# On Unsteady Three-Dimensional Axisymmetric MHD Nanofluid Flow with Entropy Generation and Thermo-Diffusion Effects on a Non-Linear Stretching Sheet

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

- (i)
- Skin-fraction coefficient:$$\begin{array}{c}{C}_{f}R{e}^{1/2}={f}^{\prime \prime}\left(0\right),\hfill \end{array}$$
- (ii)
- Nusselt number:$$\begin{array}{c}\hfill Nu=\frac{r{h}_{w}}{{k}_{f}({T}_{w}-{T}_{\infty})},\end{array}$$$$\begin{array}{c}\hfill {h}_{w}=-\left[{k}_{f}+\frac{16{\sigma}^{\ast}{T}_{\infty}^{3}}{3{K}^{\ast}}\right]{\left(\frac{\partial T}{\partial z}\right)}_{z=0}.\end{array}$$$$\begin{array}{c}\hfill -\left(1+Nr\right){\theta}^{\prime}\left(0\right)=R{e}^{-1/2}N{u}_{r}\end{array}$$
- (iii)
- The Sherwood number for solute concentration equation is:$$\begin{array}{cc}\hfill & S{h}_{r}=\frac{r{h}_{m}}{{D}_{s}({C}_{w}-{C}_{\infty})},\hfill \end{array}$$$$\begin{array}{cc}\hfill & {h}_{m}=-{D}_{s}{\left(\frac{\partial C}{\partial z}\right)}_{z=0},\hfill \end{array}$$$$\begin{array}{c}\hfill -{S}^{\prime}\left(0\right)=R{e}_{r}^{-(1/2)}S{h}_{r}\end{array}$$

## 3. Entropy Generation Analysis

## 4. Method of Solution

**D**, such as:

**D**in order to approximate derivatives of unknown functions in Equations(32)–(35) with Equation (36), which yields:

## 5. Results and Discussion

## 6. Conclusions

- The heat transfer rate increases with increasing sheet stretching.
- An increase in the Reynolds number and the Brinkman number corresponds to a significant increase in the entropy generation number. Therefore, it can be ascertained that the entropy generation number is highly affected by viscous dissipation when the nanofluid flow has a large Reynolds number.
- An increase in the Biot and Hartmann numbers corresponds to a significant increase in the entropy generation number in the vicinity of the sheet surface. The significance of the Biot and Hartmann numbers gradually fades with distance from the sheet.
- The entropy generation rate can be minimized by controlling the physical parameters.
- The number of collocation points has a significant influence on the accuracy of the solutions.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

$r,\theta ,z$ | cylindrical polar coordinate axes |

${B}_{0}$ | magnetic field strength (kg·s${}^{-2}$A${}^{-1}$) |

u | velocity in radial direction (m·s${}^{-1}$) |

$n,\lambda $ | constants |

w | velocity in axial direction (m·s${}^{-1}$) |

${\nu}_{f}$ | kinematic viscosity (m${}^{2}$s${}^{-1}$) |

T | temperature variable (K) |

$\sigma $ | electrical conductivity (${\Omega}^{-1}$m${}^{-1}$) |

${T}_{w}$ | temperature of fluid at sheet (K) |

${\rho}_{f}$ | density of fluid (kg·m${}^{-3}$) |

${T}_{\infty}$ | ambient temperature of fluid (K) |

${\alpha}_{f}$ | thermal diffusion of fluid (m${}^{2}$s${}^{-1}$) |

C | solute concentration of fluid (kg·m${}^{-3}$) |

${C}_{W}$ | solute concentration at wall (kg·m${}^{-3}$) |

${C}_{\infty}$ | solute concentration far away from the disk (kg·m${}^{-3}$) |

$\psi $ | nanoparticle concentration |

${\psi}_{W}$ | nanoparticle concentration at wall |

${\psi}_{\infty}$ | nanoparticle concentration far away from the disk |

$\tau $ | ratio of nanoparticle heat capacity |

${D}_{B}$ | Brownian mention coefficient (kg·m${}^{-1}$s${}^{-1}$) |

${D}_{T}$ | Thermophoretic diffusion coefficient (kg·m${}^{-1}$s${}^{-1}$K${}^{-1}$) |

${D}_{TC}$ | Dufour diffusion coefficient |

${c}_{p}$ | specific heat (m${}^{2}$s${}^{-2}$K${}^{-1}$) |

${k}_{f}$ | thermal conductivity (W·m${}^{-1}$K${}^{-1}$) |

${q}_{r}$ | radiative heat flux (kg·m${}^{-2}$) |

${D}_{s}$ | solute diffusion coefficient |

${D}_{CT}$ | Soret diffusion coefficient |

R | chemical reaction parameter |

${\left(\rho {c}_{p}\right)}_{s}$ | heat capacity of the nanoparticle |

${\left(\rho {c}_{p}\right)}_{f}$ | heat capacity of fluid |

${\sigma}^{\ast}$ | Stefan–Boltzmann constant |

${K}^{\ast}$ | mean absorption coefficient |

${R}_{0},a$ | constants |

${k}_{0},{b}_{0}$ | constants |

${h}_{f}$ | heat transfer coefficient (W·m${}^{-2}$ K${}^{-1}$) |

$Bi$ | Biot number |

$\eta $ | dimensionless variable |

$\theta $ | dimensionless temperature |

S | dimensionless solute concentration |

$\varphi $ | dimensionless nanoparticle concentration |

f | dimensionless velocity |

A | unsteadiness parameter |

$Ha$ | Hartmann number |

$Nr$ | thermal radiation parameter |

$Pr$ | Prandtl number |

$Nb$ | Brownian mention parameter |

$Nt$ | Thermophoresis parameter |

$Nd$ | Dufour parameter |

$Sc$ | Schmidt number |

$Ld$ | Soret parameter |

${C}_{f}$ | skin friction coefficient |

$N{u}_{r}$ | Nusselt number |

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

${u}_{w}\left(r\right)$ | velocity of the stretching sheet |

$Re$ | Reynolds number |

${h}_{w}$ | heat flux (W·m${}^{-2}$) |

${h}_{m}$ | mass flux (kg·m${}^{-2}$s${}^{-1}$) |

${E}_{gen}$ | volumetric entropy generation per unit length (W·m${}^{-3}$ K${}^{-1}$) |

${E}_{0}$ | dimensionless entropy generation rate |

$Br$ | Brinkman number |

$\Omega $ | dimensionless parameter |

$Ha$ | Hartmann number |

$\mathsf{\Sigma}$ | diffusive constant parameter |

$\Delta T$ | difference between (${T}_{W}-{T}_{\infty}$)(K) |

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**Figure 2.**Effects of the number of collocation points number on the residual error of the velocity profile ${\left|\right|Res\left(f\right)\left|\right|}_{\infty}$ when $n=3,A=0.3,Pr=7,Ha=0.5,Nr=0.2,Nb=0.5,Nt=0.5,Nd=0.02,$ $Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2$.

**Figure 3.**Effects of the number of collocation points number on the residual error of the temperature profile ${\left|\right|Res\left(\theta \right)\left|\right|}_{\infty}$ when $n=3,A=0.3,Pr=7,Ha=0.5,Nr=0.2,Nb=0.5,Nt=0.5,Nd=0.02$, $Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2$.

**Figure 4.**Effects of the number of collocation points number on the residual error of the solute concentration profile ${\left|\right|Res\left(S\right)\left|\right|}_{\infty}$ when $n=3,A=0.3,Pr=7,Ha=0.5,Nr=0.2,Nb=0.5$, $Nt=0.5,Nd=0.02,Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2$.

**Figure 5.**Effects of the number of collocation points number on the residual error of the nanoparticle concentration profile ${\left|\right|Res\left(\varphi \right)\left|\right|}_{\infty}$ when $n=3,A=0.3,Pr=7,Ha=0.5,Nr=0.2,Nb=0.5$, $Nt=0.5,Nd=0.02,Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2$.

**Figure 6.**Effects of the stretching parameter n on ${f}^{\prime}\left(\eta \right)$ when $A=0.3,Ha=0.5,Nr=0.2$, $Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 7.**Effects of the stretching parameter n on $\theta \left(\eta \right)$ where $A=0.3,Ha=0.5,Nr=0.2,Pr=10$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 8.**Effects of the stretching parameter n on $S\left(\eta \right)$ for $A=0.3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 9.**Effects of the unsteadiness parameter (A) on $\theta \left(\eta \right)$ for $n=3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 10.**Effects of the unsteadiness parameter (A) on $S\left(\eta \right)$ when $n=3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 11.**Effects of the Hartmann number ($Ha$) on ${f}^{\prime}\left(\eta \right)$ for $n=3,A=0.3,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 12.**Effects of the Hartmann number ($Ha$) on $\theta \left(\eta \right)$ when $n=3,A=0.3,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 13.**Effects of the thermal radiation parameter ($Nr$) on $\theta \left(\eta \right)$ where $n=3,A=0.3,Ha=0.5$, $Pr=7,Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 14.**Effects of the Prandtl number ($Pr$) on $\theta \left(\eta \right)$ for $n=3,A=0.3,Ha=0.5,Nr=0.2$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,{\beta}_{i}=0.2.$

**Figure 15.**Effects of the thermophoresis parameter ($Nt$) on $\theta \left(\eta \right)$ for $n=3,A=0.3,Ha=0.5,Nr=0.2$, $Pr=7,Nb=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 16.**Effects of the Schmidt number ($Sc$) on $S\left(\eta \right)$ for $n=3,A=0.3,Ha=0.5,Nr=0.2$, $Pr=7,Nb=0.5,Nt=0.5,{R}_{0}=0.3,Bi=0.2.$

**Figure 17.**Effects of the Biot number ($Bi$) on $\theta \left(\eta \right)$ for $n=3,A=0.3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02.$

**Figure 18.**Effects of the stretching parameter (n) on $\varphi \left(\eta \right)$ when $A=0.3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 19.**Effects of the unsteadiness parameter (A) on $\varphi \left(\eta \right)$ for $n=3,Ha=0.5,Nr=0.2,Pr=7$, $Nb=0.5,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 20.**Effects of the thermophoresis parameter ($Nt$) on $\varphi \left(\eta \right)$ for $n=3,A=0.3,Ha=0.5$, $Nr=0.2,Pr=7,Nb=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 21.**Effects of the Brownian motion parameter ($Nb$) on $\varphi \left(\eta \right)$ when $n=3,A=0.3$, $Ha=0.5$, $Nr=0.2,Pr=7,Nt=0.5,Nd=0.02,Sc=7,{R}_{0}=0.3,Ld=0.02,Bi=0.2.$

**Figure 22.**Effects of the Schmidt number ($Sc$) on $\varphi \left(\eta \right)$ for $n=3,A=0.3,Ha=0.5,Nr=0.2$, $Pr=7,Nb=0.5,Nt=0.5,{R}_{0}=0.3,Bi=0.2.$

**Figure 23.**Effects of the Reynolds number $Re$ on the entropy generation number ${N}_{G}$ when $Nr=0.2$, $Br=1,\Omega =1,Ha=0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Sigma}=0.5$.

**Figure 24.**Effects of Brinkman number $Br$ on the entropy generation number ${N}_{G}$ when $Re=2$, $Nr=0.2$, $\Omega =1,Ha=0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Sigma}=0.5$.

**Figure 25.**Effects of Hartmann number $Ha$ on the entropy generation number ${N}_{G}$ when $Re=2$, $Nr=0.2,Br=1,\Omega =1\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Sigma}=0.5$.

**Figure 26.**Effects of the Brinkman group parameter $Br{\Omega}^{-1}$ on the entropy generation number ${N}_{G}$ when $Re=2,Nr=0.2,Br=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Sigma}=0.5$.

**Figure 27.**Effects of Biot number $Bi$ on the entropy generation number ${N}_{G}$ when $Re=2$, $Nr=0.2$, $Br=1,Ha=0.5,\Omega =1\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Sigma}=0.5$.

**Figure 28.**Effect of the magnetic field $Ha$ and the stretching sheet parameter n on the skin friction $-{f}^{\prime \prime}\left(0\right)$ when $A=0.3,Pr=7,Nr=0.2,Nb=0.5,Nt=0.5,Nd=0.02,Ld=0.02,Sc=7$, ${R}_{0}=0.3$, $Bi=0.2$.

**Figure 29.**Effect of the thermophoresis number ($Nt$) and the stretching sheet parameter (n) on the Nusselt number ($-{\theta}^{\prime}\left(0\right)$) when $A=0.3,Ha=0.5,Pr=7,Nr=0.2,Nb=0.5,Nd=0.02$, $Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2$.

**Figure 30.**Effect of the Schmidt number $Sc$ and the stretching sheet parameter n on the Sherwood number $-{S}^{\prime}\left(0\right)$ when $A=0.3,Ha=0.5,Pr=7,Nr=0.2,Nb=0.5,Nd=0.02,Ld=0.02,Nt=0.5$, ${R}_{0}=0.3,Bi=0.2$.

**Figure 31.**Effect of the Prandtl number $Re$ and the thermal radiation parameter $Nr$ on the Nusselt number $-{\theta}^{\prime}\left(0\right)$ when $n=3,A=0.3,Ha=0.5,Sc=7,Nb=0.5,Nd=0.02,Ld=0.02,Nt=0.5$, ${R}_{0}=0.3,Bi=0.2$.

**Table 1.**Current Nusselt number $-{\theta}^{\prime}\left(0\right)$ compared with Mustafa et al. [27].

n | $\mathit{N}\mathit{t}$ | $\mathit{S}\mathit{c}$ | $\mathit{P}\mathit{r}$ | Mustafa et al. [27] | Present Results |
---|---|---|---|---|---|

0.5 | 0.1 | 20 | 5 | $1.9112911$ | $1.91068095$ |

0.5 | $1.2170065$ | $1.21659065$ | |||

0.7 | $0.9815765$ | $0.98122822$ | |||

1.0 | 0.5 | 5 | 5 | $1.6914582$ | $1.69104675$ |

10 | $1.4740787$ | $1.47375172$ | |||

20 | $1.2861370$ | $1.28590965$ | |||

2.5 | 0.5 | 20 | 0.7 | $0.6619164$ | $0.66986678$ |

5 | $1.4784288$ | $1.47847763$ | |||

7 | $1.5758736$ | $1.57604858$ |

**Table 2.**Skin friction coefficient, heat transfer coefficient and mass transfer coefficient for $Nb=0.5$, $Nt=0.5,Nd=0.02,Ld=0.02,Sc=7,{R}_{0}=0.3,Bi=0.2.$

n | A | $\mathit{H}\mathit{a}$ | $\mathit{N}\mathit{r}$ | $\mathit{P}\mathit{r}$ | $-{\mathit{f}}^{\prime \prime}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{S}}^{\prime}\left(0\right)$ |
---|---|---|---|---|---|---|---|

1 | $1.43922866$ | $-0.16417529$ | $-2.89107257$ | ||||

2 | 0.3 | 0.5 | 0.2 | 7 | $1.70034047$ | $-0.16661752$ | $-3.16939593$ |

4 | $2.12897896$ | $-0.16969131$ | $-3.66539915$ | ||||

−0.5 | $1.77701012$ | $-0.17085800$ | $-3.72686482$ | ||||

3 | 0 | 0.5 | 0.2 | 7 | $1.87092985$ | $-0.16944779$ | $-3.54329048$ |

0.5 | $1.96353732$ | $-0.16749470$ | $-3.34434681$ | ||||

1.5 | $2.17151786$ | $-0.16755905$ | $-3.37606776$ | ||||

3 | 0.3 | 2.5 | 0.2 | 7 | $2.39114443$ | $-0.16678981$ | $-3.33136800$ |

5 | $2.86708560$ | $-0.16495200$ | $-3.23526081$ | ||||

1 | $1.92657387$ | $-0.16615306$ | $-3.42533294$ | ||||

3 | 0.3 | 0.5 | 1.5 | 7 | $1.92657387$ | $-0.16464481$ | $-3.42514164$ |

2 | $1.92657387$ | $-0.16310964$ | $-3.42504767$ | ||||

4 | $1.92657387$ | $-0.16585371$ | $-3.42553409$ | ||||

3 | 0.3 | 0.5 | 0.2 | 5 | $1.92657387$ | $-0.16708464$ | $-3.42554096$ |

9 | $1.92657387$ | $-0.16894896$ | $-3.42668462$ |

Biot Number | Maximum Temperature | Change in Maximum Temperature |
---|---|---|

0.1 | 0.1264 | |

0.9 | 0.3401 | 169.07 |

2 | 0.5218 | 53.43 |

5 | 0.7377 | 41.38 |

10 | 0.853 | 15.63 |

50 | 0.9678 | 13.46 |

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Almakki, M.; Dey, S.; Mondal, S.; Sibanda, P. On Unsteady Three-Dimensional Axisymmetric MHD Nanofluid Flow with Entropy Generation and Thermo-Diffusion Effects on a Non-Linear Stretching Sheet. *Entropy* **2017**, *19*, 168.
https://doi.org/10.3390/e19070168

**AMA Style**

Almakki M, Dey S, Mondal S, Sibanda P. On Unsteady Three-Dimensional Axisymmetric MHD Nanofluid Flow with Entropy Generation and Thermo-Diffusion Effects on a Non-Linear Stretching Sheet. *Entropy*. 2017; 19(7):168.
https://doi.org/10.3390/e19070168

**Chicago/Turabian Style**

Almakki, Mohammed, Sharadia Dey, Sabyasachi Mondal, and Precious Sibanda. 2017. "On Unsteady Three-Dimensional Axisymmetric MHD Nanofluid Flow with Entropy Generation and Thermo-Diffusion Effects on a Non-Linear Stretching Sheet" *Entropy* 19, no. 7: 168.
https://doi.org/10.3390/e19070168