**Preface to "Recent Advances in Mathematical Aspect in Engineering"**

This book contains ten chapters. A total of 53 papers have been submitted for possible publication. After a comprehensive peer review, only 10 papers qualified for final publication. Extensive uses of realistic applications are given in each chapter. For the best understanding of readers, a relevant list of references is also given at the end of each chapter for further study. We wish to thank all reviewers for their excellent suggestions and critical reviews on submitted manuscripts. We were fortunate enough to have prominent scholars who contributed with their original research. We applaud all of them on the successful completion of the Special Issue "Recent Advances in Mathematical Aspects of Engineering". Errors and omissions, if any, are requested to be pointed out, which will be gratefully acknowledged in the next possible Edition. In particular, suggestions for improvement and for the scope and format of the book will be highly appreciated. We express our special gratitude to MPDI for publishing this book. We also want to express our gratitude to the entire editorial team, the contact editor Mrs. Celina Si, and our families and friends for their helpful cooperation. It is worth mentioning that this Special Issue has been cited more than 170 times and viewed more than 11,000 times in a very short span of time. We hope that this book will not only provide an overall picture and the most up-to-date findings from the scientific community working in the field, but also benefit the industrial sectors in specific market niches and end users.

> **Rahmat Ellahi, Sadiq M. Sait, and Huijin Xu** *Editors*

#### *Editorial* **Recent Advances in Mathematical Aspects of Engineering**

**Rahmat Ellahi 1,2,\*, Sadiq M. Sait <sup>3</sup> and Huijin Xu <sup>4</sup>**


**Abstract:** This special issue took this opportunity to invite researchers to contribute their latest original research findings, review articles, and short communications on advances in the state of the art of mathematical methods, theoretical studies, or experimental studies that extend the bounds of existing methodologies to new contributions addressing current challenges and engineering problems on "Recent Advances in Mathematical Aspects of Engineering" to be published in *Symmetry*.

**Keywords:** fluid mechanics; optimization; energy; heat transfer; steady and unsteady flow problems; porosity; nanofluids; particle shape effects; multiphase flow; thermodynamics; magnetohydrodynamics; electromagnetic; physiological fluid phenomena in biological systems; peristaltic; blood flow

#### **1. Introduction**

In response to a call for papers, a total of 25 papers were submitted for possible publication. After a comprehensive peer review, only 9 papers qualified for acceptance for final publication; the rest 16 papers could not be accommodated. The submissions may have been technically correct but were not considered appropriate for the scope of this special issue. The authors are from 12 geographically distributed countries: the U.S., Mexico, China, Jordan, Saudi Arabia, Pakistan, Malaysia, Vietnam, Taiwan, Thailand, Egypt, and India. This reflects the great impact of the proposed topic and the effective organization of the guest editorial team of this Special Issue. Several theoretical and experimental attempts have been devoted, and this Special Issue is one of them. We hope that this issue will not only address the current challenges but also provide an overall picture and up-to-date findings to readers of the scientific community that ultimately benefits the industrial sector regarding its specific market niches and end users.

#### **2. Methodologies and Usages**

The peristaltic flow of a Johnson–Segalman fluid in a symmetric curved channel with convective conditions and flexible walls is addressed in [1]. The channel walls are considered compliant. The main objective of this article is to discuss the effects of a curvilinear channel and heat/mass convection through boundary conditions. The constitutive equations for the Johnson–Segalman fluid are modeled and analyzed under the lubrication approach. The stream function, temperature, and concentration profiles are derived. The analytical solutions are obtained by using the regular perturbation method for a significant number, named the Weissenberg number. The influence of the parameter values on the physical level of interest is outlined and discussed. A comparison is made between Johnson–Segalman and Newtonian fluids. It is concluded that the axial velocity of a Johnson–Segalman fluid is substantially higher than that of a Newtonian fluid.

**Citation:** Ellahi, R.; Sait, S.M.; Xu, H. Recent Advances in Mathematical Aspects of Engineering. *Symmetry* **2021**, *13*, 811. https://doi.org/ 10.3390/sym13050811

Received: 26 January 2021 Accepted: 30 March 2021 Published: 6 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Bhatti et al. [2] deal with the mass transport phenomena on particle fluid motion through an annulus. A non-Newtonian fluid propagates through a ciliated annulus in the presence of two phenomenon, namely (i) endoscopy and (ii) blood clot. The outer tube is ciliated. To examine the flow behavior, the authors consider the bi-viscosity fluid model. Mathematical modeling is formulated for a small Reynolds number to examine the inertia-free flow. The purpose of this assumption is that the wavelength-to-diameter ratio is maximal, and the pressure can be considerably uniform throughout the entire cross section. The resulting equations are analytically solved, and exact solutions are given for particleand fluid-phase profiles. The computational software Mathematica is used to evaluate both closed-form and numerical results. The graphical behavior across each parameter is discussed in detail and presented with graphs. The trapping mechanism is also shown across each parameter. It is noted clearly that the particle volume fraction and the blood clot reveal converse behavior on fluid velocity; however, the velocity of the fluid reduces significantly when the fluid behaves as a Newtonian fluid. Schmidt and Soret numbers enhance the concentration mechanism. Furthermore, more pressure is required to pass the fluid when the blood clot appears.

In [3], the problem of finite-time control for nonlinear systems with time-varying delay and exogenous disturbance is studied. First, by constructing a novel augmented Lyapunov–Krasovskii functional involving several symmetric positive definite matrices, a new delay-dependent finite-time boundedness criterion is established for the considered T-S fuzzy time-delay system by employing an improved reciprocally convex combination inequality. Then, a memory state feedback controller is designed to guarantee the finitetime boundness of the closed-loop T-S fuzzy time-delay system, which is in the framework of linear matrix inequalities (LMIs). Finally, the effectiveness and merits of the proposed results are shown by a numerical example.

The maldistribution of fluid flow through multi-channels is a critical issue encountered in many areas, such as multi-channel heat exchangers, electronic device cooling, refrigeration and cryogenic devices, air separation, and the petrochemical industry. The uniformity of flow distribution in a printed circuit heat exchanger (PCHE) is investigated in [4]. The flow distribution and resistance characteristics of a PCHE plate are studied with numerical models under different flow distribution cases. The results show that a sudden change in the angle of the fluid at the inlet of the channel can be greatly reduced by using a spreader plate with an equal inner and outer radius. The flow separation of the fluid at the inlet of the channel can also be weakened, and the imbalance of flow distribution in the channel can be reduced. Therefore, flow uniformity can be improved and the pressure loss between the inlet and outlet of PCHEs can be reduced. The flow maldistribution in each PCHE channel can be reduced to ±0.2%, and the average flow maldistribution in all PCHE channels can be reduced to less than 5% when the number of manifolds reaches nine. The numerical simulation of fluid flow distribution can provide guidance for subsequent research and the design and development of multi-channel heat exchangers. In summary, the symmetry of fluid flow in multi-channels for a PCHE is analyzed in this work. This work presents the frequently encountered problem of maldistribution of fluid flow in engineering, and the performance promotion leads to symmetrical aspects in both the structure and the physical process.

The entropy generation on the asymmetric peristaltic propulsion of a non-Newtonian fluid with convective boundary conditions is presented in [5]. The Williamson fluid model is considered for the analysis of flow properties. The current fluid model has the ability to reveal Newtonian and non-Newtonian behavior. The present model is formulated via momentum, entropy, and energy equations, under the approximation of a small Reynolds number and a long wavelength of the peristaltic wave. A regular perturbation scheme is employed to obtain the series solutions up to third-order approximation. All the leading parameters are discussed with the help of graphs for entropy and temperature profiles. The irreversibility process is also discussed with the help of a Bejan number. Streamlines are plotted to examine the trapping phenomena. Results obtained provide an excellent

benchmark for further study of the entropy production with mass transfer and a peristaltic pumping mechanism.

In [6], the author examines the unsteady flow over a rotating stretchable disk with deceleration. The highly nonlinear partial differential equations of viscous fluid are simplified by existing similarity transformation. Reduced nonlinear ordinary differential equations are solved by the homotopy analysis method (HAM). The convergence of HAM solutions is also obtained. A comparison table between analytical solutions and numerical solutions is also presented. Finally, the results for useful parameters, i.e., disk stretching parameters and unsteadiness parameters, are found.

The aim of [7] is to examine the rheological significance of a Maxwell fluid configured between two isothermal stretching disks. The energy equation is extended by evaluating the heat source and sink features. The governing partial differential equations (PDEs) are converted to ordinary differential equations (ODEs) by using appropriate variables. An analytically based technique is adopted to compute the series solution of the dimensionless flow problem. The convergence of this series solution is carefully ensured. The physical interpretation of important physical parameters like the Hartmann number, Prandtl number, Archimedes number, Eckert number, heat source/sink parameter, and activation energy parameter are presented for velocity, pressure, and temperature profiles. The numerical values of different involved parameters for the skin friction coefficient and the local Nusselt number are expressed in tabular and graphical form. Moreover, the significance of an important parameter, namely Frank–Kamenetskii, is presented in both tabular and graphical form. This particular study reveals that both axial and radial velocity components decrease by increasing the Frank–Kamenetskii number and stretching the ratio parameter. The pressure distribution is enhanced with an increasing Frank–Kamenetskii number and a stretching ratio parameter. It is also observed that the temperature distribution increases with an increasing Hartmann number, Eckert number, and Archimedes number.

The key objective of the study reported in [8] is to probe the impacts of Brownian motion and thermophoresis diffusion on Casson nanofluid boundary layer flow over a nonlinear inclined stretching sheet, with the effect of convective boundaries and thermal radiations. Nonlinear ordinary differential equations are obtained from governing nonlinear partial differential equations by using compatible similarity transformations. The quantities associated with engineering aspects, such as skin friction, Sherwood number, and heat exchange, along with various impacts of material factors on the momentum, temperature, and concentration, are elucidated and clarified with diagrams. The numerical solution of the present study is obtained via the Keller-box technique and in limiting sense is reduced to the published results for accuracy purpose.

The effects of magnetohydrodynamic 3D nanofluid flow due to a rotating disk subject to Arrhenius activation energy and heat generation/absorption is examined in [9]. Flow is created due to a rotating disk. Velocity, temperature, and concentration slips at the surface of the rotating disk are considered. Effects of thermophoresis and Brownian motion are also accounted. The nonlinear expressions are deduced by the transformation procedure. The shooting technique is used to construct the numerical solution of the governing system. Plots are organized just to investigate how yjr velocity, temperature, and concentration are influenced by various emerging flow parameters. Skin-friction local Nusselt and Sherwood numbers are also plotted and analyzed. In addition, a symmetry is noticed for both components of velocity when the Hartman number increases.

#### **3. Future Trends in Fluid Mechanics**

The material that advances the state-of-the-art experimental, numerical, and theoretical methodologies or extends the bounds of existing methodologies through new contributions in symmetry is still insufficient, even with the completion of this Special Issue. The rheological characteristics with thin films under the influence of different nanoparticles and shapes can help with the development of better applications in industry. **Author Contributions:** Conceptualization, R.E.; S.M.S.; writing—original draft preparation; formal analysis, H.X. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The guest editorial team of *Symmetry* would like to thank all authors for contributing their original work to this special issue, no matter what the final decision on their submitted manuscript was. The editorial team would also like to thank all anonymous professional reviewers for their valuable time, comments, and suggestions during the review process. We also acknowledge the entire staff of the journal's editorial board for providing their cooperation regarding this Special Issue. We hope that this issue will not only provide an overall picture and most up-to-date findings to readers from the scientific community working in the field but also benefit the industrial sectors in specific market niches and end users.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


### *Article* **Convective Heat/Mass Transfer Analysis on Johnson-Segalman Fluid in a Symmetric Curved Channel with Peristalsis: Engineering Applications**

#### **Humaira Yasmin 1,\*, Naveed Iqbal 2,\* and Aiesha Hussain <sup>3</sup>**


Received: 19 August 2020; Accepted: 4 September 2020; Published: 8 September 2020

**Abstract:** The peristaltic flow of Johnson–Segalman fluid in a symmetric curved channel with convective conditions and flexible walls is addressed in this article. The channel walls are considered to be compliant. The main objective of this article is to discuss the effects of curvilinear of the channel and heat/mass convection through boundary conditions. The constitutive equations for Johnson–Segalman fluid are modeled and analyzed under lubrication approach. The stream function, temperature, and concentration profiles are derived. The analytical solutions are obtained by using regular perturbation method for significant number, named as Weissenberg number. The influence of the parameter values on the physical level of interest is outlined and discussed. Comparison is made between Jhonson-Segalman and Newtonian fluid. It is concluded that the axial velocity of Jhonson-Segalman fluid is substantially higher than that of Newtonian fluid.

**Keywords:** symmetric curved channel; Johnson-Segalman fluid; convective conditions; compliant walls

#### **1. Introduction**

The researchers have great interest in peristaltic transport of fluids due to immense applications in physiology, biomedical engineering andnindustry. Such motion is caused by a wave of expansion and contraction that propagates along the channel walls. Peristalsis includes the passage of urine from kidney to bladder, swallowing of food through oesophagus, the movement of chyme in the gastrointestinal tract, the vasomotion of small blood vessels, and many others. Blood pumps in the dialysis and heart lung machine operate on the principle of peristaltic action. The roller and finger pumps also operate according to this mechanism. In the nuclear industry, toxic materials can be moved through such a system in order to avoid contaminants from the outside area. Pioneering researches on the topic are presented by Latham [1], Shapiro et al. [2], and Yin and Fung [3]. Currently, abundant literature exists on peristaltic flows of viscous and non-Newtonian fluids under different aspects (see [4–19] and several studies there in). Amongst the several models of non-Newtonian material there is one fluid model that can describe the "spurt" phenomenon. It is subclass of integral type non-Newtonian material and is known as the Johnson–Segalman (JS) fluid. The phrase "spurt" is being used to characterize a significant volume rise to a slight rise in the moving pressure gradient. The contributions of Hayat et al. [20–22] are fundamental in this direction. Elshahed and Haroun [23] investigated the peristaltically moving Johnson–Segalman fluid together with the impact of the magnetism. Wang et al. [24] explored the peristalsis of the Johnson–Segalman fluid across a non-rigid tube. In reality, the configuration of the most physiological tubes and glandular ducts is curved. In this context, the effect of curvature appears to be meaningful. This fact gives great motivation to study peristaltic flow through curved channels. In the first place, Sato et al. [25] addressed the two-dimensional peristaltic transport of viscous liquid inside a curved channel. Ali et al. [26] revisited the analysis of Sato et al. [25] in a wave frame. Some more interesting studies for peristalsis in a curved channel can be consulted through [27–31].

The effect of heat transfer has vast applications in food processing, dilation of blood vessels, heat conduction in tissues, and its convection due to blood flow from the pores of the tissues. The impact of both heat and mass transfer plays an essential part in spreading of chemical pollutants in saturated soil, underground disposal of nuclear waste, thermal insulation, enhanced oil recovery, etc. The effects of mass transfer arose in diffusion, combustion, and distillation processes, and in many other industrial processes. Convective heat transfer through boundary conditions is used in systems, such as steam turbines, nuclear power stations, thermal energy storage, etc. In this context, Hina and Hayat [32] examined the effects of heat/mass transfer on Johnson-Segalman liquid inducing peristaltic movement in a compliant curved channel. Mehmood et al. [33], Hayat et al. [34] and Riaz et al. [35] analyzed the characteristics of heat flux in peristaltic transport with/without compliant walls. Hayat et al. [36–39] conducted an analysis of non-Newtonian fluids with peristalsis in the presence of convective constraints. Yasmin et al. [40] discussed the effects of convective conditions in peristalsis of Johnson–Segalman fluid in an asymmetric channel.

It is noted that the peristalsis of non-Newtonian fluid in a curved channel with convective mass transfer conditions at the walls is not addressed so far. Even such analysis is not carried out for viscous fluids. The current research paper varies from the existing results in terms of convective boundary conditions. The key focus of this paper is the implementation of a novel definition of convective heat and mass transfer conditions in the theory of Johnson–Segalman fluid transferred via a peristaltic motion across a curved channel. Hence, in this attempt, the convective conditions for both heat and mass transfer are considered. An incompressible Johnson–Segalman fluid is considered in a curved channel. The set of solutions for the small value of Weissenberg number are developed. The obtained results are visualized and thoroughly analyzed. Impacts reflecting the influence of pertinent parameters are pointed out physically.

#### **2. Problem Formulation**

We anticipate the peristaltic transport of the incompressible Johnson–Segalman fluid in a symmetric curved half-width (*d*1) channel clasped in a circular pattern with center *O* and radius *R*∗ (see Figure 1 and Ref. [32]).

**Figure 1.** Schematic diagram of the problem.

The flow in the channel is stimulated by small amplitude sinusoidal waves that travel along the compliant walls. The axial direction of the flow is *x* and *r* is radial direction. Here, *v* and *u* are the velocity vector components in the radial and axial directions, respectively. The wave shapes at channel walls are considered symmetric and given by

*Symmetry* **2020**, *12*, 1475

$$\tau = \pm \eta(\mathbf{x}, t) = \pm \left[ d\_1 + a \sin \left( \frac{2 \,\, \pi}{\lambda} \, \left( \mathbf{x} - \mathbf{c} \, t \right) \right) \right],\tag{1}$$

where *c* is the wave speed and *λ* is the wavelength, respectively.

The continuity and momentum equations governing the flow can be written as [32]:

$$\frac{\partial[(r+R^\*)\upsilon]}{\partial r} + R^\* \frac{\partial \mu}{\partial x} = 0,\tag{2}$$

$$\rho \left( \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial r} + \frac{R^\* u}{r + R^\*} \frac{\partial v}{\partial x} - \frac{u^2}{r + R^\*} \right) = -\frac{\partial p}{\partial r} + \frac{1}{r + R^\*} \frac{\partial}{\partial r} [(r + R^\*) \tau\_{\mathcal{V}}] + \frac{R^\*}{r + R^\*} \frac{\partial \tau\_{\mathcal{X}\mathcal{V}}}{\partial x} - \frac{\tau\_{\mathcal{X}\mathcal{X}}}{r + R^\*},\tag{3}$$

$$\rho \left( \frac{\partial \underline{u}}{\partial t} + v \frac{\partial \underline{u}}{\partial r} + \frac{R^\* \underline{u}}{r + R^\*} \frac{\partial \underline{u}}{\partial \mathbf{x}} + \frac{\mu \underline{v}}{r + R^\*} \right) = -\frac{R^\*}{r + R^\*} \frac{\partial p}{\partial \mathbf{x}} + \frac{1}{(r + R^\*)^2} \frac{\partial}{\partial r} \left[ (r + R^\*)^2 \tau\_{\rm rx} \right] + \frac{R^\*}{r + R^\*} \frac{\partial \tau\_{\rm rx}}{\partial \mathbf{x}}.\tag{4}$$

The equations for energy and concentration [32] are given by

$$\begin{split} \rho C\_p \left( \frac{\partial T}{\partial t} + v \frac{\partial T}{\partial r} + \frac{R^\* u}{r + R^\*} \frac{\partial T}{\partial \mathbf{x}} \right) &= \quad \kappa \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} \left( \frac{R^\*}{r + R^\*} \right)^2 + \frac{1}{r + R^\*} \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial r^2} \right) + (S\_{\mathcal{H}} - S\_{\mathcal{K}}) \frac{\partial v}{\partial r} \\ &+ S\_{\mathcal{H}} \left( \frac{\partial u}{\partial r} + \frac{R^\*}{r + R^\*} \frac{\partial v}{\partial \mathbf{x}} - \frac{u}{r + R^\*} \right), \end{split} \tag{5}$$

$$\begin{split} \frac{\partial \mathbb{C}}{\partial t} + v \frac{\partial \mathbb{C}}{\partial r} + \frac{R^\* u}{r + R^\*} \frac{\partial \mathbb{C}}{\partial \mathbf{x}} &= \quad D \left( \frac{\partial^2 \mathbb{C}}{\partial \mathbf{x}^2} \left( \frac{R^\*}{r + R^\*} \right)^2 + \frac{1}{r + R^\*} \frac{\partial \mathbb{C}}{\partial r} + \frac{\partial^2 \mathbb{C}}{\partial r^2} \right) \\ &+ \frac{D K\_T}{T\_m} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r + R^\*} \frac{\partial T}{\partial r} + \left( \frac{R^\*}{r + R^\*} \right)^2 \frac{\partial^2 T}{\partial \mathbf{x}^2} \right) . \end{split} \tag{6}$$

For the Johnson-Segalman fluid, the stress tensor **ø** is

$$
\mathfrak{o} = 2\mu \mathbf{D} + \mathbf{S}\_{\prime}
$$

in which the extra stress tensor **S** needs to satisfy the relationship

$$\mathbf{S} + m\left(\frac{d\mathbf{S}}{dt} + \mathbf{S}(\mathbf{W} - \xi\mathbf{D}) + (\mathbf{W} - \xi\mathbf{D})^T\mathbf{S}\right) = 2\eta\_1 \mathbf{D}\_{\text{tot}}$$

where

$$\mathbf{D} = \frac{[(grad\mathbf{V})^T + grad\mathbf{V}]}{2},$$

$$\mathbf{W} = \frac{[grad\mathbf{V} - (grad\mathbf{V})^T]}{2}.$$

The relations listed above produce the following equations:

$$S\_{rr} + m\left[\frac{dS\_{rr}}{dt} - \frac{2uS\_{rx}}{r + R^\*} + S\_{rx}\left\{(1 - \frac{\varepsilon}{\xi})\frac{\partial u}{\partial r} - \frac{1 + \frac{\varepsilon}{\xi}}{r + R^\*} [R^\* \frac{\partial v}{\partial x} - u] \right\} - 2\xi S\_{rr}\frac{\partial v}{\partial r}\right] = 2\eta\_1 \frac{\partial v}{\partial r},\tag{7}$$

$$\begin{split} S\_{rx} &+ m\frac{dS\_{rx}}{dt} + \frac{mu(S\_{rr} - S\_{xx})}{r + R^\*} + \frac{mS\_{xx}}{2}\left\{(1 - \xi)\frac{\partial u}{\partial r} - \frac{1 + \xi}{r + R^\*} \left[R^\* \frac{\partial v}{\partial x} - u\right] \right\} \\ &+ \frac{mS\_{rr}}{2}\left\{\frac{1 - \frac{\varepsilon}{\xi}}{r + R^\*} \left[R^\* \frac{\partial v}{\partial x} - u\right] - (1 + \xi)\frac{\partial u}{\partial r} \right\} \\ &= \eta\_1 \left(\frac{\partial u}{\partial r} + \frac{R^\*}{r + R^\*} \frac{\partial v}{\partial x} - \frac{u}{r + R^\*}\right),\tag{8}$$

$$S\_{xx} + m \left[ \frac{dS\_{xx}}{dt} + \frac{2\mu S\_{rx}}{r + R^\*} - S\_{rx} \left\{ (1 + \frac{\mathfrak{z}}{\mathfrak{z}}) \frac{\partial \mathfrak{u}}{\partial r} - \frac{1 - \frac{\mathfrak{z}}{\mathfrak{z}}}{r + R^\*} \left[ R^\* \frac{\partial \upsilon}{\partial \mathfrak{x}} - \mathfrak{u} \right] \right\} + 2\mathfrak{z}S\_{xx} \frac{\partial \upsilon}{\partial r} \right] = -2\eta\_1 \frac{\partial \upsilon}{\partial r}, \tag{9}$$

where *<sup>d</sup> dt* <sup>=</sup> *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>v</sup> <sup>∂</sup> <sup>∂</sup><sup>r</sup>* <sup>+</sup> *<sup>R</sup>*∗*<sup>u</sup> <sup>r</sup>*+*R*<sup>∗</sup> *<sup>∂</sup> <sup>∂</sup><sup>x</sup>* represents the material derivative with respect to time, *ρ* denotes the fluid density, *κ* the thermal conductivity of fluid, **W** and **D** skew-symmetric and symmetrical parts of the gradient of velocity, *ξ* the slip parameter, *Cp* the fluid specific heat, *T* and *C* are the fluid temperature and concentration, respectively, the thermal diffusion ratio is *KT*, *D* the mass diffusivity coefficient, *Tm* represents the mean/average temperature, *μ* and *η*<sup>1</sup> the viscosities, and *m* the relaxation time.

The appropriate boundary conditions are

$$
\mu = 0 \quad \text{at} \quad r = \pm \eta,\tag{10}
$$

$$k\frac{\partial T}{\partial r} = -h\_1(T - T\_0) \quad \text{at} \quad r = +\eta\_\prime \tag{11}$$

$$k\frac{\partial T}{\partial r}\_{r} = -h\_2(T\_0 - T) \quad \text{at} \quad r = -\eta\_\prime \tag{12}$$

$$D\frac{\partial \mathcal{C}}{\partial r}\_{r} = -h\_3(\mathcal{C} - \mathcal{C}\_0) \quad \text{at} \quad r = +\eta\_\prime \tag{13}$$

$$D\frac{\partial \mathbb{C}}{\partial r} = -h\_4(\mathbb{C}\_0 - \mathbb{C}) \quad \text{at} \quad r = -\eta\_\prime \tag{14}$$

$$R^\*[-\tau \frac{\partial^3}{\partial x^3} + m\_1 \frac{\partial^3}{\partial x \partial t^2} + d \frac{\partial^2}{\partial t \partial x}] \eta \quad = \frac{1}{r + R^\*} \frac{\partial}{\partial r} \{(r + R^\*)^2 \tau\_{\mathcal{X}}\} + R^\* \frac{\partial \tau\_{\mathcal{X}\mathcal{X}}}{\partial x} - \rho (r + R^\*)$$

$$\left[ \frac{\partial u}{\partial t} + v \frac{\partial u}{\partial r} + \frac{R^\* u}{r + R^\*} \frac{\partial u}{\partial x} + \frac{u \upsilon}{r + R^\*} \right] \text{ at } r = -\pm \eta. \tag{15}$$

Here, the pressure, the time, the fluid density, and the curvature parameters are *p*, *t*, *ρ*, and *R*∗, respectively, *T*<sup>0</sup> and *C*<sup>0</sup> the ambient temperature and concentration, *h*<sup>1</sup> and *h*<sup>2</sup> the coefficients of heat transfer at upper and lower walls, *h*<sup>3</sup> and *h*<sup>4</sup> the coefficients of mass transfer at upper and lower walls, *Srr*, *Srx*, *Sxr* and *Sxx* the components of the extra stress tensor **S**, *τ* the elastic tension, *d* the viscous damping coefficient, and *m*<sup>1</sup> the mass per unit area. Equation (10) is the no slip condition for velocity profile. Equations (11) and (12) are the convective boundary conditions for heat transfer. Analogues to the convective heat transfer at the boundary, we also use the mixed condition for the mass transfer as well (i.e., Equations (13) and (14)).

Employing the aforementioned dimensionless variables

$$\begin{array}{rcl} \chi^{\*} &=& \frac{\chi}{\lambda'}, r^\* = \frac{r}{d\_1}, u^\* = \frac{u}{c'}, v^\* = \frac{v}{c'}, \Psi^\* = \frac{\Psi}{cd\_1}, t^\* = \frac{ct}{\lambda'}, \\\eta^{\*} &=& \frac{\eta}{d\_1}, k = \frac{R^\*}{d\_1}, p^\* = \frac{d\_1^2 p}{c\lambda(\mu + \eta\_1)}, \varepsilon = \frac{a}{d}, \delta = \frac{d\_1}{\lambda}, \\\theta^{\*} &=& \frac{T - T\_0}{T\_0}, \Phi = \frac{\mathbb{C} - \mathbb{C}\_0}{\mathbb{C}\_0}, S^\*\_{ij} = \frac{d\_1 S\_{ij}}{c\eta\_1}, \mathcal{W}\varepsilon = \frac{mc}{d\_1}. \end{array}$$

Equations (7)–(9) become

$$\begin{split} 2\frac{\partial v}{\partial r} &= \, ^rS\_{rr} + \mathcal{W}e \left[ \left( \delta \frac{\partial}{\partial t} + v \frac{\partial}{\partial r} + \frac{\mu k \delta}{r+k} \frac{\partial}{\partial x} \right) S\_{rr} - \frac{2uS\_{rx}}{r+k} - 2\xi S\_{rr} \frac{\partial v}{\partial r} \right] \\ &+ \mathcal{W}eS\_{rx} \left\{ \left( 1 - \xi \right) \frac{\partial u}{\partial r} - \frac{1+\xi}{r+k} \left( k\delta \frac{\partial v}{\partial x} - u \right) \right\}, \end{split} \tag{16}$$

*Symmetry* **2020**, *12*, 1475

$$\begin{split} \left( \frac{\partial u}{\partial r} + \frac{k\delta}{r+k} \frac{\partial v}{\partial \mathbf{x}} - \frac{u}{r+k} \right) &= \, ^tS\_{rx} + \mathcal{W}e \left[ \left( \delta \frac{\partial}{\partial t} + v \frac{\partial}{\partial r} + \frac{uk\delta}{r+k} \frac{\partial}{\partial \mathbf{x}} \right) S\_{rx} + \frac{u(S\_{rr} - S\_{xx})}{r+k} \right] \\ &+ \frac{\mathcal{W}eS\_{rr}}{2} \left\{ \frac{1-\tilde{\xi}}{r+k} \left[ k \frac{\partial v}{\partial \mathbf{x}} - u \right] \right\} - (1+\tilde{\xi}) \frac{\partial u}{\partial r} \\ &+ \frac{\mathcal{W}eS\_{xx}}{2} \left\{ (1-\tilde{\xi}) \frac{\partial u}{\partial r} - \frac{1+\tilde{\xi}}{r+k} \left[ k \delta \frac{\partial v}{\partial \mathbf{x}} - u \right] \right\}, \end{split} \tag{17}$$

$$\begin{split} -2\frac{\partial v}{\partial r} &= \mathcal{S}\_{\text{xx}} + \mathcal{W}\varepsilon \left[ \left( \delta \frac{\partial}{\partial t} + v \frac{\partial}{\partial r} + \frac{uk\delta}{r+k} \frac{\partial}{\partial x} \right) S\_{\text{xx}} + \frac{2u\mathcal{S}\_{\text{rx}}}{r+k} - 2\xi S\_{\text{xx}} \frac{\partial v}{\partial r} \right] \\ &+ \mathcal{W}\varepsilon S\_{\text{rx}} \left\{ \frac{1-\xi}{r+k} \left( k\delta \frac{\partial v}{\partial x} - u \right) - \left( 1+\xi \right) \frac{\partial u}{\partial r} \right\}, \tag{18}$$

and Equations (4)–(6) are reduced to

$$\begin{split} \text{Re}\,\delta \left[ \delta \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial r} + \frac{\partial v}{\partial x} \frac{k \delta u}{r+k} - \frac{u^2}{r+k} \right] &= \quad -\frac{\eta\_1 + \mu}{\eta\_1} \frac{\partial p}{\partial r} + \frac{4\delta\mu}{\eta\_1(r+k)} \frac{\partial v}{\partial r} + \frac{k\delta^3}{r+k} \frac{\partial S\_{rv}}{\partial x} + \delta \frac{\partial S\_{rv}}{\partial r} \\ &\quad + \frac{\delta(S\_{rr} - S\_{xx})}{r+k} + \frac{\delta\mu}{\eta\_1} \frac{\partial^2 v}{\partial r^2} + \frac{\delta^2 k\mu}{\eta\_1(r+k)} \\ &\quad \times \frac{\partial}{\partial x} \left( \frac{\partial u}{\partial r} + \frac{k\delta}{r+k} \frac{\partial v}{\partial x} - \frac{u}{r+k} \right), \end{split} \tag{19}$$

$$\begin{split} \text{Re}\left[\delta\frac{\partial u}{\partial t} + v\frac{\partial u}{\partial r} + \frac{k\delta u}{r+k}\frac{\partial u}{\partial x} - \frac{uv}{r+k}\right] &= \quad -\frac{\eta\_{1}+\mu}{\eta\_{1}(r+k)}\frac{\partial p}{\partial x} + \frac{2S\_{rx}}{r+k} + \frac{\partial S\_{rx}}{\partial r} + \frac{k\delta}{r+k}\frac{\partial S\_{rx}}{\partial x} \\ &\quad - \frac{2k\delta\mu}{(r+k)\eta\_{1}} \times \frac{\partial^{2}v}{\partial x} + \frac{\mu}{\eta\_{1}}\frac{\partial}{\partial x} \left(\frac{\partial u}{\partial r} + \frac{k\delta}{r+k}\frac{\partial v}{\partial x} - \frac{u}{r+k}\right) \\ &\quad + \frac{\delta\mu}{\eta\_{1}}\frac{\partial^{2}v}{\partial r^{2}} + \frac{\delta^{2}k\mu}{\eta\_{1}(r+k)} \times \frac{\partial}{\partial r} \left(\frac{\partial u}{\partial r} + \frac{k\delta}{r+k}\frac{\partial v}{\partial x} - \frac{u}{r+k}\right) \\ &\quad + \frac{2\mu}{\eta\_{1}(r+k)} \left(\frac{\partial u}{\partial r} + \frac{\partial v}{\partial x}\frac{k\delta}{r+k} - \frac{u}{r+k}\right), \end{split} \tag{20}$$

$$\begin{aligned} \text{Re}\left[\delta\frac{\partial\theta}{\partial t} + v\frac{\partial\theta}{\partial r} + \frac{\partial\theta}{\partial x}\frac{k\delta u}{r+k}\right] &= \text{E}\left[S\_{\text{IV}}\left(\frac{\partial u}{\partial r} + \frac{k\delta}{r+k}\frac{\partial v}{\partial x} - \frac{u}{r+k}\right) + (S\_{rr} - S\_{\text{xx}})\frac{\partial v}{\partial r}\right] \\ &+ \frac{1}{\text{Pr}}\left[\frac{\partial^2\theta}{\partial r^2} + \frac{1}{r+k}\frac{\partial\theta}{\partial r} + \delta^2\frac{\partial^2\theta}{\partial x^2}\right], \end{aligned} \tag{21}$$

$$\begin{aligned} \text{Re}\left[\delta\frac{\partial\phi}{\partial t} + v\frac{\partial\phi}{\partial r} + \frac{k\delta u}{r+k}\frac{\partial\phi}{\partial x}\right] &=& \frac{1}{Sc} \left[\frac{\partial^2\phi}{\partial r^2} + \frac{1}{r+k}\frac{\partial\phi}{\partial r} + \delta^2\frac{\partial^2\phi}{\partial x^2}\right] \\ &+ \text{Sr}\left[\frac{\partial^2\theta}{\partial r^2} + \frac{1}{r+k}\frac{\partial\theta}{\partial r} + \delta^2\frac{\partial^2\theta}{\partial x^2}\right], \end{aligned} \tag{22}$$

with

$$
\mu = 0 \quad \text{at} \quad r = \pm \eta,\tag{23}
$$

$$\begin{array}{rcl}\frac{\partial\theta}{\partial r} + Bi\_1\theta &=& 0 \quad \text{at} \quad r = +\eta\_{\prime} \end{array} \tag{24}$$

$$\frac{\partial \theta}{\partial r} - Bi\_2 \theta \quad = \quad 0 \quad \text{at} \quad r = -\eta,\tag{25}$$

$$\begin{array}{rcl}\frac{\partial\varphi}{\partial r} + Bi\_3\phi &=& 0 \quad \text{at} \quad r = +\eta, \\\eta, \end{array} \tag{26}$$

$$\frac{\partial \Phi}{\partial r} - Bi\_4 \Phi \quad = \quad 0 \quad \text{at} \quad r = -\eta,\tag{27}$$

$$\begin{array}{rcl}k\left[E\_{1}\frac{\partial^{3}}{\partial x^{3}} + E\_{2}\frac{\partial^{3}}{\partial x\partial t^{2}} + E\_{3}\frac{\partial^{2}}{\partial t\partial x}\right]\eta &=& \frac{\eta\_{1}(r+k)}{\eta\_{1}+\mu}\left[\frac{\partial}{\partial r}\left(\frac{\partial u}{\partial r} + \frac{k\delta}{r+k}\frac{\partial v}{\partial x} - \frac{u}{r+k}\right) - \frac{2k\delta}{(r+k)}\frac{\partial^{2}v}{\partial r\partial x}\right] \\ & & -\frac{R\_{r}\mu(r+k)}{\eta\_{1+\mu}}\left[\delta\frac{\partial u}{\partial t} + v\frac{\partial u}{\partial r} + \frac{k\delta u}{r+k}\frac{\partial u}{\partial x} + \frac{uv}{r+k}\right] \\ & & +\frac{\eta\_{1}(r+k)}{\eta\_{1}+\mu}\left[\frac{\partial S\_{rx}}{\partial r} + \frac{\partial S\_{rx}}{\partial x}\frac{k\delta}{r+k} + \frac{2S\_{rx}}{r+k}\right] + \\ & & \frac{2\mu}{(\eta\_{1}+\mu)}\left(\frac{\partial u}{\partial r} + \frac{\partial v}{\partial x}\frac{k\delta}{r+k} - \frac{u}{r+k}\right) \quad \text{at } r \pm \eta. \end{array} \tag{28}$$

Defining the stream function *ψ*(*x*,*r*, *t*) by

$$
\mu = -\frac{\partial \psi}{\partial r}, \upsilon = \delta \frac{k}{r+k} \frac{\partial \psi}{\partial x}, \tag{29}
$$

Equation (2) is automatically satisfied and Equations (16)–(28) subject to lubrication approach become

$$0 = S\_{\mathcal{II}} + \mathcal{W}eS\_{\mathcal{IX}} \left[ -\left(1 - \xi\right)\psi\_{\mathcal{II}} - \frac{1 + \xi}{r + k}\psi\_{\mathcal{I}} + \frac{2\psi\_r}{r + k} \right],\tag{30}$$

$$\begin{aligned} \left(-\psi\_{rr} + \frac{\psi\_r}{r+k}\right)^{-} &= \left.S\_{rx} - \mathcal{W}e\frac{\psi\_r(S\_{rr} - S\_{xx})}{r+k}\right. \\ &\left. + \frac{\mathcal{W}eS\_{rr}}{2} \left\{\frac{1-\tilde{\xi}}{r+k}\psi\_r + (1+\tilde{\xi})\,\psi\_{rr}\right\} \\ &- \frac{\mathcal{W}eS\_{xx}}{2} \left\{\frac{1+\tilde{\xi}}{r+k}\psi\_r + (1-\tilde{\xi})\,\psi\_{rr}\right\}, \end{aligned} \tag{31}$$

$$0 = S\_{xx} + \mathcal{WeS}\_{rx} \left[ \left( 1 + \xi \right) \psi\_{rr} + \frac{1 - \xi}{r + k} \psi\_r - \frac{2\psi\_r}{r + k} \right],\tag{32}$$

$$\frac{\partial p}{\partial r} = 0,\tag{33}$$

$$-\frac{k(\eta\_1 + \mu)}{\eta\_1(r+k)}\frac{\partial p}{\partial x} + \frac{\partial S\_{rx}}{\partial r} + \frac{2S\_{rx}}{r+k} + \frac{\mu}{\eta\_1}\frac{\partial}{\partial r}\left(-\psi\_{\eta\tau} + \frac{\psi\_r}{r+k}\right) + \frac{2\mu}{\eta\_1(r+k)}\left(-\psi\_{\eta\tau} + \frac{\psi\_r}{r+k}\right) = 0,\tag{34}$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta = -Br \left[S\_{rx} \left(-\psi\_{rr} + \frac{\psi\_r}{k+r}\right)\right],\tag{35}
$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \phi = -ScSr \left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta,\tag{36}
$$

$$
\psi\_{\varGamma} = 0 \,\, a \text{tr} = \pm \eta = \pm \left[ 1 + \varepsilon \sin 2\pi (\mathbf{x} - t) \right],
\tag{37}
$$

$$\frac{\partial \theta}{\partial r} + Bi\_1 \theta = 0 \quad \text{at} \quad r = +\eta\_\prime \tag{38}$$

$$\frac{\partial \theta}{\partial r} - Bi\_2 \theta = 0 \quad \text{at} \quad r = -\eta,\tag{39}$$

$$\frac{\partial \overline{\phi}}{\partial r} + Bi\_3 \phi = 0 \quad \text{at} \quad r = +\eta,\tag{40}$$

$$\frac{\partial \varrho}{\partial r} - Bi\_4 \wp = 0 \quad \text{at} \quad r = -\eta,\tag{41}$$

$$\begin{aligned} k \left[ E\_1 \frac{\partial^3}{\partial x^3} + E\_2 \frac{\partial^3}{\partial x \partial t^2} + E\_3 \frac{\partial^2}{\partial x \partial t} \right] \eta &= \frac{\eta\_1 (r+k)}{\eta\_1 + \mu} \left[ \frac{\mu}{\eta\_1} \frac{\partial}{\partial r} \left( -\psi\_{rr} + \frac{\psi\_r}{k+r} \right) \right] + \frac{\partial S\_{rx}}{\partial r} + \frac{2S\_{rx}}{r+k} \\ &+ \frac{2\mu}{\eta\_1 + \mu} \left( -\psi\_{rr} + \frac{\psi\_r}{r+k} \right) \quad \text{at } r \quad = \quad \pm \eta, \end{aligned} \tag{42}$$

where the amplitude ratio is represented by (= *a*/*d*1), *δ*(= *d*1/*λ*) is the wave number, the dimensionless curvature parameter is *<sup>k</sup>*, *<sup>E</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>τ</sup>d*<sup>3</sup> 1 *λ*3*η*1*c* , *<sup>E</sup>*<sup>2</sup> <sup>=</sup> *<sup>m</sup>*1*cd*<sup>3</sup> 1 *<sup>λ</sup>*3*η*1*<sup>c</sup>* , *<sup>E</sup>*<sup>3</sup> <sup>=</sup> *dd*<sup>3</sup> 1 *<sup>λ</sup>*2*η*<sup>1</sup> the non-dimensional elasticity parameters, *Re* = *<sup>c</sup>ρd*<sup>1</sup> *<sup>η</sup>*1*λ*<sup>2</sup> the Reynolds number, *We* <sup>=</sup> *mc*/*d*<sup>1</sup> the Weissenberg number, the Prandtl number is denoted by *Pr* = *μCp*/*κ*, the Eckert number is *E* = *c*2/*CpT*0, the Schmidt numbers is *Sc* = *μ*/*ρD*, the Soret number is *Sr*(= *ρT*0*DKT*/*μTmC*0), *EPr* = *Br* is the Brinkman number, and *Bi*<sup>1</sup> = *h*1*d*/*k*, *Bi*<sup>2</sup> = *h*2*d*/*k*, *Bi*<sup>3</sup> = *h*3*d*/*D* and *Bi*<sup>4</sup> = *h*4*d*/*D* the Biot numbers for heat/mass transfer.

From Equations (30)–(32), one can get

$$S\_{rx} = \left(-\psi\_{rr} + \frac{\psi\_r}{r+k}\right) \left[1 + \mathcal{W}\epsilon^2 \left(1 - \xi^2\right) \left(-\psi\_{rr} + \frac{\psi\_r}{r+k}\right)^2\right]^{-1}.\tag{43}$$

Additionally, Equations (33) and (34) give

$$(r+k)\frac{\partial^2 S\_{rx}}{\partial r^2} + 3\frac{\partial S\_{rx}}{\partial r} + \frac{(k+r)\mu}{\eta\_1}\frac{\partial^2}{\partial r^2}\left(-\psi\_{rr} + \frac{\psi\_r}{r+k}\right) + \frac{3\mu}{\eta\_1}\frac{\partial}{\partial r}\left(-\psi\_{rr} + \frac{\psi\_r}{k+r}\right) = 0. \tag{44}$$

Heat transfer coefficient at the wall is defined by

$$Z = \eta\_{\mathbf{x}} \theta\_{\mathbf{y}}(\eta). \tag{45}$$

#### **3. Method of Solution**

We have used the standard perturbation approach relying on a small parameter to solve the strictly nonlinear differential equations, because the exact solution is not achievable. This approach is helpful in finding an approximate solution to the problem, beginning with an exact solution to a similar and simplified problem. This approach is more efficient, as it provides a solution in the form of a converging series. In order to find the series solution of the problem, we expand *ψ*, *p* and *Srx* in terms of small parameter *We*2. Therefore, we can write the flow quantities, as follows:

$$
\psi\_{-}=\quad\psi\_{0}+\mathcal{W}\varepsilon^{2}\psi\_{1}+... \tag{46}
$$

$$\mathcal{S}\_{rx} = \mathcal{S}\_{0rx} + \mathcal{W}c^2\mathcal{S}\_{1rx} + \dots \tag{47}$$

$$S\_{rr} = \quad S\_{0rr} + \mathcal{W}\varepsilon^2 S\_{1rr} + \dots \tag{48}$$

$$\mathcal{S}\_{\text{xx}} = \mathcal{S}\_{0\text{xx}} + \mathcal{W}e^2 \mathcal{S}\_{1\text{xx}} + \dots \tag{49}$$

$$
\theta\_{-}=\theta\_{0}+\mathcal{W}e^{2}\theta\_{1}+...\tag{50}
$$

$$
\phi\_- = -\phi\_0 + \mathcal{W}e^2\phi\_1 + \dots \tag{51}
$$

$$Z\_{\perp} = \quad Z\_0 + \mathcal{W}\varepsilon^2 Z\_1 + \dots \tag{52}$$

#### **4. Results**

#### *4.1. Zeroth Order System*

Using Equations (46)–(52) into Equations (35)–(45) and then equating the coefficients of *We*<sup>0</sup> we have

$$(k+r)\frac{\partial^2}{\partial r^2}\left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right) + 3\frac{\partial}{\partial r}\left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right) = 0,\tag{53}$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta\_0 = -Br \left[S\_{0rx} \left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right)\right],\tag{54}
$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \phi\_0 = -ScSr \left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta\_0 \tag{55}
$$

$$
\psi\_{0r} = 0, \text{ at } r = \pm \eta,\tag{56}
$$

$$\frac{\partial \theta\_0}{\partial r} + Bi\_1 \theta\_0 = 0 \quad \text{at} \quad r = +\eta\_\prime \tag{57}$$

$$\frac{\partial \theta\_0}{\partial r} - Bi\_2 \theta\_0 = 0 \quad \text{at} \quad r = -\eta\_\prime \tag{58}$$

$$\frac{\partial \varrho\_0}{\partial r} + Bi\_3 \wp\_0 = 0 \quad \text{at} \quad r = +\eta\_{\prime} \tag{59}$$

$$\frac{\partial \varphi\_0}{\partial r} - Bi\_4 \phi\_0 = 0 \quad \text{at} \quad r = -\eta\_{\prime} \tag{60}$$

$$k\left[E\_1\frac{\partial^3 \eta}{\partial x^3} + E\_2\frac{\partial^3 \eta}{\partial x \partial t^2} + E\_3\frac{\partial^2 \eta}{\partial x \partial t}\right] = (r+k)\frac{\partial}{\partial r}\left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right) + 2\left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right), \text{ at } r = \pm \eta,\tag{61}$$

where

$$S\_{0rr} = \left(-\psi\_{0rr} + \frac{\psi\_{0r}}{r+k}\right)^2$$

Solving Equations (53)–(55), we get

$$\Psi\_0 = \mathbb{C}\_1 + \mathbb{C}\_2 \ln(r+k) + \mathbb{C}\_3(r+k)^2 + \mathbb{C}\_4(r+k)^2 \ln(r+k),\tag{62}$$

.

$$\theta\_0 = A\_1 + A\_2 \ln(r+k) + 4Br\text{C}\_2\text{C}\_4(\ln(r+k))^2 - Br\left(\text{C}\_4(r+k)^2 + \frac{\text{C}\_2^2}{(r+k)^2}\right),\tag{63}$$

$$\phi\_0 = B\_1 \ln(r+k) + B\_2 + \frac{Br\mathbb{C}\_2^2 ScSr}{(k+r)^2} + Br\mathbb{C}\_4^2 (k+r)^2 ScSr - 4Br\mathbb{C}\_2\mathbb{C}\_4 ScSr \left(\ln(r+k)\right)^2,\tag{64}$$

and heat transfer coefficient is given by

$$Z\_0 = \eta\_x \left( \frac{A\_2}{k + \eta} + Br \left( \frac{2C\_2^2}{\left(k + \eta\right)^3} - 2C\_4^2 \left(k + \eta\right) \right) + \frac{8BrC\_2C\_4\ln(k + \eta)}{k + \eta} \right),\tag{65}$$

where

$$C\_1 = 0,$$

$$C\_2 = -L(k^2 - \eta^2)^2(\ln(k + \eta) - \ln(k - \eta)),$$

$$C\_3 = \frac{L(2k\eta + (k + \eta)^2 \ln(k + \eta) - (k - \eta)^2 \ln(k - \eta))}{16k\eta},$$

$$C\_4 = -\frac{L}{4},$$

$$A\_1 = \frac{BrM\_1(M\_{10} + M\_5 - M\_7 - C\_4^2 M\_9) - BrM(M\_4 + M\_6 + C\_4^2 M\_8 - M\_{11})}{M\_3},$$

$$A\_2 = \frac{Bi\_1 BrM\_{12} + BrM\_{13} + M\_{14} - \frac{8B \times C\_2 \mathcal{L}\_4 \ln(k + \eta)}{k + \eta} - 4Bi\_1 Br\_2 \mathcal{L}\_4 \ln(k + \eta)^2}{M},$$

$$B\_1 = \frac{BrScSr(2Bi\_1 Y\_1 + 4C\_2 \mathcal{L}\_4 (Y\_2 - Y\_4 + Y\_5) + 2Bi\_3 (C\_2^2 Y\_3 + C\_4^2 (\eta - k(1 + 2Bi\_4 \eta))))}{Y},$$

$$B\_2 = \frac{BrScSr(M\_5 + Y\_9 + \frac{2C\_2^2}{(k + \eta)^3} - \frac{Bi\_3 C\_4^2}{(k + \eta)^2} - 2C\_4^2 (k + \eta) - Bi\_3 C\_4^2 (k + \eta))}{B\_3}.$$

#### *4.2. First Order System*

The coefficients of *O*(*We*2) form the following expressions:

$$\begin{split} 0 &= -\left(r+k\right)\frac{\partial^2}{\partial r^2}\left[\left(-\psi\_{1rr}+\frac{\psi\_{1r}}{r+k}\right)-\frac{\left(1-\xi^2\right)\eta\_1}{\left(\eta\_1+\mu\right)}\left(-\psi\_{0rr}+\frac{\psi\_{0r}}{r+k}\right)^3\right] \\ &+ 3\frac{\partial}{\partial r}\left[\left(-\psi\_{1rr}+\frac{\psi\_{1r}}{r+k}\right)-\frac{\left(1-\xi^2\right)\eta\_1}{\left(\eta\_1+\mu\right)}\left(-\psi\_{0rr}+\frac{\psi\_{0r}}{r+k}\right)^3\right], \end{split} \tag{66}$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta\_1 = -Br \left[S\_{1rx} \left(-\psi\_{0rr} + \frac{\psi\_{0r}}{k+r}\right) + S\_{0rx} \left(-\psi\_{1rr} + \frac{\psi\_{1r}}{k+r}\right)\right],
\tag{67}
$$

$$
\left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \phi\_1 = -ScSr \left[\frac{\partial^2}{\partial r^2} + \frac{1}{k+r} \frac{\partial}{\partial r}\right] \theta\_{1,\prime} \tag{68}
$$

$$
\psi\_{1r} = 0, \text{ at } r = \pm \eta,\tag{69}
$$

$$\frac{\partial \theta\_1}{\partial r} + Bi\_1 \theta\_1 = 0, \text{ at } r = +\eta,\tag{70}$$

$$\frac{\partial \theta\_1}{\partial r} - Bi\_2 \theta\_1 = 0, \text{ at } r = -\eta\_{\prime} \tag{71}$$

$$\frac{\partial \phi\_1}{\partial r} + Bi\_3 \phi\_1 = 0, \text{ at } r = +\eta,\tag{72}$$

$$\frac{\partial \phi\_1}{\partial r} - Bi\_4 \phi\_1 = 0, \text{ at } r = -\eta\_\prime \tag{73}$$

$$\begin{split} 0 &= -\left(r+k\right)\frac{\partial}{\partial r}\left[\left(-\psi\_{1rr}+\frac{\psi\_{1r}}{r+k}\right)-\frac{\left(1-\xi^{2}\right)\eta\_{1}}{\left(\eta\_{1}+\mu\right)}\left(-\psi\_{0rr}+\frac{\psi\_{0r}}{r+k}\right)^{3}\right] \\ &+2\left[\left(-\psi\_{1rr}+\frac{\psi\_{1r}}{r+k}\right)-\frac{\left(1-\xi^{2}\right)\eta\_{1}}{\left(\eta\_{1}+\mu\right)}\left(-\psi\_{0rr}+\frac{\psi\_{0r}}{r+k}\right)^{3}\right], \text{ at } r=\pm\eta\_{\prime} \end{split} \tag{74}$$

with

$$S\_{1rx} = \left(-\psi\_{1rr} + \frac{\psi\_{1r}}{r+k}\right) - \left(1-\zeta^2\right)\left(-\psi\_{0rr} + \frac{\psi\_{0r}}{r+k}\right)^3. \tag{75}$$

The results corresponding to the first order are

*Symmetry* **2020**, *12*, 1475

*ψ*<sup>1</sup> = 1/3(*k* + *r*)4(*η*<sup>1</sup> + *μ*)[*C*<sup>3</sup> <sup>2</sup>*η*1(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] <sup>−</sup> 1/(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)[3*C*<sup>2</sup> <sup>2</sup>*C*4*η*1(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] +*krC*<sup>12</sup> + 1/2*r* <sup>2</sup>*C*<sup>12</sup> <sup>−</sup> 1/4(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2*C*<sup>13</sup> +*C*<sup>14</sup> + *C*<sup>11</sup> log(*k* + *r*) + 1/2[(*k* + *r*)2*C*<sup>13</sup> log(*k* + *r*)], (76) *θ*<sup>1</sup> = 1/9(*k* + *r*)6(*η*<sup>1</sup> + *μ*)[4*BrC*<sup>4</sup> <sup>2</sup> (*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] <sup>−</sup>1/(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)4(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)[4*BrC*<sup>3</sup> <sup>2</sup>*C*4(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] <sup>−</sup>*BrC*4(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2(*C*<sup>13</sup> <sup>+</sup> <sup>4</sup>*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) <sup>−</sup>1/(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)[2*BrC*2(*C*11(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) + <sup>12</sup>*C*<sup>2</sup> <sup>4</sup>*C*2*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))] +*A*<sup>12</sup> + *A*<sup>11</sup> log(*k* + *r*) +2*Br*(*C*13*C*<sup>2</sup> + 2*C*4(*C*<sup>11</sup> + 8*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))log(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2, (77) *<sup>φ</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>1/9(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)6(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)[4*BrC*<sup>4</sup> <sup>2</sup>*ScSr*(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] +1/(*k* + *r*)4(*η*<sup>1</sup> + *μ*)[4*BrC*<sup>3</sup> <sup>2</sup>*C*4*ScSr*(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)] +*BrC*4(*k* + *r*)*ScSr*(*C*<sup>13</sup> + 4*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) +1/(*k* + *r*)2(*η*<sup>1</sup> + *μ*)[2*BrC*2*ScSr*(*C*11(*η*<sup>1</sup> + *μ*) + 12*C*<sup>2</sup> <sup>4</sup>*C*2*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))] +*B*<sup>12</sup> + *B*<sup>11</sup> log(*k* + *r*) <sup>−</sup>2*BrScSr*(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))log(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)2, (78)

$$\begin{split} Z\_{1} &= \quad \eta\_{\mathrm{r}} (\frac{A\_{11}}{k+\eta} - Br \left( \frac{2C\_{2}^{2}}{\left(k+\eta\right)^{3}} - 2C\_{4}^{2} \left(k+\eta\right) \right) + \frac{8BrC\_{4}^{2} \left(\eta\_{1}-\mu\right) \left(\xi^{2}-1\right)}{3\left(k+\eta\right)^{7} \left(\eta\_{1}+\mu\right)} \\ &+ \frac{16BrC\_{3}^{2}C\_{4} \left(\eta\_{1}-\mu\right) \left(\xi^{2}-1\right)}{3\left(k+\eta\right)^{5} \left(\eta\_{1}+\mu\right)} - 2BrC\_{4} \left(k+\eta\right) \left(C\_{13}+4C\_{4}^{3} \left(\xi^{2}-1\right)\right) \\ &+ \frac{4BrC\_{2} \left(C\_{11}\left(\eta\_{1}+\mu\right)+12C\_{2}C\_{4}^{2}\mu\left(\xi^{2}-1\right)\right)}{\left(k+\eta\right)^{3} \left(\eta\_{1}+\mu\right)} \\ &+ \frac{4Br\left(C\_{13}C\_{2}+2C\_{4}\left(C\_{11}+8C\_{2}C\_{4}^{2}(\xi^{2}-1)\right)\ln(k+\eta)\right)}{k+\eta},\tag{79} \end{split} \tag{70}$$

in which

*<sup>C</sup>*<sup>11</sup> <sup>=</sup> *<sup>L</sup>*2(*L*<sup>1</sup> <sup>+</sup> <sup>2</sup>*C*<sup>2</sup> <sup>2</sup> (−9*C*4(*k*<sup>2</sup> − *<sup>η</sup>*2)2(*k*<sup>2</sup> + *<sup>η</sup>*2) + *<sup>C</sup>*2(3*k*<sup>2</sup> + *<sup>η</sup>*2)(*k*<sup>2</sup> + <sup>3</sup>*η*2))) (*k* − *η*) <sup>4</sup> (*k* + *η*) <sup>4</sup> , *<sup>C</sup>*<sup>12</sup> <sup>=</sup> *<sup>L</sup>*2(*L*<sup>3</sup> <sup>+</sup> *<sup>C</sup>*2(9*C*4(*k*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)<sup>2</sup> <sup>−</sup> <sup>4</sup>*C*2(*k*<sup>2</sup> <sup>+</sup> *<sup>η</sup>*2))) (*k*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)<sup>4</sup> , *<sup>C</sup>*<sup>13</sup> <sup>=</sup> <sup>−</sup>4*L*2*C*<sup>3</sup> 4, *C*<sup>14</sup> = 0, *<sup>A</sup>*<sup>11</sup> <sup>=</sup> (−*Br*(*Bi*2(*N*<sup>1</sup> <sup>−</sup> *<sup>N</sup>*<sup>2</sup> (*k*+*η*)7(*η*1+*μ*)) + *Bi*1( *<sup>N</sup>*<sup>3</sup> (*k*−*η*)7(*η*1+*μ*) <sup>−</sup> *<sup>N</sup>*4))) <sup>9</sup>*<sup>N</sup>* , *A*<sup>12</sup> = *Br*(*N*<sup>5</sup> <sup>+</sup> *<sup>N</sup>*<sup>6</sup> <sup>+</sup> *<sup>N</sup>*<sup>7</sup> <sup>−</sup> *<sup>N</sup>*<sup>8</sup> <sup>−</sup> *<sup>N</sup>*<sup>9</sup> <sup>+</sup> *<sup>M</sup>*(*Bi*1( *<sup>N</sup>*<sup>3</sup> (*k*−*η*)7(*η*1+*μ*) <sup>−</sup>*N*4)+(*Bi*2(*N*1<sup>−</sup> *<sup>N</sup>*<sup>2</sup> (*k*+*η*)7(*η*1+*μ*) )) *<sup>N</sup>* − *N*<sup>9</sup> −18*Bi*1(*C*13*C*<sup>2</sup> + <sup>2</sup>*C*4(*C*<sup>11</sup> + <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> + *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* + *<sup>η</sup>*)2) 9*Bi*<sup>1</sup> , *<sup>B</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup>*BrScSr*(*Bi*3(*Z*<sup>2</sup> <sup>−</sup> *<sup>Z</sup>*<sup>1</sup> (*k*−*η*)7(*η*1+*μ*)) + *Bi*4( *<sup>Z</sup>* (*k*+*η*)7(*η*1+*μ*) <sup>−</sup> *<sup>Z</sup>*3)) <sup>9</sup>*<sup>Y</sup>* , *B*<sup>12</sup> = *BrScSr*(*N*<sup>7</sup> <sup>+</sup> *<sup>N</sup>*<sup>8</sup> <sup>+</sup> *<sup>N</sup>*<sup>9</sup> <sup>+</sup> *<sup>Z</sup>*<sup>4</sup> <sup>−</sup> *<sup>Z</sup>*<sup>5</sup> <sup>+</sup> *<sup>Z</sup>*<sup>6</sup> <sup>−</sup> *<sup>Z</sup>*<sup>7</sup> <sup>−</sup> <sup>36</sup>*Bi*3*C*<sup>3</sup> <sup>2</sup>*C*4(*η*1−*μ*)(−1+*ξ*2) (*k*+*η*)4(*η*1+*μ*) <sup>+</sup> *<sup>Z</sup>*8) 9*Bi*<sup>3</sup> .

The constants appearing in these equations are written in Appendix A.

#### **5. Discussion**

The behavior of the axial velocity *u*(*y*), temperature *θ*(*y*), concentration *φ*(*y*), and heat-transfer coefficient *Z*(*x*) with respect to the influential parameters is described in this section.

#### *5.1. Axial Velocity Distribution*

Figure 2a–c examines the effect of various parameters on the axial velocity. Figure 2a clearly shows that the axial velocity increases with an increase in *We*. Such an increasing trend is due to increased relaxation time and viscosity decay. The effect of curvature parameter *k* on *u*(*y*) is depicted in Figure 2b. It is observed that the axial velocity *u*(*y*) decreases with an increase in the curvature *k* near the lower wall of the channel while the reverse situation is observed near the upper wall of the channel. Variation in *u*(*y*) for the elastic parameters *E*1, *E*2, and *E*<sup>3</sup> are shown in Figure 2c. This Figure indicates that, by increasing *E*<sup>3</sup> (which represents the oscillatory resistance), the velocity *u*(*y*) decreases and the axial velocity *u*(*y*) increases by increasing *E*1and *E*2.

**Figure 2.** (**a**) Variation of *We* on *u* when *E*<sup>1</sup> = 0.02; *E*<sup>2</sup> = 0.01; *E*<sup>3</sup> = 0.1; = 0.2; *k* = 1.5; *ξ* = 0.5; *μ* = 0.1; *η*<sup>1</sup> = 0.1; *t* = 0.1; *x* = −0.2.; (**b**) Variation of *k* on *u* when *E*<sup>1</sup> = 0.02; *E*<sup>2</sup> = 0.01; *E*<sup>3</sup> = 0.1; = 0.2; *We* = 0.01; *ξ* = 0.5; *μ* = 0.1; *η*<sup>1</sup> = 0.1; *t* = 0.1; *x* = −0.2.; (**c**) Variation of *E*1, *E*2, *E*<sup>3</sup> on *u* when = 0.2; *We* = 0.2; *k* = 1.5; *ξ* = 0.5; *μ* = 0.1; *η*<sup>1</sup> = 0.1; *t* = 0.1; *x* = 0.2.

#### *5.2. Temperature Distribution*

Figure 3a–g indicates the influence of different parameters on the fluid temperature distribution *θ*(*y*). Figure 3a demonstrates that the magnitude of the temperature profile boosts while increasing the value of *We* as Weissenberg number is the ratio of ealstic forces and viscous forces, therefore, an increase in *We* dominates the viscosity and enhance the temperature of the fluid. It reveals that temperature is higher for Johnson–Segalman fluid than that of the viscous fluid temperature. Figure 3b reflects that when Brinkman number *Br* increases, the temperature goes up. This increase in the temperature is due to the viscous dissipation effects. Figure 3c portrays the effects of elastic parameters (*E*1, *E*2, and *E*3) on the temperature profile *θ*(*y*). Increased temperature *θ*(*y*) can be seen with an increment in *E*<sup>1</sup> and *E*<sup>2</sup> and it decreases with increasing in *E*3. Figure 3d depicts that the temperature falls drastically towards the lower portion of the channel and continues to rise in the upper portion of the channel

as the curvature parameter *k* rises. Figure 3e portrays the slip parameter activity indicating that temperature declines as the slip parameter rises near the upper channel wall, while it has an opposite impact close to the lower boundary. Figure 3f illustrates that enhancing the Biot number *Bi*<sup>1</sup> reduces the temperature profile *θ*(*y*) near the upper inlet section but no impact has found in the lower inlet section. Similarly, Figure 3g reveals that the temperature profile *θ*(*y*) for Biot number *Bi*<sup>2</sup> declines near the lower inlet section and has no noticeable impact near the upper inlet section.

**Figure 3.** (**a**) effect of *We* on *θ* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 10; *Br* = 1.8; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**b**) variation of *Br* on *θ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 1.5; *We* = 0.01; *ξ* = 1.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**c**) variation of *E*1, *E*2, *E*<sup>3</sup> on *θ* when = 0.15; *We* = 0.01; *k* = 1.5; *Br* = 0.5; *ξ* = 1.9;

*μ* = 0.5; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**d**) variation of *k* on *θ* when *E*<sup>1</sup> = 0.05; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 1.8; *ξ* = 2.9; *μ* = 0.5; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**e**) variation of *ξ* on *θ* when *E*<sup>1</sup> = 0.05; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 1.5; *k* = 2.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**f**) Variation of *Bi*<sup>1</sup> on *θ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 2.5; *ξ* = 1.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *k* = 1.5 *t* = 0.1; *x* = −0.2; *Bi*<sup>2</sup> = 8. (**g**) variation of *Bi*<sup>2</sup> on *θ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 2.5; *ξ* = 1.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *k* = 1.5 *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10.

#### *5.3. Concentration Distribution*

Figure 4a–h represents the effects of emerging parameters on the fluid concentration distribution *φ*(*y*). Figure 4a depicts that concentration *φ*(*y*) increases when *We* increases due to increase in elasticity of the fluid as *We* physically represents the ratio of elastic to viscous forces. Figure 4b demonstrates that the concentration decreases when the Brinkman number intensifies. The effect of elastic parameters (*E*1, *E*2, and *E*3) are represented in Figure 4c. Here, with the increase in *E*<sup>1</sup> and *E*2, the concentration distribution decreases, while for *E*<sup>3</sup> concentration distribution *φ*(*y*) increases. Figure 4d shows the influence of slip parameter on *φ*(*y*). This Figure shows that the concentration decays in the lower half portion of the channel, while the reverse trend is seen throughout the upper half portion of the channel. Figure 4e shows that the fluid concentration reduces towards the upper wall of the channel and rises in the lower portion of the channel as the curvature parameter *k* rises. Figure 4f indicates that the concentration declines with an increase in the Schmidt number (*Sc*) which physically represents the ratio of momentum diffusivity and mass diffusivity. When we increase the value of Schmidt number, it actually dominates the mass diffusion and thus concentration of the fluid decays. The mass diffusion decays through increase in Schmidt number and, hence, concentration distribution *φ*(*y*) decreases. The effects of Biot numbers *Bi*<sup>3</sup> and *Bi*<sup>4</sup> are examined separately for the concentration profile *φ*(*y*) in the Figure 4g,h. It is found that variation of *Bi*<sup>3</sup> has significant effect near the upper wall and it hardly shows any effect near the lower wall. Similarly, the effects of *Bi*<sup>4</sup> are significant across the lower wall and the concentration profile *φ*(*y*) tends to decrease here. Biot number values are assumed to be greater than 1 and it indicates the non-uniform concentration fields inside the fluid. Also it reveals that convection is much quicker than conduction. From a realistic point of view, the parameters chosen are thus appropriate.

**Figure 4.** *Cont.*

**Figure 4.** (**a**) Variation of *We* on *φ* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 10; *Br* = 0.5; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *t* = 0.1; *x* = −0.2; *Sc* = 1; *Sr* = 1; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**b**) Variation of *Br* on *φ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 1.5; *We* = 0.01; *ξ* = 1.9; *Sc* = 1; *Sr* = 1; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**c**) Variation of *E*1, *E*2, *E*<sup>3</sup> on *φ* when = 0.15; *We* = 0.01; *k* = 1.5; *Br* = 0.5; *ξ* = 1.9; *μ* = 0.5; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Sc* = 1; *Sr* = 1; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**d**) Variation of *ξ* on *φ* when *E*<sup>1</sup> = 0.05; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 0.5; *k* = 2.5; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8; *Sc* = 1; *Sr* = 1. (**e**) variation of *k* on *φ* when *E*<sup>1</sup> = 0.05; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 1.5; *ξ* = 2.9; *μ* = 0.5; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8; *Sc* = 1; *Sr* = 1. (**f**) Variation of *Sc* on *φ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 1; *ξ* = 1.9; *k* = 1.5; *Sr* = 1; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**g**) Variation of *Bi*<sup>1</sup> on *φ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 0.5; *ξ* = 1.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *k* = 1.5; *t* = 0.1; *x* = −0.2; *Bi*<sup>2</sup> = 8; *Sc* = 1; *Sr* = 1. (**h**) Variation of *Bi*<sup>2</sup> on *φ* when *E*<sup>1</sup> = 0.04; *E*<sup>2</sup> = 0.03; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.01; *Br* = 0.5; *ξ* = 1.9; *μ* = 0.6; *η*<sup>1</sup> = 0.8; *k* = 1.5; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10; *Sc* = 1; *Sr* = 1.

#### *5.4. Coefficient of Heat-Transfer*

In Figure 5a–e, we noticed the variability of the coefficient of heat transfer *Z*(*x*) for *We*, *Br*, *k*, *Bi*1, and *Bi*2. Due to peristalsis, the nature of the heat transfer coefficient is oscillatory. The absolute value of the coefficient of overall heat transfer *Z*(*x*) falls as *We* increases (see Figure 5a). Figure 5b illustrates that the coefficient of heat transfer results in an increase when Brinkman number intensifies. Figure 5c displays the curvature parameter's behavior. This indicates that the coefficient of heat transfer *Z*(*x*) boosts with an increase in *k*. Further, Figure 5d,e analyze the effects of Biot numbers on heat transfer coefficient *Z*(*x*). Increasing *Bi*<sup>1</sup> the magnitude of heat transfer coefficient *Z*(*x*) increases and it decreases for *Bi*2.

**Figure 5.** (**a**) Variation of *We* on *Z* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 10; *Br* = 0.5; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *t* = 0.1; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**b**) Variation of *Br* on *Z* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *k* = 10; *We* = 0.005; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *t* = 0.1; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**c**) Variation of *k* on *Z* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; *We* = 0.005; = 0.15; *k* = 10; *Br* = 0.5; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *t* = 0.1; *Bi*<sup>1</sup> = 10; *Bi*<sup>2</sup> = 8. (**d**) Variation of *Bi*<sup>1</sup> on *Z* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.005; *Br* = 0.5; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *k* = 10; *t* = 0.1; *x* = −0.2; *Bi*<sup>2</sup> = 8. (**e**) Variation of *Bi*<sup>2</sup> on *Z* when *E*<sup>1</sup> = 0.5; *E*<sup>2</sup> = 0.04; *E*<sup>3</sup> = 0.01; = 0.15; *We* = 0.005; *Br* = 0.5; *ξ* = 1.8; *μ* = 0.5; *η*<sup>1</sup> = 0.6; *k* = 10; *t* = 0.1; *x* = −0.2; *Bi*<sup>1</sup> = 10.

#### **6. Conclusive Remarks**

This article addresses the peristalsis of Johnson-Segalman fluid in a circular channel with walls' compliance and convective heat and mass transfer conditions. Perturbation solution has been obtained under the long wave length and low Reynolds number approximation. The axial velocity of Johnson–Segalman is found to be greater than that of the Newtonian fluid. The velocity profile is skewed to the left because of curved channel, whereas the concentration and temperature profiles are inclined towards the right. Further, the velocity profile is not symmetric about the centre line in curved channel. At a certain level in the curved channel, the fluid approaches maximum velocity, which decreases in magnitude. The curved channel is transformed into the straight channel with relatively high value of curvature parameter. The results of this problem in Newtonian fluid model can be reduced when *m* = *μ* = 0.

**Author Contributions:** Conceptualization, H.Y.; formal analysis, N.I.; investigation, A.H.; methodology, H.Y. and A.H.; software, A.H.; validation, H.Y. and N.I.; writing—original draft preparation, H.Y., N.I. and A.H.; writing—review and editing, N.I. and H.Y.; visualization, A.H. and N.I.; supervision, H.Y. All authors have read and agreed to the published version of the manuscript.

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

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

#### **Appendix A**

The parameters appearing in the solutions are described here.

*<sup>L</sup>*<sup>1</sup> <sup>=</sup> *<sup>k</sup>*(−8*E*1*π*3 cos(2*π*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)) <sup>−</sup> <sup>8</sup>*E*2*π*3 cos(2*π*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)) + <sup>4</sup>*E*3*π*2 sin(2*π*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*), *<sup>L</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*η*1(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>3</sup>(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *L*<sup>3</sup> = 3*C*<sup>3</sup> <sup>4</sup> (−(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)<sup>2</sup> ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)+(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)<sup>2</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)), *<sup>M</sup>* <sup>=</sup> <sup>1</sup> *k* + *η* + *Bi*<sup>1</sup> ln(*k* + *η*), *<sup>M</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *<sup>k</sup>* <sup>−</sup> *<sup>η</sup>* <sup>−</sup> *Bi*<sup>2</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*), *<sup>M</sup>*<sup>2</sup> <sup>=</sup> *Bi*<sup>1</sup> −*k* + *η* + *Bi*1*Bi*<sup>2</sup> ln(*k* − *η*), *M*<sup>3</sup> = −*Bi*2*M* + *M*2, *<sup>M</sup>*<sup>4</sup> <sup>=</sup> <sup>8</sup>*C*2*C*<sup>4</sup> ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*) (*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*) , *<sup>M</sup>*<sup>5</sup> <sup>=</sup> <sup>8</sup>*C*2*C*<sup>4</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*) , *<sup>M</sup>*<sup>6</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup> <sup>2</sup> (2 + *Bi*2(*k* − *η*)) (*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)<sup>3</sup> , *<sup>M</sup>*<sup>7</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup> <sup>2</sup> (−2 + *Bi*1(*k* + *η*)) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)<sup>3</sup> , *M*<sup>8</sup> = (−2 + *Bi*2(*k* − *η*))(*k* − *η*), *M*<sup>9</sup> = (2 + *Bi*1(*k* + *η*))(*k* + *η*), *M*<sup>10</sup> = 4*Bi*1*C*2*C*<sup>4</sup> ln(*k* + *η*)2, *<sup>M</sup>*<sup>11</sup> <sup>=</sup> <sup>4</sup>*Bi*2*C*2*C*<sup>4</sup> ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)2, *<sup>M</sup>*<sup>12</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup> 2 (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*) <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* + *<sup>η</sup>*)2, *<sup>M</sup>*<sup>13</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup>*C*<sup>2</sup> 2 (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)<sup>3</sup> <sup>+</sup> <sup>2</sup>*C*<sup>2</sup> <sup>4</sup> (*k* + *η*), *<sup>M</sup>*<sup>14</sup> <sup>=</sup> *Bi*1(−*BrM*1(*M*<sup>10</sup> <sup>+</sup> *<sup>M</sup>*<sup>5</sup> <sup>−</sup> *<sup>M</sup>*<sup>7</sup> <sup>−</sup> *<sup>C</sup>*<sup>2</sup> <sup>4</sup>*M*9) + *BM*(*M*<sup>4</sup> + *<sup>M</sup>*<sup>6</sup> + *<sup>C</sup>*<sup>2</sup> <sup>4</sup>*M*<sup>8</sup> − *M*11), *<sup>N</sup>* <sup>=</sup> *Bi*<sup>1</sup> *k* − *η* + *Bi*<sup>2</sup> *k* + *η* + *Bi*1*Bi*2(− ln(*k* − *η*) + ln(*k* + *η*)), *<sup>N</sup>*<sup>1</sup> <sup>=</sup> <sup>18</sup>(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> + *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* + *<sup>η</sup>*)(<sup>2</sup> + *Bi*1(*<sup>k</sup>* + *<sup>η</sup>*))ln(*<sup>k</sup>* + *<sup>η</sup>*) *<sup>k</sup>* <sup>+</sup> *<sup>η</sup>* , *<sup>N</sup>*<sup>2</sup> = (18(*C*11*C*2(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(−<sup>2</sup> <sup>+</sup> *Bi*1(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) <sup>−</sup> <sup>4</sup>*C*<sup>4</sup> <sup>2</sup> (−<sup>6</sup> <sup>+</sup> *Bi*1(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +36*C*4*C*<sup>3</sup> <sup>2</sup> (−<sup>4</sup> <sup>+</sup> *Bi*1(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +216*C*<sup>2</sup> 2*C*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(−<sup>2</sup> <sup>+</sup> *Bi*1(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +9*C*4(*k* + *η*)8(2 + *Bi*1(*k* + *η*))(*η*<sup>1</sup> + *μ*)(*C*<sup>13</sup> + 4*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))),

*<sup>N</sup>*<sup>3</sup> = (18(*C*11*C*2(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(<sup>2</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) <sup>−</sup> <sup>4</sup>*C*<sup>4</sup> <sup>2</sup> (<sup>6</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +36*C*4*C*<sup>3</sup> <sup>2</sup> (<sup>4</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)2(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +216*C*<sup>2</sup> 2*C*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)4(<sup>2</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)4*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>+</sup>9*C*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)8(−<sup>2</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)(*C*<sup>13</sup> <sup>+</sup> <sup>4</sup>*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))), *<sup>N</sup>*<sup>4</sup> <sup>=</sup> <sup>18</sup>(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)(−<sup>2</sup> <sup>+</sup> *Bi*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*) *<sup>k</sup>* <sup>−</sup> *<sup>η</sup>* , *<sup>N</sup>*<sup>5</sup> <sup>=</sup> <sup>18</sup>*Bi*1*C*2(*C*11(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) + <sup>12</sup>*C*2*C*<sup>2</sup> <sup>4</sup>*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)2(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) <sup>+</sup> 36*Bi*1*C*<sup>3</sup> <sup>2</sup>*C*4(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *N*<sup>6</sup> = 18*C*4(*k* + *η*)(*C*<sup>13</sup> + 4*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) + <sup>9</sup>*Bi*1*C*4(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)2(*C*<sup>13</sup> <sup>+</sup> <sup>4</sup>*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) + 24*C*<sup>2</sup> <sup>2</sup> (*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)7(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *<sup>N</sup>*<sup>7</sup> <sup>=</sup> <sup>36</sup>(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*) *<sup>k</sup>* <sup>+</sup> *<sup>η</sup>* , *<sup>N</sup>*<sup>8</sup> <sup>=</sup> <sup>36</sup>*C*2(*C*11(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) + <sup>12</sup>*C*2*C*<sup>2</sup> <sup>4</sup>*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)3(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *<sup>N</sup>*<sup>9</sup> <sup>=</sup> <sup>144</sup>*C*<sup>3</sup> <sup>2</sup>*C*4(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)5(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *<sup>N</sup>*<sup>10</sup> <sup>=</sup> <sup>4</sup>*C*<sup>4</sup> <sup>2</sup> (*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)) (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)6(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) , *<sup>Y</sup>* <sup>=</sup> *Bi*<sup>3</sup> *k* − *η* + *Bi*<sup>4</sup> *k* + *η* + *Bi*3*Bi*4(− ln(*k* − *η*) + ln(*k* + *η*)), *<sup>Y</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup> 2 (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)<sup>3</sup> <sup>−</sup> *<sup>C</sup>*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*), *<sup>Y</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*Bi*<sup>3</sup> ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*) *<sup>k</sup>* <sup>−</sup> *<sup>η</sup>* , *<sup>Y</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> (*k* − *η*) <sup>3</sup> + 2*Bi*4*kη* (*k*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)<sup>2</sup> , *<sup>Y</sup>*<sup>4</sup> <sup>=</sup> *Bi*3*Bi*<sup>4</sup> ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)2, *<sup>Y</sup>*<sup>5</sup> <sup>=</sup> *Bi*<sup>4</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)( <sup>2</sup> *k* + *η* + *Bi*<sup>3</sup> ln(*k* + *η*)), *<sup>Y</sup>*<sup>6</sup> <sup>=</sup> <sup>1</sup> (−*k* + *η*) <sup>3</sup> <sup>−</sup> <sup>2</sup>*Bi*4*k<sup>η</sup>* (*k*<sup>2</sup> <sup>−</sup> *<sup>η</sup>*2)<sup>2</sup> , *Y*<sup>7</sup> = 4*C*2*C*4(*Y*<sup>4</sup> + 2*Bi*<sup>3</sup> ln(*k* − *η*) −*k* + *η* <sup>+</sup> *Bi*<sup>4</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)( <sup>−</sup><sup>2</sup> *<sup>k</sup>* <sup>+</sup> *<sup>η</sup>* <sup>−</sup> *Bi*<sup>3</sup> ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))), *<sup>Y</sup>*<sup>8</sup> <sup>=</sup> *<sup>Y</sup>*<sup>7</sup> <sup>+</sup> <sup>2</sup>*Bi*4( <sup>−</sup>*C*<sup>2</sup> 2 (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)<sup>3</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* + *<sup>η</sup>*)) + <sup>2</sup>*Bi*3(*C*<sup>2</sup> <sup>2</sup>*Y*<sup>6</sup> + *<sup>C</sup>*<sup>2</sup> <sup>4</sup> (*k* − *η* + 2*Bi*4*kη*)), *Y*<sup>9</sup> = 4*Bi*3*C*2*C*<sup>4</sup> ln(*k* + *η*)<sup>2</sup> + *Y*8( <sup>1</sup> *<sup>k</sup>*+*<sup>η</sup>* + *Bi*<sup>3</sup> ln(*k* + *η*)) *<sup>Y</sup>* , *<sup>D</sup>* = (18(*C*11*C*2(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(−<sup>2</sup> <sup>+</sup> *Bi*3(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) <sup>−</sup> <sup>4</sup>*C*<sup>4</sup> <sup>2</sup> (−<sup>6</sup> <sup>+</sup> *Bi*3(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +36*C*4*C*<sup>3</sup> <sup>2</sup> (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)2(−<sup>4</sup> <sup>+</sup> *Bi*3(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +216*C*<sup>2</sup> 2*C*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)4(−<sup>2</sup> <sup>+</sup> *Bi*3(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +9*C*4(*k* + *η*)8(2 + *Bi*3(*k* + *η*))(*η*<sup>1</sup> + *μ*)(*C*<sup>13</sup> + 4*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))), *<sup>D</sup>*<sup>1</sup> = (18(*C*11*C*2(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)4(<sup>2</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*) <sup>−</sup> <sup>4</sup>*C*<sup>4</sup> <sup>2</sup> (<sup>6</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +36*C*4*C*<sup>3</sup> <sup>2</sup> (*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)2(<sup>4</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) +216*C*<sup>2</sup> 2*C*<sup>2</sup> <sup>4</sup> (*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)4(<sup>2</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))*μ*(−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2) <sup>+</sup>9*C*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)8(−<sup>2</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))(*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*)(*C*<sup>13</sup> <sup>+</sup> <sup>4</sup>*C*<sup>3</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2))), *<sup>D</sup>*<sup>2</sup> <sup>=</sup> <sup>18</sup>(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*)(−<sup>2</sup> <sup>+</sup> *Bi*4(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*))ln(*<sup>k</sup>* <sup>−</sup> *<sup>η</sup>*) *<sup>k</sup>* <sup>−</sup> *<sup>η</sup>* , *<sup>D</sup>*<sup>3</sup> <sup>=</sup> <sup>18</sup>(*C*13*C*<sup>2</sup> <sup>+</sup> <sup>2</sup>*C*4(*C*<sup>11</sup> <sup>+</sup> <sup>8</sup>*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)(<sup>2</sup> <sup>+</sup> *Bi*3(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*))ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*) *<sup>k</sup>* <sup>+</sup> *<sup>η</sup>* , *D*<sup>4</sup> = 18*Bi*3(*C*13*C*<sup>2</sup> + 2*C*4(*C*<sup>11</sup> + 8*C*2*C*<sup>2</sup> <sup>4</sup> (−<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*2)))ln(*<sup>k</sup>* <sup>+</sup> *<sup>η</sup>*)2,

$$\begin{split} D\_5 &=& 18\mathbb{C}\_4(k+\eta)(\mathbb{C}\_{13}+4\mathbb{C}\_4^3(-1+\xi^2)+9Bi\_3\mathbb{C}\_4(k+\eta)^2(\mathbb{C}\_{13}+4\mathbb{C}\_4^3(-1+\xi^2)) \\ &+& \frac{24\mathbb{C}\_2^2(\eta\_1-\mu)(-1+\xi^2)}{(k+\eta)^5(\eta\_1+\mu)}, \\ D\_6 &=& \frac{4Bi\_3\mathbb{C}\_2^4(\eta\_1-\mu)(-1+\xi^2)}{(k+\eta)^6(\eta\_1+\mu)}, \\ D\_7 &=& \frac{18Bi\_3\mathbb{C}\_2(\mathbb{C}\_{11}(\eta\_1+\mu)+12\mathbb{C}\_2\mathbb{C}\_4^2\mu(-1+\xi^2))}{(k+\eta)^2(\eta\_1+\mu)}, \\ D\_8 &=& \frac{1}{Y}(Bi\_3(D\_2-\frac{D\_1}{(k-\eta)^7(\eta\_1+\mu)})+Bi\_4(\frac{D}{(k+\eta)^7(\eta\_1+\mu)}-D\_3))(\frac{1}{k+\eta}+Bi\_3\ln(k+\eta)). \end{split}$$

#### **References**


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