## 2 Problem Statement  
Al\({}_{2}\)O\({}_{3}\)-water fluid through two vertical permeable sheets is investigated as illustrated in Figure 1. The boundary conditions are clear in this figure. The variables are only the function of y because plates are infinite. Velocity and magnetic field vectors are considered as \(\overrightarrow{v}=\left[u,v_{0},0\right]\) and \(\overrightarrow{b}=\left[b_{x},b_{0},0\right]\) respectively. The governing equations and boundary conditions can be obtained as follows:  
\[v_{nf}\frac{d^{2}u}{dy^{2}}+\frac{\mu_{e}b_{0}}{\rho_{nf}}\frac{db_{x}}{dy}+g \beta_{nf}\left(T-T_{0}\right)+v_{0}\frac{du}{dy}=0 \tag{1}\]  
\[\frac{1}{\mu_{e}\sigma_{nf}}\frac{d^{2}b_{x}}{dy^{2}}+b_{0}\frac{du}{dy}+v_{0 }\frac{db_{x}}{dy}=0 \tag{2}\]  
\[\frac{k_{nf}}{\left(\rho C_{p}\right)_{nf}}\frac{d^{2}T}{dy^{2}}+v_{0}\frac{ dT}{dy}=0 \tag{3}\]  
\[b_{x}\left(0\right)=0,u\left(0\right)=0,\frac{dT}{dy}\left(0\right)=-\frac{q }{k_{f}} \tag{4}\]  
\[\frac{db_{x}}{dy}\left(h\right)=0,u\left(h\right)=0,T\left(h\right)=T_{0} \tag{5}\]\[ \begin{array}{l}\left(\sigma\right)_{nf^{\prime}}\left(\rho C_{p}\right)_{nf^{ \prime}}\left(\rho\beta\right)_{nf}\text{ and }\left(\rho_{nf}\right)\text{ can be introduced as }\left[3\right]\text{:}\\ \\ \frac{\sigma_{nf}}{\sigma_{f}}=1+\frac{3\left(\sigma_{p}/\sigma_{f}-1\right) \phi}{\left(\sigma_{p}/\sigma_{f}+2\right)-\left(\sigma_{p}/\sigma_{f}-1 \right)\phi}\text{,}\\ \left(\rho\text{ }C_{p}\right)_{nf}=\phi\left(\rho\text{ }C_{p}\right)_{p}+\left(\rho\text{ }C_{p}\right)_{f}\left(1-\phi\right)\text{,}\\ \end{array} \qquad\left(\rho\beta\right)_{nf}=\left(1-\phi\right)\left(\rho \beta\right)_{f}+\left(\rho\beta\right)_{p}\phi\text{,}\\ \\ \left(\rho\text{ }C_{p}\right)_{nf}=\phi\left(\rho\text{ }C_{p}\right)_{p}+\left(\rho\text{ }C_{p}\right)_{f}\left(1-\phi\right)\text{,}\\ \end{array}\qquad\rho_{nf}=\phi\rho_{p}+\rho_{f}\left(1-\phi\right) \tag{6}\]  
\(\left(k_{nf}\right)\) and \(\left(\mu_{nf}\right)\) are obtained according to Koo-Kleinstreuer-Li (KKL) model [27]:  
\[ \begin{array}{l}k_{nf}=\frac{3\left(k_{p}/k_{f}-1\right)\phi}{-\left(k_{p}/ k_{f}-1\right)\phi+\left(k_{p}/k_{f}+2\right)}+1+5\phi\times 10^{4}c_{p,f}g^{ \prime}\left(d_{p},T,\phi\right)\rho_{f}\sqrt{\frac{\kappa_{b}T}{\rho_{p}d_{p} }}\\ g^{\prime}\left(d_{p},T,\phi\right)=\\ \left(a_{1}+a_{2}Ln\left(d_{p}\right)+a_{5}Ln\left(d_{p}\right)^{2}+a_{3}Ln \left(\phi\right)+a_{4}ln\left(d_{p}\right)Ln\left(\phi\right)\right)Ln\left(T \right)+\\ \left(a_{6}+a_{7}Ln\left(d_{p}\right)+a_{10}Ln\left(d_{p}\right)^{2}+a_{8}Ln \left(\phi\right)+a_{9}ln\left(d_{p}\right)Ln\left(\phi\right)\right)\\ R_{f}=d_{p}\left(1/k_{p,eff}-1/k_{p}\right),R_{f}=4\times 10^{-8}km^{2}/W\\ \\ \mu_{nf}=\frac{\mu_{f}}{\left(1-\phi\right)^{2.5}}+\frac{k_{Brownian}}{k_{f}} \times\frac{\mu_{f}}{\Pr}\end{array}  \tag{7}\]  
All needed coefficients and properties are illustrated in Tables 1 and 2[27]. Dimensionless parameters are presented as:  
\[ \begin{array}{l}U=\frac{\
u u}{g\beta h^{2}\Delta T},B=\sqrt{\frac{\mu_{c}} {\rho_{f}}}\upsilon_{f}b_{x}\left(g\beta_{f}h^{2}\Delta T\right)^{-1},\theta= \Delta T^{-1}\left(T-T_{0}\right),\Delta T=qh/k_{f},Y=\frac{y}{h}\\ \\ \Pr=\left(\rho C_{p}\right)_{f}\frac{\mu_{f}}{k_{f}},Pm=\mu_{e}\upsilon_{f} \sigma_{f},Ha=\frac{\rho_{0}h}{\upsilon_{f}}\sqrt{\frac{\mu_{e}}{\rho}},V_{0}= \frac{\upsilon_{0}h}{\upsilon_{f}}\end{array}