. 2022 Oct 5;12:16608. doi: 10.1038/s41598-022-19625-3

Table 1.

Description of some dimensionless quantities.

For nanofluid^38,51

$ρ_{nf} = (1 - ϕ_{1}) ρ_{f} + ϕ_{1} ρ_{s 1}$

${(ρ C_{p})}_{nf} = (1 - ϕ_{1}) {(ρ C_{p})}_{f} + ϕ_{1} {(ρ C_{p})}_{s 1}$

$μ_{nf} = \frac{μ_{f}}{{(1 - ϕ_{1})}^{2.5}}$

$\frac{k_{nf}}{k_{bf}} = \frac{k_{p} + (n - 1) k_{f} - (n - 1) ϕ_{1} (k_{f} - k_{s 1})}{k_{p} + (n - 1) k_{f} + ϕ_{1} (k_{f} - k_{s 1})}$

For hybrid nanofluid³⁸

$ρ_{hnf} = (1 - ϕ_{1} - ϕ_{2}) ρ_{f} + ϕ_{1} ρ_{s 1} + ϕ_{2} ρ_{s 2}$

${(ρ C_{p})}_{hnf} = (1 - ϕ_{1} - ϕ_{2}) {(ρ C_{p})}_{f} + ϕ_{1} {(ρ C_{p})}_{s 1} + ϕ_{2} {(ρ C_{p})}_{s 2}$

$μ_{hnf} = \frac{μ_{f}}{{(1 - ϕ_{1} - ϕ_{2})}^{2.5}}$

$\frac{k_{hnf}}{k_{bf}} = \frac{k_{s 2} + (n - 1) k_{bf} - (n - 1) ϕ_{2} (k_{bf} - k_{s 2})}{k_{s 2} + (n - 1) k_{bf} + ϕ_{2} (k_{bf} - k_{s 2})}$

where $\frac{k_{bf}}{k_{f}} = \frac{k_{s 1} + (n - 1) k_{f} - (n - 1) ϕ_{1} (k_{f} - k_{s 1})}{k_{s 1} + (n - 1) k_{f} + ϕ_{1} (k_{f} - k_{s 1})}$

For tri-hybrid nanofluid^52–54

$ρ_{trihnf} = (1 - ϕ_{1} - ϕ_{2} - ϕ_{3}) ρ_{f} + ϕ_{1} ρ_{s 1} + ϕ_{2} ρ_{s 2} + ϕ_{3} ρ_{s 3}$

${(ρ C_{p})}_{trihnf} = (1 - ϕ_{1} - ϕ_{2} - ϕ_{3}) {(ρ C_{p})}_{f} + ϕ_{1} {(ρ C_{p})}_{s 1} + ϕ_{2} {(ρ C_{p})}_{s 2} + ϕ_{3} {(ρ C_{p})}_{s 3}$

$μ_{trihnf} = \frac{μ_{f}}{{(1 - ϕ_{1} - ϕ_{2} - ϕ_{3})}^{2.5}}$

$\frac{k_{trihnf}}{k_{tf}} = \frac{k_{s 3} + (n - 1) k_{tf} - (n - 1) ϕ_{3} (k_{tf} - k_{s 3})}{k_{s 3} + (n - 1) k_{tf} + ϕ_{3} (k_{tf} - k_{s 3})}$

where $\frac{k_{tf}}{k_{bf}} = \frac{k_{s 2} + (n - 1) k_{bf} - (n - 1) ϕ_{2} (k_{bf} - k_{s 2})}{k_{s 2} + (n - 1) k_{bf} + ϕ_{2} (k_{bf} - k_{s 2})}$

where $\frac{k_{bf}}{k_{f}} = \frac{k_{s 1} + (n - 1) k_{f} - (n - 1) ϕ_{1} (k_{f} - k_{s 1})}{k_{s 1} + (n - 1) k_{f} + ϕ_{1} (k_{f} - k_{s 1})}$