Table 1 Nanofluid, hybrid nanofluid, and ternary hybrid nanofluid thermophysical characteristics^3,39,40.

Density

\(\left( {\rho_{nf} } \right)\)

\(\rho_{nf} \left( { = \rho_{f} \left( {1 - \phi_{1} } \right) + \phi_{1} \left( {\rho_{{s_{1} }} } \right)} \right)\)

Viscosity

\(\left( {\mu_{nf} } \right)\)

\(\mu_{nf} \left( { = \frac{{\mu_{f} }}{{\left( {1 - \phi_{1} } \right)^{2.5} }}} \right)\)

Heat capacity

\(\left( {\rho C_{p} } \right)_{nf}\)

\(\left( {\rho C_{p} } \right)_{nf} \left( \begin{gathered} = \left( {\rho C_{p} } \right)_{f} \left( {1 - \phi_{1} } \right) \hfill \\ + \phi_{1} \left( {\rho C_{p} } \right)_{{s_{1} }} \hfill \\ \end{gathered} \right)\)

Thermal conductivity \(\left( {K_{nf} } \right)\)

\(\frac{{K_{nf} }}{{K_{f} }}\left( { = \frac{{K_{{s_{1} }} + 2K_{f} - 2\left( {K_{f} - K_{{s_{1} }} } \right)\phi_{1} }}{{K_{{s_{1} }} + 2K_{f} + \phi_{1} \left( {K_{f} - K_{{s_{1} }} } \right)}}} \right)\)

Density

\(\left( {\rho_{hnf} } \right)\)

\(\rho_{hnf} \left( \begin{gathered} = \left( {1 - \phi_{2} } \right)\left( \begin{gathered} \left( {1 - \phi_{1} } \right)\rho_{f} \hfill \\ + \phi_{1} \rho_{s1} \hfill \\ \end{gathered} \right) \hfill \\ + \phi_{2} \rho_{s2} \hfill \\ \end{gathered} \right)\)

Viscosity

\(\left( {\mu_{hnf} } \right)\)

\(\mu_{hnf} \left( { = \frac{{\mu_{f} }}{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }}} \right)\)

Heat capacity

\(\left( {\rho C_{p} } \right)_{hnf}\)

\(\left( {\rho C_{p} } \right)_{hnf} \left( { = \left( {1 - \phi_{2} } \right)\left( \begin{gathered} \left( {1 - \phi_{1} } \right)\left( {\rho C_{p} } \right)_{f} \hfill \\ + \phi_{1} \left( {\rho C_{p} } \right)_{s1} \hfill \\ \end{gathered} \right) + \phi_{2} \left( {\rho C_{p} } \right)_{s2} } \right)\)

Thermal conductivity

\(\left( {K_{hnf} } \right)\)

\(\begin{gathered} \frac{{K_{hnf} }}{{K_{nf} }}\left( { = \frac{{K_{s2} + 2K_{nf} - 2\left( {K_{nf} - K_{s2} } \right)\phi_{2} }}{{K_{s2} + 2K_{nf} + \phi_{2} \left( {K_{nf} - K_{s2} } \right)}}} \right), \hfill \\ Here \hfill \\ \,\frac{{K_{nf} }}{{K_{f} }}\left( { = \frac{{K_{s1} + 2K_{f} - 2\left( {K_{f} - K_{s1} } \right)\phi_{1} }}{{K_{s1} + 2K_{f} + \phi_{1} \left( {K_{f} - K_{s1} } \right)}}} \right) \hfill \\ \end{gathered}\)

Density

\(\left( {\rho_{thnf} } \right)\)

\(\rho_{thnf} \left( { = \left( {1 - \phi_{3} } \right)\left( \begin{gathered} \left( {1 - \phi_{2} } \right)\left[ {\left( {1 - \phi_{1} } \right)\rho_{f} + \phi_{1} \rho_{s1} } \right] \hfill \\ + \phi_{2} \rho_{s2} \hfill \\ \end{gathered} \right) + \phi_{3} \rho_{s3} } \right)\)

Viscosity

\(\left( {\mu_{thnf} } \right)\)

\(\mu_{thnf} \left( { = \frac{{\mu_{f} }}{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} \left( {1 - \phi_{3} } \right)^{2.5} }}} \right)\)

Heat capacity

\(\left( {\rho C_{p} } \right)_{thnf}\)

\(\left( {\rho C_{p} } \right)_{thnf} \left( { = \left( {1 - \phi_{3} } \right)\left( \begin{gathered} \left( {1 - \phi_{2} } \right)\left( \begin{gathered} \left( {1 - \phi_{1} } \right)\left( {\rho C_{p} } \right)_{f} \hfill \\ + \phi_{1} \left( {\rho C_{p} } \right)_{s1} \hfill \\ \end{gathered} \right) \hfill \\ + \phi_{2} \left( {\rho C_{p} } \right)_{s2} \hfill \\ \end{gathered} \right) + \phi_{3} \left( {\rho C_{p} } \right)_{s3} } \right)\)

Thermal conductivity \(\left( {K_{thnf} } \right)\)

\(\begin{gathered} \frac{{K_{thnf} }}{{K_{hnf} }}\left( { = \frac{{K_{s3} + 2K_{hnf} - 2\left( {K_{hnf} - K_{s3} } \right)\phi_{3} }}{{K_{s3} + 2K_{hnf} + \phi_{3} \left( {K_{hnf} - K_{s3} } \right)}}} \right), \hfill \\ \frac{{K_{hnf} }}{{K_{nf} }}\left( { = \frac{{K_{s2} + 2K_{nf} - 2\left( {K_{nf} - K_{s2} } \right)\phi_{2} }}{{K_{s2} + 2K_{nf} + \phi_{2} \left( {K_{nf} - K_{s2} } \right)}}} \right), \hfill \\ and \hfill \\ \,\frac{{K_{nf} }}{{K_{f} }}\left( { = \frac{{K_{s1} + 2K_{f} - 2\left( {K_{f} - K_{s1} } \right)\phi_{1} }}{{K_{s1} + 2K_{f} + \phi_{1} \left( {K_{f} - K_{s1} } \right)}}} \right) \hfill \\ \end{gathered}\)