# Radiative MHD Nanofluid Flow over a Moving Thin Needle with Entropy Generation in a Porous Medium with Dust Particles and Hall Current

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## Abstract

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## 1. Introduction

_{3}O

_{4}-H

_{2}O solution in a permeable cavity under the influence of a variable magnetic field using Control Volume Finite Element Method (CVFEM). They observed the highest heat transfer rate in the case of the platelet-shaped nanoparticles. It was further witnessed by them that the velocity of the nanofluid was on the decline once the strong magnetic field was applied. Entropy optimization for the flow of Carreau nanofluid flow with cubic auto-catalysis chemical reaction was studied by Khan et al. [22] analytically. They noticed that the sturdier magnetic field boosted the entropy generation. Sheikholeslami [23] found a numerical solution of nanofluid flow under the influence of the magnetic field in a permeable medium via the CVFEM scheme. He analyzed the influences of entropy and exergy on the presented model and reported that entropy loss enhanced in attendance of stronger magnetic field. Khan et al. [24] examined the numerical solution of 3D cross nanofluid with activation energy and binary chemical reaction with zero mass flux and convective boundary conditions. They noticed that higher estimates of activation energy boosted the concentration of the cross nanofluid. Hosseini and Sheikholeslami [25] analyzed the thermal competence of a convective nanofluid flow with entropy generation inside a microchannel under the influence of the magnetic field. Between two phases, non-equilibrium condition for a permeable media is engaged. They noticed that the entropy generation enhanced with an increase in fluid friction irreversibility. Some recent studies have also highlighted the concept of carbon nanotubes [26,27,28,29,30,31,32,33] and many therein.

## 2. Mathematical Modeling

_{2}O-CNTs-based nanofluid flow with Hall current over a moving slender needle having speed ${u}_{w}$ and radius “a” (Figure 1). The speed of fluid far away from the surface is taken as ${u}_{\infty}.$ The cylindrical coordinates (x, r) are taken in such a way that $x-$ is along the axis of the needle and $r-$ normal to the axis. The flow containing dust particles is generated in a non-Darcy absorbent media. The associated impacts affecting the flow in the heat equation are viscous dissipation and nonlinear thermal radiation. Furthermore, ${T}_{w}$ and ${T}_{\infty}$ are the constant temperatures at the wall and far off from the wall with ${T}_{\infty}>{T}_{w}.$ A magnetic field with magnetic strength ${B}_{0}$ is applied with the low Reynold number assumption [41], which eventually results in the induced magnetic field to be neglected. Two types of the equations, i.e., fluid phase and particle phase, comprising the envisioned mathematical model fulfilling laws of conservation are also laid down as given below:

- Continuity equation:Fluid phase$$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(rv\right)}{\partial r}=0,$$
- Momentum equation:Fluid phase$$\begin{array}{c}\left(1-{\varphi}_{d}\right)\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}\right)=\left(1-{\varphi}_{d}\right)\left(\frac{{\mu}_{nf}}{{\rho}_{nf}}\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right)-\frac{{\nu}_{nf}}{{k}^{*}}u\\ -\frac{{C}_{b}}{x\sqrt{{k}^{*}}}{u}^{2}+\frac{KN}{{\rho}_{nf}}\left({u}_{p}-u\right)-\frac{\sigma {B}_{0}{}^{2}}{{\rho}_{nf}\left(1+{m}^{2}\right)}u,\end{array}$$
- Continuity equation:Particle phase$$\frac{\partial \left(r{u}_{p}\right)}{\partial x}+\frac{\partial \left(r{v}_{p}\right)}{\partial r}=0,$$
- Momentum equation:Particle phase$${u}_{p}\frac{\partial {u}_{p}}{\partial x}+{v}_{p}\frac{\partial {u}_{p}}{\partial r}=\frac{K}{{m}_{d}}\left(u-{u}_{p}\right),$$
- Energy equation:Fluid phase$$\begin{array}{c}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{{k}_{nf}}{{\left(\rho {C}_{p}\right)}_{nf}}\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right)-\frac{1}{{\left(\rho {C}_{p}\right)}_{nf}}\frac{\partial {q}_{r}}{\partial r}+\frac{{\mu}_{nf}}{{\left(\rho {C}_{p}\right)}_{nf}}{\left(\frac{\partial u}{\partial r}\right)}^{2}\\ +\frac{{\mu}_{nf}}{{k}^{*}{\left(\rho {C}_{p}\right)}_{nf}}{u}^{2}+\frac{{N}_{1}}{{\tau}_{\upsilon}{\left(\rho {C}_{p}\right)}_{nf}}{\left({u}_{p}-u\right)}^{2}+\frac{{N}_{1}{\left({C}_{p}\right)}_{nf}}{{\tau}_{T}{\left(\rho {C}_{p}\right)}_{nf}}\left({T}_{p}-T\right)\\ +\frac{\sigma {B}_{0}{}^{2}}{{\left(\rho {C}_{p}\right)}_{nf}\left(1+{m}^{2}\right)}{u}^{2},\end{array}$$
- Energy equation:Particle phase$${N}_{1}{c}_{m}\left({u}_{p}\frac{\partial {T}_{p}}{\partial x}+{v}_{p}\frac{\partial {T}_{p}}{\partial r}\right)=\frac{{N}_{1}{\left({C}_{p}\right)}_{nf}}{{\tau}_{T}}\left({T}_{p}-T\right),$$

## 3. Similarity Transformation

- Momentum equation:Fluid phase$$\begin{array}{l}\left(1-{\varphi}_{d}\right){\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\varphi \frac{{\rho}_{CNT}}{{\rho}_{bf}}\right)g{g}^{\u2033}+2\left({g}^{\u2033}+\xi {g}^{\u2034}\right)-\lambda {g}^{\prime}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\varphi \frac{{\rho}_{CNT}}{{\rho}_{bf}}\right){F}_{r}{{g}^{\prime}}^{2}+{\left(1-\varphi \right)}^{2.5}\alpha \beta \left({G}^{\prime}-{g}^{\prime}\right)\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-{\left(1-\varphi \right)}^{2.5}\frac{M}{1+{m}^{2}}g\prime =0,\end{array}$$
- Momentum equation:Particle phase$${G}^{\u2033}G+\beta \left({g}^{\prime}-G\prime \right)=0,$$
- Energy equation:Fluid phase$$\begin{array}{l}\frac{{k}_{nf}}{{k}_{bf}}\left(\xi {\theta}^{\u2033}+\theta \prime \right)+0.5Pr\left(1-\varphi +\varphi \frac{{\left(\rho {C}_{p}\right)}_{CNT}}{{\left(\rho {C}_{p}\right)}_{bf}}\right)g{\theta}^{\prime}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+\frac{4}{3{N}_{r}}{\left(1+\left({\theta}_{r}-1\right)\theta \right)}^{2}\{3\xi \left({\theta}_{r}-1\right)\theta {\prime}^{2}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+\left(1+\left({\theta}_{r}-1\right)\theta \right)\left(0.5{\theta}^{\prime}+\xi \theta \u2033\right)\}+\frac{4EcPr}{{\left(1-\varphi \right)}^{2.5}}\xi {g}^{\u20332}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+2\alpha \beta EcPr{\left({G}^{\prime}-{g}^{\prime}\right)}^{2}+2EcPr\left(\frac{\lambda}{{\left(1-\varphi \right)}^{2.5}}+\frac{M}{1+{m}^{2}}\right)g{\prime}^{2}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+0.5\alpha {\beta}_{T}Pr\left(1-\varphi +\varphi \frac{{\left({C}_{p}\right)}_{CNT}}{{\left({C}_{p}\right)}_{bf}}\right)\left({\theta}_{p}-\theta \right)=0,\end{array}$$
- Energy equation:Particle phase$$G{\theta}_{p}^{\prime}-\gamma {\beta}_{T}\left(1-\varphi +\varphi \frac{{\left({C}_{p}\right)}_{CNT}}{{\left({C}_{p}\right)}_{bf}}\right)\left({\theta}_{p}-\theta \right)=0,$$$$\begin{array}{c}g\left(a\right)=\frac{a}{2}\epsilon ,{g}^{\prime}\left(a\right)=\frac{\epsilon}{2},G\left(a\right)=\frac{a}{2}\epsilon ,{G}^{\prime}\left(a\right)=\frac{\epsilon}{2},\theta \left(a\right)=1,\\ {g}^{\prime}(\infty )\to \frac{1-\epsilon}{2},\theta (\infty )\to 0,{\theta}_{p}(\infty )\to 0.\end{array}$$

## 4. Nusselt Number and Skin Friction Coefficient

## 5. Entropy Generation

## 6. Numerical Scheme

## 7. Results and Discussion

_{d}= F

_{r}= λ = Ec = 0.1, α = m = γ = 1, β = β

_{T}= 0.5, M = 0.2, N

_{r}= 6, θ

_{r}= 1.1,) and $Pr=6.8.$ Figure 2a,b exemplify the impacts of needle’s size “$a$” on the nanofluid velocity and velocity of the dust phase, respectively. It was comprehended that velocities were declining functions of the needle size in the case of CNTs of both types. Physically speaking, both velocities were highly dependent on the size of the needle. Increasing the needle’s size lowered the velocities, which was obvious. An opposite trend was witnessed in the case of Figure 2c,d. It was witnessed that temperature was dominant in the case of SWCNTs as compared to MWCNTs. This was because MWCNTs have lower thermal conductivity than SWCNTs.

## 8. Final Remarks

_{2}O-CNTs dusty nanofluid solution over a thin needle was investigated numerically. The novelty impacts of nonlinear thermal radiation with other effects were accompanied by entropy analysis. The leading outcomes of the investigation are appended as follows:

- ⮚
- Bejan number increased for larger values of Darcy–Forchheimer number.
- ⮚
- Velocity was on the decline once the size of the needle and Darcy–Forchheimer parameter’s values were enhanced.
- ⮚
- Higher estimates of Hall current parameter escalated the velocity profiles for both CNTs.
- ⮚
- An upsurge in entropy generation and the Bejan number was witnessed versus the radiation parameter.
- ⮚
- Sturdier magnetic field diminished the velocity of the fluid.
- ⮚
- Skin friction coefficient declined for growing estimates of dust particles’ mass concentration.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$r$ | Coordinate measured in radial direction |

$\left(u,v\right)$ | Velocity components along $x$ and $r$ directions |

${\mu}_{nf}$ | Effective dynamic viscosity of nanofluid |

${\rho}_{nf}$ | Density of nanofluid |

${\upsilon}_{nf}$ | Kinematic viscosity of nanofluid |

${k}^{*}$ | Darcy-permeability of the porous medium |

${C}_{b}$ | Drag coefficient |

${\varphi}_{d}$ | Volume fraction of dust particles |

$K$ | Stokes resistance |

$N$ | Number density of dust particles |

$\sigma $ | Electric conductivity |

${B}_{0}$ | Applied magnetic field |

$m$ | Hall parameter |

m_{d} | Mass concentration of the dust particles |

k_{nf} | Effective thermal conductivity of the nanofluid |

(ρC_{p})_{nf} | Effective heat capacitance of the nanofluid |

N_{1} | Density of the particle phase |

τ_{v} | Relaxation time of dust particles |

τ_{T} | Thermal equilibrium time |

τ_{w} | Shear stress at the surface |

${\dot{S}}^{\u2034}{}_{GEN}$ | Entropy generation rate per unit volume |

$\left({u}_{p},{v}_{p}\right)$ | Velocity components of particle phase in x and r directions |

${c}_{m}$ | Specific heat of the dust particles |

${u}_{w}$ | Velocity of the moving needle |

${u}_{\infty}$ | Velocity outside the boundary layer |

$T$ | Dimensional temperature of the nanofluid |

${T}_{p}$ | Temperature of the dust particle |

${T}_{w}$ | Constant surface temperature of the thin needle |

${T}_{\infty}$ | Ambient temperature |

$\lambda $ | Porosity parameter |

${F}_{r}$ | Forchheimer parameter |

$\alpha $ | Dust particles mass concentration |

$\beta $ | Fluid particle interaction parameter for velocity |

$M$ | Magnetic field parameter |

$Pr$ | Prandtl number |

${N}_{r}$ | Nonlinear radiation parameter |

${\theta}_{r}$ | Temperature ratio parameter |

$Ec$ | Eckert number |

${\beta}_{T}$ | Fluid particle interaction parameter for temperature |

$\gamma $ | Ratio of specific heat |

${q}_{w}$ | Surface heat flux |

${N}_{S}$ | Entropy generation number |

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**Figure 2.**Impact of $a$ on (

**a**) nanofluid velocity, (

**b**) the velocity of the dust phase, (

**c**) nanofluid temperature, and (

**d**) temperature of the dust phase.

**Figure 4.**Impact of ${\theta}_{r}$ on (

**a**) nanofluid temperature and (

**b**) temperature of the dust phase.

**Figure 6.**Impact of $M$ on (

**a**) nanofluid velocity, (

**b**) the velocity of the dust phase, (

**c**) nanofluid temperature, and (

**d**) temperature of the dust phase.

**Table 1.**Properties of the nanofluid defined for the presented model [46].

Properties | Nano-Fluid |
---|---|

Density | ${\rho}_{nf}=\left(1-\varphi \right){\rho}_{bf}+\varphi {\rho}_{CNT}$ |

Heat capacity | ${\left(\rho {C}_{p}\right)}_{nf}=\left(1-\varphi \right){\left(\rho {C}_{p}\right)}_{bf}+\varphi {\left(\rho {C}_{p}\right)}_{CNT}$ |

Viscosity | ${\mu}_{nf}=\frac{{\mu}_{bf}}{{\left(1-\varphi \right)}^{2.5}}$ |

Thermal conductivity | $\frac{{k}_{nf}}{{k}_{bf}}=\frac{\left(1-\varphi \right)+2\varphi \left(\frac{{k}_{CNT}}{{k}_{CNT}-{k}_{bf}}\right)ln\left(\frac{{k}_{CNT}+{k}_{bf}}{2{k}_{bf}}\right)}{\left(1-\varphi \right)+2\varphi \left(\frac{{k}_{bf}}{{k}_{CNT}-{k}_{bf}}\right)ln\left(\frac{{k}_{CNT}+{k}_{bf}}{2{k}_{bf}}\right)}$ |

Thermo-Physical Properties | H_{2}O | SWCNT | MWCNT |
---|---|---|---|

C_{p} (j/kg)K | $4179$ | 425 | 796 |

ρ (kg/m^{3}) | 997.1 | 2600 | 1600 |

k (W/mK) | 0.613 | 6600 | 3000 |

Prandtl number (Pr) | $6.8$ | $-$ | $-$ |

**Table 3.**Validation of the existing model for the values of $\text{}\mathrm{g}\text{}\u2033\left(\mathrm{a}\right)$ when $\epsilon ={\varphi}_{d}=\lambda ={F}_{r}=\alpha =\beta =M=m=0.$

$\mathit{a}$ | Ishak et al. [47] | Chen and Smith [42] | M. Idrees Afridi et al. [43] | Present Results |
---|---|---|---|---|

0.1 | 1.2888 | 1.28881 | 1.28881 | 1.28508 |

0.01 | 8.4924 | 8.49244 | 8.49233 | 8.4878 |

0.001 | 62.1637 | 62.16372 | 62.16370 | 62.1594 |

$\mathit{a}$ | ${\mathit{\varphi}}_{\mathit{d}}$ | ${\mathit{F}}_{\mathit{r}}$ | $\mathit{\beta}$ | $\mathit{M}$ | $\mathit{m}$ | Skin Friction Coefficient | |
---|---|---|---|---|---|---|---|

SWCNT | MWCNT | ||||||

0.001 | 0.00184647 | 0.00184157 | |||||

0.01 | 0.00583012 | 0.00581468 | |||||

0.2 | 0.02591430 | 0.02584880 | |||||

0.1 | 0.00583012 | 0.00581468 | |||||

2.0 | 0.00583276 | 0.00581722 | |||||

3.5 | 0.00583491 | 0.00581927 | |||||

0.10 | 0.00583012 | 0.00581468 | |||||

0.25 | 0.00644651 | 0.00640784 | |||||

0.4 | 0.00706302 | 0.00700110 | |||||

1.0 | 0.00583012 | 0.00581468 | |||||

2.0 | 0.00582586 | 0.00581043 | |||||

3.0 | 0.00582165 | 0.00580623 | |||||

0.2 | 0.00583012 | 0.00581468 | |||||

0.3 | 0.00711701 | 0.00710157 | |||||

0.4 | 0.00840431 | 0.00838888 | |||||

1.0 | 0.00583012 | 0.00581468 | |||||

1.4 | 0.00499559 | 0.00498015 | |||||

1.8 | 0.00447082 | 0.00445537 |

${\mathit{F}}_{\mathit{r}}\text{}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{m}$ | ${\mathit{N}}_{\mathit{r}}$ | ${\mathit{\theta}}_{\mathit{r}}$ | Nusselt Number | |
---|---|---|---|---|---|---|---|

SWCNT | MWCNT | ||||||

0.10 | 1.10739 | 1.05489 | |||||

0.25 | 1.12048 | 1.06695 | |||||

0.40 | 1.13570 | 1.08087 | |||||

1.0 | 1.10739 | 1.05489 | |||||

2.0 | 1.28855 | 1.22836 | |||||

3.0 | 1.44647 | 1.37965 | |||||

0.3 | 1.12089 | 1.06788 | |||||

0.5 | 1.10739 | 1.05489 | |||||

0.9 | 1.09958 | 1.04738 | |||||

1.0 | 1.10739 | 1.05489 | |||||

1.4 | 1.09857 | 1.04640 | |||||

1.8 | 1.09329 | 1.04131 | |||||

6.0 | 1.10739 | 1.05489 | |||||

9.0 | 0.80951 | 0.77264 | |||||

15.0 | 0.59158 | 0.56589 | |||||

1.1 | 1.10739 | 1.05489 | |||||

1.4 | 0.80635 | 0.77013 | |||||

1.7 | 0.71356 | 0.68292 |

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**MDPI and ACS Style**

Tlili, I.; Ramzan, M.; Kadry, S.; Kim, H.-W.; Nam, Y.
Radiative MHD Nanofluid Flow over a Moving Thin Needle with Entropy Generation in a Porous Medium with Dust Particles and Hall Current. *Entropy* **2020**, *22*, 354.
https://doi.org/10.3390/e22030354

**AMA Style**

Tlili I, Ramzan M, Kadry S, Kim H-W, Nam Y.
Radiative MHD Nanofluid Flow over a Moving Thin Needle with Entropy Generation in a Porous Medium with Dust Particles and Hall Current. *Entropy*. 2020; 22(3):354.
https://doi.org/10.3390/e22030354

**Chicago/Turabian Style**

Tlili, Iskander, Muhammad Ramzan, Seifedine Kadry, Hyun-Woo Kim, and Yunyoung Nam.
2020. "Radiative MHD Nanofluid Flow over a Moving Thin Needle with Entropy Generation in a Porous Medium with Dust Particles and Hall Current" *Entropy* 22, no. 3: 354.
https://doi.org/10.3390/e22030354