**4. Results and Discussion**

This portion comprises of graphical results and discussion of obtained results for velocity, temperature, nanoparticles, pressure gradient, and stream functions. The numerical data of the pressure rise function Δ*p* is also sketched against the domain of flow rate and found the effects of physical parameters separately. Figures 2 and 3 are sketched for the velocity profile with varying the values of (*Gr*) and (*Gc*), respectively in corresponding order. From Figure 2, it is clearly visible that velocity is decreasing in lower part and increasing in upper part of the channel and enhances its maximum peak at the center under the effect of *Gr*. One can see the similar behavior by taking increasing values of *Gc* but here the difference is that the velocity is not varying much under the effect of *Gc* in Figure 3. Figures 4–6 contain correspondingly the alteration of temperature profile θ with the variability of Biot number (*Bi*), Brinkman number (*Br*), and the Prandtl number (Pr). From Figure 4, one can notice that the temperature profile is stretched vertically with the increase in magnitudes of *Bi*. It depicts that heat convection at the boundaries enhances the temperature of the Williamson nanofluid. It is also notable here that the temperature is maximum at lower wall and minimum at the lower surface and there is much variation in temperature level at upper region as compared to lower side. Figure 5 reflects the observation that temperature is an increasing function of *Br* and the temperature gradients are prominent at the lower portions as equated with the upper ones, but the extent of heat is similar at both the surfaces as was observed for *Bi*. It can be received from Figure 6 that temperature profile is increasing in linear fashion for numerically increasing magnitudes of *Pr* but the change in heat is calculated more significantly in the central parts of the enclosure which is the totally different result than we have achieved in Figures 4 and 5. Figures 7 and 8 are presented to see the behavior of nanoparticles volume fraction ϕ with increasing magnitudes of (*Nb*) and (*Nt*). Figure 7 shows that ϕ is getting higher when someone increases *Nb*. It is also explicit here that nanoparticles are dispersed in the region between the lower and upper surfaces. On the other hand, Figure 8 revels different story, the increase in *Nt* decreases the nanoparticles concentration. Figure 9 is plotted for

pressure gradient *dp*/*dx* for *Nb*. It is seen that *dp*/*dx* is increasing as we increase *Nb* and gets maximum height at the center of the domain, i.e., *x* = 0.5. From Figure 10, we can see that pressure gradient is varying quite opposite manner for the parameter *Nt*. It can also be noticed from Figures 9 and 10 that pressure gradient gets positive values only in the central part and remains negative at the corners. Figures 11 and 12 are displaced to see the effects of parameters *M* and *We* on pressure rise Δ*p*. Here the whole area is broken into three zones, namely Region I–III. The Region I is recognized by the portion where *Q* > 0, Δ*p* > 0. Region II is named the place where *Q* > 0 and Δ*p* < 0 while Region III is composed of the part *Q*, Δ*p* < 0. Figure 11 shows that Δ*p* curves are increasing in Region I and II while decreasing in Region III with the variation of *M*. Also, the free pumping exists at *Q* ≈ 1.5. In Figure 12, it is observed that in Region I and II, Δ*p* is increasing and in Region III, it is decreasing. Also, the peristaltic pumping occurs in Regions I and II between the interval (−1.7, 0.5). The streamlines are drawn in Figures 13–15 for the parameters *Gc*, *We*, and *M*, respectively. From Figure 13, it is clear that the number of boluses is increasing, but size of the trapped bolus is decreasing in lower part of the channel, while in upper portion, the situation is totally reflected in opposite ways. Figure 14 gives the streamlines variation under the different values of *We*. It is attained here that, in the lower part, the number of boluses is increasing but size is changing randomly. The stream function for *M* has been sketched in Figure 15 and it is noted in both the lower and upper parts, the size of bolus in increasing while number is decreasing. It is also admitted by Figures 13–15 that trapped boluses are displaced towards left from upper to lower side due to asymmetric dimensions of the channel which can be made symmetric by imposing ϕ = 0.

**Figure 2.** Modification of velocity profile against *Gr* for *x* = 1, *F* = 2, *a* = 0.2, *b* = 0.1, *d* = 1.5, ϕ = 1.5, *Gc* = 0.3, *We* = 0.01, *M* = 0.1, *Bi* = 0.5.

**Figure 3.** Modification of velocity profile against *Gc* for *x* = 1, *F* = 2, *a* = 0.2, *b* = 0.1, *d* = 1.5, ϕ = 1.5, *Gr* = 0.1, *We* = 0.01, *M* = 0.1, *Bi* = 0.5.

**Figure 4.** Modification of temperature profile against *Bi* for, *n* = 2, *x* = 0.1, *F* = 5, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.01, *W* = 0.01, *M* = 0.1, *Gc* = 0.9, *Gr* = 4, *Gc* = 0.3, *Nb* = 0.5, Pr = 0.4, *Nt* = 0.2.

**Figure 5.** Modification of temperature profile against *Br* for *n* = 2, *x* = 0.1, *F* = 5, *a* = 0.2, *b* = 0.1, *d* = 0.1, ϕ = 0.1, *We* = 0.01, *Gc* = 0.9, *Gr* = 1, *Bi* = 10, *Nb* = 0.5, Pr = 0.4, *Nt* = 0.2.

**Figure 6.** Modification of temperature profile against Pr for *n* = 2, *x* = 0.1, *F* = 5, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.1, *We* = 0.01, *Gc* = 0.9, *Gr* = 1, *Bi* = 5, *Nb* = 0.5, *Gc* = 0.01, *Nt* = 0.2.

**Figure 7.** Modification of nanoparticles concentration against *Nb* for *n* = 2, *x* = 0.1, *F* = 5, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.01, *We* = 0.01, *Gc* = 0.9, *Gr* = 4, *Bi* = 0.5, Pr = 0.4, *Gc* = 0.1, *Nt* = 0.2.

**Figure 8.** Modification of nanoparticles concentration against *Nt* for, *n* = 2, *x* = 0.1, *F* = 2, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.01, *We* = 0.01, , *Gc* = 0.9, *Gr* = 4, *Bi* = 0.5, Pr = 0.4, *Gc* = 0.1, *Nt* = 0.2.

**Figure 9.** Modification of pressure gradient against *Nb* for *n* = 2, *y* = 0.1, *F* = 10, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.01, *We* = 0.1, *Gc* = 0.9, *Gr* = 4, *Bi* = 0.5, Pr = 0.4, *Gc* = 0.1, *M* = 1.5, *Nt* = 0.2.

**Figure 10.** Modification of pressure gradient against *Nt* for, *n* = 2, *y* = 0.1, *F* = 10, *a* = 0.2, *b* = 0.1, *d* = 0.51, ϕ = 0.01, *We* = 0.1, , *Gc* = 0.9, *Gr* = 4, *Bi* = 0.09, Pr = 0.4, *Gc* = 0.3, *M* = 1.5, *Nb* = 0.1.

**Figure 11.** Modification of pressure rise against *M* for, *n* = 2, *y* = 0.1, *F* = 10, *a* = 0.2, *b* = 0.3, *d* = 0.5, ϕ = 0.01, *M* = 1.3, *Gc* = 0.3, *Gr* = 0.1, *Bi* = 0.3, Pr = 0.4, *Gc* = 0.3, *Nb* = 0.3, *Nt* = 0.

**Figure 12.** Modification of pressure rise against *We* for, *n* = 0.1, *y* = 0.1, *a* = 0.2, *b* = 0.3, *d* = 0.5, ϕ = 0.01, *M* = 1.3, *Gc* = 0.3, *Gr* = 0.1, *Bi* = 0.3, Pr = 0.4, *Gc* = 0.3, *Nb* = 0.3, *Nt* = 0.2.

**Figure 13.** Modification of streamlines for *Gc* = {0.1, 0.5, 0.9} when *n* = 0.1, *y* = 0.1, *F* = 5, *a* = 0.3, *b* = 0.2, *d* = 0.1, ϕ = 0.01, *M* = 0.1, *We* = 0.1, *Gr* = 4, *Bi* = 0.09, Pr = 0.4, *Br* = 0.9, *Nb* = 0.5, *Nt* = 0.2.

**Figure 14.** Modification of streamlines for *We* = {0.1, 0.2, 0.3} when *n* = 0.1, *y* = 0.1, *a* = 0.2, *b* = 0.3, *d* = 0.5, ϕ = 0.01, *M* = 0.1, *Bi* = 0.09, Pr = 0.4, *Gc* = 0.9, *Gr* = 4, *Nb* = 0.5, *Nt* = 0.2.

**Figure 15.** Modification of streamlines for *M* = {0.1, 0.9, 1.7} when *n* = 2, *F* = 5, *a* = 0.3, *b* = 0.2, *d* = 1, ϕ = 0.01, *We* = 0.1, *Gr* = 4, *Gc* = 0.1, *Bi* = 0.09, Pr = 0.4, *Gc* = 0.9, *Nb* = 0.5, *Nt* = 0.2.

#### **5. Conclusions**

In this article, the authors have discovered the mathematical treatment of the peristaltic flow of Williamson nanofluid coated with the walls of an asymmetric heated channel. The flow has been studied analytically and graphically through variation of some pertinent parameters. From the above discussion, the main findings are given below:


**Author Contributions:** Methodology, formal Analysis and writing—original draft preparation, A.R. (Arshad Riaz); writing—review and editing, A.R. (Abdul Razaq); funding acquisition, H.A.

**Funding:** This research project was supported by a grant from the Research Center of the Center for Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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