# Theoretical and Numerical Study on Buongiorno’s Model with a Couette Flow of a Nanofluid in a Channel with an Embedded Cavity

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

- Continuity equation$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$
- Momentum equations in X and Y direction$$\rho \left(U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\right)=-\frac{\partial P}{\partial X}+\mu \left(\frac{{\partial}^{2}U}{\partial {X}^{2}}+\frac{{\partial}^{2}U}{\partial {Y}^{2}}\right)$$$$\rho \left(U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\right)=-\frac{\partial P}{\partial Y}+\mu \left(\frac{{\partial}^{2}V}{\partial {X}^{2}}+\frac{{\partial}^{2}V}{\partial {Y}^{2}}\right)$$
- Energy equation$$U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\alpha \left(\frac{{\partial}^{2}T}{\partial {X}^{2}}+\frac{{\partial}^{2}T}{\partial {Y}^{2}}\right)+\delta \left\{{D}_{B}\left(\frac{\partial \varphi}{\partial X}\frac{\partial T}{\partial X}+\frac{\partial \varphi}{\partial Y}\frac{\partial T}{\partial Y}\right)+\frac{{D}_{T}}{{T}_{c}}\left[{\left(\frac{\partial T}{\partial X}\right)}^{2}+{\left(\frac{\partial T}{\partial Y}\right)}^{2}\right]\right\}$$
- Nanoparticles equation$$U\frac{\partial \varphi}{\partial X}+V\frac{\partial \varphi}{\partial Y}={D}_{B}\left(\frac{{\partial}^{2}\varphi}{\partial {X}^{2}}+\frac{{\partial}^{2}\varphi}{\partial {Y}^{2}}\right)+\frac{{D}_{T}}{{T}_{c}}\left(\frac{{\partial}^{2}T}{\partial {X}^{2}}+\frac{{\partial}^{2}T}{\partial {Y}^{2}}\right)$$

_{c}is the cold temperature, α = k/(ρ C

_{p}) is the thermal diffusivity (k is the thermal conductivity of the fluid and C

_{p}is the fluid-specific heat constant), δ = (ρ C

_{p})

_{p}/(ρ C

_{p})

_{f}(p and f denotes nanoparticle and the base fluid). Let us introduce the Brownian motion coefficient D

_{B}, defined by the Einstein–Stokes equation, and the thermophoresis coefficient D

_{T}, that is:

_{B}is Boltzmann’s constant equal to 1.38064852 × 10

^{−23}(m

^{2}kg)/(s

^{2}K), d

_{p}is the nanoparticles diameter, k

_{p}is the particle thermal conductivity. For the thermal energy Equation (4), non-linearities take place with both Brownian diffusion and thermophoretic diffusion.

- Inlet$$U={U}_{0}\frac{Y}{H},V=0,T=\left({T}_{h}-{T}_{c}\right)\frac{Y}{H}+{T}_{c},\varphi ={\varphi}_{0}$$
- Outlet$$P=0,\frac{\partial T}{\partial X}=0$$
- Hot wall$$U={U}_{0},V=0,T={T}_{h},{D}_{B}\frac{\partial \varphi}{\partial Y}+\frac{{D}_{T}}{{T}_{c}}\frac{\partial T}{\partial Y}=0$$
- Horizontal cold walls$$U=0,V=0,T={T}_{c},{D}_{B}\frac{\partial \varphi}{\partial Y}+\frac{{D}_{T}}{{T}_{c}}\frac{\partial T}{\partial Y}=0$$
- Vertical cold walls$$U=0,v=0,T={T}_{c},{D}_{B}\frac{\partial \varphi}{\partial X}+\frac{{D}_{T}}{{T}_{c}}\frac{\partial T}{\partial X}=0$$

_{0}is the slip velocity of the hot wall and T

_{h}is the temperature of the hot wall.

_{h}on the upper wall, a uniform cold temperature T

_{c}on the lower wall and on the cavity walls, and a linearly varying temperature distribution both at the inlet and at the outlet sections. The hot and cold walls are marked with a red and blue line, respectively, in Figure 1.

_{h}− T

_{c}.

## 3. Mesh and Validation

_{B}≤ 1, and 0.1 ≤ N

_{T}≤ 1.

_{1}+ l

_{2}+ l

_{3}is the total length of the hot wall.

## 4. Results and Discussion

_{B}, or N

_{T}) while the other ones, respectively, are (100, 0, 0.1, 0.1).

_{T}, showing that also this parameter does not strongly affect the obtained solution.

_{1}+ l

_{2}/2)/l, and for Pr = 6.2, Le = 1, N

_{B}= N

_{T}= 0.1 and for different values assumed by the Reynolds number. The considered X-coordinate represents the vertical middle section of the cavity. The figure shows that while for Re = 10 the functions are monotonic, for higher values local minimum and maximum arise. Moreover, the concentration assumes value 1 for any considered Re at y = 0.75. This happens because the velocity value at that point is the same for all the conditions considered. Within the cavity, strong variations of the concentration occur. The dimensionless temperature distribution varies with the vertical coordinate only in the bottom cavity region, and for every mall value of the Reynolds number is a monotonic function.

_{B}= N

_{T}= 0.1, and for different values assumed by the parameter Re. The figure shows that the formation of a clockwise rotating cell in the cavity occurs and that this cell occupies a bigger region of the cavity for increasing values of Re. Moreover, the maximum of the dimensionless velocity increases for increasing values of Re.

_{B}are reported, showing that smaller values of N

_{B}yield higher concentrations and to less uniform behavior of this function. On the contrary, N

_{B}does not affect the trend of the dimensionless temperature.

_{T}, as shown in Figure 11.

_{B}does not affect the behavior of the Nusselt number, while N

_{T}and Le do, as evident in the right frame of Figure 12 and in Figure 13. In particular, the average Nusselt number is a decreasing function on N

_{T}, while it is an increasing function of Le. It is interesting to notice that for high values of N

_{T}, that is, N

_{T}= 0.5, the function is not monotonic, thus displaying a minimum for Re = 100.

## 5. Conclusions

_{B}, and N

_{T}and discussed. In particular, for increasing values of Re, the stream function and isotherm gradients occur only in the channel region, not involving the open cavity, while the concentration gradients appear mainly in the cavity region. For increasing values of the Lewis number, the concentration vortex tends to occupy the whole cavity. Moreover, with reference to the channel region, for increasing values of Le the concentration gradients tend to move close to the channel walls.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C_{p} | Fluid specific heat constant [J/kg K] |

D_{B} | Brownian motion coefficient |

D_{T} | Thermophoresis coefficient |

d_{P} | Nanoparticles diameter |

H | Cavity dimension |

k | Thermal conductivity [W/m K] |

k_{P} | Nanoparticles thermal conductivity [W/m K] |

Le | Lewis number |

N_{B} | Brownian diffusivity number |

N_{T} | Thermophoresis number |

Nu | Nusselt number |

P | Nanofluid pressure [Pa] |

p | Dimensionless pressure |

Pr | Prandtl number |

Re | Reynolds number |

T | Local nanofluid temperature [K] |

T_{h} | Hot wall temperature [K] |

U | x-component velocity [m/s] |

U_{0} | Slip velocity [m/s] |

V | y-component velocity [m/s] |

(u,v) | Dimensionless velocity components |

X | x-coordinate [m] |

Y | y-coordinate [m] |

(x,y) | Dimensionless coordinates |

α | Thermal diffusivity [m2/s] |

Θ | Dimensionless temperature |

µ | Nanofluid dynamic viscosity [Pa s] |

ρ | Nanofluid density [kg/m3] |

φ | Nanoparticles concentration [#/L] |

Φ | Dimensionless concentration |

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**Figure 2.**Mesh from coarse to fine: (

**left**) 1246 elements, (

**center**) 30,449 elements, (

**right**) 59,520 elements.

**Figure 3.**Stream function, Isotherms and Iso-concentration for various Reynolds number and Pr = 6.2, Le = 1, N

_{B}= N

_{T}= 0.1.

**Figure 4.**Stream function, Isotherms and Iso-concentration for various Lewis number and Pr = 6.2, Re = 100, N

_{B}= N

_{T}= 0.1.

**Figure 5.**Stream function, Isotherms and Iso-concentration for various N

_{B}number and Pr = 6.2, Re = 100, Le = 1, N

_{T}= 0.1.

**Figure 6.**Stream function, Isotherms and Iso-concentration for various N

_{T}number and Pr = 6.2, Re = 100, Le = 1, N

_{B}= 0.1.

**Figure 7.**Concentration and temperature for different Re number and Pr = 6.2, Le = 1, N

_{B}= N

_{T}= 0.1.

**Figure 9.**Concentration and temperature for different Le number and Pr = 6.2, Re = 100, N

_{B}= N

_{T}= 0.1.

**Figure 10.**Concentration and temperature for different N

_{B}number and Pr = 6.2, Re = 100, Le = 1, N

_{T}= 0.1.

**Figure 11.**Concentration and temperature for different N

_{T}number and Pr = 6.2, Re = 100, Le = 1, N

_{B}= 0.1.

**Figure 12.**Average Nusselt number as a function of Re number for different N

_{B}number and Pr = 6.2, Le = 1, N

_{T}= 0.1 (

**left**); Average Nusselt number as a function of Re number for different N

_{T}number and Pr = 6.2, Le = 1, N

_{B}= 0.1 (

**right**).

**Figure 13.**Average Nusselt number as a function of Re number for different Le number and Pr = 6.2, Le = 1, N

_{B}= N

_{T}= 0.1.

**Table 1.**Grid independence test. N is the mesh elements number, $\overline{u}$ is the mean velocity, $\mathsf{\Theta}$ is the mean temperature, $\mathsf{\Phi}$ is the mean concentration and $\overline{\mathrm{Nu}}$ is the mean Nusselt number. The mesh validation is done for Re = 100, Pr = 6.2, NB = NT = 0.1 and Le = 10.

# | N | $\overline{\mathit{u}}$ | $\overline{\mathsf{\Theta}}$ | $\overline{\mathsf{\Phi}}$ | $\overline{\mathbf{Nu}}$ |
---|---|---|---|---|---|

1 | 1246 | 0.60210 | 0.43391 | 1.04165 | 0.98558 |

2 | 30,449 | 0.60199 | 0.43461 | 1.03690 | 0.98569 |

3 | 59,520 | 0.60198 | 0.43463 | 1.03601 | 0.98571 |

4 | 77,826 | 0.60198 | 0.43463 | 1.03600 | 0.98572 |

5 | 120,360 | 0.60198 | 0.43464 | 1.03599 | 0.98573 |

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**MDPI and ACS Style**

Rossi di Schio, E.; Impiombato, A.N.; Mokhefi, A.; Biserni, C.
Theoretical and Numerical Study on Buongiorno’s Model with a Couette Flow of a Nanofluid in a Channel with an Embedded Cavity. *Appl. Sci.* **2022**, *12*, 7751.
https://doi.org/10.3390/app12157751

**AMA Style**

Rossi di Schio E, Impiombato AN, Mokhefi A, Biserni C.
Theoretical and Numerical Study on Buongiorno’s Model with a Couette Flow of a Nanofluid in a Channel with an Embedded Cavity. *Applied Sciences*. 2022; 12(15):7751.
https://doi.org/10.3390/app12157751

**Chicago/Turabian Style**

Rossi di Schio, Eugenia, Andrea Natale Impiombato, Abderrahim Mokhefi, and Cesare Biserni.
2022. "Theoretical and Numerical Study on Buongiorno’s Model with a Couette Flow of a Nanofluid in a Channel with an Embedded Cavity" *Applied Sciences* 12, no. 15: 7751.
https://doi.org/10.3390/app12157751