Table 3.

Expression for $\mathit{\alpha }$ various physical parameters.

Authors	Fluids	Value of $\mathit{\alpha }$
Crane 1970	Newtonian	$\alpha =1$
Pavlov 1974	Newtonian	$\alpha =\sqrt{1+M}$
Mahabaleshwar et al. 2014	Non-Newtonian	$\alpha =\sqrt{\frac{1+Q+K}{\left(1-k_1\right)}}$
Siddheshwar and Mahabaleshwar 2005	Non-Newtonian	$\alpha =\sqrt{\frac{1+Q}{\left(1-k_1\right)}}$
Mahabaleshwar et al. 2018,	Non-Newtonian	$\lambda {\alpha }^3+\left(1-Re{k}_1\right){\alpha }^2-\lambda \alpha -\left(1+Re\right)=0$
Amin et al. 2021	Non-Newtonian	Skin friction $f_{\eta \eta }\left(0\right)=-\sqrt{\frac{1+M+K}{\left(1-k_1\right)}}$
Amin et al. 2021	Non-Newtonian	${\theta }_{\eta }\left(0\right)=\frac{Ac{\left(\frac{c}{c-s}\right)}^B{\left(\frac{c}{c-s}\right)}^{C-A}}{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s}{\left(c-s\right)}\right)}$ $A=\frac{\mathit{Pr}}{\gamma Pr-c^2}$, $B=\frac{c\sqrt{\mathit{Pr}}+\sqrt{\gamma }\left(Pr+c^2-\gamma Pr\right)}{2\sqrt{\gamma }\left(\gamma Pr-c^2\right)},$ $C=\frac{-c\sqrt{\mathit{Pr}}+\sqrt{\gamma }\left(Pr+c^2-\gamma Pr\right)}{2\sqrt{\gamma }\left(\gamma Pr-c^2\right)}$,$s=\sqrt{\gamma Pr}$
Amin et al. 2021	Non-Newtonian	$\begin{array}{c}\theta \left(\eta \right)={\left(\frac{1+\frac{s}{\alpha }\left(1-Exp\left(-\alpha \eta \right)\right)}{Exp\left(-\alpha \eta \right)}\right)}^A\\ \frac{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s\alpha Exp\left(-\alpha \eta \right)}{\left(\alpha -s\right)\left[s\left(1-Exp\left(-\alpha \eta \right)\right)+\alpha \right]}\right)}{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s}{\left(\alpha -s\right)}\right)}\\ \end{array}$
Present work	Non-Newtonian	$\alpha =\sqrt{\frac{{\epsilon }_1+\left({\epsilon }_{3}M{\text{sin}}^2\left(\tau \right)+{\epsilon }_{2}D{a}^{-1}\right)}{\left({\epsilon }_2-{\epsilon }_1{k}_1\right)}}$
Temperature	Non-Newtonian	$\begin{array}{c}\theta \left(\eta \right)={\left(\frac{1+\frac{s}{\alpha }\left(1-Exp\left(-\alpha \eta \right)\right)}{Exp\left(-\alpha \eta \right)}\right)}^A\\ \frac{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s\alpha Exp\left(-\alpha \eta \right)}{\left(\alpha -s\right)\left[s\left(1-Exp\left(-\alpha \eta \right)\right)+\alpha \right]}\right)}{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s}{\left(\alpha -s\right)}\right)}\\ \end{array}$ $A=\frac{{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}\frac{\mathit{Pr}}{\frac{\gamma Pr{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}-{\alpha }^2}$, $B=\frac{\alpha \sqrt{Pr\frac{{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}}+\sqrt{\gamma }\left(Pr\frac{{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}+{\alpha }^2-\frac{\gamma Pr{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}\right)}{2\sqrt{\gamma }\left(\frac{{\epsilon }_4\gamma Pr}{\left({\epsilon }_5+N_R\right)}-{\alpha }^2\right)},$ $C=\frac{-\alpha \sqrt{Pr\frac{{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}}+\sqrt{\gamma }\left(Pr\frac{{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}+{\alpha }^2-\frac{\gamma Pr{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}\right)}{2\sqrt{\gamma }\left(\frac{{\epsilon }_4\gamma Pr}{\left({\epsilon }_5+N_R\right)}-{\alpha }^2\right)}$, $s=\sqrt{\frac{\gamma Pr{\epsilon }_4}{\left({\epsilon }_5+N_R\right)}}$
Nusselt number	Non-Newtonian	${\theta }_{\eta }\left(0\right)=\frac{A\alpha {\left(\frac{\alpha }{\alpha -s}\right)}^B{\left(\frac{\alpha }{\alpha -s}\right)}^{C-A}}{{}_2{F}_1\left(-A,-B;-B-C;\frac{-2s}{\left(\alpha -s\right)}\right)}$