Table 1 The physical model for trihybrid nanofluid32.

From: Energy transmission through radiative ternary nanofluid flow with exponential heat source/sink across an inclined permeable cylinder/plate: numerical computing

Viscosity

\(\frac{{\mu_{Tnf} }}{{\mu_{f} }} = \frac{1}{{(1 - \phi_{MgO} )^{2.5} (1 - \phi_{{TiO_{2} }} )^{2.5} (1 - \phi_{{CoFe_{2} O_{4} }} )^{2.5} }},\)

Density

\(\frac{{\rho_{Tnf} }}{{\rho_{f} }} = \left( {1 - \phi_{{TiO_{2} }} } \right)\left[ {\left( {1 - \phi_{{TiO_{2} }} } \right)\left\{ {\left( {1 - \phi_{{CoFe_{2} O_{4} }} } \right) + \phi_{{CoFe_{2} O_{4} }} \frac{{\rho_{{CoFe_{2} O_{4} }} }}{{\rho_{f} }}} \right\} + \phi_{{TiO_{2} }} \frac{{\rho_{{TiO_{2} }} }}{{\rho_{f} }}} \right] + \phi_{MgO} \frac{{\rho_{MgO} }}{{\rho_{f} }},\)

Specific heat

\(\left. {\frac{{(\rho cp)_{Tnf} }}{{\left( {\rho cp} \right)_{f} }} = \phi_{MgO} \frac{{\left( {\rho cp} \right)_{MgO} }}{{\left( {\rho cp} \right)_{f} }} + \left( {1 - \phi_{MgO} } \right)\left[ \begin{gathered} \left( {1 - \phi_{{TiO_{2} }} } \right)\left\{ {\left( {1 - \phi_{{CoFe_{2} O_{4} }} } \right) + \phi_{{CoFe_{2} O_{4} }} \frac{{\left( {\rho cp} \right)_{{CoFe_{2} O_{4} }} }}{{\left( {\rho cp} \right)_{f} }}} \right\} \hfill \\ + \phi_{{TiO_{2} }} \frac{{\left( {\rho cp} \right)_{{TiO_{2} }} }}{{\left( {\rho cp} \right)_{f} }} \hfill \\ \end{gathered} \right]} \right\}\)

Thermal conduction

\(\left. \begin{gathered} \frac{{k_{Tnf} }}{{k_{hnf} }} = \left( {\frac{{k_{{CoFe_{2} O_{4} }} + 2k_{hnf} - 2\phi_{{CoFe_{2} O_{4} }} \left( {k_{hnf} - k_{{CoFe_{2} O_{4} }} } \right)}}{{k_{{CoFe_{2} O_{4} }} + 2k_{hnf} + \phi_{{CoFe_{2} O_{4} }} \left( {k_{hnf} - k_{{CoFe_{2} O_{4} }} } \right)}}} \right),\frac{{k_{hnf} }}{{k_{nf} }} = \left( {\frac{{k_{{TiO_{2} }} + 2k_{nf} - 2\phi_{{TiO_{2} }} \left( {k_{nf} - k_{{TiO_{2} }} } \right)}}{{k_{{TiO_{2} }} + 2k_{nf} + \phi_{{TiO_{2} }} \left( {k_{nf} - k_{{TiO_{2} }} } \right)}}} \right), \hfill \\ \frac{{k_{nf} }}{{k_{f} }} = \left( {\frac{{k_{MgO} + 2k_{f} - 2\phi_{MgO} \left( {k_{f} - k_{MgO} } \right)}}{{k_{MgO} + 2k_{f} + \phi_{MgO} \left( {k_{f} - k_{MgO} } \right)}}} \right), \hfill \\ \end{gathered} \right\}\)

Electrical conductivity

\(\left. \begin{gathered} \frac{{\sigma_{Tnf} }}{{\sigma_{hnf} }} = \left[ {1 + \frac{{3\left( {\frac{{\sigma_{{CoFe_{2} O_{4} }} }}{{\sigma_{hnf} }} - 1} \right)\phi_{{CoFe_{2} O_{4} }} }}{{\left( {\frac{{\sigma_{{CoFe_{2} O_{4} }} }}{{\sigma_{hnf} }} + 2} \right) - \left( {\frac{{\sigma_{{CoFe_{2} O_{4} }} }}{{\sigma_{hnf} }} - 1} \right)\phi_{{CoFe_{2} O_{4} }} }}} \right],\,\frac{{\sigma_{hnf} }}{{\sigma_{nf} }} = \left[ {1 + \frac{{3\left( {\frac{{\sigma_{{TiO_{2} }} }}{{\sigma_{nf} }} - 1} \right)\phi_{{TiO_{2} }} }}{{\left( {\frac{{\sigma_{{TiO_{2} }} }}{{\sigma_{nf} }} + 2} \right) - \left( {\frac{{\sigma_{{TiO_{2} }} }}{{\sigma_{nf} }} - 1} \right)\phi_{{TiO_{2} }} }}} \right], \hfill \\ \,\,\,\frac{{\sigma_{nf} }}{{\sigma_{f} }} = \left[ {1 + \frac{{3\left( {\frac{{\sigma_{MgO} }}{{\sigma_{f} }} - 1} \right)\phi_{MgO} }}{{\left( {\frac{{\sigma_{MgO} }}{{\sigma_{f} }} + 2} \right) - \left( {\frac{{\sigma_{MgO} }}{{\sigma_{f} }} - 1} \right)\phi_{MgO} }}} \right] \hfill \\ \end{gathered} \right\}\)

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