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# Analysis of Heat Transport in a Powell-Eyring Fluid with Radiation and Joule Heating Effects via a Similarity Transformation

1 Department of Mathematics, Government Postgraduate College Haripur, 22620, Pakistan

2 Basic Sciences Department, University of Engineering and Technology, Taxila, 47050, Pakistan

3 Department of Mathematics, Government Postgraduate College, Abbottabad, 22010, Pakistan

* Corresponding Author: Tahir Naseem. Email:

(This article belongs to the Special Issue: Advances in Fluid Flow, Heat and Thermal Sciences)

*Fluid Dynamics & Materials Processing* **2023**, *19*(3), 663-677. https://doi.org/10.32604/fdmp.2022.021136

**Received** 29 December 2021; **Accepted** 31 March 2022; **Issue published** 29 September 2022

## Abstract

Heat transfer in an Eyring-Powell fluid that conducts electricity and flows past an exponentially growing sheet is considered. As the sheet is stretched in the x direction, the flow develops in the region with*y*> 0. The problem is tackled through a set of partial differential equations accounting for Magnetohydrodynamics (MHD), radiation and Joule heating effects, which are converted into a set of equivalent ordinary differential equations through a similarity transformation. The converted problem is solved in MATLAB in the framework a fourth order accurate integration scheme. It is found that the thermal relaxation period is inversely proportional to the thickness of the thermal boundary layer, whereas the Eckert-number displays the opposite trend. As this characteristic number grows, the temperature within the channel increases.

## Keywords

Nomenclature

| Velocity components, |

| Ratio of expansion rates |

| Stretching velocity, |

| Dimensionless Powell Eyring fluid parameters |

| Velocity of external flow, |

| Prandtl number |

| Characteristic length, |

| Skin friction coefficient |

| Surface temperature, |

| Local Nusselt number |

| Ambient temperature, |

| Wall shear stress |

| Stress tensor, |

| Surface heat flux |

| Kinematic viscosity, |

| Local Reynolds numbers |

| Dynamic viscosity, |

| Magnitude of magnetic field vector, |

| Density of the fluid, |

| Radiative heat flux |

| Powell-Eyring material parameter, |

| Mean absorption coefficient, |

| Powell-Eyring material parameter, |

| Stefan Boltzmann constant, |

| Specific heat, |

| Dimensionless thermal relaxation time |

| Temperature of the/fluid, |

| Thermal relaxation time |

| Thermal conductivity of the fluid, |

| Eckert number |

| Radiation parameter |

| Magnetic parameter |

Eyring-Powell fluids are a significant type of non-Newtonian fluid. Additionally, these fluids are classified as differential types, integral models, and shear rate models. The Eyring-Powell fluid has a particular advantage over the power-law model because it is based on liquid kinetic theory. For low and high shear rates, this fluid’s Newtonian behaviour decreases. These fluids are extremely valuable since they can be used in a variety of engineering, manufacturing, and industrial applications, including pulp, plasma, and other biological technology. Additionally, these fluids play a critical role in fermentation, boiling, bubble formation, column processing, and the processing of plastic foam, with substances such as mud, colours, toothpaste, blood, corn starch, custard, and honey serving as insignificant examples of non-Newtonian fluids [1].

Recent technological and engineering advancements have resulted in the development of a diverse range of non-Newtonian fluids with a number of major differences from viscous fluids. Ziegenhagen [2] explored the slow flow of a Powell-Eying type fluid and used variational techniques to obtain results. He studied the behaviour of Oldroyd and Powell Eyring fluids and discovered that both fluids behave identically in situations involving extremely slow fluid flow. Sirohi et al. [3] studied it by observing the flow of Powell-Eyring fluid around the accelerating plate. They compared three distinct techniques. Yoon et al. [4] pioneered the concept of a stretched sheet by providing a precise solution to the resulting differential system. Recent academics have investigated this topic from a variety of perspectives [5–13]. Bahia et al. [14] investigated the Powell-Eyring fluid flow and heat transport past a stretched sheet exponentially. They discovered that increasing the velocity ratio parameter results in a thinned boundary layer. Malik et al. [15] examined the Powell-Eyring fluid flow and heat transport with varying viscosity over a stretching cylinder by examining the steady condition. They concluded that as Prandtl and Reynolds numbers increase, the boundary layer shrinks. Akbar et al. [16] studied the effect of magnetic factors on Eyring-Powell fluid flow past a stretched surface. They investigated flow resistance as the magnetic and hydrodynamic properties of the fluid under study increased.

Kumar et al. [17] investigated the Powell-Eyring nanofluid passing via an inclined permeable sheet. They demonstrated that temperature increases as thermophoresis parameter values increase. While the contrary is true for nanoparticle concentration due to higher chemical reactions and Brownian parameters, increasing thermophoresis parameter values results in an increase in concentration. Pal et al. [18] demonstrated magneto-bioconvection of Powell-Eyring nanofluid via a vertical stretched sheet that is convectively heated and also contains motile gyrotactic microorganisms. They discovered that as the Schmidt number and chemical reaction parameters increase, the concentration of nanoparticles drops. Thermal relaxation time is the time required for a fluid to return to its original temperature after being heated. It is a frequently used parameter for determining the time required for heat to leave a fluid. Hayat et al. [19] investigated the effects of mass flux models on Eyring Powell fluid flow in three dimensions. They discovered that temperature and thermal-relaxation time have an inverse relationship. Reddy et al. [20] studied the effect of chemical reaction on the activation energy of Eyring Powell nanofluid flow via a stretching cylinder. They concluded that as the relaxation parameter increases, the temperature curves lose their shape. It takes a long time for an increase in the relaxation parameter assessment to transfer heat to neighbouring material particles. Additionally, the Nusselt number improves behaviour when non-dimensional thermal relaxation calculations are performed.

Mustafa [21] researched the Maxwell fluid with a generalised heat flux model for rotating flow and heat transfer. They also discovered that the thermal relaxation period is inversely proportional to temperature and thermal boundary thickness. On an unstable porous stretching sheet, Ishaq et al. [22] demonstrated the entropy production of Eyring Powell fluid flow with nanofluid thin film flow by considering the heat radiation and MHD impact. They discovered that when the Brinkmann, Hartmann, and Reynolds numbers grow, so does the entropy profile. For increasing values of the Eyring Powell and radiation parameters, the entropy profile reduces. The Eyring Powell nanofluid flow with non-linear mixed convection and entropy generation was explored by Alsaedi et al. [23]. They arrived at the conclusion that entropy generation showed a falling tendency for some fluid parameter values while increasing for others. Through a permeable stretching surface, Bhatti et al. [24] studied the irreversibility of MHD Eyring Powell nanofluid.

Ali et al. [25] used both perturbation and computational methods to examine the steady non-isothermal flow of an Eyring–Powell fluid in a conduit. The findings are provided for two viscosity models, the Reynolds and the Vogel models, which were solved using the shooting and perturbation methods, respectively. It was determined that the shooting approach outperformed the perturbation method. Nazeer et al. [26] investigated the effects of constant and space-dependent viscosity on a circular conduit filled with Eyring–Powell fluid. Additionally, heat transmission analysis is considered. The finite difference scheme is compared to the perturbation method. Numerous researchers discussed the Eyring-Powell model under a variety of scenarios and solved it analytically and numerically using a variety of numerical schemes such as the RK method, the shooting technique, and the perturbation method [27–34].

According to the existing literature, no attempt has been made to investigate the electrically conducting Eyring-Powel fluid with radiation, thermal relaxation time, and joule heating effects beyond an exponentially stretched sheet. This research fills a void in the literature and lays the groundwork for future researchers to contribute their perspectives to the open literature. This is structured as follows: Section 1 contains the literature review, Section 2 the mathematical formulation, Section 3 the methodology, Section 4 the results, and Section 5 the conclusion.

Consider an incompressible Powell Eyring fluid flowing across an exponentially stretched surface subjected to magnetic, joule heating, thermal radiation, and thermal relaxation periods, as illustrated in Fig. 1. The sheet is put on the

The governing equations so obtained are given as [35]

where

Furthermore, by means of Rosseland approximation for radiation, we get

Taking the similarity transformations as

With the above transformation the continuity equation is satisfied identically and Eqs. (2)–(4) is converted into the following form:

Here

here,

The mathematical form of local Nusselt number and skin friction coefficient are given as under

where local Reynolds numbers are

Rahimi et al. [37] solved a non-Newtonian model known as the Powell Eyring fluid model using the collocation approach. Agrawal et al. [38] solved the Eyring Powell fluid model using a fourth-order precision methodology and the homotopy analysis method (H.A.M). Jafari Moghaddam [39] studied the Eyring Powell model and described fluid flow and heat transfer over a stretching sheet. The Eyring-Powell model is also solved by using different techniques as mentioned in [40–42]. He then solved the governing PDEs by using homotopy perturbation and homotopy analysis methods to convert them to ODEs. The flow chart of the numerical scheme is presented in Fig. 2 below. The third order nonlinear ordinary differential Eq. (6) and the second order nonlinear ordinary differential Eq. (7) are expressed as difference equations and solved using BVP4C and MATLAB in this article.

The iterative approach will conclude with the required precision.

The velocity ratio parameter, the fluid parameter

4.1 Visualization of a Velocity Field

Two types of boundary layers near the sheet have evolved in a flow with exponentially changing free stream velocity over an exponentially stretched sheet. Which means that they are depending on the velocity ratio parameter

4.2 Visualization of a Temperature Field

The fluctuation of the velocity ratio parameter on the temperature profile is depicted in Fig. 7. Temperature has been discovered to be a decreasing function of

The influence of radiation on temperature distributions can be seen in Fig. 10. Increases in Rd result in an increase in heat fluxes from the sheet, which results in a rise in temperature. Ec’s effect on the temperature profile

4.3 Matching Results to Published Work

The local Nusselt number and skin friction are listed in Table 1, and were estimated using a MATLAB method with fourth-order precision (BVP4C). The skin friction coefficient increases as K increases. As a result, as

Thermal transport in the Powell-Eyring model via generalised heat flux over an exponentially stretching sheet is examined. The impact of Powell-Eyring fluid parameter, magnetic parameter M, Eckert number

• The velocity profile increases as the fluid parameter K increases, but reverse behaviour is noticed for the temperature profile.

• For increasing values of the magnetic parameter M, the velocity profile falls while the temperature rises. In addition, the resistance to flow increases as the magnetic field intensity and

• The temperature and thickness of the thermal boundary layer are inversely related to the thermal relaxation time

• Increasing values of

Simulations of local Nusselt number and skin friction/co-efficient are used to validate the published work.

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

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

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*Fluid Dynamics & Materials Processing*,

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*(3)*, 663-677. https://doi.org/10.32604/fdmp.2022.021136

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*Fluid Dyn. Mater. Proc.*, vol. 19, no. 3, pp. 663-677, 2023. https://doi.org/10.32604/fdmp.2022.021136

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