## 4 Results and Discussion  
The graphical outcomes presented in this section provide comprehensive detail about the distinct characteristics of peristaltic flow in this ciliated duct with elliptic cross section. Figures 2-4 provide the graphical outcomes of velocity, subject to the impact of distinct physical parameters. These graphs include the 2D and 3D plots of velocity profile. An increase is observed in velocity as \(Q\) increases, revealed by Figure 2a. Figure 2b provides the 3D graphical plot of velocity, which reveals flow dependency on x in addition to y. The velocity has inflative value in the middle of the conduit and a parabolic velocity profile is noted. Figure 3a,b reveal an increment in the flow profile for increasing \(\beta\). The enhancing \(\beta\) results in an increased flow profile in the middle of the duct but it declines with increasing \(\beta\) near the ciliated boundaries. An axial symmetry and parabolic velocity profile is observed. Figure 4a,b provide the 2D and 3D velocity plots for enhancing \(\alpha\). An increase in the velocity is seen for incrementing \(\alpha\). The flow profile attains high magnitude in the central region of the conduit but it shows an opposite behaviour near the ciliated boundaries with an increasing value of \(\alpha\). The increasing behaviour of velocity at the centre changes to a declining one near the ciliated walls. All the velocity graphs reveal that the flow behaviour is axially symmetric and a parabolic velocity profile is observed. Figures 5-9 provide the graphical solutions of temperature profile and its dependence on various parameters. Figure 5a,b provide the temperature plots for various numerical values of \(Q\). Temperature attains high values for increasing \(Q\). An axially symmetrical temperature behaviour is also observed in these temperature graphs. The convection rate is highest at the middle and minimum, near the ciliated boundaries. Figure 6a,b reveal the numerical solution of temperature for the uprising value of \(\lambda_{1}\). A decline in the temperature profile is observed for increasing values of \(\lambda_{1}\). If we set \(\lambda_{1}=0\) then the problem is simply a Newtonian flow problem. The increase in the value of \(\lambda_{1}\) changes the problem to a non-Newtonian flow problem. It reveals that the temperature reduces if the problem is transformed from analysis of a Newtonian to non-Newtonian flow problem. Figure 7a,b give the temperature solution for rising \(\beta\). A rise in the temperature profile is observed for upsurging \(\beta\). The convection is highest at the middle section of the conduit and then starts reducing on the way to the ciliated boundaries of the duct. Figure 8a,b provide the 2D and 3D plot of temperature for upsurging \(B_{r}\). The temperature rapidly increases for an increasing value of \(B_{r}\). An axially balanced temperature profile is revealed for the incrementing value of \(B_{r}\). The highest temperature in the middle of the conduit starts declining towards the ciliated boundaries and eventually becomes zero. Figure 9a,b depict the 2D and 3D plot of heat flux for distinct numerical entities of \(\alpha\). An increase in the convection rate is noted for upsurging \(\alpha\). The maximum temperature is noted at the centre and zero at the ciliated walls. In Figure 10a-d, \(\frac{dp}{dz}\) is plotted against the axial coordinate and its dependence on various parameters is noted. Figure 10a depicts that \(\frac{dp}{dz}\) gains higher values with increasing the value of \(\delta\). Figure 10b reveals an increment in \(\frac{dp}{dz}\) for uprising \(\lambda_{1}\). Figure 10c shows that \(\frac{dp}{dz}\) increases in the crest region of the peristaltic wave but it diminishes in the trough region of the peristaltic wave for the incrementing value of \(\phi\). Figure 10d depicts a decline in \(\frac{dp}{dz}\) for increasing \(Q\). In Figure 11a-c \(\Delta P\) against \(Q\) is drawn for distinct uprising values of the physical parameters.