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# Slip Effects on Casson Nanofluid over a Stretching Sheet with Activation Energy: RSM Analysis

1 Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari, 61100, Pakistan

2 Department of Physics, Faculty of Sciences, University of 20 Août 1955-Skikda, Skikda, 21000, Algeria

3 Department of Mechanical Engineering, University of West Attica, 12244 Athens, Greece

* Corresponding Authors: F. Mebarek-Oudina. Email: ,

(This article belongs to the Special Issue: Advances in Computational Thermo-Fluids and Nanofluids)

*Frontiers in Heat and Mass Transfer* **2024**, *22*(4), 1017-1041. https://doi.org/10.32604/fhmt.2024.052749

**Received** 13 April 2024; **Accepted** 18 June 2024; **Issue published** 30 August 2024

## Abstract

The current study is dedicated to presenting the Casson nanofluid over a stretching surface with activation energy. In order to make the problem more realistic, we employed magnetic field and slip effects on fluid flow. The governing partial differential equations (PDEs) were converted to ordinary differential equations (ODEs) by similarity variables and then solved numerically. The MATLAB built-in command ‘bvp4c’ is utilized to solve the system of ODEs. Central composite factorial design based response surface methodology (RSM) is also employed for optimization. For this, quadratic regression is used for data analysis. The results are concluded by means of tables and pictorial representations. The present study discloses that the temperature profile increases with enhancement in*Ha*, N

_{r}, N

_{b}, and N

_{t}and it shows opposite behavior for λ. The included parameters show same trend for heat transfer rate (

*Nu*

_{x}). It is also concluded that δ should be maximum for any value of N

_{b}and N

_{t}to maximize the heat transfer rate.

## Graphic Abstract

## Keywords

Nomenclature

Concentration of nanoparticles | |

The concentration of the wall | |

Stretching velocity | |

Free stream temperature | |

Temperature of the nanofluid | |

Wall temperature distribution | |

Free nanoparticle concentration | |

Strength of the uniform magnetic field | |

Acceleration due to gravity | |

Coefficient of thermophoresis | |

Brownian diffusion coefficient | |

Dimensionless stream function | |

Permeability of porous medium | |

Ratio of thermal diffusion | |

Specific heat at constant pressure | |

Cartesian coordinates along the sheet and normal to it, respectively | |

Velocity along | |

Reference velocity | |

Dimensionless temperature | |

Kinematic viscosity | |

Coefficient of viscosity | |

The density of the fluid | |

Electrical conductivity | |

Thermal conductivity | |

Dimensionless concentration | |

Ratio of effective heat capacity | |

Similarity variable |

Due to the recent advancements in nanotechnology in the last few decades, the interest of scientists in a special kind of material has developed due to its enhanced thermal transference. Due to the improved thermal efficiency, the nanofluids bring revolution in the fields of thermal engineering and computational fluid dynamics (CFD). The nanoparticles are doping of conventional materials like glycol, ethylene, and mineral oil with extraordinary thermal features. Such kind of suspension of both materials leads to enhanced thermal conductivity and viscosity. It is observed that the surface area of nanoparticles is multiple times greater than that of common fluids. Therefore, the nanoparticles’ structure is suggested as microscopic commonly between 1 and 100 nm. Such enhanced thermal materials have many practical applications like chemical processes, mechanical industries, nuclear reactions, cooling processes, solar engineering, extrusion systems, power plants, energy production, etc. Nowadays, nanoparticles are also employed in the medical field like artificial hearts, chemotherapy, cancer tissue damage, X-rays, and surgery processes.

Nanofluids were introduced by Choi et al. [1]. Buongiorno [2] claimed that Brownian and thermophoresis movement is of key importance in the slip transport characteristics. Sheikholeslami et al. [3] considered the different shapes of nanoparticles and selected the best among them in terms of heat transfer. They considered the finite element method (FEM) to compute the findings. The results depicted the increment in the velocity of the nanofluid due to the enhancement in the Darcy and Reynolds number

There are two basic kinds of nanofluid flow models. i.e., single-phase and two-phase models. Tsuji et al. [12] numerically examined the fluid and motion of particles in a channel and determined empirical parameters related to the collision of particles. Hieu et al. [13] numerically presented the two-phase flow model for simulating wave proliferation in shallow water including wave breaking, reflection, shoaling, and air movement. They demonstrated its capability in simulating wave distortion and breaking. Zeidan et al. [14] described a new solution method for a model of two-phase compressible flows consisting of multiple equations derived from principles of conservation and additional closure governing equations. Ljungqvist et al. [15] used a multi-fluid approach to simulate the problem of two-phase flow in a vessel using CFD code CFX4 and highlighted the importance of slip velocity and particle velocity. Xue et al. [16] proposed a coupling model of groundwater loss and gas drainage to examine the influence of thermally improved methane retrieval on groundwater damage and the ecological risk of coal bed methane growth. Based on the flux equivalent principle, Huang et al. [17] developed a discrete-fractured model of a single fracture, which explicitly describes macroscopic fractures as geometry elements and investigated the effect of fractures on water flooding. Riaz et al. [18] examined the nonlinear development of viscous and gravitational uncertainty in two-phase immiscible movements using numerical methods and described the physical mechanisms of finger progression and interface.

Flow on a stretching surface is extensively recognized by researchers due to its enormous industrial and engineering applications in the manufacturing of food, rubber sheets, hot rolling, glass fiber, paper production, and many others. Activation energy is the minimal energy to be possessed by the particle to endure some kind of change or chemical reaction. The activation energy is mainly applicable in food processing, geothermal engineering, chemical engineering, oil reservoirs, and mechano-chemistry. In various flow conditions like emulsions, polymer solutions, and foam, the slip effects cannot be neglected among the nanofluids and solid boundaries. This phenomenon emerges in numerous applications like nano-channels or micro-channels and in applications where the moving plates are attached to a thin film of light oil or when the surface is sheathed with a unique coating like a thick mono-layer of hydrophobic octadecyl trichlorosilane. Rana et al. [19] examined the laminar boundary layer flow resulting from the nonlinear stretching of a flat surface, taking into account thermophoresis and Brownian motion impacts using the variational FEM. Mabood et al. [20] analyzed the MHD laminar boundary layer flow of nanofluid over a stretching sheet with viscous dissipation and explored the effect of governing parameters on temperature, friction factor, nanoparticle concentration, local

Trusting the literature cited above, we came to know that the problem of Casson nanofluid for activation energy under the influence of magnetic field and slip effects has never been reported before. Therefore, our goal for this problem is to compute the approximate solution of the problem in hand and optimize

This portion presents the governing equations for Casson nanofluid based on associated laws. The influence of the magnetic field is observed by considering it vertical to the sheet. Buongiorno’s nanofluid model is considered along with thermophoresis and the Brownian impacts. The activation energy is also considered ascribed to the Arrhenius energy. The velocity components are taken as

The physical depicters appeared in the above equations are the Casson parameter

With

where

The similarity transformations are:

The transformed equations and boundary conditions are:

With

where

The dimensionless form of physical quantities involved in the study are:

MATLAB command ‘bvp4c’ is utilized to compute the numerical solutions of Eqs. (8)–(10) concerning the boundary conditions (Eqs. (11)–(12)). This is a finite difference scheme that utilizes the Lobatto-III-A formulation. It is a collocation technique with an accuracy of order four. To apply this scheme, the nonlinear ODEs are converted to a system of first-order linear ODEs by introducing the new variables. The residual of the solution is employed to control the error and mesh selection. To employ this function, nonlinear ODEs (Eqs. (8)–(10)) are replaced with the system of first-order ODEs. With the substitutions

Moreover, to evaluate the solution on specific points, the ‘deval’ command is used in MATLAB.

This portion is devoted to the demonstration of numerical outcomes of the given problem Eqs. (8)–(10) in the form of tables and pictorial structures. The given problem Eqs. (8)–(10) is solved with the help of the three-stage Labotto-III-A formula and the computed values are presented in Table 1.

Fig. 1 presents the effect of

The effect of the mixed convective parameter

The effect of

Fig. 4 presents the impact of

Fig. 6 is plotted to observe the impact of

RSM is a powerful tool to describe a broad range of variables, with partial resources, quantitative data, and the mandatory test design. The stages involved in RSM are as follows:

1. The investigation of data values is planned and executed to achieve appropriate and credible requirements for the projected response.

2. The most suitable mathematical models were described for the response surface.

3. It was defined that the mathematical models of the response surface are the most suitable.

4. To inspect the parametric direct and interaction impacts (ANOVA), variance analysis was done.

4.2 Optimization Analysis by RSM

The relation among the response variable and factor variables was computed by employing a face-centered central composite design. Tables 2–4 describe the inclusive factors and levels. The model is described in (13), in which the linear, square, and interactive terms are involved.

Response =

4.2.1 Response Surface Regression:

This section examined the response of three factors

i) Statistical Analysis

Based on the given settings. The statistical analysis executed 20 runs for

ii) Analysis of Variance and Model Estimation:

To achieve the projected regression equation, experimental model presented in Table 8 is executed under altered experimental settings. The residual plot is achieved by analysis of variance and entering the data into analytical software.

From Table 7, it is noticed that

iii) Regression Equation in Uncoded Units

From Table 8 it is noticed that the p-value of

After removing the factors which have a p-value greater than 0.05 the regression equation for

iv) Residual and Surface Plots

The residual plot is given in Fig. 9a. In the residual plot difference in observed

Fig. 9b represents

Fig. 9c observed the influence of

4.2.2 Response Surface Regression:

In this section, the response of three factors

i) Statistical Analysis:

According to the given settings, the statistical analysis executed 20 runs for

ii) Analysis of Variance and Model Estimation:

To get the approximate regression equation, the experimental model presented in Table 11 is executed under changed experimental settings. The residual plot is achieved by analysis of variance and entering the data into analytical software. By applying RSM we get Eq. (16) which is a frequent relationship between their valuable factors.

From Table 10, it is observed that the p-value of

iii) Regression Equation in Uncoded Units

From Table 11, it is observed that the p-value of

After eliminating the factors which have a p-value greater than 0.05 the regression equation for

iv) Residual and Surface Plots

The residual plot is given in Fig. 10a. In the residual plot difference observed from the scattered plot and predicted

4.2.3 Response Surface Regression:

In this section, the response of three factors of

i) Statistical Analysis:

According to specified settings. The statistical analysis executed 20 runs for

ii) Analysis of Variance and Model Estimation:

To get the approximate regression equation of the experimental model presented in Table 14 is executed under transformed experimental settings. The residual plot is achieved by analysis of variance and entering the data into analytical software. By using RSM we get Eq. (18) which is a numerous relationship between their helpful factors.

From Table 13 it is observed that the p-value of

iii) Regression Equation in Uncoded Units

From Table 14 it is observed that the p-value of

After eliminating the factors which have a p-value greater than 0.05 the regression equation for

iv) Residual and Surface Plots

The residual plot is given in Fig. 11a. In the residual plot difference observed from the scattered plot and predicted

In this study, the Casson nanofluid was examined, and its thermal analysis and optimizations were conducted. The following conclusions can be drawn:

• The velocity profile decreases with an increment in

• The temperature profile rises with enhancement in

• The concentration profile increases with rising values of

•

• For higher values of

• This work can be extended to hybrid and ternary hybrid nanofluids. Furthermore, the application of artificial neural networks could further optimize the study.

Acknowledgement: The authors express their gratitude to their affiliated universities.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm their contribution to the paper as follows: study conception and design, data collection, analysis and interpretation of results, and draft manuscript preparation: Jawad Raza, F. Mebarek-Oudina, Haider Ali, I. E. Sarris. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Data are available on request.

Conflicts of Interest: The authors certify that there are no conflicts of interest about the publication of this research.

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## Cite This Article

**APA Style**

*Frontiers in Heat and Mass Transfer*,

*22*

*(4)*, 1017-1041. https://doi.org/10.32604/fhmt.2024.052749

**Vancouver Style**

**IEEE Style**

*Front. Heat Mass Transf.*, vol. 22, no. 4, pp. 1017-1041, 2024. https://doi.org/10.32604/fhmt.2024.052749

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