# Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System

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## Abstract

**:**

_{2}O

_{3}nanofluid by using the Koo–Kleinstreuer–Li (KKL) correlation, is investigated. The impact of the magnetic field is also taken into account. KKL correlation is utilized to compute the thermal conductivity and effective viscosity. Analytical double solutions are presented for the considered axisymmetric flow model after implementing the similarity technique to transmute the leading equations into ordinary differential equations. The obtained analytic forms were used to examine and discuss the velocity profile, the temperature distribution, reduced heat transfer, and coefficient of reduced skin friction. The analytic solutions indicate that the velocity profile decreases in the branch of the first solution and uplifts in the branch of the second solution due to the presence of an aluminum particle, whereas the dimensionless temperature enhances in both solutions. In addition, the Casson parameter increases the friction factor, as well as the heat transport rate.

## 1. Introduction

_{2}O

_{3}nanofluid past a porous, radially shrinking sheet with radiation effect. In addition, the fluctuations in the velocity and thermal gradients are graphically deliberated.

## 2. Mathematical Modeling

#### 2.1. Constitutive Equation

#### 2.2. Basic Equations

_{2}O

_{3}) Casson nanofluid are investigated from a continuously non-linear, radially shrinking permeable sheet with variable wall temperature, comprising the correlation of the Koo–Kleinstreuer and Li (KKL) model. Therefore, the flow problem configuration is schematically revealed in Figure 1, where the Cartesian cylindrical coordinates $\left({r}_{b},\delta ,{z}_{b}\right)$ are taken to be in such a way that the ${r}_{b}-$axis is considered parallel to the sheet and the ${z}_{b}-$axis is measured perpendicular to it, and the flow is occurring in the $\left({r}_{b},\delta \right)-$plane. The considered flow is symmetric as well as axisymmetric about the $\left({r}_{b},\delta \right)-$plane and ${z}_{b}-$axis, respectively. Meanwhile, the partial modification in all variables in relation to $\delta $ is going to be completely terminated, and mathematically, it is expressed as $\partial /\partial \delta =0$. In addition, the surface velocity is presumed as ${u}_{w}\left({r}_{b}\right)=c{r}_{b}{}^{3}$, where $c$ signifies as an arbitrary constant. Moreover, the exterior magnetic field is considered in changeable form as $B\left({r}_{b}\right)={r}_{b}{B}_{0}$, which carries out normally to the sheet’s surface (see Appendix A for derivation of the magnetic field). Furthermore, the sheet’s surface is also assumed to be permeable, and the corresponding wall mass transfer velocity is captured as ${w}_{w}\left({r}_{b}\right)=-{r}_{b}{v}_{0}$, with ${w}_{w}<0$ and ${v}_{0}>0$ referring to mass blowing or injection, while ${w}_{w}>0$ and ${v}_{0}<0$ denote a mass suction phenomenon, respectively. Additionally, the temperature is considered as ${T}_{\infty}<{T}_{w}$, where ${T}_{\infty}$ and ${T}_{w}$ designate the steady freestream temperature and the variable or power law wall temperature, respectively.

_{2}O

_{3}) nanoparticles and the base fluid (H

_{2}O) are taken to be constant and no slip happens between them. The nanofluid thermophysical features are specified in Table 1. Under these postulations, the steady basic governing equations in the partial differential equations form are written as follows [40]:

_{2}O

_{3}nanofluids, the following format should be read as follows:

_{2}O

_{3}nanofluid is 96% and 98%, respectively [48] (Table 2). The coefficients ${b}_{i}\hspace{0.17em}(i=1\dots 10)$ are based on the type of nanoparticles, and also on these coefficients. Finally, the correlation of KKL (Koo–Kleinstreuer–Li) is given by:

#### 2.3. Non-Dimensional Equations

#### 2.4. Gradients

## 3. Analytic Solutions Methodology

## 4. Results and Discussion

_{2}O

_{3}nanofluid problem for the two distinct solution branches (first and second solutions), which may be shown visually in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. In these prepared various distinct plots, we displayed the graphical behavior of the velocity profiles, temperature distribution, heat transfer, and wall drag force with the influence of various distinct factors, such as radiation parameter ${R}_{d}$, solid nanoparticle volume fractions $\varphi $, magnetic parameter ${M}_{b}$, mass suction parameter ${S}_{b}$, and non-Newtonian or Casson fluid parameter $\chi $. The problem exhibited two distinct branch solutions (first and second), where the first branch solution in the entire paper is accessible by the solid red lines, and the second branch solution is revealed by the solid green lines. Moreover, the thermophysical properties of the aluminum oxide (Al

_{2}O

_{3}) nanoparticles and the base fluid (H

_{2}O) are arranged in Table 1, while the values of their constants of the water-Al

_{2}O

_{3}nanofluid are written in Table 2. Table 3 and Table 4 are prepared to examine the suction effect on wall drag force and heat transfer. Table 3 indicates that the value of $\left({\mu}_{nf}/{\mu}_{f}\right)G\u2033\left(0\right)$ increases for the outcome of the first branch and declines for the outcome of the second branch. This trend may imply a flow with separation. This trend may imply a flow with separation, in which the structure of a tiny wake may minimize the wall drag force past a shrinkable sheet. Thus, the values of $\left({\mu}_{nf}/{\mu}_{f}\right)G\u2033\left(0\right)$ decline. Table 4 suggests the values of $-\left({k}_{nf}/{k}_{f}\right)\theta \prime \left(0\right)$ augmenting due to greater intensity of suction in both branches of the solution. Physically, as the suction uplifts, it enhances the porosity of the sheet, which ultimately permits more nanoparticles to disperse the sheet. As a consequence, the rate of heat transfer is augmented as suction enhances.

## 5. Conclusions

_{2}O

_{3}nanofluid past a radially shrinking sheet was investigated. In addition, the effectiveness of fluid flow, viscosity, and thermal conductivity were examined using the KKL model. Closed-form double solutions of leading transport equations were presented. The effects of relevant parameters on the explained flow are discussed with the assistance of graphs. The important results of the investigations are gathered as follows:

- The suction and magnetic parameters accelerate the velocity in the first branch solution and decelerate in the second branch solution, whilst declining the temperature distribution in both branch solutions.
- The temperature profile uplifts due to the solid nanoparticle volume fraction in both solutions, while the velocity increases and decreases due to the solid nanoparticle volume fraction in the first and second branch solutions, respectively.
- The heat transfer and the friction factor increase due to the Casson parameter.
- The nanoparticle volume fraction augments the heat transfer and declines the friction factor.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${u}_{w}\left({r}_{b}\right)$ | Velocity of the shrinking disk (m/s) |

$B\left({r}_{b}\right)$ | Variable magnetic field strength (Tesla) |

${B}_{0}$ | Constant magnetic field strength (Tesla) |

${w}_{w}\left({r}_{b}\right)$ | Wall mass transfer velocity (m/s) |

${T}_{w}\left({r}_{b}\right)$ | Wall temperature (K) |

${v}_{0},c$ | Arbitrary constants |

${T}_{\infty}$ | Ambient temperature (K) |

${T}_{b}$ | Fluid temperature (K) |

${q}_{rad}$ | Radiative heat flux |

${T}_{0}$ | Constant reference temperature (K) |

$G\left(\xi \right)$ | Dimensionless velocity |

${M}_{b}$ | Magnetic parameter |

$\mathrm{Pr}$ | Prandtl number |

${S}_{b}$ | Mass transfer factor |

$N{u}_{{r}_{b}}$ | Heat transfer |

${C}_{g}$ | Skin friction coefficient |

${\mathrm{Re}}_{{r}_{b}}$ | Reynolds number |

$D$ | Denotes the roots of the equation |

$G\left(\xi \right)$ | Dimensionless velocity |

${R}_{d}$ | Radiation parameter |

$m$ | Relative factor of the power-law index |

${k}_{C}$ | Mean absorption coefficient (1/m) |

$k$ | Thermal conductivity (W/(m·K)) |

${u}_{b},{w}_{b}$ | Velocity components along the r_{b}- and z_{b}-axes (m/s) |

$\left({r}_{b},\delta ,{z}_{b}\right)$ | Cylindrical coordinates (m) |

${c}_{p}$ | Specific heat at constant pressure (J/Kg·K) |

${B}_{1}$ and ${B}_{2}$ | Arbitrary constants |

## Greek Symbols

${\nu}_{f}$ | Kinematic viscosity (m^{2}/s) |

$\gamma $ and $h$ | Empirical functions |

$\theta \left(\xi \right)$ | Dimensionless temperature |

$\chi $ | Non-Newtonian or Casson fluid parameter |

$\sigma $ | Electrical conductivity (Ω^{−1}m^{−1}) |

$\mu $ | Dynamic viscosity (N·s/m^{2}) |

$\xi $ | Pseudo-similarity variable |

$\rho $ | Density (kg/m^{3}) |

${\sigma}_{C}$ | Stefan-Boltzmann constant (W/(m^{2}·K^{4})) |

$\psi $ | Stream function |

$\eta $ | New similarity variable |

$\varphi $ | Solid nanoparticle volume fraction |

## Acronyms

KKL | Koo–Kleinstreuer–Li |

MHD | Magnetohydrodynamics |

EOF | Electro-osmotic forces |

Al_{2}O_{3} | Alumina |

2D, 3D | Two and three-dimensional |

BCs | Boundary conditions |

BLT | Boundary layer thickness |

BLF | Boundary layer flow |

VD | Viscous dissipation |

TIR | Thermal interfacial resistance |

## Subscripts

$pa$ | Nanoparticles |

$nf$ | Nanofluid |

$f$ | Regular based-fluid |

$w$ | Wall boundary condition |

$\infty $ | Far-field condition |

## Superscript

‘ | Derivative with respect to ξ |

## Appendix A

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**Figure 2.**Effect of ${S}_{b}$ on $G\prime \left(\xi \right)$ when $\varphi =0.025,\chi =1,{M}_{b}=0.5$.

**Figure 3.**Effect of ${S}_{b}$ on $\theta \left(\xi \right)$ when $\varphi =0.025,\chi =1,{M}_{b}=0.5,{R}_{d}=1.5,m=1$.

**Figure 4.**Effect of ${M}_{b}$ on $G\prime \left(\xi \right)$ when $\varphi =0.025,\chi =1,{S}_{b}=2.5$.

**Figure 5.**Effect of ${M}_{b}$ on $\theta \left(\xi \right)$ when $\varphi =0.025,\chi =1,{S}_{b}=2.5,{R}_{d}=1.5,m=1$.

**Figure 6.**Effect of $\varphi $ on $G\prime \left(\xi \right)$ when ${S}_{b}=2.5,\chi =1,{M}_{b}=0.5$.

**Figure 7.**Effect of $\varphi $ on $\theta \left(\xi \right)$ when ${S}_{b}=2.5,\chi =1,{M}_{b}=0.5,{R}_{d}=1.5,m=1$.

**Figure 8.**Effect of $\chi $ on $\left({\mu}_{nf}/{\mu}_{f}\right)G\u2033\left(0\right)\hspace{0.17em}$ versus ${S}_{b}$ when $\varphi =0.025,{M}_{b}=0.5$.

**Figure 9.**Effect of $\chi $ on $-\left({k}_{nf}/{k}_{f}\right)\theta \prime \left(0\right)$ versus ${S}_{b}$ when $\varphi =0.025,{M}_{b}=0.5,{R}_{d}=1.5,m=1$.

**Figure 10.**Effect of $\chi $ on $\left({\mu}_{nf}/{\mu}_{f}\right)G\u2033\left(0\right)\hspace{0.17em}$ versus ${S}_{b}$ when $\chi =1,{M}_{b}=0.5$.

**Figure 11.**Effect of $\varphi $ on $-\left({k}_{nf}/{k}_{f}\right)\theta \prime \left(0\right)$ versus ${S}_{b}$ when $\chi =1,{M}_{b}=0.5,{R}_{d}=1.5,m=1$.

**Figure 12.**Effect of ${R}_{d}$ on $-\left({k}_{\mathit{nf}}/{k}_{f}\right)\theta \prime \left(0\right)$ versus ${S}_{b}$ when $\varphi =0.025,\chi =1,{M}_{b}=0.5,m=1$.

**Figure 13.**Effect of ${R}_{d}$ on $\theta \left(\xi \right)$ when $\varphi =0.025,\chi =1,{M}_{b}=0.5,m=1$.

Physical Properties | Water | Al_{2}O_{3} |
---|---|---|

$k$ (Wm^{−1}K^{−1}) | 0.613 | 25 |

${c}_{p}$ (J kg^{−1} K^{−1}) | 4179 | 765 |

$\rho $ (kg m^{−3}) | 997.1 | 3970 |

$\sigma {\left(\mathsf{\Omega}\mathrm{m}\right)}^{-1}$ | 0.05 | 1 × 10^{−10} |

${d}_{\mathit{pa}}$ (nm) | - | 47 |

Pr | 6.2 | - |

Coefficient Values | Water-Al_{2}O_{3} |
---|---|

${c}_{1}$ | 52.813 |

${c}_{2}$ | 6.115 |

${c}_{3}$ | 0.695 |

${c}_{4}$ | 4.1 × 10^{−2} |

${c}_{5}$ | 0.176 |

${c}_{6}$ | −2.98.198 |

${c}_{7}$ | −34.532 |

${c}_{8}$ | −3.922 |

${c}_{9}$ | −0.235 |

${c}_{10}$ | −0.999 |

**Table 3.**Values of $\left({\mu}_{nf}/{\mu}_{f}\right)G\u2033\left(0\right)$ for distinct values of ${S}_{b},\chi ,\varphi $, and ${M}_{b}$.

$\mathit{\chi}$ | ${\mathit{M}}_{\mathit{b}}$ | $\mathit{\varphi}$ | ${\mathit{S}}_{\mathit{b}}$ | 3.5 | 4 | 4.5 | 5 |
---|---|---|---|---|---|---|---|

0.5 | 0.5 | 0.025 | First solution | 2.58605 | 3.40613 | 4.09306 | 4.73029 |

Second solution | 1.17470 | 0.89187 | 0.74219 | 0.64220 | |||

1 | 0.5 | 0.025 | First solution | 4.66415 | 5.63891 | 6.55804 | 7.44685 |

Second solution | 0.97697 | 0.80809 | 0.69483 | 0.61190 | |||

2 | 0.5 | 0.025 | First solution | 6.60110 | 7.81896 | 8.99505 | 10.14620 |

Second solution | 0.92040 | 0.77704 | 0.67544 | 0.59881 | |||

5 | 0.5 | 0.025 | First solution | 8.50938 | 9.98435 | 11.42330 | 12.83980 |

Second solution | 0.89249 | 0.76064 | 0.66483 | 0.59149 | |||

∞ | 0.5 | 0.025 | First solution | 10.4065 | 12.1435 | 13.8436 | 15.5307 |

Second solution | 0.87575 | 0.75048 | 0.65812 | 0.58680 |

**Table 4.**Values of $-\left({k}_{nf}/{k}_{f}\right)\theta \prime \left(0\right)$ for distinct values of ${S}_{b},\chi ,\varphi ,{R}_{d},m$, and ${M}_{b}$.

$\mathit{\chi}$ | ${\mathit{M}}_{\mathit{b}}$ | $\mathit{\varphi}$ | ${\mathit{R}}_{\mathit{d}}$ | $\mathit{m}$ | ${\mathit{S}}_{\mathit{b}}$ | 3.5 | 4 | 4.5 | 5 |
---|---|---|---|---|---|---|---|---|---|

0.5 | 0.5 | 0.025 | 5 | 1 | First solution | 2.82016 | 3.52056 | 4.10507 | 4.66170 |

Second solution | 2.61293 | 3.32299 | 3.99296 | 4.68492 | |||||

1 | 0.5 | 0.025 | 5 | 1 | First solution | 3.04693 | 3.63183 | 4.18821 | 4.72967 |

Second solution | 2.66077 | 3.44115 | 4.28216 | 5.18029 | |||||

2 | 0.5 | 0.025 | 5 | 1 | First solution | 3.13405 | 3.69660 | 4.24086 | 4.77445 |

Second solution | 2.70750 | 3.72440 | 4.87577 | 6.24120 | |||||

5 | 0.5 | 0.025 | 5 | 1 | First solution | 3.19054 | 3.74125 | 4.27820 | 4.80674 |

Second solution | 2.85264 | 4.31988 | 6.24757 | 9.17801 | |||||

∞ | 0.5 | 0.025 | 5 | 1 | First solution | 3.23132 | 3.77438 | 4.30633 | 4.83129 |

Second solution | 3.18356 | 5.90969 | 11.6249 | 39.6040 |

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**MDPI and ACS Style**

Khan, U.; Zaib, A.; Ishak, A.; Waini, I.; Raizah, Z.; Galal, A.M.
Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System. *Micromachines* **2022**, *13*, 1109.
https://doi.org/10.3390/mi13071109

**AMA Style**

Khan U, Zaib A, Ishak A, Waini I, Raizah Z, Galal AM.
Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System. *Micromachines*. 2022; 13(7):1109.
https://doi.org/10.3390/mi13071109

**Chicago/Turabian Style**

Khan, Umair, Aurang Zaib, Anuar Ishak, Iskandar Waini, Zehba Raizah, and Ahmed M. Galal.
2022. "Analytical Approach for a Heat Transfer Process through Nanofluid over an Irregular Porous Radially Moving Sheet by Employing KKL Correlation with Magnetic and Radiation Effects: Applications to Thermal System" *Micromachines* 13, no. 7: 1109.
https://doi.org/10.3390/mi13071109