Table 1 Thermo-physical characteristics formulas.

Properties	Nanofluid
Dynamic viscosity	${\mu }_{nf}={\mu }_{f}{\left(1-\phi \right)}^{-2.5}$
Density	${\rho }_{nf}=\left(1-\phi \right){\rho }_{f}+\phi {\rho }_{s}$
Heat capacity	$({\rho {C}_{p})}_{nf}=\left(1-\phi \right){(\rho {C}_{p})}_{f}+\phi {(\rho {C}_{p})}_{s}$
Thermal conductivity	$\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s}+\left(m-1\right){k}_{f}-(m-1)\phi ({k}_{f}-{k}_{s})}{{k}_{s}+\left(m-1\right){k}_{f}+\phi ({k}_{f}-{k}_{s})}$
Electrical conductivity	$\frac{{\sigma_{nf} }}{{\sigma_{f} }} = \left[ {1 + \frac{{3\left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} - 1} \right)\phi }}{{\left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} + 2} \right) - \left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} - 1} \right)\phi }}} \right]$

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Properties	Nanofluid
Dynamic viscosity	\({\mu }_{nf}={\mu }_{f}{\left(1-\phi \right)}^{-2.5}\)
Density	\({\rho }_{nf}=\left(1-\phi \right){\rho }_{f}+\phi {\rho }_{s}\)
Heat capacity	\(({\rho {C}_{p})}_{nf}=\left(1-\phi \right){(\rho {C}_{p})}_{f}+\phi {(\rho {C}_{p})}_{s}\)
Thermal conductivity	\(\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s}+\left(m-1\right){k}_{f}-(m-1)\phi ({k}_{f}-{k}_{s})}{{k}_{s}+\left(m-1\right){k}_{f}+\phi ({k}_{f}-{k}_{s})}\)
Electrical conductivity	\(\frac{{\sigma_{nf} }}{{\sigma_{f} }} = \left[ {1 + \frac{{3\left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} - 1} \right)\phi }}{{\left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} + 2} \right) - \left( {{\raise0.7ex\hbox{${\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{\sigma_{s} } {\sigma_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{f} }$}} - 1} \right)\phi }}} \right]\)

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