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MHD double-diffusive mixed convection of binary nanofluids through a vertical porous annulus considering Buongiorno’s two-phase model

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Abstract

Binary nanofluids are prepared using a binary liquid (e.g., salt water) instead of a pure liquid as their host fluid. Double-diffusive convection in this type of nanofluids is a kind of triple-diffusive process, which simultaneously includes diffusions of heat, nanoparticles, and solute. This paper is devoted to double-diffusive mixed convection of binary nanofluids flowing through a vertical porous annulus in the presence of an externally applied radial magnetic field. To simulate the problem, a new mathematical model following the Buongiorno’s two-phase model is proposed. The equations are solved numerically by means of the finite-volume method. Thereafter, effects of the involved dimensionless variables on the distributions of stream function, temperature, solute concentration, and nanoparticles fraction as well as the mean values of the Nusselt number and the solute Sherwood number are presented and discussed. Inspection of the results demonstrates the significant contributions of the Peclet number (Pe), the usual Lewis number (Le), and the Soret-solute Lewis number (Ld) on the simulation results. In spite of that, the effects of the Hartmann number (Ha) and the thermophoresis parameter (Nt) on the simulation results are weak, while the consequences of the nanofluid Lewis number (Ln), the double-diffusive ratio (Nc), the buoyancy ratio (Nr), and the Dufour parameter (Nd) on the Nusselt and Sherwood numbers are insignificant.

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Abbreviations

B :

Magnetic field vector (T), \({\mathbf{B}} = {\text{B}}_{0} {\mathbf{e}}_{{{\bar{\mathbf{r}}}}}\)

\(c_{\text{p}}\) :

Specific heat \(({\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} )\)

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

Brownian diffusion coefficient \(({\text{m}}^{2} \,\,{\text{s}}^{ - 1} )\)

\(D_{\text{CT}}\) :

Diffusivity of the Soret type \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)

\(D_{\text{sm}}\) :

Solute diffusivity for the porous medium \(({\text{m}}^{2} \,\,{\text{s}}^{ - 1} )\)

\(D_{\text{TC}}\) :

Diffusivity of the Dufour type \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)

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

Thermophoretic diffusion coefficient \(({\text{m}}^{2} \,\,{\text{s}}^{ - 1} )\)

\({\mathbf{e}}_{{{\bar{\mathbf{r}}}}} ,{\mathbf{e}}_{{{\bar{\mathbf{z}}}}}\) :

Unit vectors in the cylindrical coordinate system

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

Gravitational acceleration vector \(({\text{m}}\,{\text{s}}^{ - 2} )\), \({\mathbf{g}} = - {\text{g}}{\mathbf{e}}_{{{\bar{\mathbf{z}}}}}\)

H :

Height of the domain (m)

Ha :

Hartmann number

\({\mathbf{j}}_{\text{p}}\) :

Nanoparticles mass flux \(({\text{kg}}\,{\text{m}}^{ - 1} \,{\text{s}}^{ - 1} )\)

K :

Permeability (m2)

k :

Thermal conductivity \(({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} )\)

L :

Width of the domain (m)

Ld:

Soret-solute Lewis number

Le:

Usual Lewis number

Ln:

Nanofluid Lewis number

Nb:

Brownian diffusion parameter

Nc:

Double-diffusive ratio

Nd:

Dufour parameter

Nd:

Buoyancy ratio

Nt:

Thermophoresis parameter

Nu:

Local Nusselt number

\(\overline{\text{Nu}}\) :

Mean Nusselt number

p :

Pressure (Pa)

Pe:

Peclet number

Ra:

Rayleigh number

\({\bar{\text{S}}}\) :

Solute concentration

\(\overline{{{\text{S}}_{\text{c}} }}\) :

Solute concentration in the incoming flow

\({\text{S}}\) :

Dimensionless solute concentration

\({\text{Sh}}_{\text{S}}\) :

Local Sherwood number of the solute

\(\overline{\text{Sh}}_{\text{S}}\) :

Mean Sherwood number of the solute

\({\text{Sh}}_{\Phi }\) :

Local Sherwood number of the nanoparticles

\(\overline{\text{Sh}}_{\Phi }\) :

Mean Sherwood number of the nanoparticles

t :

Time (s)

T :

Temperature (K)

\(u,w\) :

Velocity components in the cylindrical coordinate system \(({\text{m}}\,{\text{s}}^{ - 1} )\)

V :

Velocity vector \(({\text{m}}\,{\text{s}}^{ - 1} )\)

\(\bar{r}, \bar{z}\) :

Cylindrical coordinates (m)

\(r,z\) :

Dimensionless cylindrical coordinates

\(\alpha\) :

Thermal diffusivity \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)

\(\beta_{\text{T}}\) :

Thermal expansion coefficient \(\left( {{\text{K}}^{ - 1} } \right)\)

\(\beta_{\text{S}}\) :

Solute concentration expansion coefficient

\(\delta\) :

Constant coefficient, \(\varepsilon \left( {\rho c} \right)_{\text{P}} /\left( {\rho c} \right)_{\text{f}}\)

ε :

Porosity

\(\bar{\varPhi }\) :

Nanoparticles fraction

\(\bar{\varPhi }_{0}\) :

Nanoparticles fraction in the incoming flow

\(\varPhi\) :

Dimensionless nanoparticles fraction

\(\mu\) :

Dynamic viscosity \(({\text{N}}\,{\text{s}}\,{\text{m}}^{ - 2} )\)

\({{\varTheta }}\) :

Dimensionless temperature

ρ :

Density \(({\text{kg}}\,{\text{m}}^{ - 3} )\)

\(\sigma\) :

Electrical conductivity \(({{\varOmega }}^{ - 1} \,{\text{m}}^{ - 1} )\)

\({{\psi }}\) :

Stream function \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)

\(\varPsi\) :

Dimensionless stream function

\(\omega\) :

Constant coefficient, \(\omega = \left( {\rho c_{\text{p}} } \right)_{\text{m}} /\left( {\rho c_{\text{p}} } \right)_{\text{f}}\)

0:

Reference value

B:

Brownian

c:

Cold

f:

Host fluid

h:

Hot

i:

Inner

o:

Outer

T:

Thermophoresis

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  1. Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Iman Zahmatkesh & Mohammad Reza Habibi Shandiz

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Zahmatkesh, I., Habibi Shandiz, M.R. MHD double-diffusive mixed convection of binary nanofluids through a vertical porous annulus considering Buongiorno’s two-phase model. J Therm Anal Calorim 147, 1793–1807 (2022). https://doi.org/10.1007/s10973-020-10439-x

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