# Induced Navier’s Slip with CNTS on a Stretching/Shrinking Sheet under the Combined Effect of Inclined MHD and Radiation

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## Abstract

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## 1. Introduction

## 2. The Theoretical Models and Solutions

#### 2.1. The Expression and Thermo-Physical Properties of the CNTs

#### 2.2. Similarity Variables

#### 2.3. Exact Solution for Velocity Equation

#### 2.4. Solution for the Temperature Equation

## 3. Results and Discussion

- In Figure 10a,b the thermal boundary layer thickness of SWCNT and MWCNT is increased with increasing the magnetic field M. Further, it is observed that SWCNT has more energy compeer to the MWCNT for both stretching and shrinking cases;
- By increasing the solid volume fraction, the thermal boundary layer thickness of SWCNT and MWCNT increases in both cases. Furthermore, it is observed that 10c SWCNT have more heat energy compered with MWCNT, and in Figure 10d SWCNT and MWCNT have some energy;
- The increasing values of ${N}_{r}$ result in greater thickness in thermal boundary as show in Figure 10e,f. Further, it is observed that SWCNT have more thermal energy compered with MWCNT.

## 4. Summary and Conclusions

- When fluid is injected into the SWCNT and MWCNT the flow of the fluid is along the direction of the motion of the surface and in the suction case the direction of the flow is changed due to the prances of mass transfer-induced slip, which has an impact on the convection process that transfers heat;
- For both stretching and shrinking cases, the impact of induced slip with mass suction significantly extend the unique and dual solution region of SWCNT and MWCNT;
- In the presence of the mass-injection slip, the velocity of fluid in SWCNT and MWCNT is increased which increases the speed of the flow along the sheet;
- By increasing the value of the ${\Gamma}_{1}$ and $\varphi $, both transverse and axial velocity increases both cases of stretching and shrinking;
- By increasing the value of the M the transverse velocity decreases and axial velocity increases both cases of stretching and shrinking;
- The thermal boundary layer thickness is increased while increasing physical parameters such as radiation $\left({N}_{r}\right)$, volume fraction, $\left(\varphi \right)$ and magnetic field $\left(M\right)$ for both cases of stretching and shrinking.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Explanations | S.I Unit |

Latin symbols | ||

${A}_{1},\hspace{0.17em}{A}_{2},\hspace{0.17em}{A}_{3},\hspace{0.17em}{A}_{4},\hspace{0.17em}{A}_{5}$ | Constant | $\left[-\right]$ |

${B}_{0}$ | Applied magnetic field | $\left[{\mathrm{wm}}^{-2}\right]$ |

${C}_{P}$ | Specific heat at constant pressure | $\left[{\mathrm{JKg}}^{-1}{\mathrm{K}}^{-1}\right]$ |

${k}^{*}$ | Mean absorption coefficient | $\left[{\mathrm{m}}^{-2}\right]$ |

M | Magnetic field | $\left[-\right]$ |

${N}_{r}$ | Radiation parameter | $\left[-\right]$ |

Pr | Prandtl number | $\left[-\right]$ |

${q}_{r}$ | Radiative heat flux | $\left[{\mathrm{Wm}}^{-2}\right]$ |

${q}_{w}$ | Local heat flux at the wall | $\left[-\right]$ |

T | Temperature | $\left[\mathrm{K}\right]$ |

${V}_{c}$ | Mass transformation | $\left[-\right]$ |

$\left(x,y\right)$ | Co-ordinate axes | $\left[\mathrm{m}\right]$ |

$\left(u,\mathrm{v}\right)$ | Velocities along x- and y- directions | $\left[{\mathrm{ms}}^{-1}\right]$ |

Greek symbols | ||

$\kappa $ | Thermal conductivity of fluid | $\left[{\mathrm{WKg}}^{-1}{\mathrm{K}}^{-1}\right]$ |

η | Similarity variable | $\left[-\right]$ |

${\mu}_{f}$ | Dynamic viscosity of fluid | $\left[-\right]$ |

$\nu $ | Kinematic viscosity | $\left[-\right]$ |

$\rho $ | Density | $\left[{\mathrm{Kgm}}^{-3}\right]$ |

${\sigma}^{*}$ | Stefan–Boltzmann constant | $\left[-\right]$ |

$\Gamma $ | Gamma function | $\left[-\right]$ |

$\varphi $ | Nanoparticle volume fraction | $\left[-\right]$ |

$\psi $ | Stream function | $\left[-\right]$ |

$\tau $ | Angle of inclination of magnetic field | $\left[-\right]$ |

Subscripts | ||

$f$ | Base fluid | $\left[-\right]$ |

$nf$ | Nanofluid | $\left[-\right]$ |

Abbreviations | ||

MHD | Magnetohydrodynamics | $\left[-\right]$ |

CNT | Carbon nanotubes | $\left[-\right]$ |

HNF | Hybrid nanofluid | $\left[-\right]$ |

SWCNT | Signal wall carbon nanotube | $\left[-\right]$ |

MWCNT | Multi-wall carbon nanotubes | $\left[-\right]$ |

## Appendix A. Roots of the Fourth Order Polynomials

## References

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**Figure 2.**Summary of the CNTs based model.from [32].

**Figure 4.**Plot ${\mathrm{Re}}_{{}_{x}}^{1/2}{C}_{f}$ verses ${\Gamma}_{1}$, (

**a**) stretching case (

**b**) shrinking case.

**Figure 5.**Plots of fluid velocity verses $\eta $ with effect of ${\Gamma}_{1}$. (

**a**) Transvers velocity (Stretching boundary), (

**b**) Transvers velocity (Shrinking boundary), (

**c**) Axial velocity (Stretching boundary) and (

**d**) Axial velocity (Shrinking boundary).

**Figure 6.**(

**a**) Plot $\alpha $ verses ${\Gamma}_{2}$, and (

**b**). (

**a**) Plot $\alpha $ verses ${\Gamma}_{1}$, on stretching case.

**Figure 7.**Plot of $f\left(\eta \right)$ and ${f}_{\eta}\left(\eta \right)$ with similarity variable $\eta $ with effect of ${\Gamma}_{1}$. (

**a**,

**c**) Stretching case, (

**b**,

**d**) shrinking case.

**Figure 8.**Plot of $f\left(\eta \right)$ and ${f}_{\eta}\left(\eta \right)$ with similarity variable $\eta $ with effect of $M$. (

**a**,

**b**) Stretching case, (

**c**,

**d**) shrinking case.

**Figure 9.**Plot of $f\left(\eta \right)$ and ${f}_{\eta}\left(\eta \right)$ with similarity variable $\eta $ with effect of $\varphi $. (

**a**,

**b**) Stretching case, (

**c**,

**d**) shrinking case.

**Figure 10.**(

**a**–

**f**) plot of $\Theta \left(\eta \right)$ with similarity variable $\eta $ with effect (

**a**) $M$ on stretching case, (

**b**) $M$ on shrinking case, (

**c**) $\varphi $ on stretching case, (

**d**) $\varphi $ on shrinking case, (

**e**) $Nr$ on stretching case, (

**f**) $Nr$ on shrinking case.

Physical Properties | Fluid Phase (Water) | SWCNT | MWCNT |
---|---|---|---|

${C}_{p}\left(\mathrm{J}/\mathrm{kgK}\right)$ | 4179 | 425 | 796 |

$\rho \left({\mathrm{Kg}/\mathrm{m}}^{3}\right)$ | 997.1 | 2600 | 1600 |

$\kappa \left(\mathrm{W}/\mathrm{mK}\right)$ | 0.613 | 6600 | 3000 |

$\sigma {\left(\mathsf{\Omega}/\mathrm{m}\right)}^{-1}$ | 0.05 | 48,000,000 | 38,000,000 |

SWCNT | $\mathit{\varphi}$ | ${\mathit{\mu}}_{\mathit{n}\mathit{f}}$ | ${\mathit{\rho}}_{\mathit{n}\mathit{f}}$ | ${\left(\mathit{\rho}\mathit{C}\mathit{p}\right)}_{\mathit{n}\mathit{f}}$ | ${\mathit{\kappa}}_{\mathit{n}\mathit{f}}$ | ${\mathit{\sigma}}_{\mathit{n}\mathsf{f}}$ |
---|---|---|---|---|---|---|

0.1 | 1.30135 | 1.16076 | 0.926519 | 2.90881 | 1.33333 | |

0.2 | 1.74695 | 1.32151 | 0.853037 | 5.29387 | 1.75000 | |

0.3 | 2.43924 | 1.48227 | 0.779556 | 0.35882 | 2.28571 | |

MWCNT | ||||||

0.1 | 1.30135 | 1.06047 | 0.926517 | 2.73334 | 1.33333 | |

0.2 | 1.74693 | 1.12093 | 0.853037 | 4.89830 | 1.75000 | |

0.3 | 2.43924 | 1.8114 | 0.779556 | 7.67900 | 2.28571 |

Authors | Fluids | The Governing Equation |
---|---|---|

Wu [23] | Newtonian | $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+{\upsilon}_{nf}\left[\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}\right],$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+{\upsilon}_{nf}\left[\frac{{\partial}^{2}v}{\partial {x}^{2}}+\frac{{\partial}^{2}v}{\partial {y}^{2}}\right],$ |

Wu [24] | Newtonian | $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha}_{nf}\frac{{\partial}^{2}T}{\partial {y}^{2}},$ |

Present work | Newtonian | $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+{\upsilon}_{nf}\left[\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}\right]-\frac{{\sigma}_{nf}{B}_{0}{}^{2}}{{\rho}_{nf}}{\mathrm{sin}}^{2}\left(\tau \right)u,$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+{\upsilon}_{nf}\left[\frac{{\partial}^{2}v}{\partial {x}^{2}}+\frac{{\partial}^{2}v}{\partial {y}^{2}}\right]-\frac{{\sigma}_{nf}{B}_{0}{}^{2}}{{\rho}_{nf}}{\mathrm{sin}}^{2}\left(\tau \right)v,$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha}_{nf}\frac{{\partial}^{2}T}{\partial {y}^{2}}-\frac{1}{{\left(\rho Cp\right)}_{nf}}\frac{\partial {q}_{r}}{\partial y}.$ |

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## Share and Cite

**MDPI and ACS Style**

Shettar, M.U.; Rudraiah, M.; Bragard, J.; Laroze, D.
Induced Navier’s Slip with CNTS on a Stretching/Shrinking Sheet under the Combined Effect of Inclined MHD and Radiation. *Energies* **2023**, *16*, 2365.
https://doi.org/10.3390/en16052365

**AMA Style**

Shettar MU, Rudraiah M, Bragard J, Laroze D.
Induced Navier’s Slip with CNTS on a Stretching/Shrinking Sheet under the Combined Effect of Inclined MHD and Radiation. *Energies*. 2023; 16(5):2365.
https://doi.org/10.3390/en16052365

**Chicago/Turabian Style**

Shettar, Mahabaleshwar Ulavathi., Mahesh Rudraiah, Jean Bragard, and David Laroze.
2023. "Induced Navier’s Slip with CNTS on a Stretching/Shrinking Sheet under the Combined Effect of Inclined MHD and Radiation" *Energies* 16, no. 5: 2365.
https://doi.org/10.3390/en16052365