# A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O–CuO for the enhancement of the solidification rate. The finite element method is engaged to obtain the numerical solution of the problem. Sheikholeslami [5] pondered over the influence of radiation and magnetohydrodynamic on the Al

_{2}O

_{3}–H

_{2}O mixture past a spongy semi-annulus. The numerical solution of the problem is witnessed by employing the Control Volume Finite Element Method (CVFEM). The flow of Casson nanofluid with inserted multi-walled carbon nanotubes past a swirling cylinder was deliberated by Ramzan et al. [6] using bvp4c MATLAB software. The said problem is pondered with impacts of entropy generation and melting heat transfer. The flow of micropolar nanofluid with binary chemical reaction, double stratification, and activation energy is also studied by Ramzan et al. [7]. The flow of viscoelastic nanofluid with analysis of entropy generation past an exponential stretched surface was discussed by Suleman et al. [8]. Farooq et al. [9] deliberated the flow of Newtonian fluid with the amalgamation of nanoparticles by utilizing the BVPh 2.0 technique and many therein [10,11,12,13,14,15].

_{3}O

_{4}, Cu, Al

_{2}O

_{3}and TiO

_{2}and water as the base fluid past an exponential stretched surface was discussed by Jusoh et al. [23]. Some recent investigations highlighting the importance of MHD may also be found in References [24,25,26,27].

## 2. Mathematical Modeling

_{2}O) are given in Table 1.

_{1}and B

_{1}are equivalent. From this assumption, it is inferring that D

_{A}and D

_{B}(diffusion coefficients) are equal i.e., δ = 1, and on account of this supposition, we have

## 3. Numerical Scheme

## 4. Results and Discussion

^{*}and M on the temperature field. It is clearly perceived that temperature enhances when both K

^{*}and M increase. The decrease in the mean absorption coefficient represents an enriched heat transfer rate and ultimately temperature is enhanced. Similarly, a stronger Lorentz force hinders the movement of the fluid, thus causing more collision between the molecules of the fluid that turns into the improved temperature. In Figure 10 and Figure 11, the behavior of the concentration profile versus h-h reactions is depicted. The concentration diminishes for growing values of h-h reactions.

_{r}and radiation parameter K

^{*}.

## 5. Final Comments

- The temperature profile is a growing function of radiation and magnetic parameters.
- For larger values of the curvature parameter, augmented velocity is observed.
- The concentration of the fluid decreases for growing values of homogeneous–heterogeneous reactions.
- For escalated values of the magnetic parameter, velocity and temperature distributions show the opposite trend.
- The skin friction and local Nusselt number show opposite behavior for curvature and nonlinearity parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

U | along x-axis fluid velocity [m/s] |

V | along r-axis fluid velocity [m/s] |

U_{w} | Stretching velocity [m/s] |

U_{e} | Free stream velocity [m/s] |

$M$ | Magnetic parameter |

${\rho}_{f}$, ${\rho}_{s}$ | Density of fluid and solid particle respectively [kg/m^{3}] |

$\varphi $ | nanofluid volume fraction |

${\tau}_{w}$ | surface shear stress [N/m^{2}] |

T_{∞} | Ambient temperature [K] |

$\lambda $ | conjugate parameter |

${\sigma}_{nf}$ | electric conductivity of fluid and nanofluid respectively [S/m] |

$\gamma $ | curvature parameter |

k_{f} | thermal conductivities of fluid [$\mathrm{J}/\mathrm{m}\mathrm{K}{\mathrm{s}}^{n-2}$] |

k_{s} | thermal conductivities of nanomaterial [$\mathrm{J}/\mathrm{mK}{\mathrm{s}}^{n-2}$] |

$\delta $ | ratio of mass diffusion coefficients |

q_{r} | radiative heat flux [$\mathrm{kg}/{\mathrm{m}}^{2}$] |

h_{f} | convective heat transfer coefficient |

C_{f} | Skin friction coefficient |

Nu_{x} | Nusselt number |

T | Temperature [K] |

T_{f} | convective fluid temperature [K] |

h_{s} | heat transfer coefficient |

${\mu}_{nf}$ | Nanofluid dynamic viscosity [$\mathrm{kg}/\mathrm{m}{\mathrm{s}}^{n-2}$] |

Re_{x} | local Reynolds number |

${k}^{*}$ | mean absorption coefficient |

${\sigma}^{*}$ | Stefan-Boltzmann constant |

D_{A}, D_{B} | diffusion coefficients $\left[{\mathrm{m}}^{2}/\text{}\mathrm{s}\right]$ |

q_{w} | surface heat flux ${[\mathrm{W}/\mathrm{m}}^{2}]$ |

S_{c} | Schmidt number |

${k}_{c},\hspace{0.17em}{k}_{s}$ | Rate constants |

K^{*} | Radiation parameter |

${\mu}_{f}$ | Fluid dynamic viscosity [$\mathrm{kg}/\mathrm{m}{\mathrm{s}}^{n-2}$] |

${\alpha}_{nf}$ | Nanofluid thermal diffusivity $\left[{\mathrm{m}}^{2}/{\mathrm{s}}^{n-2}\right]$ |

k_{1} | Strength of homogeneous reaction |

${\nu}_{nf}$ | nanofluid kinematic viscosity $\left[{\mathrm{m}}^{2}/\text{}\mathrm{s}\right]$ |

k_{2} | strength of heterogeneous reaction |

R | radius of cylinder [m] |

B | Magnetic field strength [A/m] |

N_{r} | temperature ratio parameter |

A_{1}, B_{1} | concentrations of chemical species $a,\hspace{0.17em}b$ |

$n$ | Nonlinearity exponent |

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Physical Properties | Water | Ag |
---|---|---|

C_{p} (J/kg K) | 4179 | 235.0 |

$\rho $ (kg/m^{3}) | 997.1 | 10,500.0 |

K (W/mK) | 0.61300 | 429.0 |

**Table 2.**Nusselt number (${\mathrm{Re}}_{x}{}^{-1/2}N{u}_{x}$) for numerous estimates of $\gamma $ and Pr with $M=0$, $\varphi $ = 0.0, $\lambda $ = 0.0.

$\mathit{\gamma}$ | Pr | $\mathbf{R}{\mathbf{e}}_{\mathit{x}}{}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | |
---|---|---|---|

Qasim et al. [38] | Present Result | ||

0.0 | 0.72 | 1.23664 | 1.236651 |

1.0 | 1.00000 | 1.000000 | |

6.7 | 0.33330 | 0.333310 | |

10 | 0.26876 | 0.268770 | |

1.0 | 0.72 | 0.87018 | 0.870190 |

1.0 | 0.74406 | 0.744070 | |

6.7 | 0.29661 | 0.296620 | |

10 | 0.24217 | 0.242180 |

**Table 3.**Numerical values of $-{\mathrm{Re}}_{x}{}^{1/2}{C}_{f}$ and ${\mathrm{Re}}_{x}{}^{-1/2}N{u}_{x}$ for Ag–water with $\mathrm{Pr}=6.2$.

n | $\mathit{\gamma}$ | $\mathit{\varphi}$ | M | N_{r} | K^{*} | $-\mathbf{R}{\mathbf{e}}_{\mathit{x}}{}^{1/2}{\mathit{C}}_{\mathit{f}}$ | $\mathbf{R}{\mathbf{e}}_{\mathit{x}}{}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ |
---|---|---|---|---|---|---|---|

1.0 | 0.1 | 0.1 | 1.0 | 1.2 | 1.0 | 1.92700 | 0.32149 |

2.0 | 0.1 | 0.1 | 1.0 | 1.2 | 1.0 | 2.45720 | 0.30963 |

3.0 | 0.1 | 0.1 | 1.0 | 1.2 | 1.0 | 2.89050 | 0.30305 |

2.0 | 1.0 | 0.1 | 1.0 | 1.2 | 1.0 | 2.81410 | 0.28200 |

2.0 | 2.0 | 0.1 | 1.0 | 1.2 | 1.0 | 3.18730 | 0.26410 |

2.0 | 3.0 | 0.1 | 1.0 | 1.2 | 1.0 | 3.54330 | 0.25369 |

2.0 | 1.0 | 0.1 | 1.0 | 1.2 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 0.2 | 1.0 | 1.2 | 1.0 | 3.42830 | 0.31250 |

2.0 | 1.0 | 0.3 | 1.0 | 1.2 | 1.0 | 4.04270 | 0.34319 |

2.0 | 1.0 | 0.1 | 1.0 | 1.2 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 2.0 | 1.2 | 1.0 | 2.99960 | 0.28387 |

2.0 | 1.0 | 1.0 | 3.0 | 1.2 | 1.0 | 3.17070 | 0.28553 |

2.0 | 1.0 | 1.0 | 1.0 | 0.1 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 1.0 | 0.7 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 1.0 | 1.2 | 1.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 1.0 | 1.2 | 2.0 | 2.81410 | 0.28200 |

2.0 | 1.0 | 1.0 | 1.0 | 1.2 | 3.0 | 2.81410 | 0.28200 |

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**MDPI and ACS Style**

Suleman, M.; Ramzan, M.; Ahmad, S.; Lu, D.; Muhammad, T.; Chung, J.D.
A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder. *Symmetry* **2019**, *11*, 295.
https://doi.org/10.3390/sym11020295

**AMA Style**

Suleman M, Ramzan M, Ahmad S, Lu D, Muhammad T, Chung JD.
A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder. *Symmetry*. 2019; 11(2):295.
https://doi.org/10.3390/sym11020295

**Chicago/Turabian Style**

Suleman, Muhammad, Muhammad Ramzan, Shafiq Ahmad, Dianchen Lu, Taseer Muhammad, and Jae Dong Chung.
2019. "A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder" *Symmetry* 11, no. 2: 295.
https://doi.org/10.3390/sym11020295