Table 3 Expressions for thermophysical characteristics of hybrid nanofluid.

Density

\(\rho_{{{\text{hnf}}}} = \rho_{{\text{f}}} \left( {1 - \varphi_{2} } \right)\left[ {\left( {1 - \varphi_{1} } \right) + \varphi_{1} \left( {\frac{{\rho_{{{\text{s}}_{1} }} }}{{\rho_{{\text{f}}} }}} \right)} \right] + \varphi_{2} \rho_{{{\text{s}}_{2} }}\)

Heat capacity

\(\left( {\rho {\text{c}}_{{\text{p}}} } \right)_{{{\text{hnf}}}} = \left( {\rho {\text{c}}_{{\text{p}}} } \right)_{{\text{f}}} \left( {1 - \varphi_{2} } \right)\left[ {\left( {1 - \varphi_{1} } \right) + \varphi_{1} \left( {\frac{{\left( {\rho {\text{c}}_{{\text{p}}} } \right)_{{{\text{s}}_{1} }} }}{{\left( {\rho {\text{c}}_{{\text{p}}} } \right)_{{\text{f}}} }}} \right)} \right] + \varphi_{2} \left( {\rho {\text{c}}_{{\text{p}}} } \right)_{{{\text{s}}_{2} }}\)

Viscosity

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

Thermal conductivity

\(\begin{aligned} & \frac{{\kappa_{{{\text{hnf}}}} }}{{\kappa_{{{\text{bf}}}} }} = \frac{{\kappa_{{{\text{s}}_{2} }} + \left( {{\text{s}} - 1} \right)\kappa_{{{\text{bf}}}} - \left( {{\text{s}} - 1} \right)\varphi_{2} \left( {\kappa_{{{\text{bf}}}} - \kappa_{{{\text{s}}_{2} }} } \right)}}{{\kappa_{{{\text{s}}_{2} }} + \left( {{\text{s}} - 1} \right)\kappa_{{{\text{bf}}}} + \varphi_{2} \left( {\kappa_{{{\text{bf}}}} - \kappa_{{{\text{s}}_{2} }} } \right)}}, \\ & {\text{where}}\,\,\frac{{\kappa_{{{\text{bf}}}} }}{{\kappa_{{\text{f}}} }} = \frac{{\kappa_{{{\text{s}}_{1} }} + \left( {{\text{s}} - 1} \right)\kappa_{{\text{f}}} - \left( {{\text{s}} - 1} \right)\varphi_{1} \left( {\kappa_{{\text{f}}} - \kappa_{{{\text{s}}_{1} }} } \right)}}{{\kappa_{{{\text{s}}_{1} }} + \left( {{\text{s}} - 1} \right)\kappa_{{\text{f}}} + \varphi_{1} \left( {\kappa_{{\text{f}}} - \kappa_{{{\text{s}}_{1} }} } \right)}} \\ \end{aligned}\)

Thermal expansion coefficient

\((\rho \beta )_{{{\text{hnf}}}} = (\rho \beta )_{{\text{f}}} \left[ {(1 - \phi_{1} - \phi_{2} ) + \phi_{1} \left( {\frac{{(\rho \beta )_{{{\text{s}}_{1} }} }}{{(\rho \beta )_{{\text{f}}} }}} \right)} \right] + \phi_{2} (\rho \beta )_{{{\text{s}}_{2} }}\)