\_{2}^{2}\varepsilon D\mathbf{e}\_{2}^{2}\ln r + \\ &64\mathbb{C}\_{2}^{3}\mathbb{X}\_{2}\varepsilon D\mathbf{e}\_{2}^{2}\frac{1}{r^{2}} + \mathbb{C}\_{4} \end{split} \tag{A2}$$

Volume flow rates are

$$Q\_1 = X\_1 \left( \begin{pmatrix} \mathbb{C}\_1 + 96 \mathbb{C}\_1 \mathrm{\boldsymbol{\varepsilon}} \mathrm{D} \mathrm{e}\_1^2 + \frac{1}{r} \mathbb{C}\_3 \\ - \mathbb{C} \left( \mathbb{C}\_4 + 96 \mathbb{C}\_1 \mathrm{\boldsymbol{\varepsilon}} \mathrm{D} \mathrm{e}\_1^2 \right) \left( \mathbb{I}^4 - 1 \right) - \frac{16}{3} \mathbb{\varepsilon} \mathrm{D} \mathrm{e}\_1^2 \left( \mathbb{I}^6 - 1 \right) + 64 \mathbb{C}\_1 \mathrm{\boldsymbol{\varepsilon}} \mathrm{D} \mathrm{e}\_1^2 \ln \Gamma \\ - 2 \left( \mathbb{K}\_4 + 96 \mathbb{C}\_1 \mathrm{\boldsymbol{\varepsilon}} \mathrm{D} \mathrm{e}\_1^2 \right) \mathrm{I}^2 \ln \Gamma \end{pmatrix} \right) \tag{A3}$$

$$Q\_2 = \mathbb{C}\_4 \left(\delta^2 - \Gamma^2\right) - \frac{1}{2} X\_2 \left(1 + 96 \mathbb{C}\_2 \varepsilon D \varepsilon\_2^2\right) \left(\delta^4 - \Gamma^4\right) - \frac{16}{3} X\_2 \varepsilon D \varepsilon\_2^2 \left(\delta^6 - \Gamma^6\right) - 2 \mathbb{C}\_2 X\_2 \left(1 + 192 \varepsilon D \varepsilon\_2^2\right) \times \tag{A4}$$

$$\begin{pmatrix} 8^2 \ln \delta \\ -\Gamma^2 \ln \Gamma \end{pmatrix} + 64 \mathbb{C}\_2 {}^2 \varepsilon D \varepsilon\_2^2 (\ln \delta - \ln \Gamma) \tag{A4}$$

Thickness of the coated fiber optics of both layers is [17–21]

$$\mathcal{R}\_{1} = \left[ \left[ 1 - \frac{2}{17} \mathcal{Z} \begin{pmatrix} 9 \mathfrak{s} \varepsilon \mathrm{De}\_{1}^{2} (-\Gamma + \Gamma^{6} + 10(-1 + \Gamma) \mathrm{C}\_{1} \{\Gamma + \Gamma^{2} + \Gamma^{3} + 6 \mathrm{C}\_{1} \ln \Gamma - \mathrm{C}\_{1}^{\circ} \} X\_{1}) + \\ 5 \Gamma \begin{pmatrix} -3(-1 + \Gamma) \mathrm{C}\_{3} \\ + 6 \ln \left(-1 + \Gamma^{2} \right) \\ \mathrm{C}\_{1} X\_{1} + 2 \left(-1 + \Gamma^{3} \right) \mathrm{C}\_{1} \end{pmatrix} \right] \right]^{1/2} \tag{A5}$$

$$\mathcal{R}\_2 = \left[ 1 - \frac{1}{17} \mathbb{E} \left[ 2 \left( \begin{array}{c} 5 \text{Tr} \boldsymbol{\delta} (\boldsymbol{\delta} - \boldsymbol{\Gamma}) (\boldsymbol{\delta} + \boldsymbol{\Gamma}) \mathbf{C}\_3 + \\ 16 \boldsymbol{\varepsilon} \boldsymbol{\Gamma} (-\boldsymbol{\delta} + \boldsymbol{\Gamma}) \mathbf{C}\_4 + 6 \left( \begin{array}{c} 5 \text{Tr} \boldsymbol{\delta} (\boldsymbol{\delta} - \boldsymbol{\Gamma}) \mathbf{C}\_3 + \\ 16 \boldsymbol{\varepsilon} \boldsymbol{\Omega} \mathbf{C}\_2^2 \left( \begin{array}{c} \boldsymbol{\delta}^5 \boldsymbol{\Gamma} - \boldsymbol{\delta} \boldsymbol{\Gamma}^5 + 10 (\boldsymbol{\delta} - \boldsymbol{\Gamma}) \mathbf{C}\_3 \\ \left( \begin{array}{c} \boldsymbol{\delta} \boldsymbol{\Gamma} \left( \boldsymbol{\delta}^2 + \boldsymbol{\delta} \boldsymbol{\Gamma} + \boldsymbol{\Gamma}^2 \right) + \\ 6 \boldsymbol{\Gamma} \text{K} \ln \boldsymbol{\delta}\_5 - \boldsymbol{\mathcal{C}\_3}^2 \end{array} \right) \end{array} \right) \right] \right]^{1/2} \tag{A6}$$

Temperature profiles for both layers are

$$\begin{split} \theta\_{1} = -4B\_{1}X\_{1}^{2} \begin{pmatrix} -\frac{1}{4}r^{4} - 3K\_{4}r^{2} - \frac{32}{9}\varepsilon Dc\_{1}^{2}r^{6} - 24K\_{4}\varepsilon Dc\_{1}^{2}r^{4} - 96K\_{4}^{2}\varepsilon Dc\_{1}^{2}r^{2} - 128K\_{4}^{3}X\_{1}\varepsilon Dc\_{1}^{2}\ln r - 4K\_{4}^{2}\ln r - \\ 8\varepsilon Dc\_{1}^{2}r^{4} - 96K\_{4}^{2}\varepsilon Dc\_{1}^{2}r^{2} - 384K\_{4}^{3}\varepsilon Dc\_{1}^{2}\ln r + 32K\_{4}^{3}\varepsilon Dc\_{1}^{2}\frac{1}{r^{2}} \\ + D\_{1}\ln r + D\_{2} \end{pmatrix} \tag{A7}$$

$$\begin{split} \Theta\_{2} = -4B\_{\mathsf{T}}X\_{2}^{2} \Biggl( \begin{array}{c} -\frac{1}{4}r^{4} - 3\mathsf{C}\chi^{2} - \frac{32}{\mathsf{q}}\varepsilon D\varepsilon\_{3}^{2}r^{6} - 24\mathsf{C}\chi\varepsilon D\varepsilon\_{2}^{2}r^{4} - 96\mathsf{C}\chi^{2}\varepsilon D\varepsilon\_{2}^{2}r^{2} - 128\mathsf{C}\chi^{3}\chi\_{2}\varepsilon D\varepsilon\_{2}^{2}\ln r - 4\mathsf{C}\chi^{2}\ln r \\\ 8\varepsilon\_{2}D\varepsilon\_{2}^{2}r^{4} - 96\mathsf{C}\chi^{2}\varepsilon D\varepsilon\_{2}^{2}r^{2} - 384\mathsf{C}\chi^{3}\varepsilon D\varepsilon\_{2}^{2}\ln r + 32\mathsf{C}\chi^{3}\varepsilon D\varepsilon\_{2}^{2}\frac{1}{r^{2}} \\\ + D\_{3}\ln r + D\_{4} \end{array} \right) \tag{A8}$$

where *Ka*, *Kb*, *Kc*, *Kd*, *D*1, *D*2, *D*<sup>3</sup> and *D*<sup>4</sup> are all constants given below:

$$\begin{aligned} &= -\frac{9}{3} - \frac{2}{3} \left( -\frac{2}{3} (-24h\_1^3 + 96h\_2^2 + 35\sqrt{-1}h\_1^2 + 44h\_2^3 + 44h\_3^3 - 16h\_1^2 (16h + 27h\_2^2) \right)^2 + \\ & \qquad \left( -\frac{12}{3} + 84h\_1^2 - 27h\_2h\_3 + 3\sqrt{-1}h\_1^2 + 44h\_2^3 - 44h\_3^3 + 16h\_1^2 (16h + 27h\_2^2) \right)^2 + \\ & \qquad \frac{1}{3} - \frac{1}{3} + 12\frac{1}{2} + 12h\_2^2 + 192h\_2^2 + 192h\_2^2 - 6h\_2^2 - 4h\_3^3 \ge \|h\_2\|\_2^2, \Gamma\_2 \subset \mathbb{C}\_2, \\ & \qquad \mathbb{C}\_3 = 1 + 2\frac{1}{2} + 4\mathbb{C}\_3 \mathbb{1}\Delta h\_1^2 + 3\mathbb{C}\_3 \mathbb{1}\Delta h\_2^2 - 6h\_2^2 - 4\frac{3}{4} \mathbb{1}\Delta\_h \mathbb{C}\_2 \mathbb{1}\mathcal{D}\_h^2 - \mathbb{C}\_2 \mathbb{1} \\ & \qquad \mathbb{C}\_3 = 4\mathbb{C}\_3 + \mathbb{C}\_3 \mathbb{1}\Delta\_h \$$

where

*<sup>H</sup>*<sup>1</sup> = *<sup>A</sup>*2+*B*<sup>2</sup> *A*3+*B*<sup>3</sup> , *<sup>H</sup>*<sup>2</sup> = *<sup>A</sup>*1+*B*<sup>1</sup> *A*3+*B*<sup>3</sup> , *<sup>H</sup>*<sup>3</sup> <sup>=</sup> *<sup>G</sup> <sup>A</sup>*3+*B*<sup>3</sup> , *<sup>A</sup>*<sup>1</sup> = −4*X*<sup>1</sup> ln <sup>Γ</sup> − <sup>192</sup>*X*2ε*De*<sup>2</sup> <sup>2</sup> − <sup>192</sup>*X*1ε*De*<sup>2</sup> 1Γ, *<sup>A</sup>*<sup>2</sup> = −384*X*1ε*De*<sup>2</sup> <sup>1</sup> ln <sup>Γ</sup>, *<sup>A</sup>*<sup>3</sup> = <sup>64</sup>*X*1ε*De*<sup>2</sup> 1 - 1 <sup>Γ</sup><sup>2</sup> − 1 , *B*<sup>1</sup> = 4*X*<sup>2</sup> ln Γ + 192*X*2ε*De*<sup>2</sup> <sup>2</sup>Γ2, *B*<sup>2</sup> = 384*X*2ε*De*<sup>2</sup> <sup>2</sup> ln <sup>δ</sup> + <sup>192</sup>*X*2ε*De*<sup>2</sup> <sup>2</sup>Γ2*B*<sup>3</sup> = −64*X*2ε*De*<sup>2</sup> 2Γ2 - 1 <sup>δ</sup><sup>2</sup> <sup>+</sup> <sup>1</sup> Γ2 , *A*<sup>1</sup> = −4*X*<sup>1</sup> ln Γ − 192ε1*De*<sup>1</sup> <sup>2</sup>Γ + 192ε*De*<sup>1</sup> <sup>2</sup>*A*<sup>2</sup> = −384*X*1ε*De*<sup>1</sup> <sup>2</sup> ln Γ, *A*<sup>3</sup> = 64*X*1ε*De*<sup>1</sup> 2 1 <sup>Γ</sup> − 64*X*1ε*De*<sup>1</sup> 2, *B*<sup>1</sup> = 4*X*<sup>2</sup> ln Γ + 192*X*2ε*De*<sup>2</sup> <sup>2</sup>Γ<sup>2</sup> − <sup>192</sup>*X*2ε*De*<sup>2</sup> <sup>2</sup>δ<sup>2</sup> <sup>−</sup> <sup>4</sup>*X*<sup>2</sup> ln <sup>δ</sup>, *B*<sup>2</sup> = −384*X*<sup>2</sup> ε*De*<sup>2</sup> <sup>2</sup> ln Γ + 384*X*2ε*De*<sup>2</sup> <sup>2</sup> ln <sup>δ</sup>, *<sup>B</sup>*<sup>3</sup> = −64*X*2ε*De*<sup>2</sup> 2 1 <sup>Γ</sup><sup>2</sup> − 64*X*2ε*De*<sup>2</sup> 2 1 δ2 , *G* = 1 + 2*X*<sup>1</sup> + 32*X*1ε*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> <sup>2</sup>*X*2δ<sup>2</sup> <sup>−</sup> <sup>2</sup>*X*1Γ<sup>2</sup> <sup>−</sup> <sup>32</sup>*X*2ε*De*<sup>2</sup> <sup>2</sup>δ<sup>4</sup> <sup>−</sup> <sup>32</sup>*X*1ε*De*<sup>1</sup> <sup>2</sup>Γ<sup>4</sup> − <sup>2</sup>*X*2Γ3.
