# Heat Transfer Analysis of Sisko Fluid Flow over a Stretching Sheet in a Conducting Field with Newtonian Heating and Constant Heat Flux

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Mathematical Formulation

#### 2.1. Rheological Model

#### 2.2. Governing Equations and Boundary Conditions

#### 2.3. Transformed Problem

#### 2.4. Physical Quantities of Engineering Interest

#### 2.4.1. The Coefficients of Skin Friction

#### 2.4.2. The Local Nusselt Number

## 3. Solution of the Problem

- By implementing the transformation $\frac{\partial f}{\partial {\eta}_{1}}=p\left({\eta}_{1}\right)$, the momentum equation order for $f\left({\eta}_{1}\right)$ is reduced and depicts how the actual equation for $p\left({\eta}_{1}\right)$ is displayed.
- Assume that $f\left({\eta}_{1}\right)$ is perceived here from an earlier iteration (directed by ${f}_{r}$), in order to build a scheme of iteration for $p\left({\eta}_{1}\right)$ in which, at the current iteration stage, assume that only linear terms in $p\left({\eta}_{1}\right)$ are to be estimated (directed by ${p}_{r+1}$) and for all other remaining terms that are of use, linear and nonlinear are assumed to be familiar from previous iterations. Furthermore, at the preceding iteration, nonlinear terms in $p$ are assessed.
- By implementing the transformation $\frac{\partial g}{\partial {\eta}_{1}}=q({\eta}_{1})$, the momentum equation order for $g({\eta}_{1})$ is reduced and depicts how the actual equation for $q({\eta}_{1})$ is displayed.
- Assume that $g({\eta}_{1})$ is perceived here from an earlier iteration (referred to by ${g}_{r}$), in order to build a scheme of iteration for $q({\eta}_{1})$ in which, at the current iteration stage, assume that only linear terms in $q({\eta}_{1})$ are to be estimated (referred by ${q}_{r+1}$) and for all other remaining terms that are of use, linear and nonlinear are assumed to be familiar from previous iterations. Furthermore, at the preceding iteration, nonlinear terms in $q$ are assessed.
- In a similar manner to find the remaining governing dependent variables, the iteration schemes are developed and now the variable solutions chosen in the earlier equation are used in the updated solutions.

## 4. Accelerating the Convergence of the SRM

## 5. Results and Discussion

**,**which is a Sisko fluid. This figure clearly demonstrates that when compared to the Sisko fluid, the power law fluid surface temperature is greater.

## 6. Conclusions

- By increasing the magnetic field strength, the momentum boundary layer thickness decreases, whereas the thermal boundary layer thickness increases.
- The velocity distribution in x
_{1}-direction declines, and the opposite phenomenon is observed in x_{2}-direction, while fluid temperature decreases as the stretching ratio parameter increases. - With the increase of the Sisko fluid parameter, the velocity in axial and transverse directions increases, whereas the fluid temperature reduces.
- As the Biot number increases, the fluid temperature increased.
- It was found that successive over (under) relaxation (SOR) techniques would significantly increase the convergence speed of the SRM scheme.
- In this problem, the successful performance of the SRM can be applied in fluid mechanical applications to other various related boundary layer problems.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$({x}_{1},{x}_{2},{x}_{3})$ | Space coordinates |

$({u}_{1},{u}_{2},{u}_{3})$ | Velocity components |

$\upsilon $ | Kinematic viscosity |

${B}_{0}$ | Strength of magnetic field |

$\mu $ | Coefficient of dynamic viscosity |

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

$\rho $ | Fluid density |

$\alpha $ | Stretching ratio parameter |

$\sigma $ | Electrical conductivity |

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

${a}_{1},{b}_{1}$ | Stretching constant |

${c}_{1},{d}_{1}$ | Real numbers with respect to stretchable sheet |

$M$ | Magnetic field parameter |

$k$ | Thermal conductivity |

$A$ | Material parameter of Sisko fluid |

${\mathrm{Re}}_{a},{\mathrm{Re}}_{b}$ | Local Reynolds number |

$\mathrm{Pr}$ | Prandtl number |

${h}_{s}$ | Heat transfer parameter |

${q}_{w}$ | Heat flux |

$\gamma $ | Biot number due to temperature |

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**Figure 2.**The bidirectional (axial and transverse) velocity distribution profile with the influence of $M$.

**Figure 4.**The bidirectional (axial and transverse) velocity distribution profile with the influence of $\alpha $.

**Figure 6.**The bidirectional (axial and transverse) velocity distribution profiles with the influence of A.

**Figure 9.**(

**a**): A 3D plot for $C{f}_{{x}_{1}}$-direction for different values of $Mn$. (

**b**): A 3D plot for $C{f}_{{x}_{1}}$-direction for different values of $\alpha n$.

**Figure 10.**(

**a**): Three-dimensional plot for $C{f}_{{x}_{2}}$-direction for different values of $Mn$. (

**b**): Three-dimensional plot for $C{f}_{{x}_{2}}$-direction for different values of $\alpha n$.

**Figure 11.**(

**a**): Three-dimensional plot for Nusselt number in NH case for different values of $Mn$. (

**b**): Three-dimensional plot for Nusselt number in NH case for different values of $\alpha n$.

**Figure 12.**(

**a**): Three-dimensional plot for Local Nusselt number in CHF case for different values of $Mn$. (

**b**): Three-dimensional plot for Nusselt number in CHF case for different values of $\alpha n$.

$\mathit{\alpha}$ | Iter. | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{g}}^{\u2033}(0)$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Present Study | Munir et al. [43] | Ariel [4] | Gorla et al. [47] | Present Study | Munir et al. [43] | Ariel [4] | Gorla et al. [47] | ||

0.25 | 45 | −1.04881108 | −1.048818 | −1.048813 | −1.048813 | −0.19456383 | −0.194567 | −0.194565 | −0.194564 |

5.0 | 35 | −1.09309502 | −1.093098 | −1.093096 | −1.093097 | −0.46520485 | −0.465207 | −0.465206 | −0.465205 |

0.75 | 40 | −1.13448575 | −1.134487 | −1.134486 | −1.134485 | −0.79461826 | −0.794619 | −0.794619 | −0.794622 |

1.0 | 40 | −1.17372074 | −1.173721 | −1.173721 | −1.173720 | −1.17372074 | −1.173721 | −1.173721 | −1.173720 |

**Table 2.**Numerical values for ${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ for NH at the surface of the sheet for various values of $Pr\gamma $ with $M=0.5,\alpha =0.5A=0.5$.

Pr | Γ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ for NH | ||||||
---|---|---|---|---|---|---|---|---|

n = 0.75 | ||||||||

Iter | CPU Time | Basic SRM | $\mathit{\omega}$ | Iter | CPU Time | SRM with SOR | ||

0.1 | 12 | 18.76627 | 4.77995 | 0.9 | 9 | 12.29875 | 4.77995 | |

0.71 | 12 | 20.46802 | 6.036175 | 0.9 | 9 | 10.91867 | 6.036175 | |

1 | 12 | 24.63855 | 9.396499 | 0.9 | 9 | 11.36188 | 9.396499 | |

2 | 12 | 22.21707 | 11.99444 | 0.9 | 9 | 11.10621 | 11.99444 | |

3 | 12 | 22.47097 | 14.18906 | 0.9 | 9 | 11.82956 | 14.18906 | |

4 | 12 | 22.68458 | 16.12413 | 0.9 | 9 | 11.61693 | 16.12413 | |

5 | 0.2 | 12 | 23.24728 | 2.389975 | 0.9 | 9 | 13.33771 | 2.389975 |

2 | 0.3 | 12 | 21.94909 | 1.593317 | 0.9 | 9 | 13.95093 | 1.593317 |

x | 0.4 | 14 | 21.53128 | 1.194987 | 0.9 | 9 | 16.56932 | 1.194987 |

0.5 | 17 | 28.74275 | 0.95599 | 0.85 | 10 | 14.70957 | 0.95599 | |

n = 1.75 | ||||||||

0.1 | 11 | 17.44803 | 5.716165 | 0.9 | 8 | 12.54738 | 5.716165 | |

0.71 | 11 | 20.91306 | 7.272926 | 0.9 | 8 | 11.56558 | 7.272926 | |

1 | 11 | 21.13117 | 11.38285 | 0.9 | 8 | 12.25918 | 11.38285 | |

2 | 11 | 20.89956 | 14.51542 | 0.9 | 8 | 11.95635 | 14.51542 | |

3 | 11 | 22.29397 | 17.1434 | 0.9 | 8 | 13.95373 | 17.1434 | |

4 | 11 | 20.65066 | 19.45184 | 0.9 | 8 | 12.40026 | 19.45184 | |

5 | 0.2 | 11 | 24.54107 | 2.858083 | 0.9 | 8 | 13.56013 | 2.858083 |

2 | 0.3 | 11 | 24.52556 | 1.905388 | 0.9 | 8 | 12.39242 | 1.905388 |

0.4 | 12 | 23.28151 | 1.42904 | 0.9 | 9 | 14.90034 | 1.42904 | |

0.5 | 13 | 23.59943 | 1.143232 | 0.9 | 10 | 15.30949 | 1.143232 |

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## Share and Cite

**MDPI and ACS Style**

Jayalakshmi, P.; Obulesu, M.; Ganteda, C.K.; Raju, M.C.; Varma, S.V.; Lorenzini, G. Heat Transfer Analysis of Sisko Fluid Flow over a Stretching Sheet in a Conducting Field with Newtonian Heating and Constant Heat Flux. *Energies* **2023**, *16*, 3183.
https://doi.org/10.3390/en16073183

**AMA Style**

Jayalakshmi P, Obulesu M, Ganteda CK, Raju MC, Varma SV, Lorenzini G. Heat Transfer Analysis of Sisko Fluid Flow over a Stretching Sheet in a Conducting Field with Newtonian Heating and Constant Heat Flux. *Energies*. 2023; 16(7):3183.
https://doi.org/10.3390/en16073183

**Chicago/Turabian Style**

Jayalakshmi, Pothala, Mopuri Obulesu, Charan Kumar Ganteda, Malaraju Changal Raju, Sibyala Vijayakumar Varma, and Giulio Lorenzini. 2023. "Heat Transfer Analysis of Sisko Fluid Flow over a Stretching Sheet in a Conducting Field with Newtonian Heating and Constant Heat Flux" *Energies* 16, no. 7: 3183.
https://doi.org/10.3390/en16073183