**4. Quantitative Analysis**

The governing model is tackled analytically and solutions are physically interpreted here.

The problem of rheological behavior of hybrid nanofluid flow induced by metachronical ciliary transport with heat transfer is studied. Important physical features of water and nanomaterials are represented in Table 1. Therefore, the behavior of velocity, temperature, induced magnetic field, stream function and volumetric flow rate for involving parameters is discussed in this section. Magnitudes of physical parameters are chosen corresponding to the physical situations assumed in the problem with z = 1, ε = 0.2, α = 0.05 and δ = 0.002.

Figure 5 explores the variational trend of axial velocity for escalating values of magnetic Reynolds number. As the magnetic Reynolds number rises, a high induction effects appear with the reduction in magnetic diffusion. These effects can be observed from the figure in which the velocity inside the annulus shows a decreasing behavior in the vicinity of the inner tube having radius *a1* owing to no slip velocity condition while it accelerates near the interior of outer tube with radius *a2* due to the continuous cilia beating. Moreover, a similar trend is noticed in Figures 6 and 7 for rising values of *Gr* and *M* caused by increasing buoyancy effects towards the variation in *Gr* and due to adding the flow mechanism with a rise in *M* which directly affects the flow rate. Figure 8 explicates the variational trend of magnetic induction profile against *R*. This is due to the fact that induction is directly linked with advection and the effects of *R* on the flow rate as described in Figure 5, cause magnetic induction profile to decelerate near the boundary of inner tube and accelerate in the vicinity of outer tube.

The consequences of emerging parameters on the temperature profile are inferred in Figures 9 and 10. Correspondingly, a decrease in temperature of the hybrid nanofluid for gradually mounting values of *Gr* is observed in Figure 9. The physics behind such behavior is an increase in heat-transfer rate due to enhancing buoyancy forces for rise in values of *Gr*. An increasing response of temperature of the fluid towards the magnetic Reynolds number is noted in Figure 10. This reveals that for high values of *R,* fluid particles gain more kinetic energy which is directly related to fluid temperature. All the results are plotted for mean flow rate *Q* = 2.

**Figure 5.** Variation in *w*(*r*) towards *R*.

**Figure 6.** Variation in *w*(*r*) towards *Gr*.

**Figure 7.** Variation in *w*(*r*) towards *M*.

**Figure 8.** Variation in *Hz*(*r*) towards *R*.

**Figure 9.** Variation in θ(r) towards *Gr*.

Furthermore, there is a three-dimensional physical interpretation of the velocity profile for variation in values of magnetic Reynolds number, Grashof number and Hartmann number, as displayed in Figures 11–13, respectively. The velocity maps out the parabolic trajectory for all the involving parameters. It is seen that velocity profile changes its behavior in the intervals 0.1 ≤ *r* ≤ 0.6 and 0.61 ≤ Υ ≤ 1.0 and the influences of the parameters on axial velocity are similar to the two dimensional velocity behaviors.

**Figure 11.** Three-dimensional (3-D) velocity profile towards *R*.

**Figure 12.** Three-dimensional (3-D) velocity profile towards *Gr*.

Besides this, the maximum pressure rise towards which peristalsis behaves as a pump is analyzed by means of pressure rise for one wavelength. In this regard, Figures 14–17 are prepared which exhibit the influence of embedding parameters correspond to pressure rise per wavelength towards mean flow rate. Non-linear behavior of these curves characterizes non-Newtonian fluid. All these plots contain four main parts (*a*) peristaltic pumping region, i.e., Δ*P* > 0, (*b*) free pumping region, i.e., Δ*P* = 0 (*c*) co-pumping region, i.e., Δ*P* < 0. In the region of peristaltic pumping, flow rate is positive and caused by peristalsis that occurred due to overcoming pressure difference while peristalsis of the boundaries of tube yields a free-pumping region. In the region of co-pumping, flow due to the peristalsis is assisted by negative pressure difference. The influence of the heat-generation parameter and Hartmann number are shown in Figures 14 and 15, and it is perceived that the pressure rise in co-pumping region (−1.0 ≤ *Q* ≤ −0.45) for *Gr* and (−1.0 ≤ *Q* ≤ 0.5) for *M* are decreasing. As, for Ω = 0.1, 0.3, 0.5, 0.7 and M = 0.0, 0.5, 1.0, 1.5, corresponding co-pumping regions contain (*Q* ∈ [−1.0, −0.45], *Q* ∈ [−1.0, −0.42], *Q* ∈ [−1.0, −0.4], *Q* ∈ [−1.0, −0.39]) and (*Q* ∈ [−1.0, 0.5], *Q* ∈ [−1.0, 0.48], *Q* ∈ [−1.0, −0.56], *Q* ∈ [−1.0, −0.4]), respectively. Pumping and free-pumping regions are increasing due to temperature gradient by buoyancy effects and increasing induction, correspondingly. Moreover, a similar trend for a rise in the values of *Gr* and *R* is depicted in Figures 16 and 17, and it is witnessed that the co-pumping region contains *Q* ∈ [−1.0, −0.4] but the pumping region is increasing.

**Figure 13.** Three-dimensional (3-D) velocity profile towards *M*.

**Figure 14.** Pressure rise versus *Q* for Ω.

The pressure gradient illustrates a direction and rate of rapid variation in pressure. Therefore, the pressure gradient towards embedding parameters such as the heat-generation parameter (Ω), Hartmann number (*M*), magnetic Reynolds number (*R*) and Grashof number (*Gr*) are studied and portrayed in Figures 18–21. It is perceived from these plots that the pressure gradient decreases rapidly with the variation in all the parameters. Hence, flow can easily pass through the endoscope for a small pressure gradient at *r* = 1 (near outer tube) exclusive of the imposition of the high-pressure gradient.

**Figure 15.** Pressure rise versus *Q* for *M*.

**Figure 16.** Pressure rise versus *Q* for *Gr*.

5 5

5 5

**Figure 17.** Pressure rise versus *Q* for *R*.

**Figure 18.** Pressure gradient for Ω.

**Figure 20.** Pressure gradient for *R*.

Trapping is a significant observable fact, whereby a bolus is transported with the wave speed and then a trapped bolus pressed forward along metachronical waves. These configurations are plotted in Figures 22–25 for different values of sundry parameters with panels (*a*)-(*d*) which inspect the ciliary-induced peristaltic flow pattern in the annulus. In general, the shape of streamlines is similar to the wave moving parallel to the walls of the tube. Under specific conditions, streamlines split and form a bolus which moves and circulates along the channel. The setup for the magnetic Reynolds number (*R*) is explained in Figure 22 for *M* = 1.5, *Gr* = 0.8, *Q* = 2, ε = 0.2. Higher values of R yield oscillatory motion of the fluid, and therefore a confined bolus decreases in size. Figure 23 depicts the behavior of streamlines for *M* = 1.5, *R* = 2, *Q* = 2, ε = 0.2 and it is perceived that with an increment in values of *Gr,* the confined bolus is shrinked moving towards the boundary of external tube and finally disappear due to temperature distribution caused by buoyancy forces. A similar formation of the flow pattern against rising Hartmann number is explored in Figure 24 with Gr = 0.8, R = 2, Q = 2, ε = 0.2. Physically, enhancement in magnitude of M augments fluid velocity which opposes trapping. This is in view of the fact that the magnitude of the amplitude ratio parameter (ε) indicates the length of cilia and the increment in values of ε enlarges cilia. The trapped bolus grows in size and circulates speedily as noticed in Figure 25. Thus, the presence of cilia favors trapping.

**Figure 21.** Pressure gradient for *Gr*.

**Figure 22.** *Cont*.

**Figure 22.** Behavior of streamlines for different values of magnetic Reynold number (**a**–**d**).

**Figure 23.** *Cont*.

**Figure 23.** Behavior of streamlines for different values of Grashof number (**a**–**d**).

**Figure 24.** *Cont*.

**Figure 24.** Behavior of streamlines for different values of Hartmann number (**a**–**d**).

**Figure 25.** *Cont*.

**Figure 25.** Behaviorof streamlines for different values of amplitude ratio (**a**–**d**).

Experiment-based numerical values and mathematical formulas for thermophysical characteristics of the hybrid nanofluid are expressed in Tables 1 and 2. Furthermore, the impact of engrossing parameters towards velocity and temperature profiles are presented in tabular form as shown in Tables 3 and 4. Table 3 shows that for small values of *r,* velocity decreases gradually but an increasing behavior is observed for larger radial distance against *M* and *R*. A conflicting behavior of the temperature profile towards *R* and *Gr* is depicted in Table 4. Additionally, the behaviors of velocity, temperature and magnetic induction profiles for variation in radial distance are examined and results are portrayed in Table 5. All the variations are examined for *M* = 4, *R* = 2, *Gr* = 2.5, Ω = 4, *Q* = 2, *z* = 1, ε = 0.2.

**Table 1.** Numerical values of thermal characteristics of nanomaterials and base fluid at 25 ◦C [9,12].


In addition, validation of existing results is examined by comparing them with those of Nadeem and Sadaf [50] in which an exact solution of a Newtonian Cu/blood nanofluid in the absence of magnetic induction has been studied. Table 6 shows that the two results are in good agreement. (See Table 6).


**Table 2.** Experimental relations for thermophysical characteristics of hybrid nanofluid [9,10].

**Table 3.** Numerical values of velocity profile versus *r* for variation in values of *M* and *R*.


**Table 4.** Numerical values of temperature profile versus r for variation in values of *R* and *Gr*.


**Table 5.** Variation in flow profiles for variation in radial distance *r*.



**Table 5.** *Cont*.

**Table 6.** Comparison of velocity with those of Nadeem and Sadaf [50] for n = 1, φ<sup>2</sup> = 0, M = 0 and E = 0.


#### **5. Conclusions**

This study incorporates the effects of electromagnetic induction on a rheological model of a hybrid nanofluid in ciliary-induced peristalsis through an endoscope. The major findings are summarized as:


**Author Contributions:** Conceptualization, M.A. and Z.S.; methodology, M.A., Z.S., P.K. and N.P.; software, A.A., P.K. and P.T.; validation, P.T. and and Z.S.; formal analysis, M.A., A.A., P.K. and N.P.; investigation, Z.S., P.K. and A.A.; resources, P.K. and P.T.; data curation, M.A.; writing—original draft preparation, M.A., Z.S. and A.A.; writing—review and editing, A.A., H.u.R.; visualization, Z.S. and H.u.R.; supervision, P.K. and A.A.; project administration, P.K., P.T. and Z.S.; funding acquisition, P.K., P.T. and Z.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

**Acknowledgments:** This research is supported by Postdoctoral Fellowship from King Mongkut's University of Technology Thonburi (KMUTT), Thailand.

**Conflicts of Interest:** The authors declare that they have no competing interests.
