Thermal analysis of an Eyring–Powell fluid peristaltic transport in a rectangular duct with mass transfer

Abstract

The present investigation establishes with analytical relevance for the thermal analysis in biological transport of Eyring–Powell fluid through a noncircular horizontal 3D channel. The relevant equations are physically exhibiting the problem under the lubrication mechanism. The wave frame analysis for the current study is considered to have a steady format of the problem. The resulting equations are presented with their analytic treatment by couple of series and closed-form techniques jointly. The aspects of different flow parameters are presented through plotting two- and three-dimensional curves of velocity expression, pressure gradient profile, peristaltic pressure data, temperature curves and mass distribution. The circulating bolus variation is also disclosed for the physical parameters to observe the stream lines pattern. It is observed from obtained graphical aspects that transfer of heat and mass is reduced for Eyring–Powell fluid model than a Newtonian fluid, but the same is enhanced with the aspect ratio of the three-dimensional rectangular channel.

Introduction

The most commonly used fluid models considered in engineering applications incorporate the Newtonian fluids; rather, large-scale analytical and experimental investigations have been presented for heat transfer behavior of Newtonian fluids [1,2,3]. However, in many industries including the chemical, pharmaceutical, biological and food industries, it is visualized to encounter non-Newtonian fluids depending on different stresses and viscosity models [4,5,6]. In [7], Waqas et al. have produced the theoretical data for Carreau–Yasuda nanofluid model by introducing thermal radiations and slip of the second order. They have also manipulated the impacts of chemical reaction and bioconvection and obtain the report that Weissenburg is reducing the flow velocity magnitude. Abolbashari et al. [8] talked about the modeling of Casson liquid with nanoparticles incursion and discussed the results for entropy generation through analytical approach. The study of Bhatti and Rashidi [9] came for Williamson fluid to visualize the results of diffusion of heat along with radiation in the presence of nanoparticles through a porous shrinking and stretching sheet. They have applied the numerical methods to measure the readings. The model introduced by Eyring–Powell [10] somehow contains more complex mathematical structure, but it has certain benefits as compared with other viscoelastic fluid models. The kinetic theory of liquids is the origin of this model not the empirical expressions as most of the models make base. The Newtonian characteristics can also be derived from this model at low and large shear stress.

Recently, the study of peristaltic flows has achieved an immense concentration and interest of huge number of researchers due to their significant employments and application in many useful fields like physiology, engineering and the chemical factories. In the field of fluid mechanics, peristaltic flows of various types of viscous and nonlinear physical models have been suggested by a large number of physicists, mathematicians and engineering researchers by considering peristaltic pumping. A huge bit of literature is available for the study of such fluids under peristaltic motion [11,12,13]. To incorporate the wavy mechanism of viscoelastic stress models, the researchers have been predisposed in introducing different fluid models in various types of flow geometries. Most of the containers adjusted in industries and clinical laboratories are exhibiting three dimensions in space. To discuss the behavior of peristaltic transport in certain geometries having space frame of references, a small number of researchers are motivated to deal with rectangular channels and cylinders. Reddy et al. [14] proposed the influence of moving sides on a fluid in a Cartesian peristaltic conduit and concluded the theory that the straight transactional side of the uterus is preferably observed by an enclosure of rectangular faces besides a channel with a two-coordinate system. Mekheimer et al. [15] have investigated the eccentricity of two cylinders with peristaltic transport mathematically. Stokesian peristaltic mechanism in a three-dimensional asymmetric tube is unveiled in [16]. Aranda et al. [17] talked about the Stokesian peristaltic scheme in a closed three-dimensional tube and executed the observation that 3D flexible tube is affecting the results of the peristaltic pumping through its closed ends and the way the peristaltic waves blow toward outside the cavity. Abbas et al. [18] have utilized the outcomes of nonuniform three-dimensional geometry with elastic walls for the wavy mechanism of Tangent hyperbolic fluid and developed the result that fluid travels relatively slower in due to walls compliance. It can be found in the literature that only a few studies have been reported on the pumping flow of viscoelastic fluids in three-dimensional Cartesian channels. The wide range of its applications and benefits in clinical and engineering perspectives emphasizes that this geometry must me more discussed for various models of the physical fluids.

In the field of peristaltic wonders, incorporations of thermal and mass exchange have been examined by numerous specialists. Nadeem et al. [19] introduced thermal blood analysis in a diverging tube for third-grade fluid. The relation among wavy structure with heat exchange over movement of a viscous liquid in a bidimensional magnetized and porous channel is assumed in [20]. Ogulu [21] analyzed thermal and concentration change phenomenon of blood through a solitary lymphatic vein under a uniform magnetized environment. Mekheimer and Abd Elmaboud [22] dissected the heat impact with magnetic field on pumping terminology of a viscous liquid across a vertically oriented annulus through lubrication assumptions. In the studies of three-dimensional thermally conducting mass transfer flows, most of the time the observing representative conservation equations become linear/nonlinear partial differential equations or more specifically coupled system of three equations which cannot be tackled by exact methods. In such situations, the most appropriate analytical technique to be used is homotopy perturbation method firstly introduced by He [23, 24] which was efficiently implemented for nonlinear ordinary differential equations.

To the best of authors’ information, the subject of thermal impacts and mass exchange attributes on peristaltic stream of Eyring–Powell liquid in a duct segment is yet to break down. Remembering the above conversation, we chose to explore the investigation of thermal and mass transfer examination for the peristaltic stream of Eyring–Powell liquid model in a three-dimensional cross-area of a rectangular duct. The constitutive set of equations are formatted within the sight of small wave amplitude and negligible Reynolds number presumptions with the analytical arrangements produced through homotopy perturbation method by using the method of eigenfunction. On the other hand, the data of pressure rise are interpolated through numerical treatment. The subsequent information is used to examine the graphical highlights of every relevant parameter showing up in the examination. The flow pattern is additionally shown through streamlines.

Fig. 1

Geometrical structure of the container

Fig. 2

Horizontal velocity against $\beta$ a 2D b 3D

Fig. 3

Horizontal velocity against K a 2D b 3D

Fig. 4

Horizontal velocity against $M_{1}$ a 2D b 3D

Fig. 5

Pressure rise curves $\Delta p$ against $\beta$

Mathematical formulation

Let us approach the wavy-type flow in a cross-face of Cartesian channel with three faces containing the non-Newtonian Eyring–Powell fluid under the environment of thermal and mass exchange. The width of the section is 2d, and the altitude is 2a which can be viewed in Fig. 1. The geometrical suppositions are as follows: The temperatures $T_{0}$, $T_{1}$ and mass concentrations $C_{0}$, $C_{1}$ are assigned to the horizontal sides of the channel, correspondingly. The upward and downward oriented boundaries are assumed to create long peristaltic waves as compared to the amplitude.

Mathematically, the walls suggest the relations [14]

$$\begin{aligned} Z=\digamma ^{\prime }(X,t^{\prime })=\pm \left( a+\frac{1}{b^{-1}\sec \left( \frac{2\pi }{\lambda \left( X-ct^{\prime }\right) ^{-1}}\right) }\right) . \end{aligned}$$

Here, the wave amplitude is given by b, $\lambda$ is showing the wavelength, c gives speed with which wave is propagated, t chooses the time measurements and x exhibits the wave direction. The walls normal to y-axis do not contribute in creating waves due to their stable manufacturing. The constitutive model of fluid taken is described by [10]

$$\begin{aligned}&\mathbf {S}=\mu \text {grad}\mathbf {V}^{\prime }+\frac{1}{\beta _{1}}\arcsin h\left( c_{1}^{-1}\text{grad}\mathbf {V}^{\prime }\right) , \\&\sinh ^{-1}\left( c_{1}^{-1}\text {grad}\mathbf {V}^{\prime }\right) \approx c_{1}^{-1}{\text{grad}}\mathbf {V}^{\prime } \\&\quad \mathbf {-}\frac{1}{6}\left( c_{1}^{-1}\text {grad}\mathbf {V}^{\prime }\right) ^{3}, \quad \left| c_{1}^{-1}\text {grad}\mathbf {V}^{\prime }\right| ^{5}\leqq1, \end{aligned}$$

(1)

where the viscosity is suggested by $\mu$,while $\beta$ and $c_{1}$ are the fluid material representatives. Let $\mathbf {V}^{\prime }=U\mathbf {i}+W \mathbf {k}$ be the fluid velocity vector for current flow. The flow is governed by the conservation laws described through their mathematical form below:

$$\begin{aligned}&\partial _{{\rm X}}U+\partial _{{\rm Z}}W=0, \end{aligned}$$

(2)

$$\begin{aligned}&\rho \left( \partial _{t^{\prime }}U+U*\partial _{{\rm X}}U+W*\partial _{{\rm Z}}U\right) +\partial _{{\rm X}}P=\partial _{{\rm X}}S_{{\rm XX}}+\partial _{Y}S_{{\rm XY}}+\partial _{{\rm Z}}S_{{\rm XZ}}, \end{aligned}$$

(3)

$$\begin{aligned}&\partial _{Y}P=\partial _{{\rm X}}S_{YX}+\partial _{Y}S_{{\rm YY}}+\partial _{{\rm Z}}S_{YZ}, \end{aligned}$$

(4)

$$\begin{aligned}&\rho \left( \partial _{t^{\prime }}W+U*\partial _{{\rm X}}W+W*\partial _{{\rm Z}}W\right) +\partial _{{\rm Z}}P \\&\quad =\partial _{{\rm X}}S_{{\rm ZX}}+\partial _{Y}S_{ZY}+\partial _{{\rm Z}}S_{{\rm ZZ}}, \end{aligned}$$

(5)

$$\begin{aligned}&C^{^{\prime }}\left( \partial _{t^{\prime }}T+U*\partial _{{\rm X}}T+W*\partial _{{\rm Z}}T\right) =\frac{K^{^{\prime }}}{\rho }\left( \partial _{{\rm XX}}T+\partial _{{\rm YY}}T+\partial _{{\rm ZZ}}T\right) \\&\quad \quad +\frac{1}{\rho }\left( S_{{\rm XY}}\partial _{Y}U+S_{{\rm XZ}}\partial _{{\rm Z}}U\right) , \end{aligned}$$

(6)

$$\begin{aligned}&\left( \partial _{{\rm t}^{\prime }}C+U*\partial _{{\rm X}}C+W*\partial _{{\rm Z}}C\right) =D\left( \partial _{{\rm XX}}C+\partial _{{\rm YY}}C+\partial _{{\rm ZZ}}C\right) \\&\quad \quad +\frac{DK_{{\rm T}}}{T_{{\rm m}}}\left( \partial _{{\rm XX}}T+\partial _{{\rm YY}}T+\partial _{{\rm ZZ}}T\right) . \end{aligned}$$

(7)

The above-mentioned parameters are such that $K^{^{\prime }}$ represents the thermal conductivity, $C^{^{\prime }}$ shows the specific heat, $T_{{\rm m}}$ gives the average temperature, $K_{{\rm T}}$ suggests the thermal–diffusion ratio and D predicts the mass diffusibility coefficient. Transforming the coordinates into a new coordinate system of a newly taken wave frame, we relate through the following transformations:

$$\begin{aligned} \left( x^{\prime },y^{\prime },z^{\prime },u^{\prime },w^{\prime },p^{\prime }\right) =\left( X-ct^{\prime },Y,Z,U-c,W,P\right) . \end{aligned}$$

(8)

Suggesting the subsequent dimensionless strategy, we obtain

$$\begin{aligned}&\overleftrightarrow {x}=\lambda ^{-1}x^{\prime },\quad \overleftrightarrow {y}=d^{-1}y^{\prime },\quad \overleftrightarrow {z} =a^{-1}z^{\prime },\\ &\overleftrightarrow {u}=c^{-1}u^{\prime },\quad \overleftrightarrow {w}=\left( \delta c\right) ^{-1}w^{\prime },\quad \overline{\overline{t}}=\lambda ^{-1}ct^{\prime },\quad \overline{ \overline{h}}=a^{-1}\digamma , \\&\quad \overline{\overline{p}}=\frac{a^{2}p^{\prime }}{\mu c\lambda },\theta =\frac{T-T_{0}}{T_{1}-T_{0}},\quad \sigma =\frac{C-C_{0} }{C_{1}-C_{0}}, \\&{\text{Re}}=\frac{\rho ac}{\mu },\quad \delta =\frac{a }{\lambda },\quad \phi =\frac{b}{a},\quad \overline{S}=\frac{a}{\mu c}S, \quad \beta =\frac{a}{d}, \\&\quad M_{1}=\frac{1}{\mu \beta _{1}c_{1}},K=\frac{M_{1}c^{2}}{ 6c_{1}^{2}a^{2}},\quad P_{{\rm r}}=\frac{\mu C^{^{\prime }}}{K^{^{\prime }}}, \\&E_{{\rm c}}=\frac{c^{2}}{C^{^{\prime }}\left( T_{1}-T_{0}\right) },\quad S_{{\rm c}}=\frac{\mu }{\rho D},\quad S_{{\rm r}}=\frac{\rho DK_{{\rm T}}\left( T_{1}-T_{0}\right) }{\mu T_{{\rm m}}\left( C_{1}-C_{0}\right) }. \end{aligned}$$

(9)

The amounts $M_{1}$ and K hint the non-dimensional quantities of the liquid. Joining the dimensionless amounts with the relations (2)–(7) and utilizing the requirements of laminarity of the stream $(\hbox{Re}\rightarrow 0)$ and small amplitude $(\lambda \rightarrow \infty )$ approximations, the final form of the upcoming flow equations (in the wake of skipping the bars) in newly introduced frame can be separately composed as:

$$\begin{aligned}&\partial _{{\rm x}}u+\partial _{{\rm z}}w=0, \end{aligned}$$

(10)

$$\begin{aligned}&\beta ^{2}\partial _{{\rm yy}}u+\partial _{{\rm zz}}u-\frac{3K}{1+M_{1}}\left( \beta ^{4}\left( \partial _{{\rm y}}u\right) ^{2}\partial _{{\rm yy}}u \right. \\&\quad \left. +\left( \partial _{{\rm z}}u\right) ^{2}\partial _{{\rm zz}}u\right) =\frac{1}{1+M_{1}}\partial _{{\rm x}}p, \end{aligned}$$

(11)

$$\begin{aligned}&\beta ^{2}\partial _{{\rm yy}}\theta +\partial _{{\rm zz}}\theta +P_{{\rm r}}E_{{\rm c}}\left( \left( 1+M_{1}\right) \left( \beta ^{2}\left( \partial _{{\rm y}}u\right) ^{2}\right. \right. \\&\quad \left. \left. +\left( \partial _{{\rm z}}u\right) ^{2}\right) -K\left( \beta ^{4}\left( \partial _{{\rm y}}u\right) ^{4}+\left( \partial _{{\rm z}}u\right) ^{4}\right) \right) =0, \end{aligned}$$

(12)

$$\begin{aligned}&\beta ^{2}\partial _{{\rm yy}}\sigma +\partial _{{\rm yy}}\sigma +S_{{\rm r}}S_{{\rm c}}\left( \beta ^{2}\partial _{{\rm yy}}\theta +\partial _{{\rm zz}}\theta \right) =0. \end{aligned}$$

(13)

In the above expression, $P_{{\rm r}}$ shows the Prandtl number, $E_{{\rm c}}$ gives the Eckert number, $S_{{\rm r}}$ means the Soret number and $S_{{\rm c}}$ depicts the Schmidt number. The concerning dimensionless surface conditions for the considered scenario are stated as [14]:

$$\begin{aligned}&u(\mp 1,z)+1=0\text { and }u(y,\mp h(x))+1=0\text { with }h(x)=1+\phi *\cos 2\pi x, \end{aligned}$$

(14)

$$\begin{aligned}&\theta =\left\{ \begin{array}{ll} 0&\quad {\text { at }}z=h \\ 1&\quad {\text { at }}z=-h \end{array} \right. , \end{aligned}$$

(15)

$$\begin{aligned}&\sigma =\left\{ \begin{array}{ll} 0&\quad {\text { at }}z=h \\ 1&\quad {\text { at }}z=-h \end{array} \right. . \end{aligned}$$

(16)

It is to be mentioned here that the above-discussed problem relates to the two-dimensional channel when $\beta \rightarrow 0$ and $\beta =1$ generates the square shape duct readings. It can also be analyzed that limiting K and $M_{1}$ to zero regenerates properties of linear fluid.

Solution strategy

The finalized sets of equations (11) and (15) are being handled by HPM [23, 24]. The homotopy for the unknown profiles mentioned in the above equations is structured as:

$$\begin{aligned}&L_{{\rm vel}}\left( u-\widehat{u}_{0}\right) +q\left[ L_{{\rm vel}}\left( u\right) -3K\left( 1+M_{1}\right) ^{-1}\right. \\&\quad \left. \left( \beta ^{4} \left( \frac{\partial u}{\partial y}\right) ^{2}\frac{\partial ^{2}u}{\partial y^{2}} +\left( \frac{\partial u}{\partial z}\right) ^{2}\frac{\partial ^{2}u}{ \partial z^{2}}\right) -\frac{1}{1+M_{1}}\frac{\partial p}{\partial x}\right] =0, \end{aligned}$$

(17)

$$\begin{aligned}&L_{{\rm temp}}\left( \overline{\theta }-\widehat{\theta }_{0}\right) +q \left[ L_{{\rm temp}}\left( \overline{\theta }\right) +\beta ^{2}\frac{\partial ^{2}\overline{\theta }}{\partial y^{2}}\right. \\&\quad +P_{{\rm r}}E_{{\rm c}}\left( \left( 1+M_{1}\right) \left( \beta ^{2}\left( \frac{\partial u}{\partial y}\right) ^{2}+\left( \frac{\partial u}{\partial z}\right) ^{2}\right) \right. \\&\quad \left. \left. -K\left( \beta ^{4}\left( \frac{\partial u}{\partial y }\right) ^{4}+\left( \frac{\partial u}{\partial z}\right) ^{4}\right) \right) \right] =0. \end{aligned}$$

(18)

For further analogy, we have taken $L_{{\rm vel}}=\beta ^{2} \frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial z^{2}}$ and $L_{{\rm temp}}=\frac{\partial ^{2}}{\partial z^{2}}$ as the linear operators. The initial approximations are chosen as:

$$\begin{aligned} \widehat{u}_{0}\left( y,z\right) =\frac{1-y^{2}+\beta ^{2}\left( z^{2}-h^{2}-1\right) }{\beta ^{2}},\quad \widehat{\theta }_{0}=\frac{1-z/h }{2}. \end{aligned}$$

(19)

Applying perturbation on the involved parameter q for the variables u and $\theta$, we have

$$\begin{aligned}&u=u_{0}+qu_{1}+q^{2}u_{2}+\cdots \end{aligned}$$

(20)

$$\begin{aligned}&\overline{\theta }=\theta _{0}+q\theta _{1}+q^{2}\theta _{2}+\cdots \end{aligned}$$

(21)

Injecting expressions (20) and (21) into relations (17) and (18) and then composing the terms involving coefficients of nonnegative increasing integral powers of q, we collect

$$\begin{aligned}&L_{{\rm vel}}\left( u_{0}\right) -L_{{\rm vel}}\left( \widehat{u}_{0}\right) =0, \end{aligned}$$

(22)

$$\begin{aligned}&u_{0}+1=0\text { when }y\pm 1=0,\quad u_{0}+1=0\text { when }z\pm h\left( x\right) =0, \end{aligned}$$

(23)

$$\begin{aligned}&L_{{\rm temp}}\left( \theta _{0}\right) -L_{{\rm temp}}\left( \widehat{\theta } _{0}\right) =0, \end{aligned}$$

(24)

$$\begin{aligned}&\theta _{0}=0\text { at }z=h\left( x\right) ,\quad \theta _{0}=1\text { at } z=-h\left( x\right) . \end{aligned}$$

(25)

From Eqs. (19) and (22–25), we gain the following results:

$$\begin{aligned}&u_{0}=\widehat{u}_{0}\left( y,z\right) =-1+\frac{1}{\beta ^{2}}\left( 1-y^{2}\right) +z^{2}-h^{2}\left( x\right) ,\quad \\&\quad \theta _{0}=\widehat{ \theta }_{0}=\frac{h\left( x\right) -z}{2h\left( x\right) }, \end{aligned}$$

(26)

$$\begin{aligned}&\beta ^{2}\frac{\partial ^{2}u_{1}}{\partial y^{2}}+\frac{\partial ^{2}u_{1} }{\partial z^{2}}-\frac{3K}{1+M_{1}}\left( \beta ^{4}\left( \frac{\partial u_{0}}{\partial y}\right) ^{2}\frac{\partial ^{2}u_{0}}{\partial y^{2}} \right. \\&\quad \left. +\left( \frac{\partial u_{0}}{\partial z}\right) ^{2}\frac{\partial ^{2}u_{0} }{\partial z^{2}}\right) -\frac{1}{1+M_{1}}\frac{\partial p}{\partial x}=0, \end{aligned}$$

(27)

$$\begin{aligned}&u_{1}=0\text { when }y=\pm 1\text { and }u_{1}=0\text { when }z=\pm h\left( x\right) , \end{aligned}$$

(28)

$$\begin{aligned}&\frac{\partial ^{2}\theta _{1}}{\partial z^{2}}+\beta ^{2}\frac{\partial ^{2}\theta _{0}}{\partial y^{2}}+P_{{\rm r}}E_{{\rm c}}\left( \left( 1+M_{1}\right) \left( \beta ^{2}\left( \frac{\partial u_{0}}{\partial y}\right) ^{2}+\left( \frac{\partial u_{0}}{\partial z}\right) ^{2}\right) \right. \\&\quad \left. -K\left( \beta ^{4}\left( \frac{\partial u_{0}}{\partial y}\right) ^{4}+\left( \frac{ \partial u_{0}}{\partial z}\right) ^{4}\right) \right) =0, \end{aligned}$$

(29)

$$\begin{aligned}&\theta _{1}=0\text { when }y=\pm 1,\quad \theta _{1}=0\text { when }z=\pm h\left( x\right) . \end{aligned}$$

(30)

Now, switching the eigenfunction expansion technique on Eq. (27), the closed form of $u_{1}$ can be observed as follows:

$$\begin{aligned} u_{1}=\overset{\infty }{\underset{n=1}{\sum }}a_{{\rm n}}(y)\cos \left( 2n-1\right) \frac{\pi }{2h\left( x\right) }z, \end{aligned}$$

(31)

where $a_{{\rm n}}\left( y\right)$ is summarized as:

$$\begin{aligned} a_{{\rm n}}(y)&=\left( 4\cos \left( n\pi \right) \left( \cosh \left( \lambda _{{\rm n}}^{\prime }\right) -\sinh \left( \lambda _{{\rm n}}^{\prime }\right) \right) \left( \left( -1\left( 1-2n\right) ^{2}\frac{\hbox{d}p}{\hbox{d}x}\pi ^{2}\left( \beta \lambda _{{\rm n}}\right) ^{2}-24K\right. \right. \right. \\&\left. \left( -8h^{2}\left( x\right) \left( \beta \lambda _{{\rm n}}\right) ^{2}+ \left[ \pi \left( 1-2n\right) \right] ^{2}\left( -2\beta ^{2}-\lambda _{{\rm n}}^{2}+h^{2}\left( x\right) \left( \beta \lambda _{{\rm n}}\right) ^{2}\right) \right) \right) \left( \cosh \left( \frac{\lambda _{{\rm n}}}{\beta }\right) -\sinh \left( \frac{\lambda _{{\rm n}}}{\beta }\right) \right) \\&+\left( \left( 1-2n\right) ^{2}\frac{\hbox{d}p}{\hbox{d}x}\pi ^{2}\left( \beta \lambda _{{\rm n}}\right) ^{2}+24K\left( -8h\left( x\right) ^{2}\left( \beta \lambda _{{\rm n}}\right) ^{2}+\pi ^{2}\left( 1-2n\right) ^{2}\left( -2\beta ^{2}-y^{2}\lambda _{{\rm n}}^{2}+h^{2}\left( x\right) \right. \right. \right. \\&\left. \left( \beta \lambda _{{\rm n}}\right) ^{2}\right) ))\left( \cosh \left( \lambda _{{\rm n}}^{\prime }\right) -\sinh \left( \lambda _{{\rm n}}^{\prime }\right) \right) +\left( \left( 1-2n\right) ^{2}\frac{\hbox{d}p}{\hbox{d}x}\pi ^{2}\left( \beta \lambda _{{\rm n}}\right) ^{2}+24K\left( -8h^{2}\left( x\right) \right. \right. \\&\left. \left. \left( \beta \lambda _{{\rm n}}\right) ^{2}+\pi ^{2}\left( 1-2n\right) ^{2}\left( -2\beta ^{2}-y^{2}\lambda _{{\rm n}}^{2}+h^{2}\left( x\right) \left( \beta \lambda _{{\rm n}}\right) ^{2}\right) \right) \right) \left( \cosh \left( \frac{2\lambda _{{\rm n}}}{\beta }+\lambda _{{\rm n}}^{\prime }\right) -\right. \\&\left. \sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }+\lambda _{{\rm n}}^{\prime }\right) \right) +\left( \left( 1-2n\right) ^{2}\frac{\hbox{d}p}{\hbox{d}x}\right. \pi ^{2}\left( \beta \lambda _{{\rm n}}\right) ^{2}-24K\left( -8h^{2}\left( x\right) \left( \beta \lambda _{{\rm n}}\right) ^{2}+\pi ^{2}\left( 1-2n\right) ^{2}\right. \\&\left. \left. \left( -2\beta ^{2}-\lambda _{{\rm n}}^{2}+h^{2}\left( x\right) \left( \beta \lambda _{{\rm n}}\right) ^{2}\right) \right) \right) \left( \cosh \left( \frac{\lambda _{{\rm n}}}{\beta }+2\lambda _{{\rm n}}^{\prime }\right) \right. +\left. \left. \sinh \left( \frac{\lambda _{{\rm n}}}{\beta }+2\lambda _{{\rm n}}^{\prime }\right) \right) \right) \\&/\left( \left( 1+M_{1}\right) \left( 2n-1\right) ^{3}\pi ^{3}\beta ^{2}\lambda _{{\rm n}}^{4}\left. \left( 1+\cosh \left( \frac{2\lambda _{{\rm n}}}{\beta } \right) +\sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) \right) .\right. \end{aligned}$$

(32)

Here, $\lambda _{{\rm n}}=\left( 2n-1\right) \pi /2h\left( x\right)$, $n\in Z^{+}$ suggest the eigenvalues and $\lambda _{{\rm n}}^{\prime }=\frac{\lambda _{{\rm n}}y}{ \beta }$. The expression of temperature profile is evaluated as:

$$\begin{aligned} \theta _{1}= & {} \frac{1}{105\beta ^{6}}8E_{{\rm c}}K\left( 1+M_{1}\right) P_{{\rm r}}\left( 420y^{6}\left( z^{2}-h\left( x\right) ^{2}\right) +70y^{4}\left( z^{4}-h\left( x\right) ^{4}\right) \beta ^{2}\right. \\&\left. +28y^{2}\left( z^{6}-h\left( x\right) ^{6}\right) \beta ^{4}+15\left( z^{8}-h\left( x\right) ^{8}\right) \beta ^{6}\right) . \end{aligned}$$

(33)

The particular complete solutions are achieved when $q\longrightarrow 1$ which are observed as:

$$\begin{aligned}&u=u_{0}+u_{1}+\cdots \end{aligned}$$

(34)

$$\begin{aligned}&\theta =\theta _{0}+\theta _{1}+\cdots \end{aligned}$$

(35)

The closed-form series solution of concentration profile can be calculated from Eq. (13) along with boundary conditions (16) and is found as:

$$\begin{aligned} \sigma =f\left( x,y,z\right) +\overset{\infty }{\underset{n=1}{\sum }}\left( c_{{\rm n}}\cosh \left( \frac{\lambda _{{\rm n}}}{\beta }y\right) +d_{{\rm n}}\sinh \left( \frac{\lambda _{{\rm n}}}{\beta }y\right) \right) \cos \left( \lambda _{{\rm n}}z\right) , \end{aligned}$$

(36)

where

$$\begin{aligned} f\left( x,y,z\right)= & {} \frac{1}{240h\left( x\right) \left( x,t\right) \beta ^{6}}\left( h\left( x\right) -z\right) \left( 105\beta ^{6}-64E_{{\rm c}}h\left( x\right) K\left( 1+M_{1}\right) P_{{\rm r}}S_{{\rm c}}S_{{\rm r}}\left( h\left( x\right) +z\right) \right. \\&\left( -105y^{6}+35y^{4}\left( 37h\left( x\right) ^{2}-8z^{2}\right) \beta ^{2}+7y^{2}\left( 13h\left( x\right) ^{4}-2h\left( x\right) ^{2}z^{2}-2z^{4}\right) \beta ^{4}\right. \\&\left. \left. +\left( 3h\left( x\right) ^{6}-4h\left( x\right) ^{4}z^{2}-4h\left( x\right) ^{2}z^{2}-4z^{6}\right) \beta ^{6}\right) \right) ,\\ c_{{\rm n}}= & {} \frac{-1}{\left( 2n-1\right) ^{9}\pi ^{9}\beta ^{6}}\left( 2\sec \text {h}\left( \frac{\lambda _{{\rm n}}}{\beta }\right) \left( \left( a_{2}+b_{2}-1\right) \left( 1-2n\right) ^{8}\pi ^{8}\beta ^{6}-512E_{{\rm c}}h\left( x\right) ^{2}K\left( 1+M_{1}\right) \right. \right. \\&P_{{\rm r}}S_{{\rm c}}S_{{\rm r}}\left( -49152h\left( x\right) ^{6}\beta ^{6}+768h\left( x\right) ^{4}\left( 1-2n\right) ^{2}\pi ^{2}\beta ^{4}\left( y^{2}+8h\left( x\right) ^{2}\beta ^{2}\right) +\left( 1-2n\right) ^{6}\pi ^{6}\right. \\&\left( y^{2}+h\left( x\right) ^{2}\beta ^{2}\right) \left( y^{4}+h\left( x\right) ^{4}\beta ^{4}\right) -32h\left( x\right) ^{2}\left( 1-2n\right) ^{4}\pi ^{4}\beta ^{2}\left( 4y^{4}+3h\left( x\right) ^{2}y^{2}\beta ^{2}\right. \\&+4h\left( x\right) ^{4}\beta ^{4}\cos \left( n\pi \right) \end{aligned}$$

and

$$\begin{aligned} d_{{\rm n}}=\frac{2\left( b_{2}-a_{2}\right) \cos [n\pi ]\cos \text {h}\left( \frac{ \lambda _{{\rm n}}}{\beta }\right) }{\left( 2n-1\right) \pi }. \end{aligned}$$

The volumetric flow rate $\overline{\overline{F}}$ is calculated as:

$$\begin{aligned} \overline{\overline{F}}=\underset{0}{\overset{h\left( x\right) \left( x\right) }{\int }}\underset{0}{\overset{1}{\int }}u\left( x,y,z\right) {\mathrm{d}}y{\mathrm{d}}z. \end{aligned}$$

(37)

The instantaneous flux is found by

$$\begin{aligned} \overline{Q}=\underset{0}{\overset{h}{\int }}\underset{0}{\overset{1}{\int }}\left( u+1\right) {\mathrm{d}}y{\mathrm{d}}z=\overline{\overline{F}}+h\left( x\right) . \end{aligned}$$

(38)

The average value of volumetric rate of flow per single wave circulation is defined by

$$\begin{aligned} Q=\frac{\overset{\overleftrightarrow {T}}{\underset{0}{\int }}\overline{Q}{\mathrm{d}}t }{\overleftrightarrow {T}}=\overline{\overline{F}}+1,\quad \overleftrightarrow {T}\text { being assumed the time period.} \end{aligned}$$

(39)

The pressure profile dp/dx is extracted through entertaining results (37) and (39) which is composed as:

$$\begin{aligned} \frac{\hbox{d}p}{\hbox{d}x}&=\left( -(1+M_{1})(2n-1)^{4}\pi ^{2}\lambda _{{\rm n}}^{3}\left( 1+\cosh \left( 2\lambda _{{\rm n}}\beta ^{-1}\right) +\sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) \right. \left( \frac{-2h\left( x\right) ^{3}}{3} -Q+\frac{2h\left( x\right) }{3\beta ^{2}}\right. \\&+\left[ (1+M_{1})(2n-1)^{4}\pi ^{2}\beta ^{2}\lambda _{{\rm n}}^{5}\left( 1+\left( \cosh +\sinh \right) \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) \right] ^{-1}\left( 64h\left( x\right) K\left( -24h\left( x\right) ^{2}\beta ^{2}\lambda _{{\rm n}}^{2}\right. \right. \\&\left( -\beta +\left( \beta -\lambda _{{\rm n}}\right) \cosh \left( \frac{ 2\lambda _{{\rm n}}}{\beta }\right) \right. \left. +\left( \beta -\lambda _{{\rm n}}\right) \sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) +\pi ^{2}\left( -2n+1\right) ^{2}\left( \lambda _{{\rm n}}^{3}\right. \\&\left( 1+\cosh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) +\sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) +\left( 3\beta ^{3}\left( h\left( x\right) ^{2}\lambda _{{\rm n}}^{2}-2\right) -3\beta \lambda _{{\rm n}}^{2}\right) \left( -1+\cosh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right. \\&+\left. \sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) -3\beta ^{2}\lambda _{{\rm n}}\left. \left. \left. \left( h\left( x\right) ^{2}\lambda _{{\rm n}}^{2}-2\right) \left( 1+\cosh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) +\sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) \right) \right) \right) \\&/(8h\left( x\right) (1-2n)^{2}\left( {}\right. \left. \left. \left( \beta -\lambda _{{\rm n}}\right) -\left( \beta +\lambda _{{\rm n}}\right) \cosh \left( \frac{ 2\lambda _{{\rm n}}}{\beta }\right) +\left( \beta -\lambda _{{\rm n}}\right) \sinh \left( \frac{2\lambda _{{\rm n}}}{\beta }\right) \right) \right) .\end{aligned}$$

(40)

The pressure rise $\Delta p$ is found by numerical integration on Mathematica by making use of the integral below:

$$\begin{aligned} \Delta p=\underset{0}{\overset{1}{\int }}\left( \frac{\hbox {d}p}{\hbox {d}x}\right) {\mathrm{d}}x. \end{aligned}$$

(41)

Graphical investigations

In this portion of the article, we explained the results manipulated graphically for velocity, pressure rise, pressure gradient, temperature distribution, concentration profile and stream functions to discuss the contribution of pertinent parameters. Tables are also presented to watch out the numerical variation in the profiles of velocity, temperature and concentration for Newtonian and non-Newtonian Eyring–Powell model. Tables 1, 2 and 3 correspond to the variations of velocity u, temperature $\theta$ and concentration $\sigma$, respectively. The velocity profile u is sketched in Figs. 2–4 against the space coordinate z under the alteration of the parameters $\beta$, K and $M_{1}$ for both two and three dimensions. Figures 5–7 are displayed to consider the behavior of pressure rise curve $\Delta p$ for $\beta$, $M_{1}$ and $\phi$. The variational distribution of dp/dx is revealed in Figs. 8–11 to depict the influence of the quantities $\beta$, K, $M_{1}$ and Q. The temperature distribution $\theta$ is shown in Figs. 12–14 under the variation of $\beta$, $E_{{\rm c}}$ and $M_{1}$, respectively. Figures 15–17 imply the effects of $\beta$, $S_{{\rm c}}$ and $M_{1}$ on mass concentration distribution. The stream functions are portrayed in Figs. 18–20 to examine the small rounded mass (called bolus) shape phenomenon.

Fig. 6

Pressure rise curves $\Delta p$ against $M_{1}$

Fig. 7

Pressure rise curves $\Delta p$ against $\phi$

Fig. 8

Pressure gradient dp/dx against $\beta$

Fig. 9

Pressure gradient dp/dx against K

Fig. 10

Pressure gradient dp/dx against $M_{1}$

Figure 2(a,b) shows the readings of velocity profile u under the parameter $\beta$. It is to be mentioned here that velocity is increasing when we enlarge the magnitude of $\beta$. The treatment of velocity field for the parameter K is examined from Fig. 3a, b. We can depict here that the velocity is diminishing with the monotonic changes in K by keeping all other quantities constant. From Fig. 4a, b, it is admitted that the profile is varying linearly with the small values of $M_{1}$, but as we give larger magnitude to the parameter $M_{1}$, the velocity curves are coming near to each other.

Figure 5 indicates the alteration of pressure rise curves $\Delta p$ for the parameter $\beta$. It is convincing that the pressure rise is decreasing with the increase in $\beta$ in the retrograde pumping region $\left( \Delta p>0>Q\right)$, while inverse scenario is reported in the pumping portion $\left( \Delta p,Q>0\right)$ and augmented pumping region $\left( \Delta p,Q<0\right)$. We can describe through Fig. 6 that pressure rise $\Delta p$ is varying linearly with the change of parameter $M_{1}$ in the retrograde pumping as well as peristaltic pumping areas, while reverse behavior is measured in the copumping region. The impact of pressure rise profile against the flow rate Q for the parameter $\phi$ is displayed in Fig. 7. One can observe that the attitude of pressure rise distribution for $\phi$ is quite parallel to that of the parameter $M_{1}$.

Fig. 11

Pressure gradient dp/dx against Q

Fig. 12

Temperature distribution $\theta$ against $\beta$

Fig. 13

Temperature distribution $\theta$ against $E_{{\rm c}}$

Fig. 14

Temperature distribution $\theta$ against $M_{1}$

Fig. 15

Mass concentration $\sigma$ against $\beta$

Fig. 16

Mass concentration $\sigma$ against $S_{{\rm c}}$

Fig. 17

Mass concentration $\sigma$ against $M_{1}$

Fig. 18

Streamlines against $\beta$

Fig. 19

Streamlines against K

Fig. 20

Streamlines against $M_{1}$

To see the pressure gradient distribution dp/dx against the parameter $\beta$, Fig. 8 is shown. Here, we can suggest that pressure gradient gives the inverse behavior when someone increases the contribution of $\beta$. It is also resulted that much pressure gradient is examined in the middle of the channel, while in the corner regions, the pressure gradient has a minimum width. It shows that extensive amount of pressure change is required in the central part to balance the flow relative to the walls of the channel. Figure 9 deliberates that the pressure gradient dp/dx chooses the linear relation with the parameter K and pressure gradient curves become much closer to each other imminent to the walls. The influence of parameter $M_{1}$ on pressure gradient profile dp/dx is shown in Fig. 10. One observes that the pressure gradient is increasing with the increase in $M_{1}$ in the region $x\in \left[ 0.3,0.7\right]$, while inverse investigation is made in the rest of the region. Figure 11 discloses the variation of pressure gradient dp/dx with the flow rate Q. It is measured that pressure gradient is lessening with the flow rate and it becomes nominal at the nook of the domain.

Figure 12 shows the variation of temperature distribution $\theta$ for the parameter $\beta$. It is mentioned here that temperature profile is getting larger while increasing the values of $\beta$. It is also noted here that the temperature curves are getting nearby to each other. Figure 13 ensures that the temperature distribution $\theta$ is linearly proportional to the Eckert number $\theta$ is linearly proportional to the Eckert number $E_{{\rm c}}$. One can observe the variation of temperature profile $\theta$ against the parameter $M_{1}$ from Fig. 14 and can be seen that as we increase the values of $M_{1}$, the temperature curves are reducing their height. Figure 15 shows the variation of concentration profile $\sigma$ with the effect of aspect ratio $\beta$. It is derived from this figure that the more the aspect ratio, the larger the concentration and more variation is seen in the middle of the channel as compared to the corners. It is depicted from Figs. 16 and 17 that concentration curves are varying inversely to the change in Schmidt number $S_{{\rm c}}$ and the fluid parameter $M_{1}$ but is also observed that concentration profile is depending on Schmidt number more than that of $M_{1}$.

In Fig. 18, the streamlines are appeared to observe the flow phenomenon in rectangular channel for the parameter $\beta$. It is observed that the number of trapping bolus is enhanced with rising intensity of $\beta$, while dimensions of the bolus are varying under the exceeding values of $\beta$. To see the streamlines for the parameter K, Fig. 19 is presented. It is concluded from this graph that trapping boluses are contracted in size but do not change their quantity while we give large values to K. From Fig. 20, one can estimate that trapping boluses remain the same in number and dimensions when $M_{1}\le 1$ but decrease in magnitude and volume when $M_{1}>1$.

Table 1 Data of velocity profile

Table 2 Variation of temperature factor

Table 3 Variation of mass concentration

Conclusions

In the above discussion, the authors have investigated the heat and mass transfer analysis for a viscoelastic fluid model considered in a three-dimensional rectangular peristaltic channel. The problem is handled by collaborative implementation of HPM and eigenfunction expansion method. The flow becomes steady by transforming the observations in a wave frame. The lubrication approach has been made to encounter the laminar flow through small cross sections. The obtained analytical and numerical data have been sketched against different relating coordinates to verify the physical and theoretical effects of many pertinent features of the analysis on velocity component, pressure gradient, pressure rise, thermal and mass profiles. The main points squeezed from the above observations have been labeled below:

• 1. The velocity profile is becoming larger by aspect ratio of rectangular duct, while it shows opposite facts for Eyring–Powell fluid parameters.

• 2. It is followed that peristaltic pumping is enhanced by increasing effect of aspect ratio and fluid factors.

• 3. It is evaluated from the above graphical measures that pressure gradient is varying inversely with aspect ratio and directly with non-Newtonian fluid parameter.

• 4. The temperature profile and mass concentration give linear behavior with aspect ratio and nonlinear with non-Newtonian features.

• 5. It is finalized from streamlines that trapping boluses are increasing their numbers with the variation of aspect ratio but remains constant in numbers with the fluid model.