Nonlinear peristaltic motion of a Johnson–Segalman fluid in a tapered asymmetric channel

Abstract

The peristaltic flow analysis in a tapered asymmetric channel has been made for a Johnson–Segalman fluid. The tapered asymmetric channel is assumed to be formed due to a peristaltic wave train on the non-uniform walls having different amplitudes and phase. Two-dimensional equations of a Johnson–Segalman fluid have been simplified by treating a long wavelength and low Reynolds number approximations. The reduced equations are then solved for the stream function, axial velocity and axial pressure gradient using a regular perturbation technique. The expressions for the pressure rise, axial velocity and stream function are sketched and the reasons for the variations observed in various physical parameters are interpreted with valid theory. It has been noticed that peristaltic pumping region and free pumping decrease with an increase in non-uniform parameter and the situation is quite complimentary to the case of augmented pumping. It has also been observed that the size of the tapped bolus decreases with an increase in Weissenberg number.

Keywords

Johnson–Segalman fluid; Peristaltic transport; Tapered asymmetric channel; Perturbation technique

1. Introduction

A variety of applications of a peristaltic motion in non-Newtonian fluids have led to renewed interest among the scientists and researchers. Such applications include urine transport from kidney to bladder, chime motion in the gastrointestinal tract, swallowing food through the esophagus, vasomotion of small blood vessels, movement of spermatozoa in human reproductive tract, blood pumps in dialysis and heart lung machine. Many studies are now found in the literature which includes analytical, numerical and experimental measurements [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and [12] on the peristaltic flow in different geometries.

Though the earlier studies on peristaltic motion paid much attention on Newtonian fluids there are also systems involving non-Newtonian motion in which the relation between shear stress and shear rate appears to be nonlinear in character. In fact physiological fluids frequently fall under non-Newtonian category. It is quite appropriate to observe the dynamics of the fluids by taking their non-Newtonian behavior into deliberation. Only partial information on the peristaltic transport of non-Newtonian fluids is available in the literatures [13], [14], [15], [16], [17] and [18]. Some constitutive formulation including the Johnson–Segalman model and Giesekus model [19] and [20] contains a nonmonotone shear-stress–shear-rate curve. In the present problem, we focus on the Johnson–Segalman fluid. Johnson and Seglman [21] begin with a constitutive equation that is derived systematically from both the molecular theory of Gaussian networks, and the molecular bead-spring model with Hookean springs. Johnson–Segalman fluid is considered as one among the subclass of viscoelastic fluids. This model has an ability to explain the “spurt” phenomenon. The term “spurt” has been used for the description of large increase in the volume to a small increase in the driving pressure gradient. Researchers [22], [23], [24] and [25] often deployed this model to explain the spurt phenomenon. Several experimentalists have correlated the spurt with a slip at a wall. The three distinct flows of the Johnson–Segalman type have been extensively studied by Rao and Rajagopal [26].

The quest of getting accurate methods for solving resulted nonlinear models involving higher order is of utmost concern for many researchers in this field today. Various analytical methods have been put to use successfully to obtain solutions of classical nonlinear differential equations such as the method of Perturbation method, Adomian decomposition method, Differential transform method and Homotopy perturbation method, [27], [28], [29], [30] and [31]. Hajmohammadi et al. [32] proposed based on semi-analytical methods to solve the conjugate heat transfer problems. The two semi analytical algorithms, Differential Transform Method (DTM) and Adomian Decomposition Method (ADM) are examined for solving a characteristic value problem occurring in linear stability analysis by Hajmohammadi and Nourazar [33]. The same authors discussed that the semi-analytical solution for the nonlinear integro-differential equation occurring in the problem is handled easily and accurately by implementing the Differential Transform Method (DTM) [34].

Physiologists have also observed that the intrauterine fluid flow due to myometrial contractions is peristaltic-type motion and the myometrial contractions may occur in both symmetric and asymmetric directions, De Vries et al. [35]. Eytan et al. [36] had looked at the characterization of non-pregnant woman’s uterine contractions is extremely complicated as they are composed of variable amplitudes, a range of frequencies and different wavelengths. The flows in an asymmetric channel generated by peristaltic waves propagating on the walls with different amplitudes and phase have been observed by Mishra and Rao [37]. In general, asymmetric wall oscillation in non-uniform channels may also exist in biological conduits, e.g., the uterus [38], [39], [40] and [41]. Maxey and Riley [42] disused the particle motion under non-uniform flow and analytically derived an equation for the particle motion through which many forces are measured.

The peristaltic transport of Johnson–Segalman fluid under a magnetic field has been studied by Elshahed and Haroun [43]. Nadeem and Akbar [44] have carried out a detailed analysis on the peristaltic flow of a Johnson–Segalman fluid in a non-uniform tube. The effect of magnetic field on the motion of a Johnson–Segalman fluid has been sculpted by Hayat and Ali [45]. The peristaltic transport of MHD Johnson–Segalman fluid in a planar channel has been discussed by Hayat et al. [46]. Peristaltic motion of Johnson–Segalman fluid in an asymmetric and planar channel was also reported by Hayat et al. [47]. Nadeem and Akbar [48] have made significant study on the peristaltic flow of Johnson–Segalman fluid in a non-uniform tube with heat transfer. Numerous theoretical works have been performed by many authors to investigate the various effects related to the peristaltic transport of a non-Newtonian fluid [49], [50], [51], [52], [53] and [54].

Besides on the foregoing works, we are interested to study the effect of a Johnson–Segalman fluid in the peristaltic wave train on the non-uniform walls of having different amplitudes and phase. It is worthwhile to note that such analysis does not seem to be made available in the existing literature. The governing equations have been simplified by presuming long wavelength approximation and low Reynolds number. With the aid of regular perturbation method, asymptotic solutions for stream function and pressure gradient were obtained. The expressions for pressure rise per wave length were also calculated. The effects of different physical parameters appearing in the problem have been discussed in lucid manner and shown graphically for better understanding.

2. Governing equations

Consider an incompressible, Johnson–Segalman fluid confined in a two dimensional infinite asymmetric channel. We employ a rectangular coordinate system with X parallel and Y normal to the channel walls. Moreover, we consider an infinite wave train traveling with velocity c along the channel walls. The asymmetry in the channel is induced by assuming the non-uniform peristaltic wave train on the walls to have different amplitudes and phases Fig. 1. The shape of the channel walls is

$H_{2}(X{\mbox{,}}t^{'})=d+k^{'}X+a_{2}sin\left({\frac {2\pi X}{\lambda }}-\right.$$\left.{\frac {{ct}^{'}}{\lambda }}\right){\mbox{,}}\quad \ldots \quad {\mbox{upper wall}}$

(1)

$H_{1}(X{\mbox{,}}t^{'})=-d-k^{'}X-a_{1}sin\left({\frac {2\pi X}{\lambda }}-\right.$$\left.{\frac {{ct}^{'}}{\lambda }}+\phi \right){\mbox{,}}\quad \ldots \quad {\mbox{lower wall}}$

(2)

Here ${\textstyle a_{1}}$ and ${\textstyle a_{2}}$ are the amplitudes of the waves, λ   is the wavelength, ${\textstyle k^{'}}$ is the non-uniform parameter and ${\textstyle \phi \in \left[0{\mbox{,}}\pi \right]}$ is the phase difference. Note that ${\textstyle \phi =0}$ corresponds to the channel with waves out of phase and ${\textstyle \phi =\pi }$ describes waves in phase. Moreover ${\textstyle a_{1}{\mbox{,}}\quad a_{2}{\mbox{,}}\quad d_{1}{\mbox{,}}\quad d_{2}}$ and ϕ satisfy the following inequality at the inlet of divergent or outlet of convergent channel, otherwise both walls get collided,

$a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}cos\phi \leqslant {\left(2d\right)}^{2}{\mbox{.}}$

(3)

The equations governing the flows of an incompressible fluid are

$divV=0{\mbox{,}}\quad divT=\rho {\frac {dV}{dt}}{\mbox{,}}$

(4)

where the Cauchy stress tensor for a Johnson–Segalman fluid is [19], [20], [21] and [22]

$T=-pI+\sigma {\mbox{,}}$

(5)

$\sigma =2\mu D+S{\mbox{,}}$

(6)

$S+m\left[{\frac {dS}{dt}}+S(W-eD)+{\left(W-eD\right)}^{T}S\right]=$$2\eta D{\mbox{,}}$

(7)

$D={\frac {1}{2}}\left[L+L^{T}\right]{\mbox{,}}W={\frac {1}{2}}\left[L-\right.$$\left.L^{T}\right]{\mbox{,}}L=gradV{\mbox{,}}$

(8)

Figure 1. Schematic diagram of a tapered asymmetric channel.

The above equations include the scalar pressure p, the identity tensor I, the dynamic viscosities μ and η, the relaxation time m, the slip parameter e and the respective symmetric and skew symmetric part of velocity gradient D and W. Note that model (4) reduces to the Maxwell fluid model for ${\textstyle e=1}$ and ${\textstyle \mu =0}$, we recover the classical Navier–Stokes fluid model.

For the problem under consideration, the velocity for two dimensional flows is defined

$V=\left[U(X{\mbox{,}}Y{\mbox{,}}t^{'}){\mbox{,}}V(X{\mbox{,}}Y{\mbox{,}}t^{'}){\mbox{,}}0\right]{\mbox{,}}$

(9)

From Eqs. (3), (4), (5), (6), (7) and (8), we obtain

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

(10)

$\rho \left({\frac {\partial }{\partial t^{'}}}+U{\frac {\partial }{\partial X}}+\right.$$\left.V{\frac {\partial }{\partial Y}}\right)U=-{\frac {\partial P}{\partial X}}+$$\mu \left({\frac {{\partial }^{2}}{\partial X^{2}}}+{\frac {{\partial }^{2}}{\partial Y^{2}}}\right)U+$${\frac {\partial S_{XX}}{\partial X}}+{\frac {\partial S_{XY}}{\partial Y}}{\mbox{,}}$

(11)

$\rho \left({\frac {\partial }{\partial t^{'}}}+U{\frac {\partial }{\partial X}}+\right.$$\left.V{\frac {\partial }{\partial Y}}\right)V=-{\frac {\partial P}{\partial Y}}+$$\mu \left({\frac {{\partial }^{2}}{\partial X^{2}}}+{\frac {{\partial }^{2}}{\partial Y^{2}}}\right)V+$${\frac {\partial S_{XY}}{\partial X}}+{\frac {\partial S_{YY}}{\partial Y}}{\mbox{,}}$

(12)

$2\eta {\frac {\partial U}{\partial X}}=S_{XX}+m\left[{\frac {\partial }{\partial t^{'}}}+\right.$$\left.U{\frac {\partial }{\partial X}}+V{\frac {\partial }{\partial Y}}\right]S_{XX}-$$2{emS}_{XX}{\frac {\partial U}{\partial X}}+m\left[(1-\right.$$\left.e){\frac {\partial V}{\partial X}}-(1+e){\frac {\partial U}{\partial Y}}\right]S_{XY}{\mbox{,}}$

(13)

$\eta \left({\frac {\partial U}{\partial Y}}+{\frac {\partial V}{\partial X}}\right)=$$S_{XY}+m\left[{\frac {\partial }{\partial t^{'}}}+U{\frac {\partial }{\partial X}}+\right.$$\left.V{\frac {\partial }{\partial Y}}\right]S_{XY}+$${\frac {m}{2}}\left[(1-e){\frac {\partial U}{\partial Y}}-\right.$$\left.(1+e){\frac {\partial V}{\partial X}}\right]S_{XX}+$${\frac {m}{2}}\left[(1-e){\frac {\partial V}{\partial X}}-\right.$$\left.(1+e){\frac {\partial U}{\partial Y}}\right]S_{YY}{\mbox{,}}$

(14)

$2\eta {\frac {\partial V}{\partial Y}}=S_{YY}+m\left[{\frac {\partial }{\partial t^{'}}}+\right.$$\left.U{\frac {\partial }{\partial X}}+V{\frac {\partial }{\partial Y}}\right]S_{YY}-$$2{emS}_{YY}{\frac {\partial V}{\partial Y}}+m\left[(1-\right.$$\left.e){\frac {\partial U}{\partial Y}}-(1+e){\frac {\partial V}{\partial X}}\right]S_{XY}{\mbox{.}}$

(15)

Defining the following non-dimensional quantities

${\begin{array}{rl}&x={\frac {X}{\lambda }}{\mbox{,}}\quad y={\frac {Y}{d}}{\mbox{,}}\quad t={\frac {{ct}^{'}}{\lambda }}{\mbox{,}}\quad u={\frac {U}{c}}{\mbox{,}}\quad v={\frac {V}{c}}{\mbox{,}}\quad \delta ={\frac {d}{\lambda }}{\mbox{,}}\quad h_{1}={\frac {H_{1}}{d}}{\mbox{,}}\quad h_{2}={\frac {H_{2}}{d}}{\mbox{,}}\quad p={\frac {d^{2}P}{c\lambda \mu }}{\mbox{,}}\\&S={\frac {d}{\mu c}}S(x){\mbox{,}}\quad p={\frac {d^{2}}{\lambda (\mu +\eta )c}}p(x){\mbox{,}}\quad Re={\frac {\rho cd}{\mu }}{\mbox{,}}\quad We={\frac {mc}{d}}{\mbox{,}}\quad a={\frac {a_{1}}{d}}{\mbox{,}}\quad b={\frac {a_{2}}{d}}{\mbox{,}}\quad k={\frac {\lambda k^{'}}{d}}{\mbox{.}}\end{array}}$

(16)

Using the above non-dimensional quantities, the Eqs. (10), (11), (12), (13), (14) and (15), which govern the stream function ${\textstyle \psi (x{\mbox{,}}y)}$ and by means of ${\textstyle u={\frac {\partial \psi }{\partial y}}{\mbox{,}}v=-\delta {\frac {\partial \psi }{\partial x}}}$ are

$Re\delta \left({\psi }_{y}{\psi }_{xy}-{\psi }_{x}{\psi }_{yy}\right)=$$-{\frac {\partial P}{\partial x}}\left({\frac {\mu +\eta }{\mu }}\right)+$$\delta {\frac {\partial }{\partial x}}\left(S_{xx}\right)+$${\frac {\partial }{\partial y}}\left(S_{xy}\right)+\left({\delta }^{2}{\psi }_{xxy}+\right.$$\left.{\psi }_{yyy}\right){\mbox{,}}$

(17)

$-Re{\delta }^{3}\left({\psi }_{y}{\psi }_{xx}-{\psi }_{x}{\psi }_{xy}\right)=$$-{\frac {\partial P}{\partial y}}\left({\frac {\mu +\eta }{\mu }}\right)+$${\delta }^{2}{\frac {\partial }{\partial x}}\left(S_{yx}\right)+$$\delta {\frac {\partial }{\partial y}}\left(S_{yy}\right)-$${\delta }^{2}\left({\delta }^{2}{\psi }_{xxx}+{\psi }_{xyy}\right){\mbox{,}}$

(18)

$\left({\frac {2\eta \delta }{\mu }}\right){\psi }_{xy}=$$S_{xx}+We\delta \left[{\psi }_{y}{\frac {\partial }{\partial x}}\left(S_{xx}\right)-\right.$$\left.{\psi }_{x}{\frac {\partial }{\partial y}}\left(S_{xx}\right)\right]-$$2eWe\delta {\psi }_{xy}-We\left[{\delta }^{2}(1-e){\psi }_{xx}+\right.$$\left.(1+e){\psi }_{yy}\right]S_{xy}{\mbox{,}}$

(19)

${\frac {\eta }{\mu }}\left({\psi }_{yy}-{\delta }^{2}{\psi }_{xx}\right)=$$S_{xy}+We\delta \left[{\psi }_{y}{\frac {\partial }{\partial x}}\left(S_{xy}\right)-\right.$$\left.{\psi }_{x}{\frac {\partial }{\partial y}}\left(S_{xy}\right)\right]+$${\frac {We}{2}}\left[(1-e){\psi }_{yy}+{\delta }^{2}(1+\right.$$\left.e){\psi }_{xx}\right]S_{xx}-{\frac {We}{2}}\left[{\delta }^{2}\left(1-\right.\right.$$\left.\left.e{\psi }_{xx}+\left(1+e\right){\psi }_{yy}\right)\right]S_{yy}{\mbox{,}}$

(20)

$-\left({\frac {2\eta \delta }{\mu }}\right){\psi }_{xy}=$$S_{yy}+We\delta \left[{\psi }_{y}{\frac {\partial }{\partial x}}\left(S_{yy}\right)-\right.$$\left.{\psi }_{x}{\frac {\partial }{\partial y}}\left(S_{yy}\right)\right]+$$2eWe\delta S_{yy}{\psi }_{xy}+We\left[(1-e){\psi }_{yy}+\right.$$\left.(1+e){\delta }^{2}{\psi }_{xx}\right]S_{xy}{\mbox{,}}$

(21)

where δ is wave number, Re is the Reynolds number and We is the Weissenberg number. The continuity Eq. (10) is identically satisfied.

For the lowest order in δ, Eqs. (17), (18), (19), (20) and (21) take the form

$\left({\frac {\mu +\eta }{\mu }}\right){\frac {\partial p}{\partial x}}=$${\frac {\partial S_{xy}}{\partial y}}+{\psi }_{yyy}{\mbox{,}}$

(22)

${\frac {\partial p}{\partial x}}=0{\mbox{,}}$

(23)

$S_{xx}=We\left(1+e\right){\psi }_{yy}S_{xy}{\mbox{,}}$

(24)

$\left({\frac {\eta }{\mu }}\right){\psi }_{yy}=S_{xy}+$${\frac {We}{2}}(1-e){\psi }_{yy}S_{xx}-{\frac {We}{2}}(1+$$e){\psi }_{yy}S_{yy}{\mbox{,}}$

(25)

$S_{yy}=-We(1-e){\psi }_{yy}S_{xy}{\mbox{.}}$

(26)

Also under this approximation, Eq. (23) indicates that ${\textstyle p\quad \not =\quad p(y)}$. Elimination of ψ from Eqs. (22) and (23) yields

${\frac {{\partial }^{2}}{\partial y^{2}}}S_{xy}+{\psi }_{yyyy}=$$0{\mbox{.}}$

(27)

Substituting Eqs. (24) and (26) into Eq. (25), we have

$S_{xy}={\frac {\left(\eta /\mu \right){\psi }_{yy}}{1+{We}^{2}(1-e^{2}){\left({\psi }_{yy}\right)}^{2}}}{\mbox{.}}$

(28)

Upon making use of Eq. (28) in Eqs. (22) and (27), we can write

${\frac {{\partial }^{2}}{\partial y^{2}}}\left[{\frac {\left({\frac {\eta }{\mu }}+1\right){\psi }_{yy}+{We}^{2}(1-e^{2}){\left({\psi }_{yy}\right)}^{3}}{1+{We}^{2}(1-e^{2}){\left({\psi }_{yy}\right)}^{2}}}\right]=$$0{\mbox{,}}$

(29)

$\left({\frac {\mu +\eta }{\mu }}\right){\frac {\partial p}{\partial x}}=$${\frac {\partial }{\partial y}}\left[{\frac {\left(\eta /\mu \right){\psi }_{yy}}{1+{We}^{2}(1-e^{2}){\left({\psi }_{yy}\right)}^{2}}}\right]+$${\psi }_{yyy}=0{\mbox{.}}$

(30)

The corresponding boundary conditions for the problem are

$\psi ={\frac {F}{2}}{\mbox{,}}{\frac {\partial \psi }{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{2}=1+kx+bsin(2\pi (x-$$t)){\mbox{,}}$

(31a)

$\psi =-{\frac {F}{2}}{\mbox{,}}{\frac {\partial \psi }{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{1}=-1-kx-asin(2\pi (x-$$t)+\phi ){\mbox{.}}$

(31b)

3. Perturbed systems and perturbation solutions

After using binomial expansion for small ${\textstyle {We}^{2}}$, Eqs. (29) and (30) may be simplified

${\frac {{\partial }^{2}}{\partial y^{2}}}\left[{\psi }_{yy}+\right.$$\left.{We}^{2}{\alpha }_{1}{\left({\psi }_{yy}\right)}^{3}+\right.$$\left.{We}^{4}{\alpha }_{2}{\left({\psi }_{yy}\right)}^{5}\right]=$$0{\mbox{,}}$

(32)

${\frac {\partial p}{\partial x}}={\psi }_{yyy}+{We}^{2}{\alpha }_{1}{\frac {\partial }{\partial y}}\left[{\left({\psi }_{yy}\right)}^{3}\right]+$${We}^{4}{\alpha }_{2}{\frac {\partial }{\partial y}}\left[{\left({\psi }_{yy}\right)}^{5}\right]{\mbox{,}}$

(33)

where

${\alpha }_{1}={\frac {\left(e^{2}-1\right)\eta }{\left(\eta +\mu \right)}}{\mbox{,}}\quad {\alpha }_{2}=$$\left(e^{2}-1\right){\alpha }_{1}{\mbox{.}}$

Eqs. (32) and (33) are coupled nonlinear differential equations. The exact solutions are not possible so we focus our attention to find the analytical solution. Therefore we use regular perturbation method. For perturbation solution, we express ${\textstyle \psi {\mbox{,}}\quad \partial p/\partial x}$ and F

$\psi ={\psi }_{0}+{We}^{2}{\psi }_{1}+{We}^{4}{\psi }_{2}+o\left({We}^{6}\right){\mbox{,}}$

(34a)

$F=F_{0}+{We}^{2}F_{1}+{We}^{4}F_{2}+o\left({We}^{6}\right){\mbox{,}}$

(34b)

${\frac {\partial p}{\partial x}}={\frac {\partial p_{0}}{\partial x}}+$${We}^{2}{\frac {\partial p_{1}}{\partial x}}+{We}^{4}{\frac {\partial p_{2}}{\partial x}}+$$o\left({We}^{6}\right){\mbox{,}}$

(34c)

Invoking Eqs. (34a), (34b) and (34c) in Eqs. (31a), (31b), (32) and (33), and comparing the like power of ${\textstyle {We}^{2}}$, we have the following systems.

3.1. For the system of order zero,

${\frac {{\partial }^{4}{\psi }_{0}}{\partial y^{4}}}=0{\mbox{,}}$

(35)

${\frac {\partial p_{0}}{\partial x}}={\frac {{\partial }^{3}{\psi }_{0}}{\partial y^{3}}}{\mbox{,}}$

(36)

${\psi }_{0}={\frac {F_{0}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{0}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{2}{\mbox{,}}$

${\psi }_{0}=-{\frac {F_{0}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{0}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{1}{\mbox{.}}$

(37)

3.2. For the system of order one,

${\frac {{\partial }^{4}{\psi }_{1}}{\partial y^{4}}}=-{\alpha }_{1}{\frac {{\partial }^{2}}{\partial y^{2}}}\left[{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{3}\right]{\mbox{,}}$

(38)

${\frac {\partial p_{1}}{\partial x}}={\frac {\partial }{\partial y}}\left[{\frac {{\partial }^{2}{\psi }_{1}}{\partial y^{2}}}+\right.$$\left.{\alpha }_{1}{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{3}\right]{\mbox{,}}$

(39)

${\psi }_{1}={\frac {F_{1}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{1}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{2}{\mbox{,}}$

${\psi }_{1}=-{\frac {F_{1}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{1}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{1}{\mbox{.}}$

(40)

3.3. For the system of order second,

${\frac {{\partial }^{4}{\psi }_{2}}{\partial y^{4}}}=-3{\alpha }_{1}{\frac {{\partial }^{2}}{\partial y^{2}}}\left[{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{2}\left({\frac {{\partial }^{2}{\psi }_{1}}{\partial y^{2}}}\right)\right]-$${\alpha }_{2}{\frac {{\partial }^{2}}{\partial y^{2}}}\left[{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{5}\right]{\mbox{,}}$

(41)

${\frac {\partial p_{2}}{\partial x}}={\frac {{\partial }^{3}{\psi }_{2}}{\partial y^{3}}}+$$3{\alpha }_{1}{\frac {\partial }{\partial y}}\left[{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{2}\left({\frac {{\partial }^{2}{\psi }_{1}}{\partial y^{2}}}\right)\right]+$${\alpha }_{2}{\frac {\partial }{\partial y}}\left[{\left({\frac {{\partial }^{2}{\psi }_{0}}{\partial y^{2}}}\right)}^{5}\right]{\mbox{,}}$

(42)

${\psi }_{2}={\frac {F_{2}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{2}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{2}{\mbox{,}}$

${\psi }_{2}=-{\frac {F_{2}}{2}}{\mbox{,}}{\frac {\partial {\psi }_{2}}{\partial y}}=$$0{\mbox{,}}\quad {\mbox{at}}\quad y=h_{1}{\mbox{.}}$

(43)

3.4. Zeroth-order solution

The axial velocity and axial pressure gradient at this order are, respectively, given by

${\psi }_{0}={\frac {F_{0}}{2}}-{\frac {3F_{0}{\left(h_{2}-y\right)}^{2}}{{\left(h_{1}-h_{2}\right)}^{2}}}-$${\frac {2F_{0}{\left(h_{2}-y\right)}^{3}}{{\left(h_{1}-h_{2}\right)}^{3}}}{\mbox{,}}$

(44)

${\frac {\partial p_{0}}{\partial x}}={\frac {12F_{0}}{{\left(h_{1}-h_{2}\right)}^{3}}}{\mbox{.}}$

(45)

3.5. First-order solution

The axial velocity and axial pressure gradient at this order are, respectively, given by

${\psi }_{1}={\frac {F_{1}}{2}}-{\frac {3F_{1}{\left(h_{2}-y\right)}^{2}}{{\left(h_{1}-h_{2}\right)}^{2}}}-$${\frac {2F_{1}{\left(h_{2}-y\right)}^{3}}{{\left(h_{1}-h_{2}\right)}^{3}}}+$${\frac {216F_{0}^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{2}}{5{\left(h_{1}-h_{2}\right)}^{6}}}+$${\frac {864F_{0}^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{3}}{5{\left(h_{1}-h_{2}\right)}^{7}}}+$${\frac {216F_{0}^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{4}}{{\left(h_{1}-h_{2}\right)}^{8}}}+$${\frac {432F_{0}^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{5}}{5{\left(h_{1}-h_{2}\right)}^{9}}}{\mbox{,}}$

(46)

${\frac {\partial p_{1}}{\partial x}}={\frac {12F_{1}}{{\left(h_{1}-h_{2}\right)}^{3}}}+$${\frac {1296F_{0}^{3}{\alpha }_{1}}{5{\left(h_{1}-h_{2}\right)}^{7}}}{\mbox{.}}$

(47)

3.6. Second-order solution

Substituting Eqs. (44) and (46) into Eq. (41) and solving the resulting equation subject to the boundary conditions 43 we obtain

${\psi }_{2}={\frac {F_{2}}{2}}-{\frac {{\left(h_{2}-y\right)}^{2}(27F_{1}{\alpha }_{1}F_{0}^{2}+70F_{2}h_{2}-70F_{2}y)}{35A^{3}}}-$${\frac {3F_{2}{\left(h_{2}-y\right)}^{2}}{A^{2}}}+{\frac {270F_{0}^{5}{\alpha }_{2}{\left(h_{2}-y\right)}^{2}}{1001A^{4}}}$ $\quad +{\frac {648F_{0}^{5}{\alpha }_{2}{\left(h_{2}-y\right)}^{3}}{1001{\left(h_{1}-h_{2}\right)}^{5}}}+$${\frac {324F_{0}^{5}{\left(h_{2}-y\right)}^{9}(44{\alpha }_{1}^{2}+{\alpha }_{2}h_{2}^{3}-3{\alpha }_{2}h_{2}^{2}y+3{\alpha }_{2}h_{2}y^{2}-{\alpha }_{2}y)}{11A^{14}}}$ $\quad +{\frac {648F_{0}^{5}{\left(h_{2}-y\right)}^{10}(286{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{3}-15{\alpha }_{2}h_{2}^{2}y+15{\alpha }_{2}h_{2}y^{2}-5{\alpha }_{2}y^{3})}{715A^{15}}}$ $\quad +{\frac {162F_{0}^{5}{\left(h_{2}-y\right)}^{5}(168{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{3}-75{\alpha }_{2}h_{2}^{2}y+75{\alpha }_{2}h_{2}y^{2}-25{\alpha }_{2}y^{3})}{175A^{10}}}$ $\quad +{\frac {108F_{0}^{5}{\left(h_{2}-y\right)}^{7}(792{\alpha }_{1}^{2}+35{\alpha }_{2}h_{2}^{3}-105{\alpha }_{2}h_{2}^{2}y+105{\alpha }_{2}h_{2}y^{2}-35{\alpha }_{2}y^{3})}{35A^{12}}}$ $\quad +{\frac {108F_{0}^{5}{\left(h_{2}-y\right)}^{6}(1764{\alpha }_{1}^{2}+125{\alpha }_{2}h_{2}^{3}-375{\alpha }_{2}h_{2}^{2}y+375{\alpha }_{2}h_{2}y^{2}-125{\alpha }_{2}y^{3})}{175A^{11}}}$ $\quad +{\frac {432F_{0}^{5}{\left(h_{2}-y\right)}^{8}(2277{\alpha }_{1}^{2}+70{\alpha }_{2}h_{2}^{3}-210{\alpha }_{2}h_{2}^{2}y+210{\alpha }_{2}h_{2}y^{2}-70{\alpha }_{2}y^{3})}{385A^{13}}}$ $\quad -{\frac {108F_{0}^{2}{\alpha }_{1}{\left(h_{2}-y\right)}^{2}(12{\alpha }_{1}F_{0}^{3}+35F_{1}h_{2}^{4}-140F_{1}h_{2}^{3}y+210F_{1}h_{2}^{2}y^{2}-140F_{1}h_{2}y^{3}+35F_{1}y^{4})}{175A^{7}}}$ $\quad -{\frac {54F_{0}^{2}F_{0}^{2}F_{1}{\alpha }_{1}{\left(h_{2}-y\right)}^{5}}{5A^{6}}}-$${\frac {108F_{0}^{2}F_{1}{\alpha }_{1}{\left(h_{2}-y\right)}^{7}}{7A^{8}}}-$${\frac {27F_{0}^{2}F_{1}{\alpha }_{1}{\left(h_{2}-y\right)}^{8}}{7A^{9}}}{\mbox{,}}$

(48)

${\begin{array}{rl}&{\frac {\partial p_{2}}{\partial x}}={\frac {12F_{2}}{A^{3}}}-3{\alpha }_{1}B^{2}\left({\frac {5184F_{0}^{3}{\alpha }_{1}}{5A^{7}}}-{\frac {12F_{1}}{A^{3}}}+{\frac {5184F_{0}^{3}{\alpha }_{1}h_{2}^{2}}{A^{9}}}+{\frac {5184F_{0}^{3}{\alpha }_{1}h_{2}}{A^{8}}}\right)-{\frac {3888F_{0}^{5}{\alpha }_{2}}{1001A^{5}}}\\&\quad \quad -{\frac {44{\mbox{,}}712F_{0}^{5}{\alpha }_{2}h_{2}^{5}}{7A^{10}}}-{\frac {207{\mbox{,}}360F_{0}^{5}{\alpha }_{2}h_{2}^{6}}{7A^{11}}}-{\frac {55{\mbox{,}}080F_{0}^{5}{\alpha }_{2}h_{2}^{7}}{A^{12}}}-{\frac {565{\mbox{,}}056F_{0}^{5}{\alpha }_{2}h_{2}^{8}}{11A^{13}}}-{\frac {264{\mbox{,}}384F_{0}^{5}{\alpha }_{2}h_{2}^{9}}{11A^{14}}}\\&\quad \quad -{\frac {645{\mbox{,}}408F_{0}^{5}{\alpha }_{2}h_{2}^{10}}{143A^{15}}}+{\frac {60F_{0}{\alpha }_{2}B^{4}}{A^{3}}}-{\frac {163{\mbox{,}}296F_{0}^{5}h_{2}^{6}(44{\alpha }_{1}^{2}+{\alpha }_{2}h_{2}^{3})}{11A^{14}}}-{\frac {1944F_{0}^{5}h_{2}^{2}(168{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{3})}{35A^{10}}}\\&\quad \quad -{\frac {93{\mbox{,}}312F_{0}^{5}h_{2}^{7}(286{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{3})}{143A^{15}}}-{\frac {648F_{0}^{5}h_{2}^{4}(792{\alpha }_{1}^{2}+35{\alpha }_{2}h_{2}^{3})}{A^{12}}}-{\frac {2592F_{0}^{5}h_{2}^{3}(1764{\alpha }_{1}^{2}+125{\alpha }_{2}h_{2}^{3})}{35A^{11}}}\\&\quad \quad -{\frac {20{\mbox{,}}736F_{0}^{5}h_{2}^{5}(2277{\alpha }_{1}^{2}+70{\alpha }_{2}h_{2}^{3})}{55A^{13}}}+{\frac {648F_{0}^{2}F_{1}{\alpha }_{1}h_{2}^{2}}{A^{6}}}+{\frac {2592F_{0}^{2}F_{1}{\alpha }_{1}h_{2}^{3}}{A^{7}}}+{\frac {3240F_{0}^{2}F_{1}{\alpha }_{1}h_{2}^{4}}{A^{8}}}\\&\quad \quad +{\frac {1296F_{0}^{2}F_{1}{\alpha }_{1}h_{2}^{5}}{A^{9}}}-{\frac {72F_{0}{\alpha }_{1}B}{A^{3}}}\left({\frac {432F_{0}^{3}{\alpha }_{1}}{5A^{6}}}-{\frac {6F_{1}}{A^{2}}}-{\frac {12F_{1}h_{2}}{A^{3}}}+{\frac {2592F_{0}^{3}{\alpha }_{1}h_{2}^{2}}{A^{8}}}+{\frac {1728F_{0}^{3}{\alpha }_{1}h_{2}^{3}}{A^{9}}}+{\frac {5184F_{0}^{3}{\alpha }_{1}h_{2}}{5A^{7}}}\right){\mbox{.}}\end{array}}$

(49)

where

$A=(h_{1}-h_{2}){\mbox{,}}\quad B={\frac {6F_{0}}{A^{2}}}+{\frac {12F_{0}h_{2}}{A^{3}}}{\mbox{,}}$

Now we summarize our results using Eqs. (34a), (34b) and (34c) and the relations

$F_{0}=F-{We}^{2}F_{1}-{We}^{4}F_{2}{\mbox{,}}{We}^{2}F_{1}=F-F_{0}-{We}^{4}F_{2}{\mbox{,}}{We}^{4}F_{2}=$$F-F_{0}-{We}^{2}F_{2}{\mbox{,}}$

${\begin{array}{rl}&\psi ={\frac {F}{2}}+{We}^{2}\left({\frac {216F^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{2}}{5A^{6}}}+{\frac {864F^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{3}}{5A^{7}}}+{\frac {216F^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{4}}{A^{8}}}+{\frac {432F^{3}{\alpha }_{1}{\left(h_{2}-y\right)}^{5}}{5A^{9}}}\right)\\&\quad -{\frac {3F{\left(h_{2}-y\right)}^{2}}{A^{2}}}-{\frac {2F{\left(h_{2}-y\right)}^{3}}{A^{3}}}+{\frac {270F^{5}{\alpha }_{2}{We}^{4}{\left(h_{2}-y\right)}^{2}}{1001A^{4}}}+{\frac {648F^{5}{\alpha }_{2}{We}^{4}{\left(h_{2}-y\right)}^{3}}{1001A^{5}}}\\&\quad +{\frac {324F^{5}{We}^{4}{\left(h_{2}-y\right)}^{9}(44{\alpha }_{1}^{2}+{\alpha }_{2}h_{2}^{3}-3{\alpha }_{2}h_{2}^{2}y+3{\alpha }_{2}h_{2}y^{2}-{\alpha }_{2}y^{3})}{11A^{14}}}-{\frac {1296F^{5}{\alpha }_{1}^{2}{We}^{5}(h_{2}-y)}{175A^{7}}}\\&\quad +{\frac {648F^{5}{We}^{4}{\left(h_{2}-y\right)}^{10}(286{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{3}-15{\alpha }_{2}h_{2}^{2}y+15{\alpha }_{2}h_{2}y^{2}-5{\alpha }_{2}y^{3})}{715A^{15}}}\\&\quad +{\frac {162F^{5}{We}^{4}{\left(h_{2}-y\right)}^{5}(168{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{3}-75{\alpha }_{2}h_{2}^{2}y+75{\alpha }_{2}y^{2}-25{\alpha }_{2}y^{3})}{175A^{10}}}\\&\quad +{\frac {108F^{5}{We}^{4}{\left(h_{2}-y\right)}^{7}(792{\alpha }_{1}^{2}+35{\alpha }_{2}h_{2}^{3}-105{\alpha }_{2}h_{2}^{2}y+105{\alpha }_{2}h_{2}y^{2}-35{\alpha }_{2}y^{3})}{35A^{12}}}\\&\quad +{\frac {108F^{5}{We}^{4}{\left(h_{2}-y\right)}^{6}(1764{\alpha }_{1}^{2}+125{\alpha }_{2}h_{2}^{3}-375{\alpha }_{2}h_{2}^{2}y+375{\alpha }_{2}h_{2}y^{2}-125{\alpha }_{2}y^{3})}{175A^{11}}}\\&\quad +{\frac {432F^{5}{We}^{4}{\left(h_{2}-y\right)}^{8}(2277{\alpha }_{1}^{2}+70{\alpha }_{2}h_{2}^{3}-210{\alpha }_{2}h_{2}^{2}y+210{\alpha }_{2}h_{2}y^{2}-70{\alpha }_{2}y^{3})}{385A^{13}}}{\mbox{,}}\end{array}}$

(50)

${\begin{array}{rl}&u={\frac {3F(2h_{2}-2y)}{A^{2}}}-{\frac {1944F^{5}{\alpha }_{2}{We}^{4}{\left(h_{2}-y\right)}^{2}}{1001{\left(h_{1}-h_{2}\right)}^{5}}}-{\frac {648F^{5}{We}^{4}{\left(h_{2}-y\right)}^{10}(15{\alpha }_{2}(h_{2}^{2}+y^{2})-30{\alpha }_{2}h_{2}y)}{715A^{15}}}\\&\quad -{\frac {432F^{3}{\alpha }_{1}{We}^{2}{\left(h_{2}-y\right)}^{4}}{A^{9}}}-{\frac {2916F^{5}{We}^{4}{\left(h_{2}-y\right)}^{8}(44{\alpha }_{1}^{2}+{\alpha }_{2}(h_{2}^{3}-y^{3})-3{\alpha }_{2}h_{2}^{2}y+3{\alpha }_{2}h_{2}y^{2})}{11A^{14}}}\\&\quad +{\frac {6F{\left(h_{2}-y\right)}^{2}}{A^{3}}}-{\frac {1296F^{5}{We}^{4}{\left(h_{2}-y\right)}^{9}(286{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{3}-15{\alpha }_{2}h_{2}^{2}y+15{\alpha }_{2}h_{2}y^{2}-5{\alpha }_{2}y^{3})}{143A^{15}}}\\&\quad -{\frac {162F^{5}{We}^{4}{\left(h_{2}-y\right)}^{4}(168{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{3}-75{\alpha }_{2}h_{2}^{3}y+75{\alpha }_{2}h_{2}y^{2}-25{\alpha }_{2}y^{3})}{35A^{10}}}\\&\quad -{\frac {108F^{5}{We}^{4}{\left(h_{2}-y\right)}^{6}(792{\alpha }_{1}^{2}+35{\alpha }_{2}h_{2}^{3}-105{\alpha }_{2}h_{2}^{2}y+105{\alpha }_{2}h_{2}y^{2}-35{\alpha }_{2}y^{3})}{5A^{12}}}\\&\quad -{\frac {648F^{5}{We}^{4}{\left(h_{2}-y\right)}^{5}(1764{\alpha }_{1}^{2}+125{\alpha }_{2}h_{2}^{3}-375{\alpha }_{2}h_{2}^{2}y+375{\alpha }_{2}h_{2}y^{2}-125{\alpha }_{2}y^{3})}{175A^{11}}}\\&\quad -{\frac {3456F^{5}{We}^{4}{\left(h_{2}-y\right)}^{7}(2277{\alpha }_{1}^{2}+70{\alpha }_{2}h_{2}^{3}-210{\alpha }_{2}h_{2}^{2}y+210{\alpha }_{2}h_{2}y^{2}-70{\alpha }_{2}y^{3})}{385A^{13}}}\\&\quad +{\frac {1296F^{5}{\alpha }_{1}^{2}{We}^{4}(2h_{2}-2y)}{175A^{7}}}-{\frac {324F^{5}{We}^{4}{\left(h_{2}-y\right)}^{9}(3{\alpha }_{2}h_{2}^{2}-6{\alpha }_{2}h_{2}y+3{\alpha }_{2}y^{2})}{11A^{14}}}\\&\quad -{\frac {2592F^{3}{\alpha }_{1}{We}^{2}{\left(h_{2}-y\right)}^{2}}{5A^{7}}}-{\frac {162F^{5}{We}^{5}{\left(h_{2}-y\right)}^{5}(75{\alpha }_{2}(h_{2}^{2}+y^{2})-150{\alpha }_{2}h_{2}y)}{175A^{10}}}\\&\quad -{\frac {864F^{3}{\alpha }_{1}{We}^{2}{\left(h_{2}-y\right)}^{3}}{A^{8}}}-{\frac {108F^{5}{We}^{4}{\left(h_{2}-y\right)}^{7}(105{\alpha }_{2}h_{2}^{2}-210{\alpha }_{2}h_{2}y+105{\alpha }_{2}y^{2})}{35A^{15}}}\\&\quad -{\frac {270F^{5}{\alpha }_{2}{We}^{4}(2h_{2}-2y)}{1001A^{4}}}-{\frac {432F^{5}{We}^{4}{\left(h_{2}-y\right)}^{8}(210{\alpha }_{2}(h_{2}^{2}+y^{2})-420{\alpha }_{2}h_{2}y)}{385A^{13}}}\\&\quad -{\frac {216F^{3}{\alpha }_{1}{We}^{2}(2h_{2}-2y)}{5A^{6}}}-{\frac {108F^{5}{We}^{4}{\left(h_{2}-y\right)}^{6}(375{\alpha }_{2}(h_{2}^{2}+y^{2})-750{\alpha }_{2}h_{2}y)}{175A^{11}}}{\mbox{,}}\end{array}}$

(51)

${\begin{array}{rl}&{\frac {\partial p}{\partial x}}={\frac {12F}{A^{3}}}-{\frac {7776{We}^{4}(84{\alpha }_{1}^{2}h_{2}^{3}+96{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{6}-50{\alpha }_{2})}{5A^{11}}}-{\frac {7776F^{5}h_{2}^{2}{We}^{4}(6{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{3})}{5A^{10}}}\\&\quad -{\frac {3888F^{5}{\alpha }_{2}{We}^{4}}{1001A^{5}}}-{\frac {7776F^{5}h_{2}^{4}{We}^{4}(24{\alpha }_{1}^{2}h_{2}^{3}+480{\alpha }_{1}^{2}+{\alpha }_{2}h_{2}^{6}-160{\alpha }_{2})}{A^{15}}}\\&\quad +{\frac {1296F^{3}{\alpha }_{1}{We}^{2}}{5A^{7}}}-{\frac {7776F^{5}h_{2}^{3}{We}^{4}(84{\alpha }_{1}^{2}h_{2}^{3}+960{\alpha }_{1}^{2}+5{\alpha }_{2}h_{2}^{6}-320{\alpha }_{2})}{A^{14}}}\\&\quad -{\frac {15{\mbox{,}}552F^{5}h_{2}{We}^{4}(276{\alpha }_{1}^{2}h_{2}^{3}+1692{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{6}-600{\alpha }_{2})}{5A^{13}}}\\&\quad -{\frac {15{\mbox{,}}552F^{5}h_{2}{We}^{4}(165{\alpha }_{1}^{2}h_{2}^{3}+492{\alpha }_{1}^{2}+25{\alpha }_{2}h_{2}^{6}-200{\alpha }_{2})}{5A^{12}}}{\mbox{.}}\end{array}}$

(52)

The average rise in pressure ${\textstyle \Delta p}$ over one period of wavelength is given by

$\Delta p={\int }_{0}^{1}{\int }_{0}^{1}{\frac {\partial p}{\partial x}}dxdt{\mbox{.}}$

(53)

4. Graphical results and discussion

In order to discuss the obtained results quantitatively, we assume the instantaneous volume rate of the flow ${\textstyle F(x{\mbox{,}}t)}$ periodic in ${\textstyle \left(x-t\right)}$[47], [48], [49], [50], [55], [56] and [57] as;

$F(x{\mbox{,}}t)=Q+asin2\pi \left(x-t\right)+bsin\left[2\pi \left(x-\right.\right.$$\left.\left.t\right)+\phi \right]{\mbox{.}}$

(54)

where Q is the time-average of the flow over one period of the wave.

The expression for ${\textstyle \Delta p}$ involves the integration of ${\textstyle dp/dx}$. Due to complexity of ${\textstyle dp/dx}$, Eq. (53) is not integrable analytically. Consequently, a numerical integration scheme is required for the evaluation of the integrals. MATHEMATICA and MATLAB are used to evaluate the integrals and all the plots have been then generated for the various values of the parameters of interest.

Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the graph of the average rise in pressure to against the variation of time-average flux Q  . The pumping regions such as peristaltic pumping (${\textstyle Q>0{\mbox{,}}\Delta p>0}$), augmented pumping (${\textstyle Q>0{\mbox{,}}\Delta p<0}$) and retrograde pumping (${\textstyle Q<0{\mbox{,}}\Delta p>0}$), are also depicted in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. The dependence of the average rise in pressure on a is illustrated in Fig. 2. One could notice from Fig. 2 that peristaltic pumping region becomes wider as a increases. Fig. 3 reveals that the peristaltic pumping region and free pumping ${\textstyle \left(\Delta p=0\right)}$ decrease with an increase in non-uniform parameter k and the situation is quite complimentary to the case of augmented pumping. Fig. 4 shows the variation of average rise in pressure against mean flow rate Q, for different values of Weinberg number We. It is noticed that the pumping rate decreases with an increase in Weinberg number and in the free pumping, the curves appear to be overlapped. The plot shown in Fig. 5 reveals that the mean flow rate Q   increases as ${\textstyle \Delta p}$ decreases and the pumping region decreases with the increase in the value of phase difference ${\textstyle \left(\phi \right)}$. Fig. 6 shows the plots of ${\textstyle \Delta P}$ against mean flow rate Q for various values of slip parameter e  . It indicates that dispersion of ${\textstyle \Delta p}$ in increasing trend due to slip parameter shows to be nonlinear. Fig. 7 shows the variation of the dynamic viscosity parameter on ${\textstyle \Delta p}$ against ${\textstyle \Theta }$. This graph reveals that the average rise in pressure decreases by increasing the dynamic viscosity parameter ${\textstyle \left(\eta \right)}$ in the augmented pumping region ${\textstyle \left(\Delta p<0and\Theta >0\right)}$. For large values of η, the curves of the average rise in pressure against Θ reflect the nonlinear character.

Figure 2. Effects of the upper wave amplitude of the channel ${\textstyle a}$ on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle b=0.4{\mbox{,}}\quad k=0.1{\mbox{,}}\quad \phi =\pi /6{\mbox{,}}\quad e=}$$1{\mbox{,}}\quad \eta =0.8{\mbox{,}}\quad \mu =0.5$ and ${\textstyle We=0.05}$.

Figure 3. Effects of the non-uniform of the channel k   on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle a=0.4{\mbox{,}}\quad b=0.2{\mbox{,}}\quad \phi =\pi /2{\mbox{,}}\quad e=}$$0.5{\mbox{,}}\quad \eta =0.6{\mbox{,}}\quad \mu =0.4$ and ${\textstyle We=0.15}$.

Figure 4. Effects of the weissenberg of the channel We   on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle a=0.6{\mbox{,}}\quad b=0.7{\mbox{,}}\quad \phi =\pi /3{\mbox{,}}\quad e=}$$0.4{\mbox{,}}\quad \eta =0.5{\mbox{,}}\quad \mu =0.5$ and ${\textstyle k=0.1}$.

Figure 5. Effects of the phase difference of the channel η   on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle a=0.7{\mbox{,}}\quad b=0.5{\mbox{,}}\quad k=0.05{\mbox{,}}\quad e=}$$0.7{\mbox{,}}\quad \eta =0.7{\mbox{,}}\quad \mu =0.7$ and ${\textstyle We=0.01}$.

Figure 6. Effects of the slip of the channel e   on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle a=0.5{\mbox{,}}\quad b=0.6{\mbox{,}}\quad \phi =\pi /4{\mbox{,}}\quad We=}$$0.25{\mbox{,}}\quad \eta =0.4{\mbox{,}}\quad \mu =0.5$ and ${\textstyle k=0.1}$.

Figure 7. Effects of the dynamic viscosity of the channel η   on variation of ${\textstyle \Delta p}$ with Q for ${\textstyle a=0.6{\mbox{,}}\quad b=0.7{\mbox{,}}\quad \phi =\pi /6{\mbox{,}}\quad e=}$$0.6{\mbox{,}}\quad k=0.3{\mbox{,}}\quad \mu =0.7$ and ${\textstyle We=0.2}$.

The effect of amplitude of lower wall ${\textstyle \left(a\right)}$ on the axial velocity is displayed in Fig. 8. It is clear that the axial velocity increases by increasing a  . The upshot of the slip parameter ${\textstyle \left(e\right)}$ on u is depicted in Fig. 9. It is inferred that the axial velocity u decreases in nearer part of lower and upper half on the trapped asymmetric channel, whereas a converse behavior seems to be observed in the core region of the trapped channel. Figure 10, Figure 11 and Figure 12 are plotted to study the velocity distribution of the fluid for different values of the dynamic viscosity parameter ${\textstyle {\left(\eta \right)}_{\mbox{,}}}$ non-uniform parameter ${\textstyle \left(k\right)}$ and Weinberg number ${\textstyle \left(We\right)}$. It could be perceived from the graphs that the increase in values of ${\textstyle \eta {\mbox{,}}\quad k}$ and We, decreases the axial velocity in the center region of the tapered asymmetric channel. Fig. 13 displays the influence of mean flow rate ${\textstyle \left(Q\right)}$ on the axial velocity distribution. It reveals that an increase in Q results in the increase in axial velocity.

Figure 8. Velocity profile for different values of a   when ${\textstyle b=0.2{\mbox{,}}\quad k=0.2{\mbox{,}}\quad Q=1.5{\mbox{,}}\quad We=}$$0.1{\mbox{,}}\quad \phi =\pi /3{\mbox{,}}\quad e=0.2{\mbox{,}}\quad \eta =$$0.2$ and ${\textstyle \mu =0.3}$.

Figure 9. Velocity profile for different values e   when ${\textstyle a=0.8{\mbox{,}}\quad b=0.6{\mbox{,}}\quad Q=1.9{\mbox{,}}\quad \eta =}$$0.8{\mbox{,}}\quad \phi =\pi {\mbox{,}}\quad We=0.5{\mbox{,}}\quad \mu =$$0.6$ and ${\textstyle k=0.3}$.

Figure 10. Velocity profile for different values of η   when ${\textstyle b=0.6{\mbox{,}}\quad k=0.1{\mbox{,}}\quad Q=1.4{\mbox{,}}\quad We=}$$0.7{\mbox{,}}\quad \phi =\pi {\mbox{,}}\quad e=0.5{\mbox{,}}\quad a=$$0.5$ and ${\textstyle \mu =0.3}$.

Figure 11. Velocity profile for different values k   when ${\textstyle a=0.2{\mbox{,}}\quad b=0.4{\mbox{,}}\quad Q=1.8{\mbox{,}}\quad \eta =}$$0.2{\mbox{,}}\quad \phi =\pi /2{\mbox{,}}\quad We=0.05{\mbox{,}}\quad \mu =$$0.2$ and ${\textstyle e=0.7}$.

Figure 12. Velocity profile for different values of We   when ${\textstyle a=0.7{\mbox{,}}\quad k=0.1{\mbox{,}}\quad Q=2.5{\mbox{,}}\quad \phi =}$$\pi {\mbox{,}}\quad \eta =0.5{\mbox{,}}\quad e=0.1{\mbox{,}}\quad b=$$0.5$ and ${\textstyle \mu =0.7}$.

Figure 13. Velocity profile for different values of Q   when ${\textstyle a=0.4{\mbox{,}}\quad b=0.3{\mbox{,}}\quad Q=1.5{\mbox{,}}\quad \eta =}$$0.1{\mbox{,}}\quad \phi =\pi /3{\mbox{,}}\quad We=0.1{\mbox{,}}\quad \mu =$$0.2$ and ${\textstyle e=0.6}$.

The phenomenon of trapping is another interesting area in peristaltic transport. In a reference wave frame, the formation of an internally circulating bolus of fluid by closed streamlines is called trapping and this trapped bolus is pushed ahead along with the peristaltic wave. It is interesting to notice that bolus also appears in the fixed frame which may probably be due to the influence of nonzero time-average of the flow over one period of the wave. The streamlines for different values of a are shown in Fig. 14. It is also observed that the size of bolus increases with increasing a. The sound effects of We on trapping are shown for the tapered asymmetric channel in Fig. 15. Fig. 15(a) shows that the bolus is asymmetric about the centerline and its size decreases with an increase in ${\textstyle We}$. The influence of the time average flow Q on the streamlines is displayed in Fig. 16. We observe that the trapped bolus increases in size as Q increases and more trapped bolus appears with increasing Q. Fig. 17 finally reveals that our results are in good agreement with Mishra and Rao [37] and Srivastava and Srivastava [56] in which we have taken half of mean flow rate in respect of Ref. [56].

Figure 14. Streamlines for (a) ${\textstyle a=0}$, (b) ${\textstyle a=0.3}$. (${\textstyle k=0.1{\mbox{,}}\quad \phi =\pi /3{\mbox{,}}\quad We=0.1{\mbox{,}}\quad t=}$$0.2{\mbox{,}}\quad Q=1.8{\mbox{,}}\quad b=0.2{\mbox{,}}\quad e=$$0.7{\mbox{,}}\quad \eta =0.5{\mbox{,}}\quad \mu =0.8)$.

Figure 15. Streamlines for (a) ${\textstyle We=0}$, (b) ${\textstyle We=0.6}$. (${\textstyle a=0.1{\mbox{,}}\quad b=0.2{\mbox{,}}\quad e=0.4{\mbox{,}}\quad \eta =}$$0.8{\mbox{,}}\quad \mu =0.1{\mbox{,}}\quad k=0.05{\mbox{,}}\quad t=$$0.3{\mbox{,}}\quad Q=1.2{\mbox{,}}\quad \phi =\pi /4)$.

Figure 16. Streamlines for (a) ${\textstyle Q=1.5}$, (b) ${\textstyle Q=1.6}$ (${\textstyle a=0.3{\mbox{,}}\quad b=0.2{\mbox{,}}\quad e=0.2{\mbox{,}}\quad \eta =}$$0.5{\mbox{,}}\quad \mu =0.8{\mbox{,}}\quad k=0.1{\mbox{,}}\quad t=$$0.3{\mbox{,}}\quad \phi =\pi /3{\mbox{,}}\quad We=0.2)$.

Figure 17. Comparison result of average rise in pressure versus mean flow rate.

5. Conclusions

We have developed a mathematical model to study the peristaltic transport of a Johnson–Segalman fluid in the tapered asymmetric channel. The tapered asymmetry channel is produced by choosing the peristaltic wave train on the non-uniform walls to have different amplitudes and phase. A long-wavelength and low-Reynolds number approximations have been adopted. A regular perturbation method was employed to obtain the expression for stream function, axial velocity and pressure gradient. The interaction of the rheological parameters of the fluid with peristaltic flow is discussed. The main results can be summarized as follows:

• It has been found that the average rise in pressure ${\textstyle \Delta P}$ increases with the increase of a and e and decreases with ${\textstyle k{\mbox{,}}\quad We{\mbox{,}}\quad \phi }$ and η.
• It was also observed that the axial velocity decreases with increase of ${\textstyle \eta {\mbox{,}}\quad k{\mbox{,}}\quad We}$ and ϕ.
• The velocity increases with increasing ${\textstyle a{\mbox{,}}\quad e}$ and Q.
• The size of trapped bolus decreases with an increase in the mean flow rate.
• It is noticed that our results find a very good matches with Hayat et al. [47] in the absence of non-uniform parameter ${\textstyle m=0}$.

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