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Effects of two-phase nanofluid model on MHD mixed convection in a lid-driven cavity in the presence of conductive inner block and corner heater

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Abstract

This paper investigates a steady mixed convection in a lid-driven square cavity subjected to an inclined magnetic field and heated by corner heater with an inserted square solid block. Water–Al\(_2\)O\(_3\) nanofluid fills the cavity based on Buongiorno’s two-phase model. A corner heater is configured in the left lower corner of the cavity by maintaining 40% of the bottom and vertical walls at constant hot temperature. The top horizontal wall is moving and maintained at a constant low temperature. The remainder walls are thermally insulated. The governing equations are solved numerically using the finite element method. The governing parameters are the nanoparticles volume fraction (\(0 \le \phi \le 0.04\)), Reynolds number (\(1 \le Re \le 500\)), Richardson number (\(0.01 \le Ri \le 100\)), Hartmann number (\(0 \le Ha \le 50\)) and the size of the inner solid (\(0.1 \le D \le 0.7\)). The other parameters: the Prandtl number, Lewis number, Schmidt number, ratio of Brownian to thermophoretic diffusivity and the normalized temperature parameter, are fixed at \(Pr=4.623\), \(Le=3.5\times 10^{5}\), \(Sc=3.55\times 10^{4}\), \(N_{\mathrm{BT}}=1.1\) and \(\delta =155\), respectively. The inclination of the magnetic field is fixed at \(\gamma =\frac{\pi }{4}\). Results show that at low Reynolds number, the increase in nanoparticles loading more the 2% becomes useless. It is also found that a big size of the solid block can augment heat transfer in the case of high values of both the Reynolds and Richardson numbers.

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Abbreviations

\(\overrightarrow{\mathbf{B }}\) :

Applied magnetic field

\({\mathbf B} \) :

Magnitude of magnetic field

\(C_{p}\) :

Specific heat capacity

d :

Side length of inner block

\(d_{\mathrm{f}}\) :

Diameter of the base fluid molecule

\(d_{\mathrm{p}}\) :

Diameter of the nanoparticle

D :

Dimensionless side length of the inner block, \(D=d/L\)

\(D_{\mathrm{B}}\) :

Brownian diffusion coefficient

\(D_{\mathrm{B0}}\) :

Reference Brownian diffusion coefficient

\(D_{\mathrm{T}}\) :

Thermophoretic diffusivity coefficient

\(D_{\mathrm{T0}}\) :

Reference thermophoretic diffusion coefficient

\({\mathbf {g}}\) :

Gravitational acceleration

Ha :

Hartmann number

Gr :

Grashof number

k :

Thermal conductivity

\(K_{\mathrm{r}}\) :

Square wall to nanofluid thermal conductivity ratio, \(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\)

L :

Width and height of enclosure

Le :

Lewis number

\(N_{\mathrm{BT}}\) :

Ratio of Brownian to thermophoretic diffusivity

\(\overline{Nu}\) :

Average Nusselt number

Pr :

Prandtl number

Re :

Reynolds number

\(Re_{\mathrm{B}}\) :

Brownian motion Reynolds number

Ri :

Richardson number, \(Ri=Gr/{Re}^2\)

Sc :

Schmidt number

T :

Temperature

\(T_0\) :

Reference temperature (310 K)

\(T_{\mathrm{fr}}\) :

Freezing point of the base fluid (273.15 K)

\({\mathbf {v}} \), \({\mathbf {V}} \) :

Velocity and dimensionless velocity vector, respectively

\(u_{\mathrm{B}}\) :

Brownian velocity of the nanoparticle

x, y and X, Y :

Space coordinates and dimensionless space coordinates

\(\alpha \) :

Thermal diffusivity

\(\gamma \) :

Inclination angle of magnetic field

\(\beta \) :

Thermal expansion coefficient

\(\delta \) :

Normalized temperature parameter

\(\theta \) :

Dimensionless temperature

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\rho \) :

Density

\(\sigma \) :

Electrical conductivity

\(\varphi \) :

Solid volume fraction

\(\varphi ^*\) :

Normalized solid volume fraction

\(\phi \) :

Average solid volume fraction

b :

Bottom wall

c :

Cold

f :

Base fluid

h :

Hot

nf:

Nanofluid

p :

Solid nanoparticles

t :

Top wall

w :

Solid wall

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Acknowledgements

The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010.

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Authors and Affiliations

  1. Refrigeration & Air-conditioning Technical Engineering Department, College of Technical Engineering, The Islamic University, Najaf, Iraq

    A. I. Alsabery

  2. School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600, Bangi, Selangor, Malaysia

    A. I. Alsabery & I. Hashim

  3. Mechanical Engineering Department, Engineering College, University of Basrah, Basrah, Iraq

    M. A. Ismael

  4. Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, Al-Khobar, 31952, Saudi Arabia

    A. J. Chamkha

  5. RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021, Ras Al Khaimah, United Arab Emirates

    A. J. Chamkha

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  1. A. I. Alsabery
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  4. I. Hashim
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Correspondence to A. I. Alsabery.

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Alsabery, A.I., Ismael, M.A., Chamkha, A.J. et al. Effects of two-phase nanofluid model on MHD mixed convection in a lid-driven cavity in the presence of conductive inner block and corner heater. J Therm Anal Calorim 135, 729–750 (2019). https://doi.org/10.1007/s10973-018-7377-6

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