Impact of Velocity Second Slip and Inclined Magnetic Field on Peristaltic Flow Coating with Jeffrey Fluid in Tapered Channel

Abstract

The peristaltic flow of velocity second slip boundary conditions and inclined magnetic field of Jeffrey fluid by means of heat and mass transfer in asymmetric channel was inspected in the present study. Leading equations described the existing flow were then simplified under lubrication approach. Therefore, exact solutions of stream function, concentration and temperature were deduced. Further, the numerical solutions of pressure rise and pressure gradient were computed using Mathematica software. Furthermore, the effect of the second slip parameter was argued via graphs. It has been depicted that this kind of slip is mandatory and very imperative to foresee the physical model. On the other hand, false results will be obtained.

1. Introduction

The peristaltic motion induced by channel or tube boundaries has a main role of fluid transport in living organisms and industrial pumping. Additionally, it has attracted attention in the fields of engineering and physiology. This transport is a means of fluid flow in an elastic path by the processes of contraction and expansion. In particular, such flows of viscous and non-Newtonian fluids are widely familiar in several biological systems including the human body in the transport of urine from the kidney to the bladder; chyme movement in the gastrointestinal tract, bile ducts, ureter, esophagus; spermatozoa in ducts efferent of the male reproductive tract; blood circulation in blood vessels; and movement of ovum in female fallopian tubes. Technical roller and finger pumps also function under this type of mechanism. In addition, with the existing of heat transfer, peristalsis is imperative in many processes such as oxygenation and hemodialysis. Further, heat transfer is also noteworthy in the treatment of diseased tissues in cancer. The cram of magnetohy drodynamic (MHD) peristaltic flow is useful as it is used in the reduction of bleeding during surgeries, targeted transfer of drugs via magnetic particles as drug carries, and MRI (magnetic resonance imaging) to diagnose diseases. It also has a pivotal role in the motion of physiological fluids including blood and blood pump machines. Furthermore, mass transfer in peristaltic flow occurs during the chemical breakdown of food, amalgamation of gastric juices with food, diffusive and ionic flows by means of membrane channels, diffusive oxygen transmission in tissue, drug delivery inside the body, and in other digestion processes.

There area lot of models for non-Newtonian fluids due to the complexity of fluids behaviour, like Carreau [1,2,3], Herschel [4,5], Oldroyd [6,7], Williamson [8], Johnson [9,10,11], Casson [12], Couette [13] and further as in [14,15,16,17,18,19,20,21]. Amongst these, the Jeffrey fluid model is considered as the simplest linear model that presents non-Newtonian fluid properties in a way which may enable the researchers to attain exact and analytical solutions. See, for example [22,23,24,25]. Kothandapani and Srinivas [26] have investigated peristaltic transport for Jeffrey fluid under consequences of a magnetic field in an asymmetric channel under the premise of a low Reynolds number and a long wavelength. Tripathi et al. [27] studied MHD peristaltic flow of Jeffrey fluid by means of a finite length cylindrical tube. Further, Nadeem et al. [28] examined the peristaltic flow of MHD Jeffrey fluid in eccentric cylinders. Khan et al. [29] investigated peristaltic transport for Jeffrey fluid with variable viscosity via a porous medium in an asymmetric channel. Srinivas and Pushparaj [30] have presented non-linear peristaltic flow in an inclined asymmetric channel.

In 1827, Navier [31] stated that shear stress at surface is linearly proportional to slip at surface. Fluids revealing slip effects are vital in polishing internal cavities and artificial heart valves. In particular, the application of this condition in peristaltic flows has perfect relevance in the field of polymers and physiology. Studies towards this point of research have been recently taken into account and a wide range of analytical and numerical studies have been reported in [32,33] and [34]. In a porous channel, effects of wall slip conditions and heat transfer on peristaltic transport of MHD Newtonian fluid with elastic wall properties have been discussed by Sirinivas et al. [35]. Hayat et al. [36] introduced a mathematical model in order to study the slip effects of heat and mass transfer on peristaltic transport of MHD power-law fluid and second grade fluid in the channel by flexible walls. Further, Hayat et al. [37] and [38] examined the influence of slip conditions and wall properties in the planar channel on MHD peristaltic flow of Maxwell fluid, and Williamson fluid in the non-uniform channel by heat and mass transfer, respectively. Nadeem and Akram [39] presented effects of partial slip on peristaltic flow of MHD Newtonian fluid in an asymmetric channel. They obtained the solutions using the method of Adomian decomposition and showed that trapping reduces with an increase of the velocity slip parameter, while pressure rise increases with an increase in the slip parameter. Hayat et al. [40,41] have analyzed effects of the slip condition on peristaltic flow of Phan-Thien-Tanner and of an Oldroyd 6-constant fluid, respectively. Mishra and Rao [42] investigated the effects of peristaltic flow of Newtonian fluid in an asymmetric channel. Akram and Nadeem [43] studied consequences with different waveforms of partial slip and nanofluid on peristaltic transport of non-Newtonian fluid. Recently, Hina et al. [44] investigated the peristaltic flow of pseudoplastic fluid with wall properties in a curved channel by heat or mass transfer.

In their important study, Roşca and Pop [45] showed that the second order slip flow model is essential to predict flow characteristics precisely. Very recently, Aly [46,47] and Aly and Vajravelu [48] have studied the effect of second velocity slip on fluid flow. In these studies, it was reported that these type of boundary conditions is compulsory and should be taken into consideration, otherwise, false results will be gained. As mentioned above, there are a considerable number of published papers regarding the effect of the first slip parameter, however, very less consideration has been given to peristaltic flows in the presence of the velocity second slip condition. Recently, Aly and Ebaid [49] presented an exact solution for the outcome of second slip on peristaltic flow of nanofluid in an asymmetric channel.

The intent of the current study is, therefore, to examine the effect of velocity second slip in non-Newtonian fluids by heat and mass transfer in the presence of an inclined magnetic field over an inclined tapered asymmetric channel, as many researchers have recently givenconsiderable attention to this geometry, for example [50,51,52]. As per our knowledge, no effort has been reported yet to discuss this multidimensional analysis, even in the absence of heat and mass transfer; hence, this study may be helpful in this direction of research. The present governing equations for motion, concentration and energy are simplified by assumptions of long wavelength approximation. Then, exact solutions of reduced equations are outlines. Therefore, with help of Mathematica software, many graphical outcomes are plotted and reported for various involved physical parameters of interest.

2. Mathematical Formulation

The peristaltic motion of non-Newtonian incompressible fluid in a vertical tapered asymmetric channel, under effects of a constant magnetic field is considered. It is assumed that a wave train is moving with velocity c along non-uniform walls. In addition, we also assume that the channel andmagnetic field are inclined at angles $\alpha$ and $\mathsf{\Theta }$, respectively. Further, upper and lower walls of the channel are sustained at temperature $T_{{}_0}$ and $T_{{}_1}$, respectively. For the present flow, $U$ and V are velocities in $X$ and $Y$ directions, respectively, in fixed frame. The upper and lower walls $H_1$ and $H_2,$ respectively, of tapered asymmetric channel in fixed frame are defined as:

$$\begin{array}{c}Y=H_1=d_{{}_1}+k^{\ast }X+a_{{}_1}\cos \left[\frac{2\pi }{\lambda }\left(X-ct\right)\right]\\ Y=H_2=-d_{{}_2}-k^{\ast }X-b_{{}_1}\cos \left[\frac{2\pi }{\lambda }\left(X-ct\right)+\phi \right]\end{array}$$

(1)

where $a_1$ and $b_1$ are amplitudes of waves; $\lambda$ is wave length; $d_{{}_1}+d_{{}_2}$ is width of channel; $k^{\ast }$($k^{\ast }<<1$) is non-uniform parameter; $c$ is velocity of propagation; $t$ is time; phase difference $\varphi$ varies in range $0\le \mathsf{\phi }\le \mathsf{\pi };\mathsf{\phi }=0$ corresponds to symmetric channel by waves out of phase and $\varphi =\pi$ waves are in phase; and further, $a_1,b_1,d_1,d_2$ and $\varphi$ satisfy the condition [24].

$$a_1^2+b_1^2+2a_1{b}_1\cos \varphi \le {\left(d_1+d_2\right)}^2.$$

An equation that governs flow in the presence of gravity consequences and an inclined magnetic field are defined as [25].

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(2)

$$\begin{array}{c}\rho \left(\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\right)=-\frac{\partial p}{\partial X}+\frac{\partial }{\partial X}\left(S_{XX}\right)+\frac{\partial }{\partial Y}\left(S_{XY}\right)-\sigma B_0^2\cos \theta (U\cos \theta -V\sin \theta )\\ +\rho g\sin \alpha \\ \end{array}$$

(3)

$$\begin{array}{c}\rho \left(\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\right)=-\frac{\partial p}{\partial Y}+\frac{\partial }{\partial X}\left(S_{YX}\right)+\frac{\partial }{\partial Y}{S}_{YY}+\sigma B_0^2\sin \mathsf{\Theta }(U\cos \mathsf{\Theta }-V\sin \mathsf{\Theta })\\ -\rho g\cos \alpha \end{array}$$

(4)

$$\begin{array}{c}{C}_{{}_p}\left(\frac{\partial T}{\partial t}+U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}\right)=\frac{K_{{}_1}}{\rho }\left(\frac{{\partial }^{2}T}{\partial X^2}+\frac{{\partial }^{2}T}{\partial Y^2}\right)+\upsilon (\frac{1}{1+{\lambda }_1}\left(1+{\lambda }_2\left(\frac{\partial }{\partial t}+U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y}\right)\right)\\ \left(2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right))\end{array}$$

(5)

$$\frac{\partial C}{\partial t}+U\frac{\partial C}{\partial X}+V\frac{\partial C}{\partial Y}=D_m\left(\frac{{\partial }^{2}C}{\partial X^2}+\frac{{\partial }^{2}C}{\partial Y^2}\right)+\frac{D_m{K}_{{}_T}}{T_m}\left(\frac{{\partial }^{2}T}{\partial X^2}+\frac{{\partial }^{2}T}{\partial Y^2}\right)$$

(6)

where $\rho$, $p$, $\nu$, $\sigma$, $g$, $K_{{}_1}$, $C_{{}_p}$, $T$, $D_m$, $T_m$, $K_{{}_T}$ and $C$ represent constant density, pressure, kinematic viscosity, electrical conductivity, acceleration caused by gravity, thermal conductivity, specific heat, temperature, coefficient of mass diffusivity, mean temperature, thermal diffusion ratio and concentration of fluid, respectively.

For the Jeffrey fluid model, extra stress tensor $\mathbf{S}$ is given as [26].

$$\mathbf{S}=\frac{\mu }{1+{\lambda }_{{}_1}}\left(\dot{\mathsf{\gamma }}+{\lambda }_{{}_2}\ddot{\mathsf{\gamma }}\right)$$

(7)

where ${\lambda }_{{}_1}$ is ratio of relaxation to retardation times; $\dot{\gamma }$ is shear rate; $\mu$ is viscosity of fluid; ${\lambda }_{{}_2}$ is retardation time; and dots indicate differentiation with respect to time. Extra stress tensor $\mathbf{S}$ in component form is defined as:

$$\begin{array}{c}{S}_{XX}=\frac{2\mu }{1+{\lambda }_1}\left(1+{\lambda }_2\left(\frac{\partial }{\partial t}+(U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y})\right)\right)\frac{\partial U}{\partial X}\hfill \\ S_{XY}=\frac{\mu }{1+{\lambda }_1}\left(1+{\lambda }_2\left(\frac{\partial }{\partial t}+(U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y})\right)\right)\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\hfill \\ S_{YY}=\frac{2\mu }{1+{\lambda }_1}\left(1+{\lambda }_2\left(\frac{\partial }{\partial t}+(U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y})\right)\right)\frac{\partial V}{\partial Y}\hfill \end{array}$$

Furthermore, we know that the wave frame $(x,$$y)$ and fixed frame $(X,$$Y)$ are related by the following transformations:

$$x=X-ct,y=Y,u=U-c,v=V,\mathrm{and}p\left(x\right)=p\left(X,t\right).$$

(8)

Let us define the following non-dimensional quantities:

$$\begin{gathered} \stackrel{-}{\mathrm{x}}=\frac{\mathrm{x}}{\mathsf{\lambda }},\stackrel{-}{\mathrm{y}}=\frac{\mathrm{y}}{{\mathrm{d}}_1},\stackrel{-}{\mathrm{u}}=\frac{\mathrm{u}}{\mathrm{c}},\stackrel{-}{\mathrm{v}}=\frac{\mathrm{v}}{\mathrm{c}},\mathsf{\delta }=\frac{{\mathrm{d}}_1}{\mathsf{\lambda }},\mathrm{d}=\frac{{\mathrm{d}}_2}{{\mathrm{d}}_1},\stackrel{-}{\mathrm{p}}=\frac{{\mathrm{d}}_1^2\mathrm{p}}{\mathsf{\mu }\mathrm{c}\mathsf{\lambda }},\stackrel{-}{\mathrm{t}}=\frac{\mathrm{ct}}{\mathsf{\lambda }}{,\mathrm{h}}_1=\frac{{\mathrm{H}}_1}{{\mathrm{d}}_1}, \\ {\mathrm{h}}_2=\frac{{\mathrm{H}}_2}{{\mathrm{d}}_2},\mathrm{a}=\frac{{\mathrm{a}}_1}{{\mathrm{d}}_1},\mathrm{b}=\frac{{\mathrm{b}}_1}{{\mathrm{d}}_1},\mathrm{Re}=\frac{{\mathrm{cd}}_1}{\mathrm{v}},\stackrel{-}{\mathsf{\Psi }}=\frac{\mathsf{\Psi }}{{\mathrm{cd}}_1},\mathrm{Fr}=\frac{{\mathrm{c}}^2}{{\mathrm{gd}}_1},\mathsf{\theta }=\frac{{\mathrm{T}-\mathrm{T}}_0}{{\mathrm{T}}_1{-\mathrm{T}}_0}, \\ \mathrm{Sr}=\frac{{\mathsf{\rho }\mathrm{D}}_{\mathrm{m}}{\mathrm{K}}_{\mathrm{t}}\left({\mathrm{T}}_1{-\mathrm{T}}_0\right)}{{\mathrm{T}}_{\mathrm{m}}\mathsf{\mu }\left({\mathrm{C}}_1{-\mathrm{C}}_0\right)},\mathrm{Sc}=\frac{\mathsf{\mu }}{{\mathsf{\rho }\mathrm{D}}_{\mathrm{m}}},\mathrm{Ec}=\frac{{\mathrm{c}}^2}{{\mathrm{C}}_{\mathrm{p}}\left({\mathrm{T}}_1{-\mathrm{T}}_0\right)},\stackrel{}{\mathrm{Pr}}=\frac{{\mathsf{\rho }\mathsf{\nu }\mathrm{C}}_{\mathrm{p}}}{{\mathrm{K}}_1},\stackrel{-}{\mathrm{S}}=\frac{{\mathrm{Sd}}_1}{\mathsf{\mu }\mathrm{c}}, \\ \mathrm{M}=\sqrt{\frac{\mathsf{\sigma }}{\mathsf{\mu }}}{\mathrm{B}}_0{\mathrm{d}}_1,\mathsf{\Phi }=\frac{{\mathrm{C}-\mathrm{C}}_0}{{\mathrm{C}}_1{-\mathrm{C}}_0} \end{gathered}$$

(9)

where $\mathrm{Re}$ is Reynolds number; $Fr$ is Froude number; $Sr$ is Soret number; $Sc$ is Schmidt number; $Ec$ is Eckret number; $\mathrm{Pr}$ is Prandtl number; $M$ is Hartmann number; $\theta$ is temperature of fluid in dimensionless form; and $\mathsf{\Phi }$ is concentration of fluid in dimensionless form.

With the help of Equations (7) and (8), Equations (2)–(6), in terms of stream function $\mathsf{\Psi }$ (dropping the bars, $u=\frac{\partial \mathsf{\Psi }}{\partial y},v=-\delta \frac{\partial \mathsf{\Psi }}{\partial x})$, take following form:

$$\begin{array}{c}\mathrm{Re}\delta \left({\mathsf{\Psi }}_y{\mathsf{\Psi }}_{xy}-{\mathsf{\Psi }}_x{\mathsf{\Psi }}_{yy}\right)=-\frac{\partial p}{\partial x}+\delta \frac{\partial }{\partial x}\left(S_{xx}\right)+\frac{\partial }{\partial y}\left(S_{xy}\right)-\\ M^2\cos \mathsf{\Theta }\left(\left({\mathsf{\Psi }}_y+1\right)\cos \mathsf{\Theta }+\delta {\mathsf{\Psi }}_x\sin \mathsf{\Theta }\right)+\frac{\mathrm{Re}}{Fr}\sin \alpha \end{array}$$

(10)

$$\begin{array}{c}\mathrm{Re}{\delta }^3\left(-{\mathsf{\Psi }}_y{\mathsf{\Psi }}_{xx}+{\mathsf{\Psi }}_x{\mathsf{\Psi }}_{xy}\right)=-\frac{\partial p}{\partial y}+{\delta }^2\frac{\partial }{\partial x}\left(S_{yx}\right)+\delta \frac{\partial }{\partial y}\left(S_{yy}\right)+\\ M^2\delta \sin \mathsf{\Theta }\left(\left({\mathsf{\Psi }}_y+1\right)\cos \mathsf{\Theta }+\delta {\mathsf{\Psi }}_x\sin \mathsf{\Theta }\right)-\delta \frac{\mathrm{Re}}{Fr}\cos \alpha \end{array}$$

(11)

$$\begin{array}{c}\mathrm{Re}\delta \left({\mathsf{\Psi }}_y{\theta }_x-{\mathsf{\Psi }}_x{\theta }_y\right)=\frac{1}{\mathrm{Pr}}\left({\theta }_{yy}+{\delta }^2{\theta }_{xx}\right)+\frac{Ec}{\left(1+{\lambda }_{{}_1}\right)}\left(1+\frac{{\lambda }_{{}_2}c\delta }{d_{{}_1}}\left({\mathsf{\Psi }}_y\frac{\partial }{\partial x}-{\mathsf{\Psi }}_x\frac{\partial }{\partial y}\right)\right)\\ \left(4{\delta }^2{\mathsf{\Psi }}_{xy}^2+{\left({\mathsf{\Psi }}_{yy}-{\delta }^2{\mathsf{\Psi }}_{xx}\right)}^2\right)\end{array}$$

(12)

$$\mathrm{Re}\delta \left({\mathsf{\Psi }}_y{\mathsf{\Phi }}_x-{\mathsf{\Psi }}_x{\mathsf{\Phi }}_y\right)=\frac{1}{Sc}\left({\delta }^2{\mathsf{\Phi }}_{xx}+{\mathsf{\Phi }}_{yy}\right)+Sr\left({\delta }^2{\theta }_{xx}+{\theta }_{yy}\right)$$

(13)

where extra stress tensor forJeffrey fluid in component form is defined as:

$$\begin{array}{c}{S}_{xx}=\frac{2\delta }{1+{\lambda }_{{}_1}}\left(1+\frac{{\lambda }_{{}_2}c\delta }{d_{{}_1}}\left({\mathsf{\Psi }}_y\frac{\partial }{\partial x}-{\mathsf{\Psi }}_x\frac{\partial }{\partial y}\right)\right){\mathsf{\Psi }}_{xy}\hfill \\ S_{xy}=\frac{1}{1+{\lambda }_1}\left(1+\frac{{\lambda }_{{}_2}c\delta }{d_{{}_1}}\left({\mathsf{\Psi }}_y\frac{\partial }{\partial x}-{\mathsf{\Psi }}_x\frac{\partial }{\partial y}\right)\right)\left({\mathsf{\Psi }}_{yy}-{\delta }^2{\mathsf{\Psi }}_{xx}\right)\hfill \\ S_{yy}=-\frac{2\delta }{1+{\lambda }_{{}_1}}\left(1+\frac{{\lambda }_{{}_2}c\delta }{d_{{}_1}}\left({\mathsf{\Psi }}_y\frac{\partial }{\partial x}-{\mathsf{\Psi }}_x\frac{\partial }{\partial y}\right)\right){\mathsf{\Psi }}_{xy}\hfill \end{array}$$

(14)

Using Equation (14) and assumptions of long wavelength approximation, Equations (10)–(13) reduce in the form:

$$-\frac{\partial p}{\partial x}+\frac{\partial }{\partial y}\left(\frac{1}{1+{\lambda }_{{}_1}}\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}\right)-M^2{\cos }^2\mathsf{\Theta }\left({\mathsf{\Psi }}_y+1\right)+\frac{\mathrm{Re}}{Fr}\sin \alpha =0$$

(15)

$$-\frac{\partial p}{\partial y}=0$$

(16)

$$\frac{1}{\mathrm{Pr}}\frac{{\partial }^2\theta }{\partial y^2}+\frac{Ec}{\left(1+{\lambda }_{{}_1}\right)}{\left(\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}\right)}^2=0$$

(17)

$$\frac{1}{Sc}\frac{{\partial }^2\mathsf{\Phi }}{\partial y^2}+Sr\frac{{\partial }^2\theta }{\partial y^2}=0.$$

(18)

Elimination of pressure from Equation (15) to (16) gives:

$$\frac{{\partial }^2}{\partial y^2}\left(\frac{1}{1+{\lambda }_{{}_1}}\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}\right)-M^2{\cos }^2\mathsf{\Theta }\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}=0$$

(19)

$$\frac{1}{\mathrm{Pr}}\frac{{\partial }^2\theta }{\partial y^2}+\frac{Ec}{\left(1+{\lambda }_{{}_1}\right)}{\left(\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}\right)}^2=0$$

(20)

$$\frac{1}{Sc}\frac{{\partial }^2\mathsf{\Phi }}{\partial y^2}+Sr\frac{{\partial }^2\theta }{\partial y^2}=0.$$

(21)

The system of PDEs given above in Equation (19) through (21) is solved subject to the following boundary conditions:

$$\begin{array}{c}\mathsf{\Psi }=\frac{F}{2}\mathrm{at}y=h_{{}_1}=1+kx+a\cos 2\pi x\hfill \\ \mathsf{\Psi }=-\frac{F}{2}\mathrm{at}y=h_{{}_2}=-d-kx-b\cos (2\pi x+\phi )\hfill \\ \frac{\partial \mathsf{\Psi }}{\partial y}=-\frac{{\eta }_{{}_1}^{\ast }}{(1+{\lambda }_{{}_1})}\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}-\frac{{\eta }_{{}_2}^{\ast }}{\left(1+{\lambda }_{{}_1}\right)}\frac{{\partial }^3\mathsf{\Psi }}{\partial y^3}-1\mathrm{at}y=h_{{}_1}\hfill \\ \frac{\partial \mathsf{\Psi }}{\partial y}=\frac{{\eta }_{{}_1}^{\ast }}{\left(1+{\lambda }_{{}_1}\right)}\frac{{\partial }^2\mathsf{\Psi }}{\partial y^2}+\frac{{\eta }_{{}_2}^{\ast }}{\left(1+{\lambda }_{{}_1}\right)}\frac{{\partial }^3\mathsf{\Psi }}{\partial y^3}-1\mathrm{at}y=h_{{}_2}\hfill \end{array}$$

(22)

$$\begin{array}{c}\theta +\beta \frac{\partial \theta }{\partial y}=0\mathrm{at}y=h_{{}_1}\\ \theta -\beta \frac{\partial \theta }{\partial y}=1\mathrm{at}y=h_{{}_2}\end{array}$$

(23)

$$\begin{array}{c}\mathsf{\Phi }+\gamma \frac{\partial \mathsf{\Phi }}{\partial y}=0\mathrm{at}y=h_{{}_1}\\ \mathsf{\Phi }-\gamma \frac{\partial \mathsf{\Phi }}{\partial y}=1\mathrm{at}y=h_{{}_2}\end{array}$$

(24)

where $F$ is flux in wave frame; ${\eta }_{{}_1}^{\ast },$${\eta }_{{}_2}^{\ast },$$\beta$ and $\gamma$ represent 1st-order slip parameter, 2nd-order slip parameter, thermal slip parameter, and concentration slip parameter, respectively; $h_{{}_1}$ and $h_{{}_2}$ are thedimensionless form of surfaces of peristaltic walls.

3. Exact Solution of Problem

Exact solution of Equation (19) satisfying boundary conditions (22) can be deduced as:

$$\begin{array}{cc}\hfill \mathsf{\Psi }=& \frac{1}{L_{{}_{\mathrm{\infty }}}}(2\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)\left(\left(-h_{{}_1}-h_{{}_2}+2y\right)\left(F{\eta }_{{}_1}{m}^2+1\right)-{\eta }_{{}_2}{m}^2\left(F+h_{{}_1}-h_{{}_2}\right)\right)\hfill \\ & +2({\eta }_{{}_2}{m}^2(F-h_{{}_2})(\cosh \left(m\left(y-h_{{}_1}\right)\right)+\cosh \left(m\left(y-h_{{}_2}\right)\right)-1)-F{\eta }_{{}_1}m(\sinh \left(m\left(y-h_{{}_1}\right)\right)\hfill \\ & +\sinh \left(m\left(y-h_{{}_2}\right)\right))+F\cosh \left(m\left(y-h_{{}_1}\right)\right)-F\cosh \left(m\left(y-h_{{}_2}\right)\right)+h_{{}_1}({\eta }_{{}_2}{m}^2(\cosh \left(m\left(y-h_{{}_1}\right)\right)\hfill \\ & +\cosh \left(m\left(y-h_{{}_2}\right)\right)-1)-{\eta }_{{}_1}m\left(\sinh \left(m\left(y-h_{{}_1}\right)\right)+\sinh \left(m\left(y-h_{{}_2}\right)\right)\right)+\cosh \left(m\left(y-h_{{}_1}\right)\right)\hfill \\ & -\cosh \left(m\left(y-h_{{}_2}\right)\right)+1)+h_{{}_2}({\eta }_{{}_1}m\left(\sinh \left(m\left(y-h_{{}_1}\right)\right)+\sinh \left(m\left(y-h_{{}_2}\right)\right)\right)-\cosh \left(m\left(y-h_{{}_1}\right)\right)\hfill \\ & +\cosh \left(m\left(y-h_{{}_2}\right)\right)+1)-2y)+m\left(h_{{}_1}+h_{{}_2}-2y\right)(F{\eta }_{{}_2}^2{m}^4-{\eta }_{{}_1}\left(F{\eta }_{{}_1}{m}^2+2\right)\hfill \\ & -F)\sinh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right),\hfill \end{array}$$

(25)

where $m=$$M\cos \mathsf{\Theta }$, ${\eta }_{{}_1}=\frac{{\eta }_{{}_1}^{\ast }}{(1+{\lambda }_{{}_1})},$${\eta }_{{}_2}=\frac{{\eta }_{{}_1}^{\ast }}{(1+{\lambda }_{{}_1})}$ and $L_{{}_{\infty }}$ is a function of $x$ defined in the Appendix A.

Now making use of Equation (25) in Equation (20), the exact solution of Equation (20) is derived as:

$$\begin{array}{cc}\hfill \theta =& \frac{m^2\mathrm{Pr}Ec{\left(F+h_{{}_1}-h_{{}_2}\right)}^2}{L_0}(2\left(2m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)-1\right)+\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right)\right)\hfill \\ & +m({\eta }_{{}_1}({\eta }_{{}_1}m(4m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right)-\cosh \left(2m\left(y-h_{{}_1}\right)\right)-\cosh \left(2m\left(y-h_{{}_2}\right)\right)\hfill \\ & -2\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))+2(4m^2{y}^2\sinh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+\sinh \left(2m\left(y-h_{{}_1}\right)\right)\hfill \\ & -\sinh \left(2m\left(y-h_{{}_2}\right)\right)))+{\eta }_{{}_2}^2{m}^3(-(2(2m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right)\hfill \\ & +\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))+\cosh \left(2m\left(y-h_{{}_1}\right)\right)+\cosh \left(2m\left(y-h_{{}_2}\right)\right)))\hfill \\ & +2{\eta }_{{}_2}m({\eta }_{{}_1}m(\sinh \left(2m\left(y-h_{{}_1}\right)\right)+\sinh \left(2m\left(y-h_{{}_2}\right)\right)+2\sinh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))\hfill \\ & -\cosh \left(2m\left(y-h_{{}_1}\right)\right)+\cosh \left(2m\left(y-h_{{}_2}\right)\right)))-\cosh \left(2m\left(y-h_{{}_1}\right)\right)\hfill \\ & -\cosh \left(2m\left(y-h_{{}_2}\right)\right))+A_{{}_1}y+A_{{}_0},\hfill \end{array}$$

(26)

where $A_{{}_0},$$A_{{}_1}$ are functions of x and their values are computed by means of Equation (23) as:

$$\begin{array}{cc}\hfill A_{{}_0}=& \frac{1}{8\left(2\beta +h_{{}_1}-h_{{}_2}\right)\left({\lambda }_{{}_1}+1\right)}(8\left(\beta +h_{{}_1}\right)\left({\lambda }_{{}_1}+1\right)+\hfill \\ & \frac{1}{L_{{}_5}}(m^{2}Ec\mathrm{Pr}{\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_9}{L}_{{}_6}{}^2-2m^{2}Ec\mathrm{Pr}{L}_{{}_{11}}\left(\beta -h_{{}_2}\right){\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_6}{L}_{{}_4}\hfill \\ & +m^{2}Ec\mathrm{Pr}{\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_{10}}{L}_{{}_4}{}^2)),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill A_{{}_1}=& \frac{4m^{4}Ec\mathrm{Pr}{\left(F+h_{{}_1}-h_{{}_2}\right)}^2}{8\left({\lambda }_{{}_1}+1\right)\left(2\beta +h_{{}_1}-h_{{}_2}\right)L_8}({\eta }_{{}_2}^2\left(h_{{}_1}+h_{{}_2}\right)m^4\left(2\beta +h_{{}_1}-h_{{}_2}\right)\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right)\hfill \\ & -2\left(h_{{}_1}+h_{{}_2}\right)\left(2\beta +h_{{}_1}-h_{{}_2}\right){({\eta }_{{}_1}m\cosh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)+\sinh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m))}^2\hfill \\ & -4{\eta }_{{}_2}\cosh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)\left(2\beta m\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+\sinh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)\right)\hfill \\ & ({\eta }_{{}_1}m\cosh \left((\left(h_{{}_1}-h_{{}_2}\right)m)\right)+\sinh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)))-\frac{8\left({\lambda }_1+1\right)}{8\left({\lambda }_{{}_1}+1\right)\left(2\beta +h_{{}_1}-h_{{}_2}\right)}.\hfill \end{array}$$

(28)

With the help of Equation (26), exact solution of the concentration profile in Equation (21) is concluded as:

$$\begin{array}{cc}\hfill \mathsf{\Phi }=& \frac{m^2\mathrm{Pr}\mathrm{ScSr}Ec{\left(F+h_{{}_1}-h_{{}_2}\right)}^2}{L_0}(-2(2m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)-1\right)+\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))\hfill \\ & +m({\eta }_{{}_1}({\eta }_{{}_1}m(2(\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right)-2m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right))+\cosh \left(2m\left(y-h_{{}_1}\right)\right)\hfill \\ & +\cosh \left(2m\left(y-h_{{}_2}\right)\right))-2(4m^2{y}^2\sinh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+\sinh \left(2m\left(y-h_{{}_1}\right)\right)-\sinh \left(2m\left(y-h_{{}_2}\right)\right)))\hfill \\ & +{\eta }_{{}_2}^2{m}^3(2(2m^2{y}^2\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right)+\cosh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))+\cosh \left(2m\left(y-h_{{}_1}\right)\right)\hfill \\ & +\cosh \left(2m\left(y-h_{{}_2}\right)\right))-2{\eta }_{{}_2}m({\eta }_{{}_1}m(\sinh \left(2m\left(y-h_{{}_1}\right)\right)+\sinh \left(2m\left(y-h_{{}_2}\right)\right)\hfill \\ & +2\sinh \left(m\left(-h_{{}_1}-h_{{}_2}+2y\right)\right))-\cosh \left(2m\left(y-h_{{}_1}\right)\right)+\cosh \left(2m\left(y-h_{{}_2}\right)\right)))\hfill \\ & +\cosh \left(2m\left(y-h_{{}_1}\right)\right)+\cosh \left(2m\left(y-h_{{}_2}\right)\right))+A_{{}_3}y+A_{{}_2}\hfill \end{array}$$

(29)

where $A_{{}_2},$$A_{{}_3}$ are functions of x and their values are computed by means of Equation (24) as:

$$\begin{array}{cc}\hfill A_{{}_2}=& \frac{1}{8\left(2\gamma +h_{{}_1}-h_{{}_2}\right)\left({\lambda }_{{}_1}+1\right)}(8\left(\gamma +h_{{}_1}\right)\left({\lambda }_{{}_1}+1\right)+\frac{1}{L_5}(m^2\mathrm{Pr}\mathrm{ScSr}Ec{\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_2}{L}_{{}_4}{}^2\hfill \\ & +m^2\mathrm{Pr}\mathrm{ScSr}Ec{\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_1}{L}_{{}_6}^2-2m^2\mathrm{Pr}\mathrm{ScSr}Ec{L}_{{}_7}{\left(F+h_{{}_1}-h_{{}_2}\right)}^2{L}_{{}_3}{L}_{{}_4}))\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill A_{{}_3}=& \frac{4m^4\mathrm{Pr}\mathrm{ScSr}Ec{\left(F+h_{{}_1}-h_{{}_2}\right)}^2}{8\left({\lambda }_{{}_1}+1\right)\left(2\gamma +h_{{}_1}-h_{{}_2}\right)L_{{}_8}}({\eta }_{{}_2}^2\left(h_{{}_1}+h_{{}_2}\right)m^4\left(2\gamma +h_{{}_1}-h_{{}_2}\right)\left(-\left(\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+1\right)\right)\hfill \\ & +2\left(h_{{}_1}+h_{{}_2}\right)\left(2\gamma +h_{{}_1}-h_{{}_2}\right){({\eta }_{{}_{{}_1}}m\cosh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)+\sinh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m))}^2\hfill \\ & +4{\eta }_{{}_2}\cosh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)\left(2\gamma m\cosh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)+\sinh \left(\left(h_{{}_1}-h_{{}_2}\right)m\right)\right)\hfill \\ & ({\eta }_{{}_1}m\cosh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)+\sinh (\frac{1}{2}\left(h_{{}_1}-h_{{}_2}\right)m)))-\frac{8\left({\lambda }_{{}_1}+1\right)}{8\left({\lambda }_{{}_1}+1\right)\left(2\gamma +h_{{}_1}-h_{{}_2}\right)}.\hfill \end{array}$$

(31)

It should be noted that $L_{{}_0}-L_{{}_{11}}$ appeared in Equation (26) through (31) and are functions of $x$ defined in the Appendix A.

4. Different 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:

$$h_{{}_1}(x)=1+kx+a\sin 2\pi x,h_{{}_2}(x)=-d-kx-b\sin (2\pi x+\phi )$$

Multisinsoidal wave:

$$h_{{}_1}(x)=1+kx+a\sin 2n\pi x,h_{{}_2}(x)=-d-kx-b\sin (2n\pi x+\phi )$$

Triangular wave:

$$\begin{array}{c}{h}_{{}_1}(x)=1+kx+a\left[\frac{8}{{\pi }^3}{\displaystyle \sum _{m=1}^{\infty }}\frac{{\left(-1\right)}^{m+1}}{{\left(2m-1\right)}^2}\sin \left(2\pi \left(2m-1\right)x\right)\right]\hfill \\ h_{{}_2}(x)=-d-kx-b\left[\frac{8}{{\pi }^3}{\displaystyle \sum _{m=1}^{\infty }}\frac{{\left(-1\right)}^{m+1}}{{\left(2m-1\right)}^2}\sin \left(2\pi \left(2m-1\right)x+\phi \right)\right]\hfill \end{array}$$

Trapezoidal wave:

$$\begin{array}{c}{h}_{{}_1}(x)=1+kx+a\left[\frac{32}{{\pi }^2}{\displaystyle \sum _{m=1}^{\infty }}\frac{\sin \frac{\pi }{8}\left(2m-1\right)}{{\left(2m-1\right)}^2}\sin \left(2\pi \left(2m-1\right)x\right)\right]\hfill \\ h_{{}_2}(x)=-d-kx-b\left[\frac{32}{{\pi }^2}{\displaystyle \sum _{m=1}^{\infty }}\frac{\sin \frac{\pi }{8}\left(2m-1\right)}{{\left(2m-1\right)}^2}\sin \left(2\pi \left(2m-1\right)x+\phi \right)\right]\hfill \end{array}$$

Square wave:

$$\begin{array}{c}{h}_{{}_1}(x)=1+kx+a\left[\frac{4}{\pi }{\displaystyle \sum _{m=1}^{\infty }}\frac{{\left(-1\right)}^{m+1}}{\left(2m-1\right)}\cos \left(2\left(2m-1\right)\pi x\right)\right]\hfill \\ h_{{}_2}(x)=-d-kx-b\left[\frac{4}{\pi }{\displaystyle \sum _{m=1}^{\infty }}\frac{{\left(-1\right)}^{m+1}}{\left(2m-1\right)}\cos \left(2\left(2m-1\right)\pi x+\phi \right)\right]\hfill \end{array}$$

Sawtooth wave:

$$\begin{array}{c}{h}_{{}_1}(x)=1+kx+a\left[\frac{8}{{\pi }^3}{\displaystyle \sum _{m=1}^{\infty }}\frac{\sin (2\pi mx)}{m}\right]\hfill \\ h_{{}_2}(x)=-d-kx-b\left[\frac{8}{{\pi }^3}{\displaystyle \sum _{m=1}^{\infty }}\frac{\sin (\left(2\pi mx)+\phi \right)}{m}\right]\hfill \end{array}$$

5. Special Cases

If ${\eta }_{{}_{{}_1}}={\eta }_{{}_2}=\beta =\gamma =k=0,$ 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:

• Kothandapani and Srinivas [26], when $\mathsf{\Theta }=\alpha =0,$
• Srinivas and Pushparaj [30], for $\mathsf{\Theta }={\lambda }_{{}_1}=0$ and
• Mishra and Rao [42], at $\mathsf{\Theta }=M=\alpha ={\lambda }_{{}_1}=0.$

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.

Figure 1, Figure 2, Figure 3 and Figure 4 are displayed to observe behaviour of pressure rise for diverse values of Jeffrey parameter ${\lambda }_{{}_1}$, non-uniform parameter $k$, Reynolds number $\mathrm{Re}$ and inclination angle $\mathsf{\Theta }$. It is noted from Figure 1; Figure 2 that behaviour of pressure rise decreases in retrograde pumping $\left(\mathsf{\Delta }p>0,Q<0\right),$ peristaltic pumping $\left(\mathsf{\Delta }p>0,Q>0\right)$ and free pumping $\left(\mathsf{\Delta }p=0\right)$ regions with an increase in ${\lambda }_{{}_1}$ and $k$, whereas the behaviour of pressure rise is quite opposite in the co-pumping region $\left(\mathsf{\Delta }p<0,Q>0\right).$ In this region, with an increase in ${\lambda }_{{}_1}$ and $k$, pressure rise increases. Figure 3 presents the behaviour of pressure rise for diverse values of $\mathrm{Re}$. From this figure, we depicted that pressure rise increases in all pumping regions with an increase in values of $\mathrm{Re}$. It is shown from Figure 4 that in the retrograde pumping $\left(\mathsf{\Delta }p>0,Q<0\right)$ region, pressure rise increases with an increase in $\mathsf{\Theta }$, whereas in the co-pumping region $\left(\mathsf{\Delta }p<0,Q>0\right)$, behaviour of pressure rise decreases with an increase in $\mathsf{\Theta }.$

Figure 5, Figure 6, Figure 7 and Figure 8 are plotted in order to notice the behaviour of pressure gradient for various values of $\alpha ,$ Jeffrey parameter ${\lambda }_{{}_1}$, Hartmann number $M$ and non-dimensional slip parameters ${\eta }_{{}_1}$ and ${\eta }_{{}_2}$. It is illustrated that for $x\in [0,0.2]$ and $x\in [0.8,1],$ the pressure gradient is small so that flow can easily pass without the compulsion of a large pressure gradient, whereas in region $x\in [0.2,0.8],$ the pressure gradient increases with an increase in $\alpha$, and it decreases with an increase in ${\lambda }_{{}_1}$, $M$, ${\eta }_{{}_1}$ and ${\eta }_{{}_2}$, 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.

The behaviour of temperature profiles for diverse values of $Ec$, ${\lambda }_{{}_1}$ and $\mathrm{Pr}$ are shown in Figure 10, Figure 11 and Figure 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 ${\lambda }_{{}_1}.$ It has been observed from Figure 11 that the temperature profile decreases with an increase in ${\lambda }_{{}_1}.$ It has been observed from Figure 12 that the temperature profile increases with an increase in values of $\mathrm{Pr}$. This happens due to the direct relation of $\mathrm{Pr}$ with the temperature profile.

Figure 13, Figure 14 and Figure 15 demonstrate theconcentration profile for diverse values of $Ec$, ${\lambda }_{{}_1}$, $Sr$ 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 ${\lambda }_{{}_1}$ 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 $Sc$.

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 ${\lambda }_{{}_1}$, $M$, ${\eta }_{{}_1}$ and ${\eta }_{{}_2}$. It has been observed from Figure 16, Figure 17 and Figure 18 that the size of the trapping bolus decreases with an increase in values of ${\lambda }_{{}_1}$, $M$, ${\eta }_{{}_1}$ and ${\eta }_{{}_2}.$ 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. Shows the comparison of the present work with existing literature in tabular form.

y	Present Work	Nadeem andAkram [24]	Kothandapani and Srinivas [26]	Srinivas and Pushparaj [30]	Mishra and Rao [42]
−1.109	−1.000	−1.000	−1.000	−1.000	−1.000
−1.009	−0.932753	−0.714278	−0.678228	−0.757937	−0.862651
−0.909	−0.8743	−0.505606	−0.459713	−0.565103	−0.736612
−0.809	−0.825682	−0.352378	−0.310456	−0.410812	−0.621188
−0.709	−0.785355	−0.239918	−0.208522	−0.287568	−0.516378
−0.609	−0.752035	−0.157452	−0.138929	−0.189381	−0.422183
−0.509	−0.724664	−0.097082	−0.0914497	−0.11148,	−0.338602
−0.409	−0.702372	−0.0530242	−0.0591067	−0.0500785	−0.265636
−0.309	−0.684449	−0.0210585	−0.0371466,	−0.00219338	−0.203285
−0.209	−0.670326	0.00187701	−0.0223419	0.0345026	−0.151548
−0.109	−0.659554	0.0179793	−0.0125168	0.0617927	−0.110426
−0.009	−0.65179	0.0287909	−0.00622739	0.081003	−0.0799192
0.091	−0.646787	0.0353474	−0.00254937	0.0930671	−0.0600266
0.191	−0.644387	0.0382767	−0.000942165	0.0985713	−0.0507487
0.291	−0.644513	0.0378597	−0.00116958	0.0977831	−0.0520854
0.391	−0.647169	0.0340561	−0.00326504	0.0906641	−0.0640367
0.491	−0.65244	0.0265019	−0.00753651	0.0768684	−0.0866028
0.591	−0.660492	0.0144732	−0.0146117	0.0557255	−0.119783
0.691	−0.671583,	−0.00318205	−0.0255305	0.0262081	−0.163579
0.791	−0.686065	−0.0281551	−0.0418976	−0.0131183	−0.217989
0.891	−0.704397	−0.0628381	−0.0661183	−0.0641648	−0.283014
0.991	−0.727164	−0.110553	−0.101752	−0.129412	−0.358653
1.091	−0.755089	−0.175871	−0.154037	−0.212031	−0.444907
1.191	−0.789059	−0.265049	−0.230655	−0.316035	−0.541775
1.291	−0.830155	−0.386629	−0.342868	−0.446481	−0.649259
1.391	−0.879684	−0.552257	−0.507167	−0.609706	−0.767357
1.491	−0.939219	−0.777798	−0.747697	−0.813642	−0.896069
1.591	−1.000	−1.000	−1.000	−1.000	−1.000

is constructed.

7. Conclusions

In this research, we have investigated the effect of velocity second slip on non-Newtonian fluids by heat and mass transfer in the presence of an inclined magnetic field over an inclined tapered asymmetric channel. Governing equations of motion, energy and concentration were simplified via assumptions of long wavelength approximation. Then, exact solutions of reduced equations were calculated. Graphical results were plotted and reported for various involved physical parameters of concern. The main results can be typified as:

• The results presented in [24,26,30,42] were considered as special cases of the present work.
• The pressure rise decreases in retrograde, peristaltic and free pumping regions and increases in co-pumping regions, with an increase in relaxation to retardation times ${\lambda }_{{}_1}$ and non-uniform parameter $k$.
• The pressure rise increases in all pumping regions with an increase in Reynolds number $\mathrm{Re}$.
• The pressure gradient increases with an increase in $\alpha$ and decreases with an increase in ${\lambda }_{{}_1}$, Hartmann number $M$, slip parameter ${\eta }_{{}_1}$ and ${\eta }_{{}_2}.$
• The temperature profile increases with an increase in values of Eckret number $Ec$ and decreases with an increase in relaxation to retardation times ${\lambda }_{{}_1}.$
• The concentration profile decreases with an increase in Soret number $Sr$ and Schmidt number $Sc.$
• The size of the trapping bolus decreases with an increase in values of relaxation to retardation times ${\lambda }_{{}_1},$ Hartmann number $M$, slip parameter ${\eta }_{{}_1}$ and ${\eta }_{{}_2}.$