## 4 Results and Discussion  
The present steady flow problem represented by the system of nonlinear ODEs in Equations (15)-(17) with the conditions (see Equation (18) and (19)) is carried out using the bvp4c (boundary value problem with fourth order accuracy) function in MATLAB software. To solve the suitable boundary value problems, 3-stage Lobatto IIIa (finite difference method) is used in the built-in-bvp4c function [66; 67; 68]. The main interest in the present study is to examine the effect of the pertinent parameters namely suction \(S\), thermal relaxation \(\gamma_{T}\), thermal stratification \(\delta_{1}\) and solutal stratification \(\delta_{2}\) on the dimensionless velocity, temperature and concentration profiles. The authors are also concerned with the existence of dual solutions since suction and shrinking parameters are taken into account. In the present work, \(M=Le=1\), \(Nb=0.3\), \(Nt=0.1\), \(\lambda=-0.5\), \(N=0.5\), \(\delta_{1}=0.1\), \(\delta_{2}=0.2\), \(\varepsilon=-1\), \(S=2.5\) and \(\gamma_{T}=0.01\) are considered as the main value for the numerical computations whereas \(\Pr=6.2\) is selected to symbolize the water as the base fluid. It is worth to mention here that the results are new, hence, the velocity, temperature and concentration profiles with the inflation of control parameters may be different from the existing studies. However, a few of comparisons is made with the previously published results for limiting cases to validate the bvp4c method used in the current work as tabulated in Tables 1 and 2. The approximate relative error \(\varepsilon_{a}\), is also calculated and it shows that \(\varepsilon_{a}\) between present and previous results are adequately small (0-0.3694%). It is clear from Table 2 that the thermal relaxation parameter \(\gamma_{T}\) which resulting from the non-Fourier energy model can intensify the rate of heat transfer while reduce the rate of mass transfer. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Pr** & **Present** & **[PERSON] and [PERSON] [18]** & \(\varepsilon_{a}=|\frac{a-b}{a}|\times 100\%\) & **[PERSON] et al. [24]** & \(\varepsilon_{a}=|\frac{a-b}{a}|\times 100\%\) \\ \hline
0.07 & 0.066056 & 0.0663 & 0.3694\% & 0.0663 & 0.3694\% \\
0.2 & 0.169089 & 0.1691 & 0.0065\% & 0.1691 & 0.0065\% \\
0.7 & 0.453916 & 0.4539 & 0.0035\% & 0.4539 & 0.0035\% \\
2 & 0.911358 & 0.9113 & 0.0064\% & 0.9114 & 0.0046\% \\
7 & 1.895403 & 1.8954 & 0.0001\% & 1.8954 & 0.0001\% \\
20 & 3.353904 & 3.3539 & 0.0001\% & 3.3539 & 0.0001\% \\
70 & 6.462199 & 6.4621 & 0.0015\% & 6.4622 & 0.0000\% \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison data of \(-\theta^{\prime}(0)\) for the case of unstratified viscous fluid when \(M=\lambda=N=Nb=Nt=\gamma_{T}=0\), \(\varepsilon=1\) and various values of Pr. Table 3 compares the numerical values of the heat and mass transfer rates for the problem of dual stratified nanofluid over a shrinking and stretching sheet. For the shrinking flow, \(-\theta^{\prime}(0)\) approximately increases 5.83% and 11.59% when \(\gamma_{T}=0.005\) and 0.01, respectively. In contrast, \(-\phi^{\prime}(0)\) decreases 1.13% when \(\gamma_{T}=0.005\) and 2.42% when \(\gamma_{T}=0.01\). However, the percentage of increment for \(-\theta^{\prime}(0)\) in the case of stretching flow is slightly higher as compared to the shrinking flow. In addition, the assisting flow case (\(\lambda=0.3\)) and opposing flow case (\(\lambda=-0.5\)) in Tables 2 and 3, correspondingly have same pattern (increment/decrement) of the heat and mass transfer rates. Figures 2-4 exhibit the variations of \(f^{\prime\prime}(0)\), \(-\theta^{\prime}(0)\) and \(-\phi^{\prime}(0)\) towards \(\varepsilon\) within specific range of \(S\). First and second solutions are designated by straight and dashed lines, respectively. Both solutions are found and remain up to a critical value, \(\varepsilon_{c}\) whereas no possible solution is obtained for \(\varepsilon>|\varepsilon_{c}|\). A critical or turning point is interpreted as a point where first and second (dual) solutions meet.