Table 1 Description of some dimensionless quantities.

From: Hall current and morphological effects on MHD micropolar non-Newtonian tri-hybrid nanofluid flow between two parallel surfaces

For nanofluid38,51

\({\rho }_{nf}=\left(1-{\phi }_{1}\right){\rho }_{f}+{\phi }_{1}{\rho }_{s1}\)

\({(\rho {C}_{p})}_{nf}=\left(1-{\phi }_{1}\right){(\rho {C}_{p})}_{f}+{\phi }_{1}{(\rho {C}_{p})}_{s1}\)

\({\mu }_{nf}= \frac{{\mu }_{f}}{{\left(1-{\phi }_{1}\right)}^{2.5}}\)

\(\frac{{k}_{nf}}{{k}_{bf}}=\frac{{k}_{p}+(n-1){k}_{f}-(n-1){\phi }_{1}( {k}_{f}- {k}_{s1} )}{{k}_{p}+\left(n-1\right){k}_{f} + {\phi }_{1}( {k}_{f}- {k}_{s1 } )}\)

For hybrid nanofluid38

\({\rho }_{hnf}=\left(1-{\phi }_{1}-{\phi }_{2}\right){\rho }_{f}+{\phi }_{1}{\rho }_{s1}+{\phi }_{2}{\rho }_{s2}\)

\({(\rho {C}_{p})}_{hnf}=\left(1-{\phi }_{1}-{\phi }_{2}\right){(\rho {C}_{p})}_{f}+{\phi }_{1}{(\rho {C}_{p})}_{s1}+{\phi }_{2}{(\rho {C}_{p})}_{s2}\)

\({\mu }_{hnf}= \frac{{\mu }_{f}}{{\left(1-{\phi }_{1}-{\phi }_{2}\right)}^{2.5}}\)

\(\frac{{k}_{hnf}}{{k}_{bf}}=\frac{{k}_{s2}+(n-1){k}_{bf}-(n-1){\upphi }_{2}( {k}_{bf}- {k}_{s2} )}{{k}_{s2}+\left(n-1\right){k}_{bf} + {\upphi }_{2}( {k}_{bf}- {k}_{s2 } )}\)

where \(\frac{{k}_{bf}}{{k}_{f}}=\frac{{k}_{s1}+(n-1){k}_{f}-(n-1){\upphi }_{1}( {k}_{f}- {k}_{s1} )}{{k}_{s1}+\left(n-1\right){k}_{f} + {\upphi }_{1}( {k}_{f}- {k}_{s1 } )}\)

For tri-hybrid nanofluid52,53,54

\({\rho }_{\mathrm{trihnf}}=\left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right){\rho }_{f}+{\phi }_{1}{\rho }_{s1}+{\phi }_{2}{\rho }_{s2}+{\phi }_{3}{\rho }_{s3}\)

\({(\rho {C}_{p})}_{trihnf}=\left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right){(\rho {C}_{p})}_{f}+{\phi }_{1}{(\rho {C}_{p})}_{s1}+{\phi }_{2}{(\rho {C}_{p})}_{s2}+{\phi }_{3}{(\rho {C}_{p})}_{s3}\)

\({\mu }_{trihnf}= \frac{{\mu }_{f}}{{\left(1-{\phi }_{1}-{\phi }_{2}-{\phi }_{3}\right)}^{2.5}}\)

\(\frac{{k}_{trihnf}}{{k}_{tf}}=\frac{{k}_{s3}+(n-1){k}_{tf}-(n-1){\upphi }_{3}( {k}_{tf}- {k}_{s3} )}{{k}_{s3}+\left(n-1\right){k}_{tf} + {\upphi }_{3}( {k}_{tf}- {k}_{s3 } )}\)

where \(\frac{{k}_{tf}}{{k}_{bf}}=\frac{{k}_{s2}+(n-1){k}_{bf}-(n-1){\upphi }_{2}( {k}_{bf}- {k}_{s2} )}{{k}_{s2}+\left(n-1\right){k}_{bf} + {\upphi }_{2}( {k}_{bf}- {k}_{s2 } )}\)

where \(\frac{{k}_{bf}}{{k}_{f}}=\frac{{k}_{s1}+(n-1){k}_{f}-(n-1){\upphi }_{1}( {k}_{f}- {k}_{s1} )}{{k}_{s1}+\left(n-1\right){k}_{f} + {\upphi }_{1}( {k}_{f}- {k}_{s1 } )}\)

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