Table 2 Effective properties of nanofluid^35,40,41.

Viscosity

\(\frac{\mu _{nf}}{\mu _{f}}=1+2.5\phi +4.5\left[ \frac{1}{\tfrac{h_p}{d_p}(2+\tfrac{h_p}{d_p})(1+\tfrac{h_p}{d_p})^2}\right]\)

Thermal expansion

\(\frac{(\rho \check{\beta }_0)_{nf}}{(\rho \check{\beta }_0)_{f}} = (1-\phi ) + \phi \frac{(\rho \beta _0)_s}{(\rho \beta _0)_f}\)

Electrical conductivity

\(\frac{\sigma _{nf}}{\sigma _{f}}=1+\frac{3\left( \tfrac{\sigma _s}{\sigma _f}-1\right) \phi }{\left( \tfrac{\sigma _s}{\sigma _f}+2\right) -\left( \tfrac{\sigma _s}{\sigma _f}-1\right) \phi }\)

Thermal conductivity

\(\frac{k_{nf}}{k_{f}}=\frac{k_s+2k_f-2\phi (k_f-k_s)}{k_s+2k_f+\phi (k_f-k_s)}\)

Density

\(\frac{\rho _{nf}}{\rho _{f}}=1-\phi +\phi \frac{\rho _s}{\rho _f}\)

Heat capacity

\(\frac{(\rho C_p)_{nf}}{(\rho C_p)_{f}}=1-\phi +\phi \frac{(\rho C_p)_{s}}{(\rho C_p)_{f}}\)