# Nonlinear Mixed Convection in a Reactive Third-Grade Fluid Flow with Convective Wall Cooling and Variable Properties

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Solution by Weighted Residual Method

## 4. Solution by Shooting-Runge–Kutta Method

## 5. Results and Discussion

## 6. Concluding Remarks

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- Flow, temperature, and entropy generation are enhanced with increasing values of the Grashof number, the quadratic component of buoyancy and Frank-Kameneskii parameter but reduces with increasing third-grade material parameter.
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- Increasing values of third-grade parameter encourages the thermal stability of the flow, while increasing values of the linear and nonlinear buoyancy parameter destabilizes the flow.
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- The Grashof number increase encourages the early occurrence of thermal runaway and exergy loss in the flow domain.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\left(x,y{}^{\prime}\right)$ | dimensional Cartesian coordinates, (m) |

$y$ | dimensional Cartesian coordinates |

${T}_{0}$ | referenced temperature, (K) |

T | dimensional fluid temperature (K) |

$\theta $ | dimensionless fluid temperature |

$u{}^{\prime}$ | the dimensional flow velocity (m/s) |

$u$ | the dimensionless flow velocity |

$E$ | activation energy (E/mols) |

$Q$ | reaction heat (J) |

${C}_{0}$ | initial specie concentration (mol) |

$A$ | reaction rate constant |

$m$ | reaction exponent |

$\epsilon $ | dimensionless activation energy |

$R$ | universal rate constant J/(K.mol) |

${\mu}_{0}$ | constant dynamic viscosity (Poise) |

$\rho $ | fluid density (Kg/m^{3}) |

$g$ | gravitational acceleration m/s^{2} |

$\hslash $ | Planck’s constant (Js) |

$\upsilon $ | frequency of vibration N.s/m^{2} |

$P$ | fluid pressure (N/m^{2}) |

${k}_{0}$ | referenced thermal conductivity J/(mK), |

$\left(\overline{\alpha},\overline{\eta}\right)$ | viscosity and thermal conductivity, respectively(1/K). |

$\left(\alpha ,\eta \right)$ | dimensionless variation parameters for viscosity and thermal conductivity, respectively. |

$Gr$ | modified Grashof number |

$\sigma $ | coefficient of the quadratic thermal expansion, |

$\lambda $ | Frank-Kameneskii parameter |

$\delta $ | Viscous dissipation parameter |

$Bi$ | Biot number |

$\gamma $ | non-Newtonian material parameter |

${h}_{1,2}-$ | coefficient of heat transfer 1/K |

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**Figure 6.**Bifurcation plots when $Gr=7,\sigma =0.1=\epsilon =\eta =\alpha ,\delta =0.2,\gamma =0.3,B{i}_{1,2}=10$.

**Table 1.**Validation of numerical results for velocity $\delta =0.5,\eta =0.1,\alpha =0.3$, $\alpha =0.1,N=30,$ $\gamma =0.3,Gr=1,$ $B{i}_{1}=B{i}_{2}=10,$ $\lambda =0.4,\epsilon =0.1,m=0.5$.

$y$ | $u{\left(y\right)}_{\mathbf{SCCM}}$ | $u{\left(y\right)}_{\mathbf{RK}4}$ | $\left|u{\left(y\right)}_{\mathbf{CWRM}}-u{\left(y\right)}_{\mathbf{RK}4}\right|$ |
---|---|---|---|

0 | −6.96195549259×10^{−18} | 0.000000000000000 | 6.961955492590054 × 10^{−18} |

0.1 | 0.04313941885340904 | 0.04313941993774605 | 1.084337016010739 × 10^{−9} |

0.2 | 0.07813634077297339 | 0.07813635190993772 | 1.11369643229775 × 10^{−8} |

0.3 | 0.10406977160813229 | 0.10406978921035206 | 1.760221977897824 × 10^{−9} |

0.4 | 0.12006990360382683 | 0.12006992567673559 | 2.207290876465872 × 10^{−8} |

0.5 | 0.12548527366302917 | 0.12548528888798596 | 1.522495679529001 × 10^{−9} |

0.6 | 0.12006990360382683 | 0.12006990533241987 | 1.728593046479432 × 10^{−9} |

0.7 | 0.1040697716081323 | 0.10406977022133693 | 1.386795372981808 × 10^{−9} |

0.8 | 0.07813634077297338 | 0.07813634228509685 | 1.512123468105919 × 10^{−9} |

0.9 | 0.043139418853409044 | 0.04313941955035403 | 6.96944987832459 × 10^{−10} |

1.0 | −7.69196285588 × 10^{−18} | −9.12736796 × 10^{−10} | 9.12736788740352 × 10^{−10} |

$y$ | $\theta {\left(y\right)}_{\mathbf{SCCM}}$ | $\theta {\left(y\right)}_{\mathbf{RK}4}$ | $\left|\theta {\left(y\right)}_{\mathbf{SCCM}}-\theta {\left(y\right)}_{\mathbf{RK}4}\right|$ |
---|---|---|---|

0 | 0.022151849024484603 | 0.022151849374097372 | 3.496127692903528 × 10^{−10} |

0.1 | 0.04208489447368267 | 0.04208489042284373 | 4.050838935121259 × 10^{−9} |

0.2 | 0.05751544434774708 | 0.057515440006678895 | 4.341068185476082 × 10^{−9} |

0.3 | 0.06849268440471615 | 0.06849268031195316 | 4.092762989627019 × 10^{−9} |

0.4 | 0.07505930588225566 | 0.07505930197957113 | 3.902684536649659 × 10^{−9} |

0.5 | 0.07724467496042003 | 0.07724467142322092 | 3.537199116943057 × 10^{−9} |

0.6 | 0.07505930588225566 | 0.07505930282648429 | 3.055771372051374 × 10^{−9} |

0.7 | 0.06849268440471615 | 0.06849268167736071 | 2.727355438714163 × 10^{−9} |

0.8 | 0.05751544434774707 | 0.05751544180643439 | 2.541312688064678 × 10^{−9} |

0.9 | 0.04208489447368267 | 0.04208489220244641 | 2.271236268502896 × 10^{−9} |

1.0 | 0.022151849024484603 | 0.02215184702121924 | 2.003265362621187 × 10^{−9} |

**Table 3.**Fast convergence of critical values by weighted residual method when $Gr=3,\delta =0.2,B{i}_{1}=10=B{i}_{2},m=0.5,\epsilon =0.1$.

$N$ | $\alpha $ | $\sigma $ | $\gamma $ | $\eta $ | ${\lambda}_{c}$ |
---|---|---|---|---|---|

5 | 0.1 | 0.1 | 0.3 | 0.1 | 3.369629994329486 |

10 | 0.1 | 0.1 | 0.3 | 0.1 | 3.329979429762467 |

15 | 0.1 | 0.1 | 0.3 | 0.1 | 3.3377915382719980 |

20 | 0.1 | 0.1 | 0.3 | 0.1 | 3.3375451276368135 |

25 | 0.1 | 0.1 | 0.3 | 0.1 | 3.3376391452777585 |

30 | 0.1 | 0.1 | 0.3 | 0.1 | 3.3376334297519272 |

35 | 0.1 | 0.1 | 0.3 | 0.1 | 3.3376356319779727 |

$\sigma $ | $\alpha $ | $\gamma $ | $\eta $ | $Gr$ | ${\lambda}_{c}$ |
---|---|---|---|---|---|

0.1 | 0.1 | 0.3 | 0.1 | 3 | 3.337791538271998 |

0.3 | 0.1 | 0.3 | 0.1 | 3 | 3.294547099649542 |

0.5 | 0.1 | 0.3 | 0.1 | 3 | 3.251342585235665 |

0.1 | 0.3 | 0.3 | 0.1 | 3 | 3.327113416799457 |

0.1 | 0.5 | 0.3 | 0.1 | 3 | 3.3180031889548034 |

0.1 | 0.1 | 0.5 | 0.1 | 3 | 3.354693697420452 |

0.1 | 0.1 | 0.8 | 0.1 | 3 | 3.370237078964153 |

0.1 | 0.1 | 0.3 | 0.01 | 3 | 3.5150936154732557 |

0.1 | 0.1 | 0.3 | 0.001 | 3 | 3.5341579072708558 |

0.1 | 0.1 | 0.3 | 0.1 | 5 | 3.1911369048073452 |

0.1 | 0.1 | 0.3 | 0.1 | 7 | 3.0387490555917243 |

**Table 5.**Validation with previous result when $B{i}_{1}=B{i}_{2}=10,Gr=0=\gamma =\eta ,\alpha =0.1$.

$y$ | ${u}_{1}\left(y\right)$ Makinde & Aziz [34] | ${u}_{2}\left(y\right)$ Present Result | $\left|{u}_{1}\left(y\right)-{u}_{2}\left(y\right)\right|$ |
---|---|---|---|

0 | 0.0000000000000000 | 1.9996146847 × 10^{−18} | 1.999614684734152 × 10^{−18} |

0.1 | 0.04500260897270728 | 0.04500261769036607 | 8.717658789292315 × 10^{−9} |

0.2 | 0.08000542199231503 | 0.08000543151384462 | 9.521529592548816 × 10^{−9} |

0.3 | 0.10500767322725557 | 0.10500768307611696 | 9.848861393102482 × 10^{−9} |

0.4 | 0.12000907000446212 | 0.12000908068447211 | 1.068000998749596 × 10^{−8} |

0.5 | 0.12500954265746725 | 0.12500954933196154 | 6.674494290592747 × 10^{−9} |

0.6 | 0.12000907650344211 | 0.12000908068447211 | 4.181029994443364 × 10^{−9} |

0.7 | 0.10500768080412677 | 0.10500768307611698 | 2.271990207081131 × 10^{−9} |

0.8 | 0.08000543108796186 | 0.08000543151384461 | 4.258827457359615 × 10^{−10} |

0.9 | 0.045002617803451495 | 0.04500261769036607 | 1.130854229702826 × 10^{−10} |

1.0 | 8.38288175864 × 10^{−10} | 2.13968421631 × 10^{−18} | 8.38288173724192 × 10^{−10} |

**Table 6.**Validation with previous result when $B{i}_{1}=B{i}_{2}=10,Gr=0=\gamma =\eta ,\alpha =0.1$.

$y$ | ${\theta}_{1}\left(y\right)-SRK4$ Makinde & Aziz [34] | ${\theta}_{2}\left(y\right)-SCCM$ Present Result | $\left|{\theta}_{1}\left(y\right)-{\theta}_{2}\left(y\right)\right|$ |
---|---|---|---|

0 | 0.0004166959732611298 | 0.00041669643079810236 | 4.57536972555978 × 10^{−10} |

0.1 | 0.0007242078568868499 | 0.0007242204230993113 | 1.256621246132922 × 10^{−8} |

0.2 | 0.0008700589609001264 | 0.0008700665516844415 | 7.59078431502528 × 10^{−9} |

0.3 | 0.0009242312498706879 | 0.0009242382030372055 | 6.953166517597101 × 10^{−9} |

0.4 | 0.0009367330606715411 | 0.0009367393712081855 | 6.310536644453524 × 10^{−9} |

0.5 | 0.0009375671371315745 | 0.0009375727826577018 | 5.645526127313893 × 10^{−9} |

0.6 | 0.000936734339945098 | 0.0009367393712081855 | 5.031263087466702 × 10^{−9} |

0.7 | 0.0009242337880723514 | 0.0009242382030372052 | 4.414964853774461 × 10^{−9} |

0.8 | 0.0008700627611325915 | 0.0008700665516844411 | 3.790551849616394 × 10^{−9} |

0.9 | 0.0007242172747296661 | 0.0007242204230993107 | 3.148369644634038 × 10^{−9} |

1.0 | 0.0004166939431660092 | 0.0004166964307981019 | 2.487632092722583 × 10^{−9} |

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**MDPI and ACS Style**

Adesanya, S.O.; Yusuf, T.A.; Lebelo, R.S.
Nonlinear Mixed Convection in a Reactive Third-Grade Fluid Flow with Convective Wall Cooling and Variable Properties. *Mathematics* **2022**, *10*, 4276.
https://doi.org/10.3390/math10224276

**AMA Style**

Adesanya SO, Yusuf TA, Lebelo RS.
Nonlinear Mixed Convection in a Reactive Third-Grade Fluid Flow with Convective Wall Cooling and Variable Properties. *Mathematics*. 2022; 10(22):4276.
https://doi.org/10.3390/math10224276

**Chicago/Turabian Style**

Adesanya, Samuel Olumide, Tunde Abdulkadir Yusuf, and Ramoshweu Solomon Lebelo.
2022. "Nonlinear Mixed Convection in a Reactive Third-Grade Fluid Flow with Convective Wall Cooling and Variable Properties" *Mathematics* 10, no. 22: 4276.
https://doi.org/10.3390/math10224276