## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**y** & **U (Second-Grade)** & **V (Second-Grade)** & **U (Newtonian)** & **V (Newtonian)** \\ \hline
0 & 1.167076 & 0.00000 & 1.201914 & 0.00000 \\ \hline
0.1 & 1.1497845 & \(-\)0.081264 & 1.180927 & \(-\)0.08822 \\ \hline
0.2 & 1.0949367 & \(-\)0.152951 & 1.115670 & \(-\)0.16739 \\ \hline
0.3 & 0.9933403 & \(-\)0.204728 & 0.998994 & \(-\)0.22740 \\ \hline
0.4 & 0.8287558 & \(-\)0.224632 & 0.818103 & \(-\)0.256048 \\ \hline
0.5 & 0.5765326 & \(-\)0.197962 & 0.553115 & \(-\)0.237581 \\ \hline
0.6 & 0.20171104 & \(-\)0.105730 & 0.174831 & \(-\)0.151016 \\ \hline
0.7 & \(-\)0.3433545 & \(-\)0.105730 & \(-\)0.35855 & 0.0322200 \\ \hline
0.8 & \(-\)1.1216959 & \(-\)0.105730 & \(-\)1.10617 & 0.3514422 \\ \hline
0.9 & \(-\)1.1216959 & \(-\)0.105730 & \(-\)2.15119 & 0.8610299 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of velocity distribution for a second-grade fluid when \(Re=0.1\) and for a Newtonian fluid obtained from Reference [25] for fixed values of \(M=1\), \(K=0.1\)\(\alpha=0.4\), \(\beta=0.4\), \(\epsilon=0.1\), \(Q=1.5\). volume to the bulk volume (\(K\)), the viscosity of the fluid (\(\lambda_{1}\)) and the Reynolds number \(Re\) in the upper part of the conduit. The same behaviour is observed in the lower part of the channel in the opposite direction, due to the symmetry of the ciliated conduit. Figure 3: \(Cont.\)  
Figure 4a,d illustrate that the horizontal pressure gradient along the flow exhibits a periodic behaviour. A favourable pressure gradient (\(\frac{\partial p}{\partial x}<0\)) surges when the intensity of the magnetic field and/or the porosity increase. The cilia require more pressure to induce the fluid flow in the presence of a magnetic field applied in the transverse direction. On the contrary, the porous medium assists the fluid flow in the flow direction; therefore, the horizontal pressure gradient is a reducing function of the porosity parameter. Figure 4c,d show the dual behaviour of the horizontal pressure gradient by developing fluid viscosity \(\lambda_{1}\) and inertial forces \(Re\). The pressure gradient decays in the regions \(-1<x<-0.5\) and \(0<x<0.5\), while it rises in the regions \(-0.5<x<0\) and \(0.5<x<1\) by increasing \(\lambda_{1}\) and \(Re\). The behaviour of the vertical pressure gradient is presented in Figure 5a,d. It is symmetric about the centreline and behaves like a sinusoidal wave. The vertical pressure gradient is favourable (negative pressure gradient) in the region \(-0.5<x<0\) and adverse (positive pressure gradient) in the region \(0<x<0.5\). Increasing the values of \(\lambda_{1}\) and \(K\) leads to an increase of the pressure gradient amplitude, the Hartmann number \(M\) having an opposite influence. It is also observed that near the upper wall (\(y=0.5\)), the pressure gradient amplitude increases in the flow direction due to the recovery and effective stroke of the cilia beating. Figure 4: _Cont._  
The streamlines are displayed in Figures 6-9 for different values of the Hartmann number \(M\), porosity parameter \(K\), fluid parameter \(\lambda_{1}\) and Reynolds number \(Re\). Figure 6a-c demonstrate that increasing the Hartmann number \(M\) leads to more resistance for the inertial flow; therefore, the bolus size reduces. It can be observed from Figure 7a-c that the bolus size rises with increased values of the porosity parameter \(K\), as it allows the fluid to flow more easily through the pores. Figure 8a-c indicate that, by increasing the fluid parameter \(\lambda_{1}\), the size of the trapped bolus rises. As shown in Figure 9a-c, the bolus size and the number of streamlines become larger when the inertial forces are dominant over the viscous forces (increasing the Re number). Figure 6: Influence of the Hartmann number \(M\) on the streamline patterns for \(\alpha=0.4\), \(\beta=0.4\), \(\varepsilon=0.1\), \(Q=1.5,\lambda_{1}=1\), \(K=0.1\), \(Re=0.1\).