Table 1 The thermophysical properties of hybrid nanofluid⁴⁵.

Density

\(\frac{{\tilde{\rho }_{hnf} }}{{\tilde{\rho }_{f} }} = \left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} } \right)\left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \frac{{\tilde{\rho }_{z1} }}{{\tilde{\rho }_{f} }}} \right) + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \frac{{\tilde{\rho }_{z2} }}{{\tilde{\rho }_{f} }},\)

Heat capacitance

\(\frac{{\left( {\tilde{\rho }C_{p} } \right)_{hnf} }}{{\left( {\tilde{\rho }C_{p} } \right)_{f} }} = \left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} } \right)\left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \frac{{\left( {\tilde{\rho }C_{p} } \right)_{z1} }}{{\left( {\tilde{\rho }C_{p} } \right)_{f} }}} \right) + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \frac{{\left( {\tilde{\rho }C_{p} } \right)_{z2} }}{{\left( {\tilde{\rho }C_{p} } \right)_{f} }},\)

Dynamic viscosity

\(\frac{{\mu_{hnf} }}{{\mu_{f} }} = \frac{1}{{\left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} } \right)^{\frac{10}{4}} \left( {1 - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} } \right)^{\frac{10}{4}} }},\)

Thermal conductivity

\(\begin{gathered} \frac{{\tilde{k}_{hnf} }}{{\tilde{k}_{f} }} = \left[ {\frac{{\tilde{k}_{z2} + 2\,\tilde{k}_{bf} + 2\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \left( {\tilde{k}_{z2} - \tilde{k}_{f} } \right)}}{{\tilde{k}_{s2} + 2\,\tilde{k}_{bf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \left( {\tilde{k}_{z2} - \tilde{k}_{f} } \right)}}} \right], \hfill \\ \frac{{\tilde{k}_{bf} }}{{\tilde{k}_{f} }} = \left[ {\frac{{\tilde{k}_{z1} + 2\tilde{k}_{f} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \left( {\tilde{k}_{z1} - \tilde{k}_{f} } \right)}}{{\tilde{k}_{z1} + 2\tilde{k}_{f} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \left( {\tilde{k}_{z1} - \tilde{k}_{f} } \right)}}} \right] \hfill \\ \end{gathered}\)

Electrical conductivity

\(\begin{gathered} \frac{{\ddddot \sigma_{hnf} }}{{\ddddot \sigma_{f} }} = \left[ {\frac{{\ddddot \sigma_{z2} + 2\,\ddddot \sigma_{bf} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \left( {\ddddot \sigma_{s2} - \ddddot \sigma_{f} } \right)}}{{\ddddot \sigma_{z2} + 2\,\ddddot \sigma_{bf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{2} \left( {\ddddot \sigma_{s2} - \ddddot \sigma_{f} } \right)}}} \right], \hfill \\ \frac{{\ddddot \sigma_{bf} }}{{\ddddot \sigma_{f} }} = \left[ {\frac{{\ddddot \sigma_{s1} + 2\,\ddddot \sigma_{f} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \left( {\ddddot \sigma_{s1} - \ddddot \sigma_{f} } \right)}}{{\ddddot \sigma_{z1} + 2\,\ddddot \sigma_{f} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\chi }_{1} \left( {\ddddot \sigma_{z1} - \ddddot \sigma_{f} } \right)}}} \right] \hfill \\ \end{gathered}\)