# Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling and Formulation

- Oldroyd-B nanofluid when ${\lambda}_{1}\ne 0,{\lambda}_{2}=0\text{}\mathrm{and}\text{}{\lambda}_{3}\ne 0$.
- Maxwell nanofluid when ${\lambda}_{1}\ne 0,{\lambda}_{2}=0\text{}\mathrm{and}\text{}{\lambda}_{3}=0$.
- Jeffrey nanofluid when ${\lambda}_{1}=0,{\lambda}_{2}\ne 0\text{}\mathrm{and}\text{}{\lambda}_{3}\ne 0$.

## 3. Solution by Homtopy Analysis Method (HAM)

^{th}-order problem for Equations (14), (15) and (20) were:

## 4. HAM Solution Convergences

## 5. Results and Discussion

#### Tables Discussion

## 6. Comparison of Analytical Solutions and Numerical Solutions

## 7. Conclusions

- ➢
- The upsurges in magnetic field diminishes the velocity field.
- ➢
- The upsurges in Prandtl number and thermal relaxation parameters diminish the temperature field.
- ➢
- The upsurges in Schmidt number upsurges the concentration field.
- ➢
- The larger homogeneous reaction and heterogeneous reaction strengths falloff from the concentration field.

## Author Contributions

## Conflicts of Interest

## Nomenclature

${\mathrm{B}}_{0}$ | Magnetic field strength (NmA^{−1}) |

$C{f}_{x}$ | Skin friction coefficient |

${D}_{I}$, ${D}_{J}$ | Diffusion coefficients |

$F$ | Velocity profile |

$G$ | Temperature profile |

$I$, $J$ | Chemical species |

i, j | Concentration |

$K$ | Strength of homogenous reaction |

${K}_{s}$ | Strength of heterogeneous reaction |

$k$ | Thermal conductivity (Wm^{−1}K^{−1}) |

$M$ | Magnetic parameter |

$\mathrm{N}{u}_{x}$ | Nusselt number |

$\mathrm{Pr}$ | Prandtl number |

$q$ | Heat flux (Wm^{−2}) |

${\mathrm{Re}}_{x}$ | Local Reynolds number |

$Sc$ | Schmidt number |

$S{h}_{x}$ | Sherwood number |

$T$ | Fluid temperature (K) |

${\mathrm{T}}_{w}$ | Surface temperature (K) |

${\mathrm{T}}_{\infty}$ | Temperature at infinity (K) |

$u$, $v$ | Velocity components (ms^{−1}) |

$x,y$ | Coordinates |

$\alpha $ | Thermal diffusivity (m^{2}s^{−1}) |

$\eta $ | Similarity variable |

$\mu $ | Dynamic viscosity (mPa) |

${\upsilon}_{f}$ | Kinematic viscosity (mPa) |

${\rho}_{f}$ | Density (Kgm^{−3}) |

${\lambda}_{1}$ | Relaxation time |

${\lambda}_{2}$ | Relaxation to retardation time |

${\lambda}_{3}$ | Retardation time |

$\zeta $ | Stretching rate |

$\kappa $ | Deborah number |

$\Omega $ | Thermal relaxation parameter |

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

$\varphi $ | Dimensional concentration profile |

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**Figure 1.**The combined $\hslash $-curves for ${F}^{\u2033}(0),{G}^{\prime}(0)\mathrm{and}{\varphi}^{\prime}(0)$.

**Figure 2.**Impact of $M$ on ${F}^{\prime}(\eta )$, when $\Omega =0.5,Sc=0.6,\mathrm{K}=0.8,\mathrm{Pr}=0.7,{\mathrm{K}}_{s}=0.9$.

**Figure 3.**Impact of $\mathrm{Pr}$ on $G(\eta )$, when $\Omega =0.5,Sc=0.6,\mathrm{K}=0.8,\mathrm{M}=0.1,{\mathrm{K}}_{s}=0.9$.

**Figure 4.**Impact of $\Omega $ on $G(\eta )$, when $Sc=0.6,\mathrm{Pr}=0.7,\mathrm{K}=0.8,\mathrm{M}=0.1,{\mathrm{K}}_{s}=0.9$.

**Figure 5.**Impact of $Sc$ on $\varphi (\eta )$, when $\Omega =0.1,\mathrm{Pr}=0.7,\mathrm{K}=0.8,\mathrm{M}=0.1,{\mathrm{K}}_{s}=0.9$.

**Figure 6.**Impact of $\mathrm{K}$ on $\varphi (\eta )$, when $\Omega =0.1,\mathrm{Pr}=0.7,Sc=0.6,\mathrm{M}=0.1,{\mathrm{K}}_{s}=0.9$.

**Figure 7.**Impact of ${K}_{s}$ on $\varphi (\eta )$, when $\mathrm{Pr}=0.7,\Omega =0.1,Sc=0.6,\mathrm{M}=0.1,\mathrm{K}=0.8$.

M | Ref. [47] | Present Results for Jeffrey Nanofluid | Ref. [47] | Present Results for Maxwell Nanofluid | Ref. [47] | Present Results for Oldroyd-B Nanofluid |
---|---|---|---|---|---|---|

1.0 | 1.210458 | 0.210462 | 1.504151 | 1.504153 | 1.071019 | 1.071022 |

2.0 | 1.431584 | 1.431587 | 1.804788 | 1.804791 | 1.248081 | 1.248084 |

Ω | Pr | Ref. [47] | Jeffrey Nanofluid | Ref. [47] | Maxwell Nanofluid | Ref. [47] | Oldroyd-B Nanofluid |
---|---|---|---|---|---|---|---|

1.0 | ---------- | 0.610394 | ---------- | 0.595298 | ---------- | 0.610846 | |

1.2 | ---------- | 0.607503 | ---------- | 0.593311 | ---------- | 0.607993 | |

6.0 | 0.418081 | 0.513786 | 0.421167 | 0.511247 | 0.426476 | 0.5154367 | |

7.0 | 0.439695 | 0.626865 | 0.441919 | 0.548966 | 0.447670 | 0.5477974 |

Sc | K | K_{s} | Jeffrey | Maxwell | Oldroyd-B |
---|---|---|---|---|---|

1.2 | −0.096477 | −0.095593 | −0.096771 | ||

1.5 | −0.096782 | −0.095890 | −0.097081 | ||

1.5 | −0.058030 | −0.047238 | −0.049135 | ||

1.7 | −0.056699 | −0.037230 | −0.039262 | ||

0.5 | −0.018933 | −0.037233 | 0.012399 | ||

0.8 | −0.160028 | 0.046603 | −0.125205 |

**Table 4.**Symmetry of HAM versus numerical solutions for ${F}^{\prime}(\eta )$, when $Sc=\mathrm{Pr}={\mathrm{K}}_{2}=1.0$, $\kappa ={\kappa}_{1}={\kappa}_{2}={\lambda}_{2}=\mathrm{M}=\mathrm{K}=0.1$.

$\mathit{\eta}$ | HAM ${\mathit{F}}^{\mathbf{\prime}}\mathbf{(}\mathit{\eta}\mathbf{)}$ | Numerical ${\mathit{F}}^{\mathbf{\prime}}\mathbf{(}\mathit{\eta}\mathbf{)}$ | Absolute Error AE |
---|---|---|---|

0.0 | 3.33067 × 10^{−}^{16} | 0.000000 | 3.33067 × 10^{−}^{16} |

0.1 | 0.0950433 | 0.093334 | 0.002876 |

0.2 | 0.180827 | 0.173972 | 0.009647 |

0.3 | 0.258260 | 0.242791 | 0.016488 |

0.4 | 0.328165 | 0.300578 | 0.020276 |

0.5 | 0.391281 | 0.348033 | 0.019089 |

0.6 | 0.448277 | 0.385777 | 0.011828 |

0.7 | 0.499753 | 0.414357 | 0.085396 |

0.8 | 0.546251 | 0.434262 | 0.111989 |

0.9 | 0.588258 | 0.445923 | 0.142335 |

1.0 | 0.626213 | 0.449730 | 0.176483 |

**Table 5.**Symmetry of HAM versus numerical solutions for $G(\eta )$, when $Sc=\mathrm{Pr}={\mathrm{K}}_{s}=1.0$, $\kappa ={\kappa}_{1}={\kappa}_{2}={\lambda}_{3}=\mathrm{M}=\mathrm{K}=0.1$.

$\mathit{\eta}$ | HAM $\mathit{G}(\mathit{\eta})$ | Numerical $\mathit{G}(\mathit{\eta})$ | Absolute Error AE |
---|---|---|---|

0.0 | 1.000000 | 1.00000 | 0.000000 |

0.1 | 0.915165 | 0.887477 | 0.002876 |

0.2 | 0.835725 | 0.775917 | 0.009647 |

0.3 | 0.761845 | 0.666195 | 0.016488 |

0.4 | 0.693506 | 0.559077 | 0.020276 |

0.5 | 0.630563 | 0.45522 | 0.019089 |

0.6 | 0.572791 | 0.355165 | 0.011828 |

0.7 | 0.519912 | 0.259342 | 0.250570 |

0.8 | 0.471618 | 0.168073 | 0.303545 |

0.9 | 0.427594 | 0.0815811 | 0.346013 |

1.0 | 0.387518 | 5.60459 × 10^{−9} | 0.387517 |

**Table 6.**Symmetry of HAM versus numerical solutions for $\varphi (\eta )$, when $Sc=\mathrm{Pr}={\mathrm{K}}_{s}=1.0$, $\kappa ={\kappa}_{1}={\kappa}_{2}={\lambda}_{2}=\mathrm{M}=\mathrm{K}=0.1$.

$\mathit{\eta}$ | HAM $\mathit{\varphi}(\mathit{\eta})$ | Numerical $\mathit{\varphi}(\mathit{\eta})$ | Absolute Error AE |
---|---|---|---|

0.0 | 0.396762 | 0.404080 | 0.007318 |

0.1 | 0.429884 | 0.436425 | 0.006841 |

0.2 | 0.464667 | 0.468606 | 0.003939 |

0.3 | 0.499830 | 0.500349 | 0.000519 |

0.4 | 0.534490 | 0.531418 | 0.003072 |

0.5 | 0.568056 | 0.561618 | 0.006438 |

0.6 | 0.600146 | 0.590789 | 0.009357 |

0.7 | 0.630533 | 0.618810 | 0.011723 |

0.8 | 0.659102 | 0.645589 | 0.013513 |

0.9 | 0.685811 | 0.671065 | 0.014746 |

1.0 | 0.710675 | 0.695202 | 0.015473 |

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## Share and Cite

**MDPI and ACS Style**

Saeed, A.; Islam, S.; Dawar, A.; Shah, Z.; Kumam, P.; Khan, W.
Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction. *Symmetry* **2019**, *11*, 439.
https://doi.org/10.3390/sym11030439

**AMA Style**

Saeed A, Islam S, Dawar A, Shah Z, Kumam P, Khan W.
Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction. *Symmetry*. 2019; 11(3):439.
https://doi.org/10.3390/sym11030439

**Chicago/Turabian Style**

Saeed, Anwar, Saeed Islam, Abdullah Dawar, Zahir Shah, Poom Kumam, and Waris Khan.
2019. "Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction" *Symmetry* 11, no. 3: 439.
https://doi.org/10.3390/sym11030439