**4. Results and Discussion**

The objective of the current analysis is to present the closed form solutions of MHD Newtonian fluid with variable viscosity. The expression for pressure rise per wavelength and frictional forces are difficult to integrate analytically; therefore, numerical integration is used to evaluate the integrals. Figures 2–4 are plotted for pressure rise and friction force against flow rate *Q* when viscosity is constant. In Figure 2, it is observed that pressure rise increases with an increase of *M* up to *Q* < 1.7, after which the curves intersect each other and, finally, it gives an opposite behavior. The effects *M* on *F*(0)(for outer tube) and *F*(1) (for inner tube) are presented in Figures 3 and 4. It is depicted from Figures 3

and 4 that with an increase in *M*, both *F*(0) and *F*(1) decrease *M* for small *Q* and, finally, the behavior is reversed at the end. A comparison of the velocity field for constant viscosity case is made between the Adomian decomposition solution and perturbation solutions obtained by [6]. (see Figure 5). Figures 6–9 are prepared when (viscosity) μ = *r*. It is observed from Figure 6 that in the retrograde (Δ*p* > 0, *Q* < 0) and peristaltic pumping (Δ*p* > 0, *Q* > 0) regions, the pressure rise decreases with an increase in amplitude ratio ϕ. Figures 7 and 8 show that *F*(0) and *F*(1) give an opposite behavior as compared with Δ*p*. The velocity field increases with the increase in *M* and the maximum value of the velocity is at the center (see Figure 9). Figures 10–13 are prepared when the viscosity μ = <sup>1</sup> *r* . It is observed from Figure 10 that with an increase in *r*1, the pressure rise decreases in the retrograde (Δ*p* > 0, *Q* < 0), peristaltic pumping (Δ*p* > 0, *Q* > 0), and copuming (Δ*p* < 0, *Q* > 0) regions. It is depicted from Figures 11 and 12 that with an increase in *r*1, both *F*(0) and *F*(1) decrease for small *Q* and, finally, the behavior is reversed at the end. The velocity profile for different values of *M* for the case when viscosity is μ = <sup>1</sup> *<sup>r</sup>* is shown in Figure 13. It is observed from Figure 13 that the magnitude value of the velocity profile decreases with an increase in *M*.

**Figure 2.** The variation of Δ*p* with *Q* for different values of *M* at *r*<sup>1</sup> = 0.4, ϕ = 0.4, when μ = 1.

**Figure 3.** The variation of friction force *F*(0) (outer tube) with *Q* for different values of *M* at *r*<sup>1</sup> = 0.4, ϕ = 0.4, when μ = 1.

**Figure 4.** The variation of friction force *F*(1) (inner tube) with *Q* for different values of *M* at *r*<sup>1</sup> = 0.4, φ = 0.4, when μ = 1.

**Figure 5.** Comparison with the existing literature.

**Figure 6.** The variation of Δ*p* with *Q* for different values of ϕ at *M* = 3, *r*<sup>1</sup> = 0.1, when μ = *r*.

**Figure 7.** The variation of friction force *F*(0) (outer tube) with *Q* for different values of φ at *r*<sup>1</sup> = 0.1, *M* = 3, when μ = *r*.

**Figure 8.** The variation of friction force *F*(1) (inner tube) with *Q* for different values of ϕ at *r*<sup>1</sup> = 0.1, *M* = 3, when μ = *r*.

**Figure 9.** Velocity profiles for different values of *M* at *t* = 1, *z* = 1, ϕ = 1, when μ = *r*.

**Figure 10.** The variation of Δ*p* with *Q* for different values of *r*<sup>1</sup> at *M* = 3, ϕ = 0.4, when μ = <sup>1</sup> *r* .

**Figure 11.** The variation of friction force *F*(0) (outer tube) with *Q* for different values of *r*<sup>1</sup> at ϕ = 0.4, *M* = 3, when μ = <sup>1</sup> *r* .

**Figure 12.** The variation of friction force *F*(1) (inner tube) with *Q* for different values of *r*<sup>1</sup> at ϕ = 0.4, *M* = 3, when μ = <sup>1</sup> *r* .

**Figure 13.** Velocity profiles for different values of *M* at *t* = 1, *z* = 1, ϕ = 1, when μ = <sup>1</sup> *r* .

#### *Trapping Phenomenon*

The trapping phenomenon is an interesting phenomenon in peristaltic motion, which is discussed in Figures 14–18 for the case when μ = 1, μ = *r*, and μ = <sup>1</sup> *<sup>r</sup>* . Stream lines for different values of ϕ for the case when μ = 1 are shown in Figure 14. It is observed from Figure 6 that with an increase in amplitude ratio ϕ, the size of the trapping bolus increases. Stream lines for different values of *M* and ϕ for the case when μ = *r* are shown in Figures 15 and 16. It is observed from Figure 15 that the size of the trapping bolus decreases with an increase in Hartmann number *M*. The size of the trapping bolus increases with an increase in amplitude ratio ϕ (see Figure 16). Stream lines for different values of *M* and ϕ for the case when μ = <sup>1</sup> *<sup>r</sup>* are shown in Figures 17 and 18. It is observed from the Figures that the size of the trapping bolus increases with an increase in Hartmann number *M* and amplitude ratio ϕ.

**Figure 14.** Streamlines for two different values of ϕ for (**a**) ϕ = 0.1 and (**b**) ϕ = 0.101. The other parameters are chosen as *M* = 2, *Q* = 0.41, *r*<sup>1</sup> = 0.65 when μ = 1.

**Figure 15.** Streamlines for two different values of *M* for (**a**) *M* = 1 and (**b**) *M* = 0.6. The other parameters are chosen as ϕ = 0.2, *Q* = 0.45, *r*<sup>1</sup> = 1 when μ = *r*.

**Figure 16.** Streamlines for two different values of ϕ for (**a**) ϕ = 0.29 and (**b**) ϕ = 0.3. The other parameters are chosen as *M* = 1, *Q* = 0.45, *r*<sup>1</sup> = 1.1 when μ = *r*.

**Figure 17.** Streamlines for two different values of *M* for (**a**) *M* = 0.2 and (**b**) *M* = 0.1. The other parameters are chosen as ϕ = 0.1, *Q* = 0.4, *r*<sup>1</sup> = 1.1 when μ = <sup>1</sup> *r* .

**Figure 18.** Streamlines for two different values of *r*<sup>1</sup> for (**a**) *r*<sup>1</sup> = 1.11 and (**b**) *r*<sup>1</sup> = 1.13. The other parameters are chosen as *M* = 0.2, *Q* = 0.4, ϕ = 0.1 when μ = <sup>1</sup> *r* .

#### **5. Conclusions**

In the present analysis, peristaltic flow was discussed for MHD Newtonian fluid through the gap between two coaxial tubes, where the fluid viscosity was treated as variable. In addition, the inner tube was considered to be at rest, while the outer tube has the sinusoidal wave travelling down its motion. Further, the governing equations are simplified under the assumptions of long wavelength and low Reynolds number. The solution of the problem under discussion is computed analytically using the Adomian decomposition method. The results of the proposed problem are discussed through graphs. The main findings are summarized as follows:


**Author Contributions:** Conceptualization, S.A. and E.H.A.; Methodology, S.A. and F.A.; Software, S.A. and E.H.A., F.A.; Validation, S.A., F.A. and S.N.; Formal Analysis, E.H.A.; Investigation, S.A.; Resources, S.N.; Data Curation: F.A.; Writing—Original Draft Preparation, S.A.; Writing—Review and Editing: S.A. and F.A.; Visualization, E.H.A.; Supervision, S.N.; Project Administration, S.A. and F.A.; Funding Acquisition, S.A. and F.A.

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

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