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Marangoni natural convection in a cubical cavity filled with a nanofluid

Buongiorno’s nanofluid model

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Abstract

Natural convection in a differentially heated cubical cavity filled with a water-based nanofluid under the Marangoni effect from the upper free surface is studied numerically. It is supposed that the Brownian diffusion and thermophoresis are the major slip mechanisms for nanoparticles. Dimensionless governing equations formulated using vector potential functions, vorticity vector and temperature have been solved by the finite difference method of the second-order accuracy. It should be noted that the Marangoni effect is owing to the dependences of surface tension on the temperature and nanoparticles concentration. The effects of the Marangoni number, Lewis number, buoyancy ratio parameter, Brownian diffusion parameter and thermophoresis parameter on nanofluid flow, heat and mass transfer have been analyzed. It has been revealed that a growth of Marangoni number results in the heat transfer rate reduction.

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Abbreviations

C :

Nanoparticle volume fraction (−)

C 0 :

Characteristic nanoparticle volume fraction (−)

D B :

Brownian diffusion coefficient (m2 s−1)

D T :

Thermophoretic diffusion coefficient (m2 s−1)

g :

Gravitational acceleration vector (m s−2)

j p :

Nanoparticles mass flux (kg m−2 s−1)

L :

Size of the cavity (m)

Le :

Lewis number (−)

Ma C :

Marangoni number due to concentration difference effect (−)

Ma T :

Marangoni number due to temperature difference effect (−)

Nb :

Brownian diffusion parameter (−)

Nr :

Buoyancy ratio parameter (−)

Nt :

Thermophoresis parameter (−)

Nu :

Local Nusselt number (−)

\( \overline{Nu} \) :

Average Nusselt number (−)

p :

Dimensional pressure (kg m−1 s−2)

Pr :

Prandtl number (−)

Ra :

Rayleigh number (−)

t :

Dimensional time (s)

T :

Dimensional temperature (K)

T c :

Cooled temperature of the vertical surface at \( \bar{x} = L \) (K)

T h :

Heated temperature of the vertical surface \( \bar{x} = 0 \) (K)

T 0 :

Mean temperature of the heated and cooled surfaces (K)

V :

Dimensional velocity vector (m s−1)

\( \bar{u},\;\bar{v},\;\bar{w} \) :

Dimensional velocity components (m s−1)

x, y, z :

Dimensionless Cartesian coordinates (−)

\( \bar{x},\;\bar{y},\;\bar{z} \) :

Dimensional Cartesian coordinates (m)

α :

Thermal diffusivity (m2 s−1)

β :

Volumetric expansion coefficient (K−1)

δ :

Heat capacitance ratio (−)

θ :

Dimensionless temperature (−)

μ :

Dynamic viscosity (kg m−1 s−1)

\( \rho_{\text{f0}} \) :

Fluid density (kg m−3)

ρ p :

Nanoparticle mass density (kg m−3)

σ :

Surface tension (kg s−2)

ϕ :

Nanoparticles volume fraction (−)

\( \bar{\psi }_{\text{x}} ,\;\,\bar{\psi }_{\text{y}} ,\;\,\bar{\psi }_{\text{z}} \) :

Dimensional vector potential functions (m2 s−1)

\( \psi_{\text{x}} ,\;\,\psi_{\text{y}} ,\;\,\psi_{\text{z}} \) :

Dimensionless vector potential functions (−)

\( \bar{\omega }_{\text{x}} ,\;\bar{\omega }_{\text{y}} ,\;\bar{\omega }_{\text{z}} \) :

Dimensional vorticity vector components (s−1)

\( \omega_{\text{x}} ,\;\omega_{\text{y}} ,\;\omega_{\text{z}} \) :

Dimensionless vorticity vector components (−)

c:

Cold

f:

Fluid

h:

Hot

p:

Nanoparticle

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Acknowledgements

This work of Mikhail A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation (Project Number 13.6542.2017/6.7). The work of Ioan Pop has been supported from the Grant PN-III-P4-ID-PCE-2016-0036, UEFISCDI, Romania.

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Authors and Affiliations

  1. Laboratory on Convective Heat and Mass Transfer, Tomsk State University, Tomsk, Russia, 634050

    Mikhail A. Sheremet

  2. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

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  1. Mikhail A. Sheremet
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Correspondence to Mikhail A. Sheremet.

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Sheremet, M.A., Pop, I. Marangoni natural convection in a cubical cavity filled with a nanofluid. J Therm Anal Calorim 135, 357–369 (2019). https://doi.org/10.1007/s10973-018-7069-2

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