## 3 Solution of the Problem
### Volume Flow Rate  
The impacts of the key parameters \(\lambda_{1}\), \(H_{a}\), \(D_{a}\), \(S_{t}\), \(g_{t}\), and \(M\) on the volume flow rate \(Q\) were examined and the results are presented in this subsection. Figures 4 and 5 were plotted to see the variations in \(Q\) versus \(\varepsilon\). The graphs in Figure 4a,b shows that higher values of \(\lambda_{1}\) and \(H_{a}\) tended to diminish the volume flow rate along the channel length. However, Figure 4c shows an opposite behavior of \(Q\) under the influence of the porosity parameter \(D_{a}\), i.e., the increasing values of \(D_{a}\) assisted the flow rate. This was physically verified because increasing values of \(D_{a}\) means enlarging the porosity of the medium, which allows more fluid to pass through it (as observed in References [24; 25; 26; 27; 28; 29]). We see in Figure 5a that the higher values of the Grash of number \(g_{t}\) tended to enhance \(Q\). In fact, as the Grashof number became larger, the impact of the buoyancy force became stronger, which supported the flow of the fluid and caused an enhancement in the volume flow rate. We can notice from Figure 5b that with increasing the values of \(S_{t}\), the flow rate increased. It is interesting to note in these graphs that the volume flow rate was an increasing function of the cilia length parameter \(\varepsilon\) over the range \(0<\varepsilon<1\). When \(\varepsilon=0\), we obtain a peristaltic pumping problem, which is not under discussion in this study. The variations in \(Q\) versus \(\delta\) (the eccentricity parameter) are sketched in Figures 6 and 7. In Figure 6a, we see that the flow rate declined when we increased the values of \(M\) (chemical reaction parameter). Physically, the chemical reactions could cause an increase in the internal resistance within the fluid, which resulted in the reduction of the flow rate. Figure 6b shows that the graphs of \(Q\) versus \(\delta\) were straight lines with positive slopes, which indicated that the flow rate increased in a linear way with the eccentricity parameter \(\delta\). These graphs also indicate that the flow rate gradually rose with the pressure drop (i.e., \(\Delta p<0\)) and declined with the pressure rise (i.e., \(\Delta p>0\)). This is a well-known observation in a peristaltic pumping system that also remains valid in ciliary pumping. Figure 7a,b shows that the volume flow rate reduced with increasing the values of \(H_{a}\), whereas it increased with increasing values of \(D_{a}\). These graphical results were in accord with our previous results, as shown in Figures 4 and 5. It is worthwhile to note that these observations were in close agreement with the previous studies of References [23; 24; 25; 26; 27; 28; 29; 30; 31; 32], where the authors noted that the effects of heat transfer may cause an increase in the volume flow rate.  
Figure 5: Volume flow rate \(Q\) versus \(\varepsilon\) for different values of: (**a**) \(g_{t}\) and (**b**) \(S_{t}\), keeping other parameters fixed as \(\lambda_{1}=1.5\), \(H_{a}=0.3\), \(D_{a}=0.3\), \(M=0.1\), \(g_{c}=0.5\), \(S_{r}=0.5\), \(S_{c}=0.1\), \(\Delta p=0.5\), \(\beta=0.4\), and \(\delta=0.3\).