# Computational Analysis on Magnetized and Non-Magnetized Boundary Layer Flow of Casson Fluid Past a Cylindrical Surface by Using Artificial Neural Networking

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## Abstract

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## 1. Introduction

- Formulation of Casson fluid flow towards cylindrical surfaces with pertinent physical effects.
- Comparative examination of Casson velocity for magnetized and non-magnetized flow fields.
- Examination of Casson concentration for chemically reactive and non-reactive flow fields.
- Evaluation of the SFC at the cylindrical surface for non-magnetized and magnetized flow fields.
- Estimation of the SFC by using an artificial neural networking model.

## 2. Mathematical Formulation

## 3. Non-Magnetized Mathematical Model

## 4. Solution Methodology

## 5. Numerical Outcomes

## 6. Artificial Neural Networking Outcomes

## 7. Conclusions

- The margin of deviation and difference values reveals that both ANN models can predict SFC values with relatively low error values.
- The error levels in the error histograms are also quite low. Furthermore, for both ANN models, we noticed that the data points were inside the +10% error band range.
- The coefficient of determination values’ proximity to one and the low mean squared error values demonstrate that each ANN model can carry out predictions with high accuracy.
- The magnitude of velocity is higher for the case of non-magnetized Casson fluid flow as compared to non-magnetic flow.
- For both chemically reactive and non-reactive flows, the concentration profiles show a declining nature towards the Schmidt number and curvature parameter.
- The SFC is found to be the decreasing function of the Casson fluid parameter and the velocities ratio parameter while the opposite is the case for the curvature parameter.
- For variation in the Casson fluid parameter, thermal Grashof number, and curvature parameter, the magnitude of SFC is higher for the case of magnetized flow as compared to the non-magnetized flow regime.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\tilde{X},\tilde{R}$ | Cylindrical coordinates |

$\tilde{U},\tilde{V}$ | Velocity components |

$\nu $ | Kinematic viscosity |

$\beta $ | Casson fluid parameter |

${\beta}_{T}$ | Thermal expansion coefficient |

${g}_{0}$ | Gravitational acceleration |

$\alpha $ | Angle of inclination |

${\beta}_{C}$ | Solutal expansion coefficient |

${\tilde{T}}_{\infty}$ | Ambient temperature |

$\tilde{T}$ | Temperature of fluid |

${B}_{0}$ | Magnetic field constant |

$\tilde{C}$ | Concentration of fluid |

${\tilde{C}}_{\infty}$ | Ambient concentration |

${\tilde{U}}_{e}$ | Free stream velocity |

$\sigma $ | Fluid electrical conductivity |

${c}_{p}$ | Specific heat at constant pressure |

$\rho $ | Fluid density |

$\overline{q}$ | Radiative heat flux |

$\kappa $ | Variable thermal conductivity |

$\overline{\mu}$ | Dynamic viscosity |

${Q}_{0}$ | Heat generation coefficient |

$L$ | Characteristic length |

$\epsilon $ | Small parameter |

R_{1} | Radius of cylinder |

${k}_{c}$ | Chemical reaction rate |

${\tilde{C}}_{w}$ | Surface concentration |

${U}_{0}$ | Reference velocity |

${\tilde{T}}_{w}$ | Surface temperature |

${D}_{m}$ | Mass diffusivity |

${F}_{C}\prime (\eta )$ | Fluid velocity |

${\theta}_{C}(\eta )$ | Fluid temperature |

${\varphi}_{C}(\eta )$ | Fluid concentration |

${G}_{T}$ | Temperature Grashof number |

${G}_{C}$ | Concentration Grashof number |

Pr | Prandtl number |

A | Velocities ratio parameter |

$R$ | Radiation parameter |

$\gamma $ | Curvature parameter |

${k}^{\ast}$ | Coefficient of mean absorption |

E | Eckert number |

$M$ | Magnetic field parameter |

Rc | Chemical reaction parameter |

Sc | Schmidt number |

${\sigma}^{\ast}$ | Stefan–Boltzmann constant |

H | Heat generation parameter |

## References

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**Figure 1.**(

**a**) Geometry of the problem. (

**b**) Impact of $\beta $ on ${f}^{\prime}(\eta )$. (

**c**) Impact of $\gamma $ on ${f}^{\prime}(\eta )$. (

**d**) Impact of A on ${f}^{\prime}(\eta )$.

**Figure 3.**(

**a**) The layered structure of the developed ANN models for SFC. (

**b**) Structural topology of the ANN model for the non-magnetized flow field. (

**c**) Structural topology of the ANN model for the magnetized flow field.

**Figure 8.**(

**a**,

**b**). The differences between the target values and the ANN outputs for each set of data when M = 0 and M = 0.2.

$\mathit{\beta}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0 | M = 0 | M = 0 | |

0.2 | −0.8224 | −4.9344 | 5.205703 |

0.3 | −0.8633 | −3.7409 | 3.64113 |

0.4 | −0.8965 | −3.1377 | 3.207106 |

0.5 | −0.9241 | −2.7723 | 2.685965 |

0.6 | −0.9474 | −2.5264 | 2.468694 |

0.7 | −0.9673 | −2.3492 | 2.21511 |

0.8 | −0.9846 | −2.2154 | 2.107406 |

0.9 | −0.9998 | −2.1106 | 2.041828 |

1.0 | −1.0131 | −2.0262 | 1.926957 |

2.0 | −1.0928 | −1.6392 | 1.526076 |

$\mathit{\beta}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0.2 | M = 0.2 | M = 0.2 | |

0.2 | −0.8517 | −5.1102 | 5.127272 |

0.3 | −0.9017 | −3.9074 | 3.906281 |

0.4 | −0.9420 | −3.1377 | 3.142552 |

0.5 | −0.9752 | −2.9256 | 2.913623 |

0.6 | −1.0039 | −2.6771 | 2.689821 |

0.7 | −1.0271 | −2.4944 | 2.491419 |

0.8 | −1.0478 | −2.3575 | 2.352806 |

0.9 | −1.0659 | −2.2503 | 2.252857 |

1.0 | −1.0818 | −2.1639 | 2.16358 |

2.0 | −1.1763 | −2.3526 | 2.370091 |

A | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0 | M = 0 | M = 0 | |

0.2 | −0.4365 | −1.8913 | 1.798471 |

0.3 | −0.4153 | −1.7995 | 1.775878 |

0.4 | −0.3859 | −1.6721 | 1.761253 |

0.5 | −0.3486 | −1.5105 | 1.47786 |

0.6 | −0.3036 | −1.3154 | 1.397436 |

0.7 | −0.2511 | −1.0881 | 1.114926 |

0.8 | −0.1914 | −0.8293 | 0.84088 |

0.9 | −0.1245 | −0.5395 | 0.560701 |

1.0 | −0.0497 | −0.2154 | 0.221825 |

2.0 | −0.0097 | −0.0421 | 0.044358 |

A | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0.2 | M = 0.2 | M = 0.2 | |

0.2 | −0.4757 | −2.0612 | 2.061184 |

0.3 | −0.4457 | −1.9312 | 1.937046 |

0.4 | −0.4081 | −1.7683 | 1.776405 |

0.5 | −0.3633 | −1.5742 | 1.580536 |

0.6 | −0.3114 | −1.3494 | 1.351205 |

0.7 | −0.2526 | −1.0945 | 1.090686 |

0.8 | −0.1871 | −0.8107 | 0.81182 |

0.9 | −0.1146 | −0.4965 | 0.498124 |

1.0 | −0.0344 | −0.1491 | 0.150031 |

2.0 | −0.0067 | −0.0291 | 0.028967 |

$\mathit{\gamma}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0 | M = 0 | M = 0 | |

0.2 | −0.5089 | −2.2052 | 2.054256 |

0.3 | −0.5595 | −2.4243 | 2.433774 |

0.4 | −0.6068 | −2.6293 | 2.688039 |

0.5 | −0.6523 | −2.8264 | 2.789083 |

0.6 | −0.6965 | −3.0179 | 3.168692 |

0.7 | −0.7396 | −3.2046 | 3.04553 |

0.8 | −0.7820 | −3.3884 | 3.265665 |

0.9 | −0.8236 | −3.5686 | 3.535665 |

1.0 | −0.8647 | −3.7467 | 3.75129 |

2.0 | −1.2520 | −5.4249 | 5.194255 |

$\mathit{\gamma}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0.2 | M = 0.2 | M = 0.2 | |

0.2 | −0.5547 | −2.4035 | 2.397728 |

0.3 | −0.6038 | −2.6163 | 2.618119 |

0.4 | −0.6499 | −2.8161 | 2.820131 |

0.5 | −0.6944 | −3.0088 | 3.010963 |

0.6 | −0.7377 | −3.1964 | 3.195347 |

0.7 | −0.7800 | −3.3797 | 3.376525 |

0.8 | −0.8216 | −3.5599 | 3.55692 |

0.9 | −0.8626 | −3.7376 | 3.738581 |

1.0 | −0.9030 | −3.9126 | 3.923491 |

2.0 | −1.2857 | −5.5709 | 5.600951 |

${\mathit{G}}_{\mathit{T}}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0 | M = 0 | M = 0 | |

0.2 | −0.3822 | −1.6561 | 1.545852 |

0.3 | −0.3107 | −1.3463 | 1.296924 |

0.4 | −0.2319 | −1.0048 | 1.027428 |

0.5 | −0.1363 | −1.0016 | 0.938263 |

0.6 | −1.1515 | −4.9894 | 5.305239 |

0.7 | −1.1315 | −4.9027 | 4.950683 |

0.8 | −1.1115 | −4.8162 | 4.683114 |

0.9 | −1.0715 | −4.6428 | 4.742156 |

1.0 | −1.0915 | −4.7295 | 4.449012 |

2.0 | −2.0515 | −8.8892 | 8.987435 |

${\mathit{G}}_{\mathit{T}}$ | ${\mathit{f}}^{\u2033}(0)$ | $(1+1/\mathit{\beta}){\mathit{f}}^{\u2033}(0)$ | ANN Values |
---|---|---|---|

M = 0.2 | M = 0.2 | M = 0.2 | |

0.2 | −0.8802 | −3.8139 | 3.786153 |

0.3 | −0.8563 | −3.7103 | 3.708539 |

0.4 | −0.8324 | −3.6067 | 3.609377 |

0.5 | −0.8084 | −3.5027 | 3.498244 |

0.6 | −0.7845 | −3.3993 | 3.403159 |

0.7 | −0.7606 | −3.2956 | 3.298737 |

0.8 | −0.7367 | −3.1921 | 3.187463 |

0.9 | −0.7128 | −3.0885 | 3.082916 |

1.0 | −0.6889 | −2.9850 | 2.986601 |

2.0 | −0.4509 | −1.9537 | 1.955074 |

Model | Inputs | Output | |||
---|---|---|---|---|---|

Model 1 (M = 0) | A | γ | G_{T} | β | SFC |

Model 2 (M = 0.2) | A | γ | G_{T} | β | SFC |

Model | Number of Neuron | Training | Validation | Test | Total |
---|---|---|---|---|---|

Model 1 (M = 0) | 20 | 28 | 6 | 6 | 40 |

Model 2 (M = 0.2) | 10 | 28 | 6 | 6 | 40 |

Model | MSE | R | MoD_{min} (%) | MoD_{max} (%) |
---|---|---|---|---|

Model 1 (M = 0) | 5.28 × 10^{−2} | 0.96081 | −0.12 | 6.9 |

Model 2 (M = 0.2) | 2.39 × 10^{−5} | 0.99998 | 0.0008 | −0.74 |

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## Share and Cite

**MDPI and ACS Style**

Rehman, K.U.; Shatanawi, W.; Çolak, A.B.
Computational Analysis on Magnetized and Non-Magnetized Boundary Layer Flow of Casson Fluid Past a Cylindrical Surface by Using Artificial Neural Networking. *Mathematics* **2023**, *11*, 326.
https://doi.org/10.3390/math11020326

**AMA Style**

Rehman KU, Shatanawi W, Çolak AB.
Computational Analysis on Magnetized and Non-Magnetized Boundary Layer Flow of Casson Fluid Past a Cylindrical Surface by Using Artificial Neural Networking. *Mathematics*. 2023; 11(2):326.
https://doi.org/10.3390/math11020326

**Chicago/Turabian Style**

Rehman, Khalil Ur, Wasfi Shatanawi, and Andaç Batur Çolak.
2023. "Computational Analysis on Magnetized and Non-Magnetized Boundary Layer Flow of Casson Fluid Past a Cylindrical Surface by Using Artificial Neural Networking" *Mathematics* 11, no. 2: 326.
https://doi.org/10.3390/math11020326