**4. Graphical Analysis**

This study describes a critical analysis with which to approach two different fluid models that can disclose the properties of Synovial fluid when there is no slip at the boundaries and thin-film coating with non-Newtonian thick fluid (Synovial) is applied the walls. A non-linear coupled system of partial differential equations subject to boundary conditions is solved for shear-thinning and thickening models (Models 1 and 2). The complicated equations are solved by a regular perturbation method. To analyze graphically, Figures 1–11 have been sketched to measure the behavior of emerging factors on velocity distribution, pressure gradient profile, pressure rise, and trapping phenomena. Figures 1 and 2 show the effect of concentration parameter α and Weissenberg number We on the velocity component *u* for both models, respectively. It is extracted that velocity behaves in an opposite manner to shear-thinning and thickening models against multiple values of α. The Weissenberg number is helpful to analyze viscoelastic flows. It is the ratio of elastic forces and viscous forces. In Figure 2, we can understand that the velocity distribution of Model 1 behaves as an increasing quantity for higher values of Weissenberg number. This behavior reveals that elastic forces are dominant over viscous forces. However, the reaction of Model 2 is opposite as matched to Model 1. In Model 2, it can be noticed that viscous forces are dominant over elastic forces. This implies that the nature of shear thinning (Model 1) and the thickening (Model 2) are entirely different. Figure 3 displays the dependence of velocity on the average volume flow rate, as expected increase in the value of *Q* increases the flow velocity in both models.

**Figure 1.** Effects of α on velocity profile with *n* = −0.28, *Q* = 2, *x* = 0, *a* = 0.1, *b* = 0.1, *d* = 0.8, ϕ = 0.1, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

**Figure 2.** Effects of We on velocity profile with *n* = −0.28, *Q* = 1, *x* = 0, *a* = 0.1, *b* = 0.1, *d* = 0.8, ϕ = 0.1, α = 0.9 for (**a**) Model 1 and (**b**) Model 2.

**Figure 3.** Effects of *Q* on velocity profile with *n* = −0.28, α = 0.5, *x* = 0, *a* = 0.1, *b* = 0.1, *d* = 0.8, ϕ = 0.1, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

To compare the differences between two models, we include Figures 4–6 for pressure gradient d*p*/d*x*. In Figures 4 and 5, it is noted that with an excess of α and *Q* pressure gradient rises. As one can see, the prediction of the viscosity magnitude gets much larger values for the Model 2, unlike Model 1, whereas Weissenberg number We acts in an opposite way, that is, the change in pressure becomes larger throughout the flow and smaller for Model 2 than for Model 1 (see Figure 6).

**Figure 4.** Effects of α on pressure gradient with *n* = −0.28, *Q* = 0.1, *a* = 0.1, *b* = 0.1, *d* = 2, ϕ = 0.1, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

**Figure 5.** Effects of *Q* on pressure gradient with *n* = −0.28, α = 0.5, *a* = 0.5, *b* = 0.1, *d* = 2, ϕ = 0.1, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

**Figure 6.** Effects of We on pressure gradient with *n* = −0.28, *Q* = 0.1, *a* = 0.5, *b* = 0.1, *d* = 2, ϕ = 0.1, α = 0.5 for (**a**) Model 1 and (**b**) Model 2.

Figures 7–9 are plotted to determine the behavior of pumping rate in different regions. The pumping features can be examined by the pressure rise (Δ*p*) versus the average volume flow rate/mean flux *Q*. The complete area is divided into four quarters [13]. Figure 7a describes the pressure rise Δ*p* under the variety in values of α. It is observed that pressure rise is linearly dependent on flow rate, and free pumping is attained at *Q* = 0. It is evaluated here that while increasing α, the pressure rise Δ*p* decreases in Region II, whereas it increases in Region III. Figure 7b is plotted for Model 2, and

one can easily infer from it that dependence is not linear other than in α = 0.1. This figure indicates that with an increase in α, magnitude of Δ*p* decreases in Region III and has opposite behavior in other two regions. The effects of phase angle ϕ on Δ*p* are depicted in Figure 8. For Model 1, we can visualize that there is an increase of pressure rise in Region II when ϕ increases, while the reverse situation is found in Region II and remains consistent in Region I. It is entirely possible that the opposite behavior of Model 1 and Model 2 is due to the curvature in the flow domain. Figure 9a examines the influence of Weissenberg number We on Δ*p* for Model 1. It is noticed that Δ*p* increases by increasing We in Regions I and II, while the reduction in pressure rise is seen in Region III. On the other hand, the behavior of pressure gradient for We is also noted in Figure 9b. Model 2 shows a continuous increase in the Region I, hasty fall in Region II, and a drastic increase in Region III.

**Figure 7.** Effects of α on pressure rise with *n* = −0.28, *a* = 0.1, *b* = 0.1, *d* = 2, ϕ = 0.1, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

**Figure 8.** Effects of ϕ on pressure rise with *n* = −0.28, *a* = 0.1, *b* = 0.1, *d* = 0.1, α = 0.4, We = 0.05 for (**a**) Model 1 and (**b**) Model 2.

Trapping scheme is another important mechanism for analyzing flow pattern. However, in peristaltic (or sinusoidal) motion, a closed contour of streamlines can be examined at time-averaged flow rate and different values of amplitude. This phenomenon is known as trapping. According to the physiological point of view, the fluids can be trapped due to continuing movements of smooth boundaries, which are beneficial to adequately propel the working biological liquid from one point to another point. Due to proper prorogation, the working organs can stay alive for a long time without any difficulty. Therefore, the trapping phenomena can be observed by sketching stream functions against the concentration parameter α and the volume flow rate *Q.* Figures 10 and 11 are drawn to show the trapping phenomena. Figure 10a–c is illustrated for Model 1. It is observed that for changing values of α, a large bolus is formed at the center that decreases in size and increases in α. For Model 2, Figure 10d–f shows as α increases the bolus formed above *y* = 0 decreases in size, whereas below *y* = 0 it increases, and more boluses are obtained with large values of α. Figure 11 shows the effect of variation of *Q* on trapping. It can be analyzed that with an increase in *Q*, bolus decreases and increases in size above and below *y* = 0, respectively. The present investigation is also suggested for three-dimensional flow configuration with appropriate assumptions and modifications.

**Figure 9.** Effects of We on pressure rise with *n* = −0.28, *a* = 0.1, *b* = 0.1, *d* = 0.1, ϕ = 0.1, α = 0.4 for (**a**) Model 1 and (**b**) Model 2.

**Figure 10.** Stream lines for different values of α, α = 3, 3.2, 3.4: (**a**–**c**) for Model 1, (**d**–**f**) for Model 2. The other parameters are *n* = −0.2, *a* = 0.05, *b* = 0, *d* = 0.1, ϕ = 0.1, We = 0.05, *Q* = 5.

**Figure 11.** Stream lines for different values of *Q* = 2, 3, and 4: (**a**–**c**) for Model 1, (**d**–**f**) for Model 2. The other parameters are *n* = −0.2, *a* = 0.05, *b* = 0, *d* = 0.1, ϕ = 0.1, We = 0.05, α = 3.

#### **5. Conclusions**

In the current analysis, we examined theoretically the peristaltic motion of Synovial fluid in the two-dimensional asymmetric channel in the presence of coating on the walls exposing thin-film layers. The Synovial fluid has viscoelastic material; it can be described under specific physical conditions such as non-Newtonian fluid. We have considered two models for viscosity to capture shear-thinning properties and viscosity dependence on the concentration of hyaluronic acid. Analytic solutions for velocity, concentration, and pressure gradient are first produced using the regular perturbation method, and then the behavior of pertinent parameters is examined and discussed graphically. The expression of pressure rise is obtained numerically. The contours have also been drawn to explain the action of the trapping bolus phenomenon. The model with the shear-thinning index is directly dependent on the concentration of hyaluronic acid, which seems to be appropriate. According to our knowledge, no studies have been presented before that can describe the concentration effects on shear-thinning and thickening models for the peristaltic flow of Synovial fluid. Solutions are carried out for velocity, concentration field, and pressure gradient. The behavior of all the governing parameters is shown and scrutinized. The present analysis is also applicable for experimental investigation and assurance to give reliance for the significance of the governing nonlinear-boundary value problem.

**Author Contributions:** Data Curation, M.M.B.; Formal Analysis, A.R.; Validation, H.A.A.-O.; Funding Acquisition, H.A.A.-O.; Methodology, A.R.; Resources, A.Z.

**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.

**Acknowledgments:** The authors are very thankful to University of Education, Lahore, Jauharabad Campus for administrative and technical support.

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