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# Casson Nanofluid Flow with Cattaneo-Christov Heat Flux and Chemical Reaction Past a Stretching Sheet in the Presence of Porous Medium

1 Department of Mathematics, Capital University of Science and Technology, Islamabad, 45710, Pakistan

2 Department of Mathematics, University of Azad Jammu and Kashmir, Muzzaffarabad, Azad Kashmir, 13100, Pakistan

3 Department of Mathematics, University of Gujrat, Gujrat, 50700, Pakisan

4 Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China

5 Department of Mathematics, JNTUH College of Engineering, Science & Technology Hyderabad, Telangana, 500085, India

* Corresponding Author: Ali Hassan. Email:

(This article belongs to the Special Issue: Computational and Numerical Advances in Heat Transfer: Models and Methods I)

*Frontiers in Heat and Mass Transfer* **2024**, *22*(4), 1261-1276. https://doi.org/10.32604/fhmt.2024.048091

**Received** 27 November 2023; **Accepted** 19 July 2024; **Issue published** 30 August 2024

## Abstract

In the current work, inclined magnetic field, thermal radiation, and the Cattaneo-Christov heat flux are taken into account as we analyze the impact of chemical reaction on magneto-hydrodynamic Casson nanofluid flow on a stretching sheet. Modified Buongiorno’s nanofluid model has been used to model the flow governing equations. The stretching surface is embedded in a porous medium. By using similarity transformations, the nonlinear partial differential equations are transformed into a set of dimensionless ordinary differential equations. The numerical solution of transformed dimensionless equations is achieved by applying the shooting procedure together with Rung-Kutta 4th-order method employing MATLAB. The impact of significant parameters on the velocity profile , temperature distribution , concentration profile , skin friction coefficient , Nusselt number and Sherwood number are analyzed and displayed in graphical and tabular formats. With an increase in Casson fluid , the motion of the Casson fluid decelerates whereas the temperature profile increases. As the thermal relation factor expands , the temperature reduces, and consequently thermal boundary layer shrinks. Additionally, by raising the level of thermal radiation the temperature profile significantly improves, and an abrupt expansion has also been observed in the associated thermal boundary with raise thermal radiation strength. It was observed that higher permeability hinders the acceleration of Casson fluid. Higher Brownian motion levels correspond to lower levels of the Casson fluid concentration profile. Moreover, it is observed that chemical reaction has an inverse relation with the concentration level of Casson fluid. The current model’s significant uses include heat energy enhancement, petroleum recovery, energy devices, food manufacturing processes, and cooling device adjustment, among others. Furthermore, present outcomes have been found in great agreement with already published work.## Keywords

Nomenclature

µf | Viscosity of the fluid ( |

ρf | Density of the fluid |

νf | Kinematic viscosity |

k | Thermal conductivity |

α | Thermal diffusivity |

σ | Electrical conductivity |

u,v | x,y-component of fluid velocity (m/s) |

B0 | Magnetic field constant |

k1 | Permeability constant |

qr | Radiative heat flux |

q | Heat generation constant |

σ∗ | Stefan Boltzmann constant |

k∗ | Absorption coefficient |

Cf | Skin friction coefficient |

Casson fluid parameter | |

R | Thermal radiation parameter |

M | Magnetic parameter |

K | Permeability parameter |

Pr | Prandtl number |

Nb | Brownian motion parameter |

Nt | Thermophoresis parameter |

Sc | Schmidt number |

Relaxation time parameter | |

Chemical reaction parameter | |

Nu | Nusselt number |

Sh | Sherwood number |

f | Dimensionless velocity |

θ | Dimensionless temperature |

Colloidal suspension of nanoparticles into base fluid has introduced a new class of fluids called nanofluids. Nanofluid passes remarkable properties that the technology was unlikely to attain through conventional fluids. When the nano-meter-sized nanoparticles are dispersed in the convectional fluid, the formed mixture exhibit enhanced chemical reactivity, electrical conductivity characteristics and in particular heat transfer and thermal conductivity. Applications of nanofluids in sectors like aeronautics, medicine, and pharmaceutics have produced numerous innovative products. These products include brake fluids, nuclear reactions, improvements in cooling transformer oil, power plants, and space technologies. Choi et al. [1] is the person who introduced the term nanofluids through his experimental work. This invention opened doors for other researchers and provided humanity with a platform to extract more out of it. The earliest works done on nanofluids were by Wang et al. [2] and Jahani et al. [3]. Buongiorno [4] introduced the nanofluid model. Later, Hussain et al. [5] extended the model for exponentially expanding surfaces.

Khan et al. [6] were able to generate the first-ever paperwork on the laminar flow of nanofluids over a stretching surface emphasizing that behavior can also be well observed in nanofluids. Noghrebatadi et al. [7] and Hady et al. [8] performed similar experiments depicting the behavior of nanofluids. Wang [9] discovered theoretically and experimentally the flow towards a shrinking sheet. Out of many significant characteristics, the most advanced to grasp interest are MHD and thermal radiation effects. Nadeem et al. [10] used the Homotopy method to investigate the two-dimensional flow of heat transfer considering Williamson nanofluids. His work was followed by Prasannakumara et al. [11] analysing chemical activity over a porous medium. Danish et al. [12] provided a thorough extension to this phenomenon. More work on Williamson nanofluids was presented by Srinivasulu et al. [13] who studied MHD and the thermal effects of Williamson flow. The presentation of heat transfer on a hybrid nanofluid model with effects of MHD and thermal radiation was made in its earliest form by Zainal et al. [14]. Reddy et al. [15] investigated thermal radiation improvement on stagnation point flow. Mondal et al. [16] who performed comparative studies keeping in view heat transfer under thermal radiation impact. Further, many researchers have investigated the thermal radiation regimes under the effect of distinct external forces [17–20].

Multiple slips influence on MHD with chemical reaction with heat flux was studied by Gul et al. [21]. Moreover, Pramanik [22] explored heat transfer in the Casson nanofluid flow with thermal radiation. Mahanthesh et al.'s [23] analysis of the flow through an elongating surface was motivated by many physical factors. The difficult problem was reduced to a simpler one by utilizing the boundary layer approach before being resolved using the shooting method. In their analysis, they established a comparative study. Mohyud-Din et al. [24] studied the compressed flow of gas using Non-Newtonian fluid. A thorough explanation of the MHD Casson fluid including the properties of Hall and Dufour was conducted by Vijayaragavan et al. [25]. Yousef et al. [26] examined the dissipative Casson-Williamson fluid under the influence of the chemical reaction. Mukhopadhyay [27] investigated the Casson fluid with heat transfer over a nonlinear stretching surface. Dero et al. [28] explored the impact of viscous dissipation on the Casson fluid over the nonlinear stretching and shrinking surface. Recent research [29–31] has described various elements of these flows utilizing the Casson fluid.

Khan et al. [32] discussed heat transfer in nanotechnology with Casson fluid flow. Mabood et al. [33] investigated boundary layer flow over a nonlinear stretching sheet. Zhang et al. [34] performed a similar investigation of events but using a porous medium, whereas Krishna et al. [35] performed Newtonian heating on MHD hybrid nanofluid, and Nadeem et al.'s [36] work targeted a porous stretching sheet. Bhatti et al. [37] critically evaluated Reynolds number in relation to magnetic field. Manvi et al. [38] studied MHD Casson fluid with boundary layer and Brownian motion, heat production, and thermal profile which were later validated by Popey et al. [39] under the effects of MHD. Chamkha et al. [40] successfully described MHD boundary layer flow with convective slip flow under the effects of heat. Malik et al. [41] unlike others chose a non-Newtonian fluid for instant Casson nanofluid to discuss velocity changes under MHD effects. Ganga et al. [42] and Waheed et al. [43] also contributed significantly by considering unsteady MHD in the fluid flow problems. As discussed by Biswal et al. [44], most chemical reaction processes are determined by the presence of species. Chamkha et al. [45] analyzed heat produced or absorbed by a uniform vertical permeable surface with MHD effects.

In the present work, we discuss the steady 2D MHD flow of Casson nanofluid past a stretching sheet with the boundary conditions by using the thermal radiation. The impact of the inclined magnetic field, Cattaneo-Christov heat flux, and chemical reaction field have also been discussed. For the proposed problem, we utilized the well-known shooting technique, the shooting method is implemented in MATLAB to obtain the solution of a reduced system of nonlinear ODEs with the boundary conditions. The current model’s significant uses include heat energy enhancement, petroleum recovery, energy devices, food manufacturing processes, and cooling device adjustment, among others. The numerical solution for various parameters is discussed for the dimensionless velocity, temperature, and concentration. Investigation of achieved numerical outcomes is given through tables and graphs.

Consider steady 2D non-Newtonian MHD Casson nanofluid flow in a porous medium past a stretching sheet with

2.2 Problem Governing Equations

In this section, a mathematical model has been developed using the constitutive relation. Casson fluid constitutive relations have been used for formulation and Cattaneo-Christov model has been used to formulate the energy equation. The modified Buongiorno nanofluid model has been implemented in the present formulation. Fig. 1 illustrates the coordinate system and problem schematic. Flow governing PDE’s are given as follows:

Associated boundary conditions have been taken as

In the above model,

where

The modified BC’ss are as follows:

Different dimensionless variables are formulated as

2.3 Skin Friction, Nusselt and Sherwood Numbers

The important parameters of interest include skin friction coefficient, local Nusselt number, and local Sherwood number, which are formulated as follows:

Here, the skin friction or shear stress is represented by

Dimensionless formations of friction, Nusselt & Sherwood numbers are

To numerically solve ODEs ((7)–(9)) subject to the boundary circumstances ((10)), the shooting technique has been used in MATLAB. The notations listed below have been taken into consideration.

Transformed first ODEs is created by converting the momentum (Eqs. (7)–(9)).

The RK-4 method will be used to numerically solve the above mentioned initial value problem. The bounded domain

Newton’s system updates the missing condition

where

The following set of first order coupled ODE’s may be used to represent the system of Eqs. (9) and (10):

The RK-4 technique will be used to numerically solve the initial value problem mentioned above. The missing conditions p and q in the above system of equations must be selected in such a way that the following condition is satisfied:

Fig. 2 illustrates the computational approach for solving our problem. Using Newton’s technique and the following stopping criteria, the two equations above are resolved:

In this section, physical interpretations are provided for the influence of flow parameters such as Casson fluid parameter

4.1 Code Validation and Analysis of Results

In this subsection, the validation of the presented outcomes has been presented and analyzed in comparison with Reddy et al. [46]. Reddy et al. [46] employed Buongiorno nanofluid model with heat generation/absorption effect in the presence of chemical reaction over the porous medium for non-Newtonian Casson fluid. Additionally, they ignored the Cattaneo-Christov heat flux while modeling the problem. Whereas, in this work, we have addressed the Cattaneo-Christov heat flux with in the presence of thermal radiation, chemical reaction, and Buongiorno nanofluid model. The outcomes in the present study have been obtained using MATLAB using the shooting method. We reproduce Reddy et al.'s [46] skin friction coefficient to ensure the accuracy of our findings. The comparison presented in Table 1 for this comparison we have chosen

4.2 Velocity, Temperature, and Concentration Profiles

In this section, the outcomes of the present study have been presented and discussed under varying influence of the different study parameters such as Casson fluid parameter

Fig. 3a,b represents the impact of Casson parameter β on the velocity profile

Fig. 4a,b depicts effects of the permeability parameter K on the temperature distribution and velocity field. These outcomes indicate that when the porosity K of material is raised, the velocity profile drops. This outcome is attributed to the fact that when K is raised, the porous layer is amplified, reducing the thickness of the momentum boundary layer. Similarly, a rise in K improves the boundary layer region’s temperature of the fluid. Darcian’s body force is transferring that heat from the solid wall to the stream zone. Fig. 4c,d indicates the impact of

Fig. 5a,b represents the impact of thermal radiation R and thermal relaxation time factor

4.3 Nusselt Number and Skin Friction

The validation of our results has been presented for the skin friction coefficient in Table 1 under varying effects of magnetization force, a good agreement has been found with already published results and present outcomes. In this section, numerical outcomes of the skin friction coefficient, local Nusselt number, and Sherwood number for the distinct values of parameters namely, magnetization force, Casson fluid parameter, permeability parameter, thermal radiation, and chemical reaction parameter are shown in Tables 2 and 3.

Table 2 shows the numerical outcomes for the skin friction coefficient under the influence of the magnetization force, Casson fluid, and permeability parameters. It can be noted from the outcomes of skin friction that when permeability levels are increased the skin friction is enhanced whereas reduced skin friction rates have been obtained under the varying impact of magnetic field and Casson fluid parameter. Table 3 demonstrates Nusselt number and Sherwood number outcomes under the effect of different varying study parameters. It is worth mentioning here that raising the levels of permeability and chemical reaction decreases the Nusselt number whereas raising the levels of thermal radiation, magnetization force, and Casson fluid parameter enhances the rate of heat transfer coefficient. Moreover, raising the Schmidt number reduce Sherwood’s number as compared to other study parameters.

In this paper, two-dimensional Casson nanofluid flow across a stretched sheet under the impact of the inclined magnetic field, Cattaneo-Christov heat flux, and first-order chemical reaction has been investigated numerically using the shooting method in MATLAB. The results of the current investigation can be categorized as follows:

• The temperature profile of the fluid and its velocity are directly and inversely proportional in Casson fluid, respectively.

• Magnetization force is inversely proportional to the velocity of the fluid and directly proportional to the fluid temperature.

• The temperature distribution gets larger with increasing values of thermal radiation.

• Enhancing the magnetic parameter M results in a rise in the skin friction coefficient.

• The behavior of the temperature profile decreases as the thermal relaxation time parameter

• The Nusselt number decreases as the value of the chemical reaction parameter rises.

• An increment is noticed in the temperature distribution by raising the values of Brownian motion

• The concentration profile can be reduced by raising the values of the chemical reaction parameter

• Due to the increasing values of the thermal radiation R, the values of

Acknowledgement: The authors would like to thank the editors and worthy reviewers for the constructive suggestions to enhance the overall presentation of this version of the article.

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

Author Contributions: Study conception and design: Mahzad Ahmed, Raja Mussadaq Yousaf; Data collection: Mahzad Ahmed, Raja Mussadaq Yousaf, B. Shankar Goud; Analysis and interpretation of results: Mahzad Ahmed, Raja Mussadaq Yousaf; Draft manuscript preparation: Mahzad Ahmed, Raja Mussadaq Yousaf, B. Shankar Goud; Project administration, Supervision, Sources and writing—review and editing: Ali Hassan. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: This research has no unavailable data.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Frontiers in Heat and Mass Transfer*,

*22*

*(4)*, 1261-1276. https://doi.org/10.32604/fhmt.2024.048091

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