# MHD Flow and Heat Transfer Analysis in the Wire Coating Process Using Elastic-Viscous

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^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Modeling the Problem

_{w}is dragged with velocity v through a pressure-type coating die of length L and radius R

_{d}. The coordinate system is taken at the center of the wire, in which r is taken perpendicular to the flow direction and the z-axis is along the flow. Here

**Θ**

_{w}and

**Θ**

_{d}represents the wire and die temperature, respectively. A constant pressure gradient acts upon the fluid direction and the magnetic field of strength transversely along the axial direction.Due to a small magnetic Reynolds number, the induced magnetic field is negligible, which is also a valid assumption on a laboratory scale.

**S**=

**S**(r),

**Θ**=

**Θ**(r)

_{w}and w = 0 at r = R

_{d}

**Θ**=

**Θ**

_{w}at r =R

_{w}and

**Θ**=

**Θ**

_{d}at r = R

_{d}

**S**is the extra stress tensor,

**A**

_{1}is the Rivlin–Ericksen tensor, and γ

_{i}(i = 1–7) are the material constants.

**A**

_{n}=

**A**

_{n − 1}

**L**

^{T}+

**LA**

_{n – 1}+D

**A**

_{n – 1}/Dt, n = 2, 3,…

_{1}−γ

_{7}= 0.

- For the Newtonian fluid model, all γ
_{1}− γ_{7}= 0. - For the second-grade fluid model, all γ
_{1}= γ_{3}= γ_{5}= γ_{6}= γ_{7}= 0. - For the Oldroyd-B model, all γ
_{3}− γ_{7}= 0. - For the Maxwell model, all γ
_{2}− γ_{7}= 0. - For the Johnson–Segalman model, all γ
_{5}= γ_{6}= γ_{7}= 0. - For the Oldroyd-6model, all γ
_{6}= γ_{7}= 0.

_{p}, D/Dt, k, Θ, are the velocity of the fluid, density of the fluid, shear stress, specific heat, material derivative, thermal conductivity, temperature, and velocity gradient, respectively.

**J**·

**B**as given in Equation (8). The electrostatic force produced due to charge density is negligible and we only consider the applied magnetic field

**B**

_{0}normal to the flow direction.

## 3. Solution of the Modeled Problem

## 4. Analysis of the Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Analysis of the Adomian Decomposition Method (ADM)

_{0}, w

_{1}, w

_{2},…, w

_{n}, separately, a decomposition method is used.

_{0}, w

_{1}, w

_{2}, w

_{3},…, it should be noted that ADM suggests that f(r) in fact describes the zeroth componentw

_{0}.

#### Appendix A.2. Analysis of Optimal Homotopy Asymptotic Method (OHAM)

_{1}, C

_{2}, C

_{3}are auxiliary constants.

_{1}, C

_{2},…C

_{m}can be obtained from the following relation:

## Appendix B

r | First Order | Second Order |
---|---|---|

1 | 0 | 0 |

1.1 | 3.90 × 10^{−9} | 2.0 × 10^{−10} |

1.2 | 8.44 × 10^{−9} | 3.0 × 10^{−10} |

1.3 | 3.74 × 10^{−10} | 9.2 × 10^{−10} |

1.4 | 6.70 × 10^{−10} | 1.4 × 10^{−12} |

1.5 | 8.22 × 10^{−10} | 1.0 × 10^{−12} |

1.6 | 8.58 × 10^{−11} | 2.0 × 10^{−12} |

1.7 | 8.22 × 10^{−11} | 1.2 × 10^{−13} |

1.8 | 6.70 × 10^{−11} | 7.0 × 10^{−13} |

1.9 | 3.74 × 10^{−11} | 2.0 × 10^{−15} |

2 | 8.44 × 10^{−14} | −5.0 × 10^{−17} |

r | First Order | Second Order |
---|---|---|

1 | 0 | 0 |

1.1 | 7.51 × 10^{−14} | 7.93 × 10^{−16} |

1.2 | 2.77 × 10^{−12} | 2.21 × 10^{−14} |

1.3 | 1.73 × 10^{−11} | 1.11 × 10^{−13} |

1.4 | 5.02 × 10^{−11} | 2.46 × 10^{−13} |

1.5 | 9.34 × 10^{−11} | 3.12 × 10^{−13} |

1.6 | 1.28 × 10^{−10} | 2.43 × 10^{−13} |

1.7 | 1.39 × 10^{−10} | 1.15 × 10^{−13} |

1.8 | 1.23 × 10^{−10} | 1.40 × 10^{−14} |

1.9 | –7.50 × 10^{−11} | 1.97 × 10^{−14} |

2 | 1.95 × 10^{−11} | 2.26 × 10^{−13} |

r | First Order | Second Order |
---|---|---|

1 | 0 | 0 |

1.1 | 3 × 10−11 | 2.64 × 10−09 |

1.2 | 0 | 5.03 × 10−09 |

1.3 | –1 × 10−10 | 6.92 × 10−09 |

1.4 | 2 × 10−10 | 8.14 × 10−09 |

1.5 | 1.1 × 10−09 | 8.55 × 10−09 |

1.6 | 4.4 × 10−09 | 8.14 × 10−09 |

1.7 | 1.35 × 10−08 | 6.92 × 10−08 |

1.8 | 3.68 × 10−08 | 5.03 × 10−10 |

1.9 | 9.01 × 10−08 | 2.64 × 10−11 |

2 | 2.027 × 10−07 | –9.53 × 10−13 |

**Table B4.**Numerical comparison of OHAM and ADM when β = 0.2, α = 0.3, δ = 2, M = 0.1, C

_{1}= −0.001652328, C

_{2}= −0.00173421, C

_{3}= 0.0010243621, C

_{4}= 0.0001825341.

r | OHAM | ADM | Absolute Error |
---|---|---|---|

1 | 1 | 1 | 0 |

1.1 | 0.001524394 | 0.001524371 | 0.0125 × 10^{−5} |

1.2 | 0.001352091 | 0.001352171 | 0.004 × 10^{−5} |

1.3 | 0.006210390 | 0.006230392 | 0.872 × 10^{−5} |

1.4 | 0.011607241 | 0.011606221 | 0.101 × 10^{−5} |

1.5 | 0.010442045 | 0.010442141 | 0.712 × 10^{−5} |

1.6 | 0.001520519 | 0.001522512 | 0.101 × 10^{−5} |

1.7 | 0.006014981 | 0.007214980 | 0.106 × 10^{−5} |

1.8 | 0.000304513 | 0.000304511 | 0.103 × 10^{−5} |

1.9 | 0.0000114221 | 0.0000114221 | 0.001 × 10^{−5} |

2.0 | 0.00001 × 10^{−18} | 0.00013 × 10^{−19} | 0.001 × 10^{−18} |

**Table B5.**Velocity comparison of the present work with published work [20] when α = 0.2, β = 0.1, M= 0, δ = 2.

r | OHAM | Reference [20] | Absolute Error |
---|---|---|---|

1 | 1 | 1 | 0 |

1.1 | 0.0011703 | 0.0011712 | 0.0000009 |

1.2 | 0.0002104 | 0.0002125 | 0.0000021 |

1.3 | 0.0300722 | 0.0300710 | 0.0000012 |

1.4 | 0.0216071 | 0.0216012 | 0.0000059 |

1.5 | 0.0104212 | 0.0104221 | 0.0000009 |

1.6 | 0.0015412 | 0.0054533 | 0.0039121 |

1.7 | 0.0071200 | 0.0071401 | 0.0000201 |

1.8 | 0.0035020 | 0.0035013 | 0.0000007 |

1.9 | 0.0137500 | 0.0137521 | 0.0000021 |

2 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Khan, Z.; Shah, R.A.; Islam, S.; Jan, H.; Jan, B.; Rasheed, H.-U.; Khan, A.
MHD Flow and Heat Transfer Analysis in the Wire Coating Process Using Elastic-Viscous. *Coatings* **2017**, *7*, 15.
https://doi.org/10.3390/coatings7010015

**AMA Style**

Khan Z, Shah RA, Islam S, Jan H, Jan B, Rasheed H-U, Khan A.
MHD Flow and Heat Transfer Analysis in the Wire Coating Process Using Elastic-Viscous. *Coatings*. 2017; 7(1):15.
https://doi.org/10.3390/coatings7010015

**Chicago/Turabian Style**

Khan, Zeeshan, Rehan Ali Shah, Saeed Islam, Hamid Jan, Bilal Jan, Haroon-Ur Rasheed, and Aurangzeeb Khan.
2017. "MHD Flow and Heat Transfer Analysis in the Wire Coating Process Using Elastic-Viscous" *Coatings* 7, no. 1: 15.
https://doi.org/10.3390/coatings7010015