**5. Results and Discussion**

The authors obtained the quantitative analysis of nanoparticles in Jeffery fluid flowing past eccentric annuli having peristaltic waves at the outer surface. Heat and mass transfer phenomenon was also taken under consideration by the law of conservation of mass and energy. Lubrication theory was utilized to make the assumptions about laminar flow through arteries. Moreover, the effects of entropy generation and Bejan number were observed, which affect the flow due to irreversibility mechanism of temperature distribution, viscous dissipation, and nanoparticles' concentration. In this section, we describe the effects of emerging physical parameters of obtaining quantities through figures which are drawn on Mathematica and ordered in a subsequent manner. Numerical data were achieved for the expression of pressure rise by using built-in commands in mathematical software. Table 1 is placed to find the variation of pressure rise data Δ*p*, for a flow rate domain *Q* from the interval [–1, 1], by varying the parameters δ and *Gr* under the constant values of other factors. This table suggests that peristaltic pumping occurs at *Q* = 0. Figures 2 and 3 show the residual error curves, which clearly reflects the highly convergent solution of temperature distribution and nanoparticles' concentration, respectively, by keeping the rest of the quantities numerically fixed. Moreover, the values used in the graph emphasized that we can assign these numerical values of the parameters involved. Figure 4 confirms the validation of current analysis by comparing the present analysis with the study Nadeem et al. [32], which was published for viscous fluid. From this figure it is quite obvious that the current

study's results were similar to the results obtained in [32] when we neglected the non-Newtonian effects by assigning a zero value to the Jeffrey fluid parameter λ1. It was also found from this graph that for Jeffrey fluid the radial velocity reduces. This is due to the increase in shearing stress as λ<sup>1</sup> grows, which was introduced into the boundary layer, which can cause loss of speed.

**Figure 2.** Residual error curves of temperature distribution θ.

**Figure 3.** Residual error curves of nanoparticles' concentration σ

**Figure 4.** Comparison of present study with [32].

Figures 5–7 were plotted for pressure rise profile Δ*p* against the flow rate domain *Q*. Figure 5 evaluates the effects of two parameters, the inner radius of the annuli δ, and the local temperature Grashof number *Gr* on peristaltic pressure rise curves. From this figure we can imagine that the lines of Δ*p* are declining from left to right and intersecting each other at *Q* = –0.2. It can also be concluded here that pumping rate is increasing with both the parameters on the negative side of the domain, but reducing its inclination on the region of positive interval [0, 2]. This is due to the fact that increasing the radius of the inner cylinder exerts greater pressure on the flow on the left side, as compared to the other one, due to the eccentricity of the two annuli. Moreover, an increase in the local temperature Grashof number is produced due to the increase in outer cylinder radius, thus producing more pumping on the left side, whilst keeping the other parameters uniform. Figure 6 is sketched for Δ*p* to estimate the influence of eccentricity parameter ε and the local nanoparticles Grashof number *Br*. One can observe clearly that a similar behavior is shown with ε and *Br* in comparison to δ and *Gr*. The velocity profile can be considered in Figures 7 and 8. Figure 7 discloses the variation of axial velocity *w* against the radial coordinate *r*, which is plotted for increasing numerical values of eccentricity factor ε and inner cylinder velocity *V*. It is shown from this graph that when we speed up the inner cylinder, the maximum velocity of fluid gets reduced near the outer annulus surface, while an increase is noticed near the walls of the inner cylinder; also, under the impact of eccentricity of two cylinders, fluid enhances its speed, but near the lower walls it becomes stable, which is very much in line with the experimental and physical results. From Figure 8, we can predict that by enlarging the nanoparticle Grashof number *Gr* and temperature Grashof number *Br*, the fluid travels rapidly in the space away from the lower surface, which is not closer to the lower boundary. Figures 9 and 10 were included to find the theoretical characteristics of temperature distribution under the alteration of Brownian motion parameter *Nb* and thermophoresis parameter *Nt*, correspondingly. It is obvious that, by raising the amount of both parameters, temperature of the liquid varied directly and the maximum temperature gradient was seen in the middle part of the space. This behavior clearly notifies that, in the presence of nanoparticles, the thermal conductivity of the fluid enhances significantly, which is also evident from the pioneer study on nanofluids [1]. This also suggests that the thermal conduction is caused by Brownian diffusion and thermophoresis diffusion in the rise of flow, which leads to an increase in flow temperature distribution.

**Figure 5.** Variation of pressure rise Δ*p* with δ and *Gr* for fixed θ = 0.8, ϕ = 0.1, ε = 0.1, *Br* = 0.2, *Nb* = 0.5, *Nt* = 0.2, λ<sup>1</sup> = 5, *V* = 0.1, *t* = 0.05..

**Figure 6.** Variation of pressure rise Δ*p* with ε and *Br* for fixed θ = 0.8, ϕ = 0.1, δ = 0.1, *Gr* = 0.2, *Nb* = 0.5, *Nt* = 0.2, λ<sup>1</sup> = 5, *V* = 0.1, *t* = 0.05..

**Figure 7.** Variation of velocity profile *w* with ε and *V* for fixed θ = 0.8, ϕ = 0.1, δ = 0.1, *Br* = 0.3, *Gr* = 1, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, *Q* = 1. .

**Figure 8.** Variation of velocity profile w with *Br* and *Gr* for fixed θ = 0.1, ϕ = 0.5, δ = 0.5, ε = 0.1, *V* = 0.3, λ<sup>1</sup> = 0.7, *z* = 0, *t* = 0.1, *Q* = 1. .

**Figure 9.** Variation of temperature profile θ with *Nb* for fixed θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.1, *V* = 0.3, λ<sup>1</sup> = 0.7, *z* = 0, *t* = 0.3, *Nt* = 0.2, *Br* = 0.5..

**Figure 10.** Variation of temperature profile θ with *Nt* for fixed θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.1, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.3, *Nb* = 0.2, *Br* = 0.5.

The profile of nanoparticles σ ismentioned in the diagrams labeled as Figures 11 and 12. In Figure 11, we can see the effects of Brownian motion parameter *Nb* on nanoparticles' concentration. It is clearly seen from this graph that the amount of nanoparticles is lowered with the variation of *Nb*. Figure 12 reflects the curves of nanoparticles' profile for the parameter *Nt* and it can be suggested that nanoparticles' concentration gets enlarged.

**Figure 11.** Variation of nanoparticles phenomenon σ with *Nb* for fixed θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.1, *z* = 0, *t* = 0.3, *Nt* = 0.2..

**Figure 12.** Variation of temperature profile σ with *Nt* for fixed θ = 0.1, ϕ = 0.5, δ = 0.2, ε = 0.1, *z* = 0, *t* = 0.2, *Nb* = 0.2..

Figures 13–18 exhibit the influence of emerging parameters on entropy function *Ns*. Figure 13 contains the graph of *Ns* against the Brinkman number *Gc*. This figure implies that entropy generation increases near the lower surface of the space with the increasing effects of *Gc*, but in the wider part it gets lowered with the varying factor. It was noticed that the entropy of the system increases with the incursion in *Nb* in most of the region, but near the walls it is almost stable (see Figure 14). Figure 15 concludes that the entropy shows similar characteristics with *Nt*, as seen for *Nb*, but an opposite result can be seen near the lower wall. This is because rising of *Nt* involves larger viscous dissipation effects, due to energy production generating more entropy. From Figure 16 it can be visualized that entropy is proportional to the concentration difference parameter Ω in the interval *r* > 0.5, but for 0 < *r* < 0.5 an inverse relation is shown, but the ratio of temperature to concentration parameters Γ and the temperature difference parameter Λ showed increasing effects on the entropy generation, which can be confirmed from Figures 17 and 18, accordingly.

**Figure 13.** Curves of *Ns* with fixed *Gc* where θ = 0.1, ϕ = 0.5, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Γ = 0.4, *Nb* = 0.9, *Nt* = 0.5, *Br* = 1, *Gr* = 3, Λ = 0.4, Ω = 0.3, *Q* = 1. .

**Figure 14.** Curves of *Ns* with fixed *Nb* where θ = 0.1, ϕ = 0.8, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Γ = 0.4, *Gc* = 0.01, *Nt* = 0.9, *Br* = 1, *Gr* = 3, Λ = 1, Ω = 0.3, *Q* = 1. .

**Figure 15.** Curves of *Ns* with fixed *Nt* where θ = 0.1, ϕ = 0.8, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Γ = 0.4, *Gc* = 0.05, *Nb* = 0.9, *Br* = 1, *Gr* = 3, Λ = 1, Ω = 0.3, *Q* = 1. .

**Figure 16.** Curves of *Ns* with fixed Ω where θ = 0.1, ϕ = 0.8, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Γ = 0.4, *Gc* = 0.01, *Nt* = 0.5, *Br* = 1, *Gr* = 5, Λ = 1, *Nb* = 0.9, *Q* = 1. .

**Figure 17.** Curves of *Ns* with fixed Γ where θ = 0.3, ϕ = 0.8, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Ω = 0.3, *Gc* = 0.01, *Nt* = 0.5, *Br* = 1, *Gr* = 3, Λ = 0.4, *Nb* = 0.9, *Q* = 1..

**Figure 18.** Curves of *Ns* with fixed Λ where θ = 0.3, ϕ = 0.8, δ = 0.01, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 1, *z* = 0, *t* = 0.1, Ω = 0.3, *Gc* = 0.01, *Nt* = 0.5, *Br* = 1, *Gr* = 3, Γ = 0.4, *Nb* = 0.9, *Q* = 1..

Figures 19–22 are established to show the effects of physical factors on Bejan number *Be*, which is the ratio of two entropy generations. Figure 19 elucidates that the increase in *Gc* imposes an increase in Bejan number, which reflects the aspect that entropy due to heat transfer is less than that of total entropy in the lower region, but totally inverse readings are noted in the rest of the space. With the growing effects of Ω, Bejan number *Be* enhances through the flow domain, which can be found in Figure 20. From Figures 21 and 22 it is evident that *Be* decreases with increments in Γ and Λ, which indicates that the total entropy leads the same because of heat irreversibility.

**Figure 19.** Curves of *Be* with fixed *Gc* where θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.01, *V* = 0.3, λ<sup>1</sup> = 0.1, *z* = 0, *t* = 0.1, Ω = 0.3, Λ = 1, *Nt* = 0.5, *Br* = 1, *Gr* = 5, Γ = 0.4, *Nb* = 0.9, *Q* = 1..

**Figure 20.** Curves of *Be* with fixed Ω where θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.2, *V* = 0.3, λ<sup>1</sup> = 0.1, *z* = 0, *t* = 0.1, *Gc* = 0.01, Λ = 1, *Nt* = 0.5, *Br* = 1, *Gr* = 5, Γ = 0.4, *Nb* = 0.9, *Q* = 1..

**Figure 21.** Curves of *Be* with fixed Γ where θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.2, *V* = 0.3, λ<sup>1</sup> = 0.1, *z* = 0, *t* = 0.1, *Gc* = 0.01, Λ = 1, *Nt* = 0.5, *Br* = 1, *Gr* = 5, Ω = 0.3, *Nb* = 0.9, *Q* = 1..

**Figure 22.** Curves of *Be* with fixed Λ where θ = 0.1, ϕ = 0.5, δ = 0.1, ε = 0.2, *V* = 0.3, λ<sup>1</sup> = 0.1, *z* = 0, *t* = 0.1, *Gc* = 0.01, Γ = 0.4, *Nt* = 0.5, *Br* = 1, *Gr* = 5, Ω = 0.3, *Nb* = 0.9, *Q* = 1..

#### **6. Conclusions**

In the current article, entropy generation analysis and Bejan number characteristics were investigated for peristaltic propulsion of Jeffrey fluid by introducing nanoparticles passing through two eccentric asymmetric annuli. Analytical solutions for velocity, temperature, and nanoparticles' concentration were summarized. The pressure rise expression was evaluated numerically. Equations representing the laws of conservation were manipulated through the lubrication approach. The dimensionless phenomenon was also taken into account by incorporating some suitable transformations. Entropy generation number and Bejan number were achieved by substituting

the obtained values of temperature distribution, velocity profile, and nanoparticles' concentration. Effects of appertaining parameters were achieved by sketching diagrams. From the graphical features of the analysis, we gathered the following key observations:


**Author Contributions:** Conceptualization and methodology, A.R. and A.G.; software, I.K.; validation, S.U.K.; formal analysis and investigation, D.B.; writing—original draft preparation, K.S.N.; writing—review and editing, K.R.; funding acquisition, I.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number (RGP-2019-3).

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