# Mathematical Analysis of the Coating Process over a Porous Web Lubricated with Upper-Convected Maxwell Fluid

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

_{0}as the separation at the nip. During the procedure, the suction velocity v

_{0}is assumed to remain constant. Moreover, the x-axis is taken along flow motion, whereas y-axis is assumed to be transversal to the flow direction, as depicted by Figure 1. The symmetry of the model is pursued for the analysis. The motion of the upper-convected Maxwell fluid model is discussed using the following equations:

_{0}represents viscoelastic parameter and suction velocity, respectively.

## 3. Solution of the Problem

## 4. Operating Variables

#### 4.1. Separating Force

#### 4.2. Power Input

#### 4.3. Adiabatic Temperature

## 5. Results and Discussion

_{s}, the separation force, and the power input are highlighted in Table 1, which is generated through a variation of B. The maximum coating thickness of up to four decimal places is observed which can be as high as 1.5590. Beyond this point by increasing the value of B, the coating thickness up to four decimal places remains the same. The highest separation point is detected at B = 10 which is 2.0630. It is observed that coating thickness increases by increasing B. Physically, it is quite evident that by increasing the viscoelastic parameter, the fluid becomes more viscous. The minimum thickness of coating was observed at B = 1.1, which is 1.1728. It was observed that by setting B → 0, no significant change in coating thickness is found, whereas by setting B → ∞, it is examined that λ → 1.3, as found in the literature by Greener [5] in 1979. It is worth mentioning that with the variation in B, one can really control the coating thickness. Similarly, Table 2 and Table 3 are generated for various values of Re and N

_{Ca₂}, respectively. It is quite interesting that by increasing the Reynold’s number the coating thickness, both the separation point and separating force decrease as expected. However, with the increase of Reynolds number, the power transferred to fluid by roll also increases. This means that by increasing Reynolds number, the fluid penetration increases, which physically causes a reduction in the engineering quantities. As seen in Table 2, an increase in coating thickness and detachment point is experienced by increasing modified capillary number. Since the capillary number N

_{Ca₂}represents the relative effect of the viscous drag force versus the surface tension forces acting across an interface between a liquid and a gas, increasing capillary number viscous forces dominates over interfacial forces; this effect can be seen in Table 3, as by increasing the capillary number separation point, the coating thickness increases as well.

## 6. Conclusions

- In the case of the upper-convected Maxwell fluid model, which is a class of viscoelastic material, a theoretical study was carried out, as most of the material in the coating industry is viscoelastic. Hence the theoretical results for these industries are presented in this study so that they can set their engineering quantities numerically according to the theoretical findings listed in Table 1, Table 2 and Table 3;
- Coating thickness, separation region/separation point, roll separation force, power input, and pressure can be controlled through Reynolds number $\mathrm{Re}$ and fluid parameter $B$;
- Separating point and coating thickness increases by increasing Capillary number;
- Viscous force has a dominant role on coating thickness, separation force, and power input;
- The outcomes of Middleman [5] are obtained when B → 0 and Re → 0;
- The nip region demonstrates the highest velocity and pressure gradient.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

τ | Extra stress tensor |

ρ | Density |

^{B} | Viscoelastic parameter |

μ | Viscosity |

^{v} | Kinematic viscosity |

Re | Reynolds number |

γ | Surface tension |

λ | Coating thickness |

N_{Ca₂} | Modified capillary number |

## References

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**Figure 3.**Influence of B on velocity profile at x = 0. (

**a**) $\lambda =\mathrm{1.3050..1.4522}$, (

**b**) $\lambda =\mathrm{1.4789..1.5590}$.

**Figure 4.**Influence of B on velocity profile at x = 0.25. (

**a**) $\lambda =\mathrm{1.3050..1.4522}$, (

**b**) $\lambda =\mathrm{1.4789..1.5590}$.

**Figure 5.**Influence of B on velocity profile at x = 0.5. (

**a**) $\lambda =\mathrm{1.3050..1.4522}$, (

**b**) $\lambda =\mathrm{1.4789..1.5590}$.

**Figure 6.**Influence of B on velocity profile at x = 0.75. (

**a**) $\lambda =\mathrm{1.3050..1.4522}$, (

**b**) $\lambda =\mathrm{1.4789..1.5590}$.

**Figure 7.**Influence of B on velocity profile at x = 1. (

**a**) $\lambda =\mathrm{1.3050..1.4522}$, (

**b**) $\lambda =\mathrm{1.4789..1.5590}$.

**Figure 8.**Influence of Re on velocity profile at x = 0. (

**a**) $\lambda =\mathrm{1.2027..1.0721}$, (

**b**) $\lambda =\mathrm{1.0271..0.9332}$.

**Figure 9.**Influence of Re on velocity profile at x = 0.25. (

**a**) $\lambda =\mathrm{1.2027..1.0721}$, (

**b**) $\lambda =\mathrm{1.0271..0.9332}$.

**Figure 10.**Influence of Re on velocity profile at x = 0.5. (

**a**) $\lambda =\mathrm{1.2027..1.0721}$, (

**b**) $\lambda =\mathrm{1.0271..0.9332}$.

**Figure 11.**Influence of Re on velocity profile at x = 0.75. (

**a**) $\lambda =\mathrm{1.2027..1.0721}$, (

**b**) $\lambda =\mathrm{1.0271..0.9332}$.

**Figure 12.**Influence of Re on velocity profile at x = 1. (

**a**) $\lambda =\mathrm{1.2027..1.0721}$, (

**b**) $\lambda =\mathrm{1.0271..0.9332}$.

**Table 1.**Influence of Maxwell parameter on separation point, coating thickness, separating force, and power input by fixing Re = 10, N

_{Ca₂}= 1.

B | x_{s} | λ | F | P_{w} |
---|---|---|---|---|

1.1 | 1.8000 | 1.3050 | 7.8107 | −0.2335 |

2 | 1.8425 | 1.3437 | 10.3295 | −0.3614 |

3 | 1.8883 | 1.3864 | 13.8451 | −0.5327 |

4 | 1.9254 | 1.4218 | 17.7476 | −0.7010 |

5 | 1.9567 | 1.4522 | 22.0537 | −0.8803 |

6 | 1.9838 | 1.4789 | 26.7183 | −0.9980 |

7 | 2.0075 | 1.5025 | 31.6752 | −1.1240 |

8 | 2.0282 | 1.5234 | 36.8622 | −1.2360 |

9 | 2.0466 | 1.5421 | 42.2522 | −1.3367 |

10 | 2.0630 | 1.5590 | 47.8226 | −1.4263 |

**Table 2.**Effect of Reynolds Number on separation point, coating thickness, separating force and power input fixing B = 10, N

_{Ca₂}= 1.

Re | x_{s} | λ | F | P_{w} |
---|---|---|---|---|

2 | 2.2351 | 1.2027 | 5.1937 | −0.0433 |

4 | 2.1744 | 1.1553 | 0.8853 | 0.1910 |

6 | 2.1282 | 1.1203 | −2.9233 | 0.3726 |

8 | 2.0919 | 1.0933 | −6.3312 | 0.4912 |

10 | 2.0630 | 1.0721 | −9.4709 | 0.5914 |

15 | 1.9999 | 1.0271 | −17.6440 | 0.7867 |

20 | 1.9774 | 1.0114 | −23.5755 | 0.8553 |

30 | 1.9362 | 0.9831 | −37.1632 | 0.9527 |

50 | 1.8952 | 0.9556 | −65.0365 | 1.0213 |

90 | 1.8612 | 0.9332 | −122.8755 | 1.0673 |

**Table 3.**Effect of modified capillary number on separation point, coating thickness by fixing Re = 1 and B = 7.

N_{Ca₂} | x_{s} | λ |
---|---|---|

1 | 2.2474 | 1.2303 |

2 | 2.2563 | 1.2376 |

3 | 2.2592 | 1.2400 |

4 | 2.2607 | 1.2412 |

5 | 2.2615 | 1.2418 |

6 | 2.2619 | 1.2422 |

7 | 2.2625 | 1.2427 |

8 | 2.2629 | 1.2430 |

9 | 2.2631 | 1.2432 |

10 | 2.2633 | 1.2433 |

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**MDPI and ACS Style**

Zafar, M.; A. Rana, M.; Zahid, M.; Ahmad, B. Mathematical Analysis of the Coating Process over a Porous Web Lubricated with Upper-Convected Maxwell Fluid. *Coatings* **2019**, *9*, 458.
https://doi.org/10.3390/coatings9070458

**AMA Style**

Zafar M, A. Rana M, Zahid M, Ahmad B. Mathematical Analysis of the Coating Process over a Porous Web Lubricated with Upper-Convected Maxwell Fluid. *Coatings*. 2019; 9(7):458.
https://doi.org/10.3390/coatings9070458

**Chicago/Turabian Style**

Zafar, Muhammad, Muhammad A. Rana, Muhammad Zahid, and Babar Ahmad. 2019. "Mathematical Analysis of the Coating Process over a Porous Web Lubricated with Upper-Convected Maxwell Fluid" *Coatings* 9, no. 7: 458.
https://doi.org/10.3390/coatings9070458