**4. Di**ff**erent Wave Shapes**

Non-dimensional expressions for six considered wave forms are given as [43]. Expressions for sinusoidal, multisinsoidal, triangular, square, trapezoidal and sawtooth waves are derived from the Fourier series.

Sinusoidal wave:

$$h\_1(\mathbf{x}) = \ 1 + k\mathbf{x} + a\sin 2\pi \mathbf{x},\\ h\_2(\mathbf{x}) = \ -d - k\mathbf{x} - b\sin(2\pi \mathbf{x} + q).$$

Multisinsoidal wave:

$$h\_1(\mathbf{x}) = \ 1 + k\mathbf{x} + a\sin 2n\pi\mathbf{x},\\ h\_2(\mathbf{x}) = \ -d - k\mathbf{x} - b\sin(2n\pi\mathbf{x} + \varphi).$$

Triangular wave:

$$\begin{aligned} h\_1(\mathbf{x}) &= \ 1 + k\mathbf{x} + a \Big[ \frac{8}{\pi^3} \sum\_{m=1}^{\infty} \frac{(-1)^{m+1}}{(2m-1)^2} \sin(2\pi(2m-1)\mathbf{x}) \Big] \\ h\_2(\mathbf{x}) &= \ -d - k\mathbf{x} - b \Big[ \frac{8}{\pi^3} \sum\_{m=1}^{\infty} \frac{(-1)^{m+1}}{(2m-1)^2} \sin(2\pi(2m-1)\mathbf{x} + q) \Big] \end{aligned}$$

Trapezoidal wave:

$$\begin{array}{rcl}h\_1(\mathbf{x}) &=& 1 + k\mathbf{x} + a\Big[\frac{32}{\pi^2} \sum\_{m=1}^{\infty} \frac{\sin\frac{\pi}{8}(2m-1)}{\left(2m-1\right)^2} \sin\left(2\pi(2m-1)\right)\mathbf{x}\Big] \\ h\_2(\mathbf{x}) &=& -d - k\mathbf{x} - b\Big[\frac{32}{\pi^2} \sum\_{m=1}^{\infty} \frac{\sin\frac{\pi}{8}(2m-1)}{\left(2m-1\right)^2} \sin\left(2\pi(2m-1)\right)\mathbf{x} + \boldsymbol{\varphi}\big] \end{array}$$

Square wave:

$$\begin{array}{rcl}h\_1(\mathbf{x})&=&\mathbf{1}+k\mathbf{x}+a\Big[\frac{4}{\pi}\sum\_{m=1}^{\infty}\frac{(-1)^{m+1}}{(2m-1)}\cos(2(2m-1)\,\pi\mathbf{x})\Big]\\h\_2(\mathbf{x})&=&-d-k\mathbf{x}-b\Big[\frac{4}{\pi}\sum\_{m=1}^{\infty}\frac{(-1)^{m+1}}{(2m-1)}\cos(2(2m-1)\,\pi\mathbf{x}+\varrho)\Big]\end{array}$$

Sawtooth wave:

$$\begin{array}{rcl} h\_1(\mathbf{x}) &=& 1 + k\mathbf{x} + a \Big[ \begin{array}{cc} \frac{\mathbf{8}}{\pi \overline{\mathbf{x}}^3} & \sum\_{m=1}^{\infty} \frac{\sin(2\pi mx)}{m} \end{array} \Big] \\ h\_2(\mathbf{x}) &=& -d - k\mathbf{x} - b \Big[ \begin{array}{cc} \frac{\mathbf{8}}{\pi \overline{\mathbf{x}}^3} & \sum\_{m=1}^{\infty} \frac{\sin((2\pi mx) + \varphi)}{m} \end{array} \Big] \end{array}$$

#### **5. Special Cases**

If η<sup>1</sup> = η<sup>2</sup> = β = γ = *k* = 0, results of Nadeem and Akram [24] can be recovered as a special case for present study. In addition to the vanishing of these values and in absence of heat and mass transfer, the following results can be obtained as further special cases:


### **6. Results and Discussion**

In the proceeding section, numerical results of current problems are conferred through graphs. Mathematica software is utilized to analyze expressions for pressure gradient and pressure rise numerically.

Figures 1–4 are displayed to observe behaviour of pressure rise for diverse values of Jeffrey parameter λ<sup>1</sup> , non-uniform parameter *k*, Reynolds number Re and inclination angle Θ. It is noted from Figure 1; Figure 2 that behaviour of pressure rise decreases in retrograde pumping (Δ*p* > 0, *Q* < 0), peristaltic pumping (Δ*p* > 0, *Q* > 0) and free pumping (Δ*p* = 0) regions with an increase in λ<sup>1</sup> and *k*, whereas the behaviour of pressure rise is quite opposite in the co-pumping region (Δ*p* < 0, *Q* > 0). In this region, with an increase in λ<sup>1</sup> and *k*, pressure rise increases. Figure 3 presents the behaviour of pressure rise for diverse values of Re. From this figure, we depicted that pressure rise increases in all pumping regions with an increase in values of Re. It is shown from Figure 4 that in the retrograde pumping (Δ*p* > 0, *Q* < 0) region, pressure rise increases with an increase in Θ, whereas in the co-pumping region (Δ*p* < 0, *Q* > 0), behaviour of pressure rise decreases with an increase in Θ.

**Figure 1.** Variation of Δ*p* with *Q* for different values of λ<sup>1</sup> for fixed *a* = 0.7, α = 0.2, *b* = 0.7, *d* = 1.5, Θ = <sup>π</sup> <sup>6</sup> , *<sup>M</sup>* <sup>=</sup> 0.5, Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.3, η<sup>2</sup> = 0.4, *k* = 0.5.

**Figure 2.** Variation of Δ*p* with *Q* for different values of *k* for fixed *a* = 0.7, α = 0.2, *b* = 0.7, *d* = 1.5, Θ = <sup>π</sup> <sup>6</sup> , *<sup>M</sup>* <sup>=</sup> 0.5, Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>2</sup> = 0.4, η<sup>1</sup> = 0.5, λ<sup>1</sup> = 0.1.

**Figure 3.** Variation of Δ*p* with *Q* for different values of Re for fixed *a* = 0.7, α = 0.2, *b* = 0.7, *d* = 1.5, Θ = <sup>π</sup> <sup>6</sup> , <sup>λ</sup><sup>1</sup> <sup>=</sup> 0.4, *<sup>M</sup>* <sup>=</sup> 0.5, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.3, η<sup>2</sup> = 0.4, *k* = 0.5.

**Figure 4.** Variation of Δ*p* with *Q* for different values of Θ for fixed *a* = 0.7, α = 0.2, *b* = 0.7, *d* = 1.5, λ<sup>1</sup> = 0.7, *M* = 1, *R* = 0.3, ϕ = <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.3, η<sup>2</sup> = 0.4, *k* = 0.5.

Figures 5–8 are plotted in order to notice the behaviour of pressure gradient for various values of α, Jeffrey parameter λ<sup>1</sup> , Hartmann number *M* and non-dimensional slip parameters η<sup>1</sup> and η<sup>2</sup> . It is illustrated that for *x* ∈ [0, 0.2] and *x* ∈ [0.8, 1], the pressure gradient is small so that flow can easily pass without the compulsion of a large pressure gradient, whereas in region *x* ∈ [0.2, 0.8], the pressure

gradient increases with an increase in α, and it decreases with an increase in λ<sup>1</sup> , *M*, η<sup>1</sup> and η<sup>2</sup> , so more pressure gradient is necessary to maintain the flux to pass. Figure 9 shows the behaviour of the pressure gradient for diverse wave forms. It has been observed from Figure 9 that pressure gradient is maximum for square waves.

**Figure 5.** Variation of *dp*/*dx* with *x* for different values of α for fixed *a* = 0.7, *b* = 0.7, *d* = 1.5, *Fr* = 0.6, Θ = <sup>π</sup> <sup>3</sup> , *<sup>M</sup>* <sup>=</sup> 0.5, Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , η<sup>1</sup> = 0.4 η<sup>2</sup> = 0.5, λ<sup>1</sup> = 0.3, *k* = 0.1, *Q* = 1.

**Figure 6.** Variation of *dp*/*dx* with *x* for different values of λ<sup>1</sup> for fixed *a* = 0.7, α = 0.3, *b* = 0.7, *d* = 1.5, Θ = <sup>π</sup> <sup>3</sup> , Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.5, η<sup>2</sup> = 0.4, *k* = 0.1, *M* = 0.5, *Q* = 0.8..

**Figure 7.** Variation of *dp*/*dx* with *x* for different values of *M* for fixed *a* = 0.7, α = 0.3, *b* = 0.5, *d* = 1.8, Θ = <sup>π</sup> <sup>3</sup> , Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.5, η<sup>2</sup> = 0.4, *k* = 0.5, λ<sup>1</sup> = 0.1, *Q* = 1.

**Figure 8.** Variation of *dp*/*dx* with *x* for different values of η<sup>1</sup> and η<sup>2</sup> for fixed *a* = 0.7, α = 0.2, *b* = 0.7, *d* = 1.5, *Fr* = 0.6, Θ = <sup>π</sup> <sup>3</sup> , *<sup>M</sup>* <sup>=</sup> 0.5, Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , λ<sup>1</sup> = 0.3, *k* = 0.1, *Q* = 1.

**Figure 9.** *Cont.*

**Figure 9.** Variation of *dp*/*dx* with *x* for different wave forms for fixed *a* = 0.9, α = 0.3, *b* = 0.1, *d* = 2, Θ = <sup>π</sup> <sup>3</sup> , *<sup>M</sup>* <sup>=</sup> 0.5, Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.5, η<sup>2</sup> = 0.4, λ<sup>1</sup> = 0.5, *k* = 0.1.

The behaviour of temperature profiles for diverse values of *Ec*, λ<sup>1</sup> and Pr are shown in Figures 10–12. It has been observed from Figure 10 that the temperature profile increases with an increase in *Ec*. This phenomena is physically valid as *Ec* shows a direct connection with temperature profile. Figure 11 depicts variation of the temperature profile for diverse values of λ<sup>1</sup> . It has been observed from Figure 11 that the temperature profile decreases with an increase in λ<sup>1</sup> . It has been observed from Figure 12 that the temperature profile increases with an increase in values of Pr. This happens due to the direct relation of Pr with the temperature profile.

Figures 13–15 demonstrate theconcentration profile for diverse values of *Ec*, λ<sup>1</sup> , *Sr* and *Sc*. It has been observed from Figure 13, Figure 14 that concentration profiles show opposite behaviour in comparison with the temperature profile. This observable fact physically holds as the temperature profile shows its inverse relationship with the concentration profile. It has been observed from Figure 13 that the concentration profile decreases with an increase in values of *Ec*. It has been depicted from Figure 14 that with an increase in λ<sup>1</sup> that the concentration profile increases. Figure 15 shows the concentration profile for diverse values of *Sr* and *Sc*. It has been shown in Figure 15 that the concentration profile decreases with an increase in *Sr* and *Sc*.

**Figure 10.** Temperature profile for different values of *Ec* for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> 4 , β = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, *k* = 0.2, λ<sup>1</sup> = 0.2, *M* = 0.1, Pr = 1, *Q* = 4, *x* = 1.

**Figure 11.** Temperature profile for different values of λ<sup>1</sup> for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> 4 , β = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, *k* = 0.2, *M* = 0.1, Pr = 1, *Q* = 4, *Ec* = 0.2, *x* = 1.

**Figure 12.** Temperature profile for different values of Pr for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> 4 , β = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, λ<sup>1</sup> = 1, *k* = 0.2, *M* = 0.1, *Q* = 4, *Ec* = 0.2, *x* = 1.

**Figure 13.** Concentration profile for different values of *Ec* for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> 4 , γ = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, *k* = 0.2, λ<sup>1</sup> = 0.2, *M* = 0.1, Pr = 1, *Q* = 4, *Sc* = 0.3, *Sr* = 0.4, *x* = 1.

**Figure 14.** Concentration profile for different values of λ<sup>1</sup> for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> 4 , γ = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, *k* = 0.2, *M* = 0.9, Pr = 1, *Q* = 5, *Sc* = 0.6, *Sr* = 0.4, *Ec* = 0.8, *x* = 1.

**Figure 15.** Concentration profile for different values of *Sr* and *Sc* for fixed *a* = 0.5, *b* = 1.2, *d* = 1.5, ϕ = <sup>π</sup> <sup>4</sup> , γ = 0.0009, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, λ<sup>1</sup> = 0.9, *k* = 0.2, *M* = 0.8, Pr = 0.5, *Q* = 5, *Ec* = 0.8, *x* = 1.

In addition, an interesting observable fact in peristaltic flow is trapping. This is basically a pattern of an internally circulating bolus of fluid via closed stream lines. The trapping phenomena is discussed for different values of λ<sup>1</sup> , *M*, η<sup>1</sup> and η<sup>2</sup> . It has been observed from Figures 16–18 that the size of the trapping bolus decreases with an increase in values of λ<sup>1</sup> , *M*, η<sup>1</sup> and η<sup>2</sup> . Figure 19 shows the behaviour of stream lines for diverse wave forms. It has been observed that in all considered wave forms that the trapped bolus increases in size and its size is smaller in the case of the triangular wave when compared with the other three wave forms. Figure 20 shows comparison of the present work with existing literature. It is observed in this figure that the exact solution of the present work and existing literature satisfies the boundary conditions. Moreover, the magnitude value of the velocity profile is maximum in the case of the present work and Nadeem and Akram [24]. In order to show the comparison of the present work with existing literature in tabular form, Table 1 is constructed.


**Table 1.** Shows the comparison of the present work with existing literature in tabular form.

**Figure 16.** Stream lines for different values of λ<sup>1</sup> . (**a**) for λ<sup>1</sup> = 0.1, (**b**) for λ<sup>1</sup> = 1.6. The other parameters are *a* = 0.7, *b* = 0.7, *d* = 1, Θ = <sup>π</sup> <sup>5</sup> , *M* = 2.2, ϕ = 0.01, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, *k* = 0.1, *Q* = 1.5.

**Figure 17.** Stream lines for different values of *M*. (**a**) for *M* = 2.2. (**b**) for *M* = 2.54. The other parameters are *a* = 0.7, *b* = 0.7, *d* = 1, Θ = <sup>π</sup> <sup>5</sup> , *M* = 2.2, ϕ = 0.01, η<sup>1</sup> = 0.009, η<sup>2</sup> = 0.001, λ<sup>1</sup> = 1, *k* = 0.1, *Q* = 1.5.

**Figure 18.** Stream lines for different values of η<sup>1</sup> and η2.(**a**) for η<sup>1</sup> = 0.01 and η<sup>2</sup> = 0.001,(**b**) for η<sup>1</sup> = 0.09 and η<sup>2</sup> = 0.002. The other parameters are *a* = 0.7, *b* = 0.7, *d* = 1, Θ = <sup>π</sup> <sup>5</sup> , *M* = 1.8, ϕ = 0.01, λ<sup>1</sup> = 1.2, *k* = 0.1, *Q* = 1.5.

**Figure 19.** *Cont.*

**Figure 19.** Stream lines fordifferent wave forms. (**a**) for multisinsoidal wave, (**b**) for trapezoidal wave, (**c**) for triangular wave, (**d**) for square wave, (**e**) for sawtooth wave, for fixed *a* = 0.9, α = 0.3, *b* = 0.1, *d* = 2, Θ = <sup>π</sup> <sup>3</sup> , *<sup>M</sup>* <sup>=</sup> 0.5 Re <sup>=</sup> 0.4, <sup>ϕ</sup> <sup>=</sup> <sup>π</sup> <sup>4</sup> , *Fr* = 0.6, η<sup>1</sup> = 0.5, η<sup>2</sup> = 0.4, λ<sup>1</sup> = 0.5, *k* = 0.1.

**Figure 20.** Comparison of the present work with existing literature.
