Table 1 Thermo-physical characteristics54.

From: Numerical simulation on energy transfer enhancement of a Williamson ferrofluid subjected to thermal radiation and a magnetic field using hybrid ultrafine particles

Characteristics of the mono nanofluid

Characteristics of the hybrid nanofluid

\(\rho_{F - nf} = \left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)\rho_{f} + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \rho_{{\text{Fe}_{3} {\text{O}}_{4} }}\)

\(\rho_{F - Hnf} = \left( {1 - \chi_{{\text{Al}_{2} {\text{O}}_{3} }} } \right)\left[ {\left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)\rho_{f} + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \rho_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right] + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} \rho_{{\text{Al}_{2} {\text{O}}_{3} }}\)

\(\begin{aligned} \left( {\rho c_{p} } \right)_{F - nf} = & \left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)\left( {\rho c_{p} } \right)_{f} \\ & + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \left( {\rho c_{p} } \right)_{{\text{Fe}_{3} {\text{O}}_{4} }} \\ \end{aligned}\)

\(\begin{aligned} \left( {\rho c_{p} } \right)_{F - Hnf} = & \left( {1 - \chi_{{\text{Al}_{2} {\text{O}}_{3} }} } \right)[\left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)(\rho c_{p} )_{f} \\ & + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} (\rho c_{p} )_{{\text{Fe}_{3} {\text{O}}_{4} }} ] + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} (\rho c_{p} )_{{\text{Al}_{2} {\text{O}}_{3} }} \\ \end{aligned}\)

\(\beta_{F - nf} = \left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)\beta_{f} + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \beta_{{\text{Fe}_{3} {\text{O}}_{4} }}\)

\(\beta_{F - Hnf} = \left( {1 - \chi_{{\text{Al}_{2} {\text{O}}_{3} }} } \right)\left[ {\left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)\beta_{f} + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \beta_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right] + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} \beta_{{\text{Al}_{2} {\text{O}}_{3} }}\)

\(\mu_{F - nf} = \frac{{\mu_{f} }}{{\left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)^{2.5} }}\)

\(\mu_{F - Hnf} = \frac{{\mu_{f} }}{{\left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)^{2.5} \left( {1 - \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)^{2.5} }}\)

\(\frac{{k_{F - nf} }}{{k_{f} }} = \frac{{\left( {k_{{\text{Fe}_{3} {\text{O}}_{4} }} + 2k_{f} } \right) - 2\chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \left( {k_{f} - k_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)}}{{\left( {k_{{\text{Fe}_{3} {\text{O}}_{4} }} + 2k_{f} } \right) + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} \left( {k_{f} - k_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)}}\)

\(\,\frac{{k_{F - Hnf} }}{{k_{bf} }} = \frac{{k_{{\text{Al}_{2} {\text{O}}_{3} }} + 2k_{bf} - 2\chi_{{\text{Al}_{2} {\text{O}}_{3} }} \left( {k_{bf} - k_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)}}{{k_{{\text{Al}_{2} {\text{O}}_{3} }} + 2k_{bf} + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} \left( {k_{bf} - k_{{\text{Fe}_{3} {\text{O}}_{4} }} } \right)}}\)

\(\,\frac{{k_{bf} }}{{k_{f} }} = \frac{{k_{{\text{Al}_{2} {\text{O}}_{3} }} + 2k_{f} - 2\chi_{{\text{Al}_{2} {\text{O}}_{3} }} \left( {k_{f} - k_{{\text{Al}_{2} {\text{O}}_{3} }} } \right)}}{{k_{{\text{Al}_{2} {\text{O}}_{3} }} + 2k_{f} + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} \left( {k_{f} - k_{{\text{Al}_{2} {\text{O}}_{3} }} } \right)}}\)

\(\alpha_{F - nf} = \frac{{k_{F - nf} }}{{\left( {\rho c_{p} } \right)_{F - nf} }}\)

\(\alpha_{F - Hnf} = \frac{{k_{F - Hnf} }}{{\left( {\rho c_{p} } \right)_{F - Hnf} }}\)

\(\begin{array}{*{20}l} {\frac{{\sigma_{F - nf} }}{{\sigma_{f} }} = 1 + \frac{{3(\sigma - 1)\chi_{{\text{Fe}_{3} {\text{O}}_{4} }} }}{{(\sigma + 2) - (\sigma - 1)\chi_{{\text{Fe}_{3} {\text{O}}_{4} }} }}} \hfill \\ {\sigma = \frac{{\sigma_{{\text{Fe}_{3} {\text{O}}_{4} }} }}{{\sigma_{f} }}} \hfill \\ \end{array}\)

\(\begin{gathered} \frac{{\sigma_{F - Hnf} }}{{\sigma_{bf} }} = \frac{{\sigma_{{\text{Al}_{2} {\text{O}}_{3} }} + 2\sigma_{bf} - 2\chi_{{\text{Al}_{2} {\text{O}}_{3} }} (\sigma_{bf} - \sigma_{{\text{Al}_{2} {\text{O}}_{3} }} )}}{{\sigma_{{\text{Al}_{2} {\text{O}}_{3} }} + 2\sigma_{bf} + \chi_{{\text{Al}_{2} {\text{O}}_{3} }} (\sigma_{bf} - \sigma_{{\text{Al}_{2} {\text{O}}_{3} }} )}} \hfill \\ \frac{{\sigma_{bf} }}{{\sigma_{f} }} = \frac{{\sigma_{{\text{Fe}_{3} {\text{O}}_{4} }} + 2\sigma_{f} - 2\chi_{{\text{Fe}_{3} {\text{O}}_{4} }} (\sigma_{f} - \sigma_{{\text{Fe}_{3} {\text{O}}_{4} }} )}}{{\sigma_{{\text{Fe}_{3} {\text{O}}_{4} }} + 2\sigma_{f} + \chi_{{\text{Fe}_{3} {\text{O}}_{4} }} (\sigma_{f} - \sigma_{{\text{Fe}_{3} {\text{O}}_{4} }} )}} \hfill \\ \end{gathered}\)

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