## 3 Results and Discussion  
Table 1 shows the results of horizontal and vertical velocity for a second-grade and Newtonian fluid in the upper half of channel under the effect of inertial force, magnetic field and porous medium. The negative and positive values of the velocity profile show that ciliary flow is generated by the to-and-fro motion of cilia. Numerical results show that  
horizontal velocity is maximum at the centre of the channel, and vertical velocity shows increasing and decreasing effects, which generates the wavy movement. It is also observed that the horizontal flow of the Newtonian fluid is larger than that of the second-grade fluid because the second-grade fluid has high viscosity compared with the Newtonian fluid. In this section, the effects of the Hartmann number \(M\), porosity parameter \(K\), fluid parameter \(\lambda_{1}\) and Reynolds number \(Re\) on the velocity field, pressure gradient and stream function are investigated by fixing the values as mentioned in [2; 31]. Note that the fluid parameter \(\lambda_{2}\) could be used instead of \(\lambda_{1}\), as the second-grade fluid model satisfies the thermodynamic condition \(\lambda_{1}+\lambda_{2}=0\). Thus, one will quantify the effect of \(\lambda_{1}\) on the velocity profile, pressure distribution and stream function. The implementation of the HPM solution can be validated by looking at the graphs: the horizontal velocity component is high at the centre of the channel because the flow is along the direction of the metachronal waves, and the vertical velocity component exhibits a sinusoidal behaviour which is assumed in the boundary conditions. Figure 2a,d illustrate the effects of the operating parameters \(M\), \(K\), \(Re\) and \(\lambda_{1}\) on the horizontal velocity component. Their effects on the ciliated flow are prominent in the middle of the channel and insignificant nearby the channel walls, which generates the parabolic behaviour of the flow due to the force caused by the tips of the cilia bed. Figure 2a shows that the Hartmann number \(M\), which is the ratio of the electromagnetic to the viscous forces, weakens the horizontal velocity because the electromagnetic force is applied in the transverse direction of the flow. In the present study, the magnetic force is dominant over the viscous one, which helps to slow the ciliary flow and can be used to observe the activity of ciliary motion by resonance imaging for diagnostic purposes. Figure 2b shows that the porosity parameter \(K\) helps to assist the axial flow. The porosity parameter is the ratio of the pore volume to the bulk volume, which is generally less than one. It shows that if the bulk volume of dense cilia reduces and the pore volume grows, then a parallel flow rises near the middle of the channel, because the fluid has more space to flow through the pores. Figure 2c shows that the velocity profile in the horizontal direction increases with the increasing value of the second-grade fluid parameter \(\lambda_{1}\) because of the presence of a magnetic field and the Reynolds number, which causes a reduction in fluid viscosity. Figure 2d displays that the Reynolds number supports the accelerating flow of the second-grade viscous fluid. The behaviour of the vertical velocity component is displayed in Figure 3a,d. It vanishes in the centre of the channel and behaves like a sinusoidal wave. The vertical velocity rises by increasing the strength of the magnetic field (\(M\)), the ratio of the pores