Optimizing the Entropy of the Unsteady Flow of Ternary Nanofluids in an Inclined Conduit: Smart Pumping Using Electro-Osmotic Flow

Abstract

The purpose of this study is to investigate the flow of liquid (specifically water) supported by three different types of nanoparticles (copper, silver, and aluminum oxide) in a slanted microcorrugated pipe with varying diameters and wave displacements. This model fulfills several key uses in the fields of environmental and water treatment, such as improving fluid mixing within inclined microchannels to boost the effectiveness of filtration and separation processes and designing inclined channels with ripples to better mix waste materials and more effectively separate different components. In this perception, we analyzed a model for the fluids flow inside the microchannel with the electromagnetic field (EMF) effects and pressure variation in the conduit under the external influence of thermal radiation and heat source which did not appear in the last published literature. The analytical techniques with assistance of mathematical software were used to solve the main governing equations such as Poisson, momentum, and energy equations and then deduce the heat transfer rate at the peristaltic conduit surfaces and the system’s ideal entropy. The results of the model simulation suggested that a number of factors could have a big influence on how thermal systems are built. It was claimed that the irreversibility resulting from friction and Joule heating, as opposed to thermal irreversibility, controls the system’s entropy buildup more tightly. Furthermore, the alteration of the conduit’s form and geometry resulted in a substantial enhancement in the heat transfer rate at its bottom wall, ranging from 20% to 600%. Despite numerous studies, an accurate model regarding the mechanism of the liquid’s flow supported by tri-nanoparticles in an inclined microcorrugated conduit enclosing variable diameters and wave shifts is far from being understood.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The nanofluids are characterized by the uniform mixes of various nanoparticles (for example carbon, carbides, oxides, and metals) containing a basic fluid (such as oil, water, and ethylene glycol). Earlier study revealed that [1] due to the inclusion of these nanoparticles in the fluid, its effective thermal conductivity can appreciably increase. In recent years, this idea has been used to improve the heat transport performance of various engineering liquids used for biomedical (such as cancer diagnosis, neuroelectronic interfaces, drug deliveries, and photodynamic therapies) and other purposes [2–4]. A branch of continuum fluid mechanics called the magneto-hydrodynamics (MHD) involves the electrically conducting fluid motion in the presence of applied electromagnetic fields (EMFs). These EMFs can originate from natural resources (including storms and magnetic field of earth) or can be produced artificially. Some of the real-like uses include geophysics, astrophysics (stellar interiors and interstellar medium), aerodynamic boundary layers control, heat exchanger pump design, and propulsion of space vehicle, thermal shield, magnetic data storage, and power generation devices [5]. Moatimid et al. [6] examined the EM nanofluid’s (made of copper and water) peristaltic flow in a wall-compliance channel. Haroun [7] simplified an electrohydrodynamic oscillatory flow via the proper nondimensional parameters to obtain a correlation amid moving and fixed frame coordinates. In addition, many studies were conducted to evaluate the impact of EMFs on various physiological fluids’ peristaltic flow [8–14].

Thermal radiation, exchange, and transfer processes are utilized in several technologies, including jet propulsion, solar concentrators, combustion, materials processing, and plume dynamics. The mechanics of heat transfer in many industrial processes are complicated, necessitating accurate assessment for energy optimization of the system. Recent advancements in analytical methods, computer modeling, and simulations have facilitated accurate predictions of thermal convection with significant radiative flux. It has been observed that in many conductive and convective fluxes, the energy transfer between two locations is predominantly determined by the temperature differential. Nonetheless, the absolute temperature differential between two bodies determines their radiation-mediated thermal energy transfer rate, with radiation effects becoming pronounced at elevated absolute temperatures [15]. A novel approach was developed to determine the convective mass and heat radiation transfer for the peristaltic transport in fluids [16]. Meanwhile, the effects of heat radiation on the mixed convection peristaltic flow of a third grade nanofluid in a curved channel was examined [17]. Hayat et al. [18] investigated the nonlinear radiation-driven peristaltic fluid flow in an inclined channel. Peristalsis describes the wave-like motion that arises from the swift expansion and contraction of an elongated channel, moving along its length. The functions of various physiological systems, such as the movement of urine from the human kidney to the bladder, muscle contraction and expansion in the digestive system, chyme waves in the gastrointestinal tract, locomotion in worms, transport of sanitary fluids, and food swallowing via the esophagus, involve these transport mechanisms. The concept of peristaltic transport was first introduced by Latham [19] and later the practical significance of this transport mechanism was emphasized [20–26].

A single theoretical framework that unifies magnetism and electricity has had a significant impact on science and technology. It clarifies a wide range of natural phenomena, including as how light behaves, the workings of electric circuits, and how motors and generators operate. Modern technologies such as renewable energy systems, medical imaging, and telecommunications have all benefited greatly from EM progress. The versatility of electromagnetism makes it a foundational element in the scientific research, driving advancements in both understanding and application. Jayavel, Katta, and Lodhi [27] discussed the impact of EM force on the squeezing flow between two parallel porous medium plates. Prakash et al. [28] studied the dual effect of electric and magnetic field on the unsteady squeezing flow. Vijayaragavan, Bharathi, and Prakash [29] investigated the impact of electro-osmosis and chemical reactions on the behavior of convective flow over an accelerating plate. Rajaram, Varadharaj, and Jayavel [30–32] investigated the use of EM in a variety of models and applications, such as Casson and nanofluids, under the influence of external factors, such as chemical reactions or heat radiation, are present. Elsaid, Sayed, and Abdel-Wahed [33, 34] examined the effects of EM hydrodynamic forces on peristaltic artery blood flow in both vertical and horizontal orientations. Sayed et al. [35] investigated the impact of EM hydrodynamic on the flow and temperature behavior inside a corrugated 3D microchannel.

In every facet of life, the transport of heat and fluid are involved inherently. Lately, dedicated efforts have been made to determine the heat transfer process, particularly for the biological/physiological fluid transport wherein the entropy production due to heat transfer processes is focused. The entropy production related to the heat transfer and fluid flow were analyzed by Bejan [36]. Diverse elements, including viscosity, chemical reactions, and frictional forces, have shown to contribute to energy loss in a thermodynamic system, resulting in entropy formation. Consequently, entropy minimization became crucial for enhancing the performance of diverse thermal engineering systems. To attain this objective, numerous solutions have been implemented in machine design, including electronic cooling, chemical vapor deposition, and solar collector components. The efficiency of any thermal system can be affected by the irreversibility due to temperature gradient, thus reducing the energy quality of the system. Various studies have indicated that the second law of thermodynamics is more efficient for the energy optimization of a thermal system than first law, wherein the irreversibility analysis of the entropy generated by a system can be useful to determine its performance [37]. The entropy produced by a MHD peristaltic flow of copper–water nanofluid under the influence of slip was determined by Elsaid and Abdel-Wahed [38]. In addition, the entropy generated due to the peristaltic flow of nanofluid was analyzed by Elsaid and Abdel-Wahed [39]. Al Qarni et al. [40] investigated the entropy production due to MHD cilia-driven transport of Prandtl fluid. Meanwhile, the entropy production due to the peristaltic flow in a rotating medium having generalized complaint walls was examined by Ahmed et al. [41]. Lately, Sridhar and Ramesh [42] analyzed the entropy production due to peristaltic transport of hybrid nanofluid through a two-dimensional vertical channel in the presence of temperature-dependent thermal conductivity. Despite numerous studies, an accurate model regarding the mechanism of the liquids flow supported by tri-nanoparticles in an inclined microcorrugated conduit enclosing variable diameters and wave shifts is far from being understood. In this perception, we solved some model equations for the fluids flow inside the microchannel with the EMF effects and pressure variation in the conduit under the external influence of radiation and heat source. Various quantities such as the rate of heat transfer at the peristaltic conduit surfaces and optimal entropy generated by the system were calculated for the first time. The results were analyzed, discussed, and interpreted using different mechanisms.

2. Issue Modeling

Suppose an inclined irregular conduit contains unsteady flow of three types of nanoparticles (copper, Cu; silver, Ag; and aluminum oxide, Al₂O₃) in water (as base fluid) at a velocity $c$. Table 1 shows the physical characteristics of water and three types of nanoparticles used in the model. The motion is assumed to occur at the long wavelength and low Reynolds number that is subjected to the EMF (axial uniform electric field, $E_X$, and normal magnetic field, $B_0$), external thermal force (heat flux of $q_r$), and heat source of $Q_0$. The conduit walls are considered as a sinusoidal wavy shape with phase difference $\gamma$ with the upper and lower wall defined in Figure 1.$$\begin{array}{c}\text{(1)}& Y_{1}X,\overline{t}=l_0+aX+l_1\cos \frac{\pi }{\lambda }X-c\overline{t},Y_{2}X,\overline{t}=-l_0-aX-l_1\cos \frac{\pi }{\lambda }X-c\overline{t}+\gamma .\end{array}$$

Table 1

Physical attributes of water (base fluid) and three types of nanoparticles [38, 39].

Properties	Base fluid (water)	$\text{Cu}$	$\text{Ag}$	${\text{Al}}_2{\mathrm{O}}_3$
$\text{Cp}\mathrm{j}/\text{kgK}$	4179	385	235	765
$\rho \text{kg}/\mathrm{m}³$	997.1	8933	10,500	3971
$\mathrm{k}\mathrm{W}/\text{mK}$	0.613	401	429	40
$\sigma \mathrm{s}/\mathrm{m}$	$0.05$	$5.96\times {10}^7$	$3.6\times {10}^7$	$1\times {10}^{-5}$
$\gamma \mathrm{I}/\mathrm{K}$	$21\times {10}^{-5}$	1.66$\times {10}^{-5}$	1.88$\times {10}^{-5}$	0.86$\times {10}^{-5}$

[figure(s) omitted; refer to PDF]

The concentration of Cu, Ag, and Al₂O₃ nanoparticles are ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$, respectively. The equations of continuity, momentum, heat, and Poisson–Boltzmann used in the model are as follows [33, 34]:

The governing equations of the model built according to the following assumptions:

• The flow within the inclined conduit assumed to be Newtonian and laminar

• The base fluid and the suspended particles both in thermal equilibrium and single-phase status

• An external electric and magnetic field are considered along the length of the conduit, and gravitational effects considered due to the inclination of the conduit

• Effects of viscous dissipation and Joule heating were neglected as they were deemed minimal compared to the primary heat transfer mechanisms

• No chemical reactions occur

• The motion with long wavelength and low Reynold number and no-slip conditions applied at the conduit walls

• The fluid is assumed to be gray absorbing–emitting radiation$$\begin{array}{}\text{(2)}& \frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,\text{(3)}& {\rho }_{thnf}\frac{\partial U}{\partial \overline{t}}+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+{\mu }_{thnf}\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}+g{\rho \gamma }_{thnf}\overline{T}-T_w\sin \alpha -{\sigma }_{thnf}{B}_0^{2}U+{\mu }_e{E}_X,\text{(4)}& {\rho }_{thnf}\frac{\partial V}{\partial \overline{t}}+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+{\mu }_{thnf}\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}-g{\rho \gamma }_{thnf}\overline{T}-T_w\cos \alpha ,\text{(5)}& {\rho c_p}_{thnf}\frac{\partial T}{\partial \overline{t}}+U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=k_{thnf}\frac{{\partial }^{2}T}{\partial X^2}+\frac{{\partial }^{2}T}{\partial Y^2}+Q_0-\frac{\partial q_r}{\partial Y},\text{(6)}& {\nabla }^2\overline{\beta }=-\frac{{\mu }_e}{{ϵ}_0},\end{array}$$

where $\overline{\beta }=\overline{\beta }X,Y$ is the electric double layer (EDL) potential function, ${\mu }_e$ is the net charge density, ${ϵ}_0$ is the solution’s permittivity constant, $\mathrm{U}=\mathrm{U}X,Y,\overline{t}$ is the axial velocity, $\mathrm{V}=\mathrm{V}X,Y,\overline{t}$ is the transverse velocity, $P=PX$ is the pressure, $T=\mathrm{T}X,Y,\overline{t}$ is the flow temperature, $q_r=-4{\sigma }^{\mathrm{\ast }}/3K^{\mathrm{\ast }}\partial T^4/\partial Y$ is the radiative heat flux, ${\sigma }^{\mathrm{\ast }}$ is the Stefan–Boltzmann constant, and $K^{\mathrm{\ast }}$ is the coefficient of absorption [23].

The electrical field is defined by $E=-\nabla \overline{\beta }$, so $\nabla ·E={\mu }_e/{ϵ}_0$.

From equation (6), we get$$\begin{array}{}\text{(7)}& {\mu }_e=-{ϵ}_0{\nabla }^2\overline{\beta }=-2e{c}_0{n}_0\text{sin}h\frac{e{c}_0\overline{\beta }}{k_B{T}_a},\end{array}$$where $n_0$ is the ion density, $e$ is the electronic charge, and $c_0$ is the valence.

Based on the Debye–Hückle linearization principle which assume $\overline{\beta }$ is very small, then $e{c}_0\overline{\beta }/k_B{T}_a\ll 1$ so the term $e{c}_0\overline{\beta }/k_B{T}_a$ can be used to approximate the term $\sinh e{c}_0\overline{\beta }/k_B{T}_a$.

To summarize, the linearized version of the Poisson equation can be found using the following form:$$\begin{array}{}\text{(8)}& {\nabla }^2\overline{\beta }=\frac{2n_0{e}^2{c}_0^2}{{ϵ}_0{k}_B{T}_a}\overline{\beta }.\end{array}$$

Referring to the Debye–Hückle linearization principle [36], equation (8) can be expressed as$$\begin{array}{}\text{(9)}& {\nabla }^2\overline{\beta }={\vartheta }^2\overline{\beta },\end{array}$$where $\vartheta =e{c}_0\sqrt{2n_0/{ϵ}_0{k}_B{T}_a}$ is Debye–Hückle parameter.

Introduction of various dimensionless variables ($\xi$ and $\alpha$ are the wave number and irregularity parameter) in equations (2)–(6) yields the following [37]:$$\begin{array}{c}\text{(10)}& \beta =\frac{\overline{\beta }}{\zeta },x=\frac{X}{\lambda },\\ y=\frac{Y}{l_0},\\ t=\frac{c\overline{t}}{\lambda },\\ h=\frac{H}{l_0},\\ \xi =\frac{l_0}{\lambda },\\ \alpha =\frac{a}{\xi },\\ \text{(11)}& U=cu,V=c\xi v,\\ P=-\frac{{\mu }_{f}c\lambda }{l_0^2}p,\\ \theta =\frac{T-T_w}{T_w},\\ \text{(12)}& \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\text{(13)}& \xi Re\frac{{\rho }_{thnf}}{{\rho }_f}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{\partial p}{\partial x}+\frac{{\mu }_{thnf}}{{\mu }_f}{\xi }^2\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{thnf}}{{\sigma }_f}{M}^{2}u+\sin \alpha G{r}_T\frac{{\rho \gamma }_{thnf}}{{\rho \gamma }_f}\theta +m^2\omega \beta ,\text{(14)}& {\xi }^{3}Re\frac{{\rho }_{thnf}}{{\rho }_f}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{\partial p}{\partial y}+\frac{{\mu }_{thnf}}{{\mu }_f}{\xi }^4\frac{{\partial }^{2}v}{\partial x^2}+{\xi }^2\frac{{\partial }^{2}v}{\partial y^2}-\xi \cos \alpha G{r}_T\frac{{\rho \gamma }_{thnf}}{{\rho \gamma }_f}\theta ,\text{(15)}& \frac{{\rho c_p}_{thnf}}{{\rho c_p}_f}\frac{k_f}{k_{thnf}}\mathrm{Pr}Re\xi \frac{\partial \theta }{\partial t}+u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}={\xi }^2\frac{{\partial }^2\theta }{\partial x^2}+\frac{{\partial }^2\theta }{\partial y^2}+Rd\frac{k_f}{k_{thnf}}\frac{{\partial }^2\theta }{\partial y^2}+\frac{k_f}{k_{thnf}}Q,\text{(16)}& {\xi }^2\frac{{\partial }^2\beta }{{\partial x}^2}+\frac{{\partial }^2\beta }{{\partial y}^2}=m^2\beta .\end{array}$$

The corresponding electro-osmotic velocity ($\omega$), the number for electro-osmotic flow ($m$), Hartman number ($Ha$), Prandtl ($\mathrm{Pr}$), Reynolds ($Re$), Grashof ($Gr$), the parameter for radiation ($Rd$), and heat generation ($Q$) are defined as [37]$$\begin{array}{}\text{(17)}& \begin{array}{c}m=\frac{\vartheta l_0}{\zeta }\\ \omega =-\frac{{ϵ}_{0}E\zeta l_0^2}{c{\mu }_f}\\ Ha=\frac{B_0^2{l}_0^2{\sigma }_f}{{\mu }_f}\\ \mathrm{Pr}=\frac{{c_p}_f{\mu }_f}{k_f}\\ Re=\frac{{\rho }_{f}c{l}_0}{{\mu }_f}\\ Gr=\frac{g{\rho \gamma }_f{l}_0^2}{c{\mu }_f}{T}_w\\ Rd=\frac{16{\sigma }^{\mathrm{\ast }}{∆T}^3}{3{\mu }_f{c_p}_f{K}^{\mathrm{\ast }}}\\ Q=\frac{l_0^2}{k_f{T}_w}{Q}_0\end{array}.\end{array}$$

Both wave number ($\xi$) and Reynolds number tend to zero under the long wavelength approximation $\lambda \gg l_0$ with $l_0$ as the initial breadth of the channel. Thus, equations (12)–(16) transform to$$\begin{array}{}\text{(18)}& \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\text{(19)}& \frac{\partial p}{\partial x}+\frac{{\mu }_{thnf}}{{\mu }_f}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{thnf}}{{\sigma }_f}Hau+Gr\frac{{\rho \gamma }_{thnf}}{{\rho \gamma }_f}\sin \alpha \theta +m^2\omega \mathrm{\beta }=0,\text{(20)}& \frac{\partial p}{\partial y}=0,\text{(21)}& \frac{{\partial }^2\theta }{\partial y^2}+Rd\frac{k_f}{k_{thnf}}\frac{{\partial }^2\theta }{\partial y^2}+\frac{k_f}{k_{thnf}}Q=0,\text{(22)}& \frac{{\partial }^2\beta }{{\partial y}^2}=m^2\mathrm{\beta }.\end{array}$$

Now, the task boils down in solving these equations subjected to the following boundary conditions:$$\begin{array}{}\text{(23)}& \begin{array}{c}ux,y,t=0,\theta x,y,t=0,\beta x,y,t=1,aty=h_{1,2}\\ vx,y,t=\frac{\partial h}{\partial t},aty=h_{1,2}\end{array},\end{array}$$where $h_1=1+\alpha x+\epsilon \cos \pi x-t,h_2=-1-\alpha x-\epsilon \cos \pi x-t+\gamma$, and $\epsilon =l_1/l_0$ is the wave amplitude ratio

3. Solutions of the System

The closed form solution of the longitudinal velocity ($u$), temperature ($T$), and the electrical potential ($\beta$) given in equations (19), (21), and (22), respectively, under the boundary conditions (23) were solved using Mathematica software thereby obtaining the following:$$\begin{array}{}\text{(24)}& \theta x,y,t=\frac{Q{h}_1-y-h_2+y}{2A_3+Rd},\text{(25)}& ux,y,t=\frac{\begin{array}{c}-2\cos h\sqrt{A_4}\sqrt{Ha}{h}_1+h_2-2y/2\sqrt{A_1}\mathrm{sech}\sqrt{A_4}{h}_1-h_2\sqrt{Ha}/2\sqrt{A_1}\\ \begin{array}{c}{A}_{4}Ha{A}_3+Rd-A_1{m}^{2}Px+A_{4}HaPx+m^2\omega \\ \begin{array}{c}+A_1{A}_{2}Gr-A_{4}Ha+A_1{m}^{2}QSin\alpha \\ \begin{array}{c}\sec h1/2h_1-h_{2}m2A_{4}Ha{A}_{4}Ha-A_1{m}^2\times \\ \begin{array}{c}Px{A}_3+Rd\cosh 1/2h_1-h_{2}m+\\ 2{A_4}^2{Ha}^2{m}^2{A}_3+Rd\omega \cosh 1/2m{h}_1+h_2-2y\\ \begin{array}{c}{A}_{2}Gr-A_{4}Ha+A_1{m}^{2}Q2A_1+A_{4}Ha{h}_1-y{h}_2-y\times \\ \cosh 1/2h_1-h_{2}m\sin \alpha \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}}{2{A_4}^2{Ha}^2{A}_{4}Ha-A_1{m}^2{A}_3+Rd}\text{(26)}& \beta x,y,t=\frac{\cos hmy-\sin hmy\cos h{h}_1+h_{2}m+\cos h2\text{my}+\sin h{h}_1+h_{2}m+\sin h2my}{\cos h{h}_{1}m+\cos h{h}_{2}m+\sin h{h}_{1}m+\sin h{h}_{2}m},\end{array}$$where$$\begin{array}{c}\text{(27)}& A_1=\frac{{\mu }_{thnf}}{{\mu }_f},A_2=\frac{{\rho \gamma }_{thnf}}{{\rho \gamma }_f},\\ A_3=\frac{k_{thnf}}{k_f},\\ A_4=\frac{{\sigma }_{thnf}}{{\sigma }_f},\\ Px=\frac{\partial p}{\partial x}.\end{array}$$

Using equations (18)–(22) were solved to get the transverse velocity of flow $vx,y,t$ written as$$\begin{array}{}\text{(28)}& vx,y,t=\begin{array}{c}-12\cosh \frac{\sqrt{A_4}\sqrt{Ha}{h}_1+h_2}{2\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{A_4}\sqrt{Ha}{h}_1-h_2}{2\sqrt{A_1}}{A}_{4}Ha{A}_3+Rd\\ -A_1{m}^{2}Px+A_{4}Ha{m}^2\omega +Px\\ +A_1{A}_{2}Gr-A_{4}Ha+A_1{m}^{2}Q\sin \alpha \sinh \frac{\sqrt{A_4}\sqrt{Ha}y}{\sqrt{A_1}}+\dots \end{array}.\end{array}$$

The complete expression of $vx,y,t$ is presented in Appendix A.

The streamline function can be defined as$$\begin{array}{c}\text{(29)}& \psi ={\displaystyle \int }ux,y,t\partial y,\psi =-{\displaystyle \int }vx,y,t\partial x.\end{array}$$

Thus, obtaining$$\begin{array}{}\text{(30)}& \psi x,y,t=\frac{1}{2{A_4}^2{Ha}^2{A}_{4}Ha-A_1{m}^2{A}_3+Rd}\begin{array}{c}\frac{1}{6}{A}_{4}Ha-A_1{m}^{2}y12A_{4}HaPx{A}_3+Rd+A_{2}GrQ-12A_1+A_{4}Ha-6h1h2+3h1+h2y-2y^2\sin \alpha \\ \begin{array}{c}+\frac{1}{\sqrt{A_4}\sqrt{Ha}}2\sqrt{A_1}\mathrm{sech}\frac{\sqrt{A_4}h1-h2\sqrt{Ha}}{2\sqrt{A_1}}{A}_{4}Ha{A}_3+Rd-A_1{m}^{2}Px+A_{4}HaPx+m^2\omega \\ \begin{array}{c}+A_{1}A2Gr-A_{4}Ha+A_1{m}^{2}Q\sin \alpha \times \sinh \frac{\sqrt{A_4}\sqrt{Ha}h1+h2-2y}{2\sqrt{A_1}}\\ -2{A_4}^2{Ha}^{2}m{A}_3+Rd\omega \\ \times \mathrm{sech}\frac{1}{2}h1-h2m\sinh \frac{1}{2}mh1+h2-2y.\end{array}\end{array}\end{array}.\end{array}$$

The physical properties of the mixture could be taken using the following equations [40]:$$\begin{array}{}\text{(31)}& \begin{array}{c}{\rho }_{\text{thnf}}=1-{\varphi }_{1}1-{\varphi }_{2}1-{\varphi }_3{\rho }_f+{\varphi }_3{\rho }_{\mathrm{s}3}+{\varphi }_2{\rho }_{\mathrm{s}2}+{\varphi }_1{\rho }_{\mathrm{s}1}\\ {\sigma }_{nf}={\sigma }_f\frac{1+2{\varphi }_3{\sigma }_{s3}+1-2{\varphi }_3{\sigma }_f}{1-{\varphi }_3{\sigma }_{s3}+1+{\varphi }_3{\sigma }_f}\\ \begin{array}{c}{\sigma }_{hnf}={\sigma }_{nf}\frac{1+2{\varphi }_2{\sigma }_{s2}+1-2{\varphi }_2{\sigma }_{nf}}{1-{\varphi }_2{\sigma }_{s2}+1+{\varphi }_2{\sigma }_{nf}}\\ {\sigma }_{thnf}={\sigma }_{hnf}\frac{1+2{\varphi }_1{\sigma }_{s1}+1-2{\varphi }_1{\sigma }_{hnf}}{1-{\varphi }_1{\sigma }_{s1}+1+{\varphi }_1{\sigma }_{hnf}}\\ \begin{array}{c}{k}_{nf}=k_f\frac{k_{S3}+2k_f-2{\varphi }_3{k}_f-k_{S3}}{k_{S3}+2k_f+{\varphi }_3{k}_f-K_{S3}}\\ k_{hnf}=k_{nf}\frac{k_{S2}+2k_{nf}-2{\varphi }_2{k}_{nf}-k_{S2}}{k_{S2}+2k_{nf}+{\varphi }_2{k}_{nf}-k_{S2}}\\ k_{thnf}=k_{hnf}\frac{k_{S1}+2k_{hnf}-2{\varphi }_1{k}_{hnf}-k_{S1}}{k_{S1}+2k_{hnf}+{\varphi }_1{k}_{hnf}-k_{S1}}\end{array}\end{array}\end{array}.\end{array}$$

4. Coefficient of Heat Transfer

The heat transfer coefficient at the conduit upper ($Z_1$) and lower ($Z_2$) wall can be defined as [41]$$\begin{array}{}\text{(32)}& Z_1=\frac{\partial h_1}{\partial x}.\frac{\partial \theta }{\partial y},\text{(33)}& Z_1=\frac{Q2y-\epsilon \cos \pi x-t+\epsilon \cos \pi x-t+\gamma -\delta +\pi \epsilon \sin \pi x-t}{2A3+Rd},\text{(34)}& Z_2=\frac{\partial h_2}{\partial x}.\frac{\partial \theta }{\partial y},\text{(35)}& Z_2=-\frac{Q2y-\epsilon \cos \pi x-t+\epsilon \cos \pi x-t+\gamma -\delta +\pi \epsilon \sin \pi x-t+\gamma }{2A3+Rd}.\end{array}$$

5. Entropy Optimization

Following the second law of thermodynamics, the volumetric entropy generated inside the peristaltic conduit was determined in terms of the thermal, viscous, and joule heating irreversibility [42–44].$$\begin{array}{}\text{(36)}& G_{\text{gen}}^{‴}=\frac{k_{thnf}}{{T_1}^2}+\frac{16{\sigma }^{\mathrm{\ast }}{∆T}^3}{3{T_1}^2{K}^{\mathrm{\ast }}}{\frac{\partial T}{\partial Y}}^2+\frac{{\mu }_{thnf}}{T_1}{\frac{\partial U}{\partial Y}}^2+\frac{{\sigma }_{thnf}}{T_1}{B}_0^2{U}^2+\frac{{\sigma }_e}{T_1}{E}_x^2.\end{array}$$

Using equations (10) and (11), one obtains$$\begin{array}{}\text{(37)}& G_{\text{gen}}^{‴}=\frac{k_f}{{l_1}^2}\frac{{∆T}^2}{{T_1}^2}\frac{k_{thnf}}{k_f}+PrRd{\frac{\partial \theta }{\partial y}}^2+\frac{{\mu }_{thnf}}{{\mu }_f}\mathrm{Pr}Ec{\frac{\partial u}{\partial y}}^2+\frac{{\sigma }_{thnf}}{{\sigma }_f}\mathrm{Pr}EcHa{u}^2+\tau T_r,\end{array}$$where $Ec=c^2/{T_1{c}_p}_f$ is the Eckert number, $\tau ={{l_1}^2\sigma }_e/k_f{T}_1{E}_x^2$ is joule heating parameter, and $T_r=T_1/∆T$ temperature ratio.

The characteristic rate of entropy production can be defined as$$\begin{array}{}\text{(38)}& {\dot{G}}_0^{‴}=\frac{k_f}{{l_1}^2}\frac{{∆T}^2}{{T_1}^2}.\end{array}$$

By simplifying equations (37) and (38), one gets the total entropy number ($Es$):$$\begin{array}{}\text{(39)}& Es=\underset{E_{th}}{\underbrace{\frac{k_{thnf}}{k_f}+\mathrm{Pr}Rd{\frac{\partial \theta }{\partial y}}^2}}+\underset{E_{fr}}{\underbrace{\frac{{\mu }_{thnf}}{{\mu }_f}\mathrm{Pr}Ec{\frac{\partial u}{\partial y}}^2}}+\underset{E_{jo}}{\underbrace{\frac{{\sigma }_{thnf}}{{\sigma }_f}\mathrm{Pr}EcHa{u}^2+\tau T_r}},\end{array}$$where the first ($E_{th}$), second ($E_{fr}$), and third ($E_{jo}$) terms signify the corresponding thermal (due to both convection and radiation), friction, and joule heating irreversibility. Bejan number ($Be$) being the ratio of thermal to total irreversibility can take a value in the range of 0–1, wherein $Be=0$ and $Be=1$ refer to the predominance of the viscous dissipation and thermal irreversibility, respectively, over all other irreversible processes. The values of $Be$ were estimated via$$\begin{array}{}\text{(40)}& Be=\frac{E_{th}}{E_{th}+E_{fr}+E_{jo}}.\end{array}$$

6. Results and Graphs

The effects of various embedded parameters on the flow characteristics and temperature with the conduit were evaluated (Table 2). The fluid flow was determined in terms of the variation of axial inclined and transverse velocity, as well as the 2D and 3D temperature profiles that could influence the conduit’s irregularity. In addition, the streamline profiles were drawn using the contour lines.

Table 2

The embedded parameter-dependent heat transfer coefficient at the upper and lower walls.

${\varphi }_1={\varphi }_2={\varphi }_3$	$Z_1$		$Z_2$
	$t=0$	$t=1$	$t=0$	$t=1$
0, 0, 0	−0.00507322	−0.00492678	0.0163431	−0.00601777
0.05, 0, 0	−0.00470367	−0.00456789	0.0151526	−0.00557941
0, 0.05, 0	−0.00470356	−0.00456778	0.0151522	−0.00557928
0, 0, 0.05	−0.00471818	−0.00458199	0.0151994	−0.00559663
0.05, 0.05, 0	−0.00433816	−0.00421293	0.0139751	−0.00514585
0.05, 0, 0.05	−0.00435255	−0.00422691	0.0140215	−0.00516292
0, 0.05, 0.05	−0.00435243	−0.00422679	0.0140211	−0.00516277
0.05, 0.05, 0.05	−0.00399456	−0.00387926	0.0128683	−0.00473828
ε	${\mathit{Z}}_{\mathbf{1}}$		${\mathit{Z}}_{\mathbf{2}}$
	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$
0	−0.00393691	−0.00393691	0.00393691	0.00393691
0.1	−0.00399456	−0.00387926	0.0128683	−0.00473828
0.2	−0.00405222	−0.0038216	0.0220558	−0.0131573
0.3	−0.00410987	−0.00376395	0.0314994	−0.0213202
γ	${\mathit{Z}}_{\mathbf{1}}$		${\mathit{Z}}_{\mathbf{2}}$
	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$
0	−0.00393691	−0.00393691	0.00393691	0.00393691
$\pi /12$	−0.00394362	−0.00393020	0.00715019	0.00073453
$\pi /6$	−0.00396328	−0.00391054	0.01018880	−0.00223212
$\pi /4$	−0.00399456	−0.00387926	0.01286830	−0.00473828
$\mathit{\alpha }$	${\mathit{Z}}_{\mathbf{1}}$		${\mathit{Z}}_{\mathbf{2}}$
	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{1}$
0	0	0	0.00887369	−0.008617540
0.1	−0.00399456	−0.00387926	0.01286830	−0.004738280
0.2	−0.00798913	−0.00775851	0.01686280	−0.000859028
0.3	−0.01198370	−0.01163780	0.02085740	0.003020230

6.1. Axial Inclined Velocity

Figures 2, 3, 4, 5, and 6 illustrate the dependence of axial inclined velocity on various parameters of the model, wherein solid and dashed lines signify the corresponding unsteady and steady fluid flow. The inclusion of nanoparticles in the fluid was observed to reduce its steady state speed in the conduit (Figure 2). With the variation of conduit’s diameter (irregular parameter), the axial flow for the unsteady state was significantly affected (Figure 3), wherein the obtained incensement may be due to the increase of conduit slut. The flow behavior was changed with the type of motion (Figure 4) in which the unsteady flow velocity was increased, and the steady velocity was reduced with the increase in the wave amplitude ratio. These results demonstrated an appreciable effect of the wave amplitude ratio on the fluid flow through the conduit. In addition, the value of the maximum speed for the steady state could approach more towards the upper wall of the conduit. Conversely, the maximum speed for the unsteady state flow displayed similar profile. The maximum velocity started to diminish when the walls are in-phase and then shifted towards the bottom half of the conduit with the increase of the phase angle. For steady state flow, this phenomenon became more obvious. In general, the velocity was reduced with the increase of the phase angle (Figure 5). Gravity could significantly affect the rapidity of the fluid flow downward along the channel axis except for the magnetic field perpendicular to the flow. In this case, a Lorentz force was responsible for the slowing down of the fluid in this direction, especially for the steady state motion (Figure 6). The axial velocity showed a monotonic increase due to the electric field generation on the walls (Figure 7), wherein the fluid particles tend to move faster, especially in the unsteady state.

[figure(s) omitted; refer to PDF]

6.2. Transversal Velocity Behavior

Figures 7, 8, 9, 10, and 11 illustrate the dependence of transverse velocity on various parameters of the model. Generally, the velocity was wavy and increased in the first half of the channel and then decreased in the other half accompanied by a shift of the inflection point to one of the two halves depending on the parameter values. The insertion of nanoparticles was found to reduce the velocity in the upper two-third part of the conduit then in the last third (Figure 7), wherein the unsteady flow was most affected. The variation of conduit geometry was found to improve the velocity, wherein an increase in the irregularity parameter could affect significantly the unsteady flow in the upper half of the conduit compared to the lower half (Figures 8, 9, and 10). Furthermore, with the increase of the both wave amplitude ratio and phase angle, the velocity for unsteady flow was reduced. With the increase of the phase angle, the unsteady flow velocity was increased (Figures 9 and 10). The applied magnetic field had more significant effect (Figure 11) on the transverse speed of the steady flow than the unsteady one, indicating a decrease in the value of the speed than the one observed in the absence of a magnetic field. In short, a noticeable improvement in the flow velocity was obtained with the increase of the magnetic field strength.

[figure(s) omitted; refer to PDF]

6.3. Conduit Temperature

Figures 12, 13, and 14 show the influence of channel geometry, shape, and corrugation types on the fluid temperature within the channel. Clearly, irregularity parameter and wave amplitude ratio were found to boost the fluid flow temperature. Conversely, the temperature for the steady flow was dropped with the increasing phase angle. In addition, with the increase of the wave amplitude ratio, the temperature for the unstable flow was increased, indicating a reverse effect on the fluid temperature for the steady flow. Figures 15 and 16 depict the influence of thermal forces (thermal radiation/heat source) on the channel temperature. As predicted, the temperature was increased with the increase of the heat source, in which the uneven flow became clearer and stronger. However, the influence of radiation was very different from the expected one, wherein the fluid flow temperature was reduced in the presence of thermal radiation. This observation can be ascribed to the more efficient transport of the thermal energy at higher temperatures. Thus, an increasing thermal radiation caused more thermal energy transfer to the cold section of the conduit thereby lowering the temperature of the airflow inside the conduit.

[figure(s) omitted; refer to PDF]

6.4. Pressure Raise Behavior

Figure 17 illustrates how the pressure gradient rises as the Hartmann number $Ha$ does, reflecting the flow’s reaction to the growing magnetic field. As a result, the flow becomes less smooth and takes more pressure to sustain, which raises the hydrodynamic resistance. The flow becomes steadier but has a greater pressure gradient with time as a result of the magnetic field’s growing impact, which is a modest but detectable effect of the temporal variation.

[figure(s) omitted; refer to PDF]

As we explore Figure 18, we see that angle zero, or black, denotes a horizontal channel. It also seems that the pressure gradient in this instance is lower than it is at other angles. The primary forces operating are viscosity and flow at $\alpha =0$, as there is no gravity influence in the flow direction. Green lines denote a channel that is entirely vertical (90 degrees). It is the case that we observe the highest pressure gradient. This is due to the fact that gravity directly influences flow, raising hydraulic resistance and producing a higher pressure gradient. Red and blue, the other colors, stand for average tilt angles ranging from 0 to 90 degrees. The curves suggest that as gravity starts to exert more of an influence on the flow, raising the angle gradually raises the pressure gradient. Furthermore, lines that have been dashed $t=0$ represent the pressure gradient at the start of time, when motion’s effects are the least. Furthermore, it can be inferred that the pressure gradient is increasing due to flow effects and accelerating velocity with time since the solid line at time $t=0.5$ is depicted later and seems to be slightly larger than the dashed line.

[figure(s) omitted; refer to PDF]

As depicted in Figure 19, the electro-osmotic forces operating on the system get stronger as the electro-osmotic number $m$ increases, which causes the pressure gradient along the x-axis to increase. This increase is indicative of the enhanced flow produced by electro-osmosis, in which there are more pressure changes within the system as a result of an increase in the fluid flow rate.

[figure(s) omitted; refer to PDF]

6.5. Streamlines Distribution

Figures 20, 21, 22, 23, and 24 present the effects of various key factors on the fluid flow streamlines distribution inside the conduit. Typically, the flow lines were observed to separate into two sections, wherein the top section of the conduit has positive flow streamlines, while the lower section has negative flow lines. Figure 20 demonstrates the impact of nanoparticles’ contents on the streamlines distribution. The streamlines density was diminished with the increase of nanoparticles contents. When the streamlines traveled further downward along the conduit, the quantity of the trapped boluses was increased, appearing more concentrated on the walls. In addition, the flow streamlines converge was substantially increased in the absence of magnetic field accompanied by more trapped boluses on the walls. The flow speed was improved with the increase of the Hartmann number (Figure 21). With the increase in the electro-osmotic number, a little rise in the concentration of the flow streamlines with a convergence of the flow lines and the concentration of trapped boluses on the walls were observed (Figure 22). Figures 23 and 24 illustrate the impact of increasing heat source parameters and Grashof number on the streamlines’ contour of the flow. A more streamlined flow with fewer boluses was found to develop in the absence of both heat source parameters and the Grashof number, albeit with lower values. This is because increasing these parameters causes boluses to develop, which might obstruct fluid flow.

[figure(s) omitted; refer to PDF]

6.6. Entropy Generation

Figures 25, 26, 27, 28, 29, and 30 show the influence of various key factors and parameters on the Bejan number. As mentioned before, the thermal and viscous dissipation irreversibility dominance of the entropy generation depended on the Bejan number. The Bejan number was less than 0.5 when the temperature parameters were varied, indicating the dominance of the viscous irreversibility on the entropy production. The Bejan ratio was increased due to the application of both magnetic field and heat source (Figures 25 and 26). As shown in Figures 27–30, with the increase of thermal forces, such as radiation, Eckert number, and electro-osmotic number, the value of the Bejan ratio was dropped, indicating a greater dominance of the viscous irreversibility on the entropy production. These results clearly demonstrated that the steady flow had a favorable influence in boosting the Bejan ratio, thus the system’s entropy.

[figure(s) omitted; refer to PDF]

6.7. Heat Transfer Rate at Each Wall

Table 2 shows the embedded parameter-dependent heat transfer coefficient at the upper and lower walls. To ascertain the cooling efficiency of a thermal system, it is important to analyze the heat transfer rate at the walls of the system. In the current study, the heat transfer rate for the steady flow was higher compared to the unsteady flow especially at the lower wall. The heat transfer rate at the upper and lower walls of the conduit was reduced by 21.26% with the increase of nanoparticles content of 5%. Conversely, the variation of the wave amplitude ratio produced a positive impact on the heat transfer rate, which was increased up to 4.4% at the upper wall of the conduit, and the increased considerably up to 700% at the lower wall of the conduit with the increase in the ratio from zero to 0.3. A slight increase on the heat transfer rate of approximately 1.5% was observed with the increase in the channel inclination angle. In addition, an increase in the heat transfer rate about 1.2% at the upper wall was evidenced with the increase of the irregularity parameter from zero to 0.3, and then the heat transfer rate was increased significantly as much as 135% at the lower wall of the conduit.

7. Conclusion

Based on the model simulation results, the following conclusions were drawn:

• The conduit shape had a considerable impact on the proposed tri-nanoparticles–embedded fluid flow speed and temperature inside the conduit.

• The axial flow speed was reduced due to the difference in the phase of the walls, applied EMF, and nanoparticles, indicating the impact of the fluid viscosity on the entropy production. In contrast, the flow speed was enhanced due to the conduit’s irregularity and the electro-osmatic velocity generated by the electric field.

• The viscous dissipation continued to be the dominating source for the entropy production in the studied system despite a rise in the levels of thermal forces in the conduit.

• The external thermal forces such as magnetic and electric fields showed a considerable influence of the streamline’s distribution accompanied by a divergence in the flow streamlines and increase in the trapped boluses at the walls and an increase in the Grashof number.

• In short, it is established that the heat transfer rate at the walls can appreciably be improved by controlling the channel geometry including the phase angle, wave amplitude ratio, and irregular parameters.

• Present systematic study can be useful for the development of the high-performance cooling systems in the diverse thermal technologies needed for the practical applications.

Funding

The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project number 445-9-620.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project number 445-9-620.

Appendix

$$\begin{array}{}\text{(A.1)}& v=\frac{1}{24{\text{Ha}}^2\text{Rd}+A_3{A}_4^2-m^2{A}_1+\text{Ha}{A}_4}\mathrm{\ast }\begin{array}{c}4\sqrt{\text{Ha}}\cosh \frac{\sqrt{\text{Ha}}y\sqrt{A_4}}{\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{\text{Ha}}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}sinh\frac{\sqrt{\text{Ha}}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\sqrt{A_1}\text{Rd}+A_3\sqrt{A_4}{m}^2{A}_1-\text{Ha}{A}_4{\text{Px}}^{\prime }\\ +24\sqrt{Ha}sinh\frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}\sqrt{A_1}Rd+A_3\sqrt{A_4}-m^2{A}_1+Ha{A}_4{Px}^{\prime }\\ +24\sqrt{\text{Ha}}\cosh \frac{\sqrt{\text{Ha}}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{\text{Ha}}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}sinh\frac{\sqrt{\text{Ha}}y\sqrt{A_4}}{\sqrt{A_1}}\sqrt{A_1}\text{Rd}+A_3\sqrt{A_4}-m^2{A}_1+\text{Ha}{A}_4{\text{Px}}^{\prime }\\ -24\text{Ha\hspace{0.17em}}y\text{Rd}+A_3{A}_4-m^2{A}_1+\text{Ha}{A}_4{\text{Px}}^{\prime }+2\text{GrHa\hspace{0.17em}}mQ{y}^3\sin \alpha A_2{A}_4-m^2{A}_1+\text{Ha}{A}_4\tanh \frac{1}{2}m{h}_1-h_2{h_1}^{\prime }-{h_2}^{\prime }\\ -m-12\text{Gr}{m}^{2}Qy\sin \alpha A_1^2{A}_2+12\text{GrHa}Qy\sin \alpha A_1{A}_2{A}_4-2\text{GrHa}{m}^{2}Q{y}^3\sin \alpha A_1{A}_2{A}_4\\ -m-12\text{Gr}{m}^{2}Qy\sin \alpha A_1^2{A}_2+12\text{GrHa}Qy\sin \alpha A_1{A}_2{A}_4-2\text{GrHa}{m}^{2}Q{y}^3\sin \alpha A_1{A}_2{A}_4\\ -12{\text{Ha}}^{2}m\text{Rd}\omega \cosh \frac{1}{2}m{h}_1+h_2\mathrm{sech}\frac{1}{2}m{h}_1-h_{2}sinhmy{A}_4^2\\ +12{\text{Ha}}^{2}m\text{Rd}\omega \cosh mys\text{ech}\frac{1}{2}m{h}_1-h_{2}s\text{inh}\frac{1}{2}m{h}_1+h_2{A}_4^2+2\text{Gr}{\text{Ha}}^{2}Q{y}^3\sin \alpha A_2{A}_4^2\\ -12{Ha}^{2}m\omega \cosh \frac{1}{2}m{h}_1+h_2\mathrm{sech}\frac{1}{2}m{h}_1-h_{2}sinhmy{A}_3{A}_4^2\\ +12{Ha}^{2}m\omega \cosh my\mathrm{sech}\frac{1}{2}m{h}_1-h_{2}sinh\frac{1}{2}m{h}_1+h_2{A}_3{A}_4^2-12HaPxyRd+A_3{A}_4-m^2{A}_1+Ha{A}_4\\ -3GrHaQysin\alpha A_2{A}_4-m^2{A}_1+Ha{A}_4{h}_{1}y-2h_2+3GrHa{m}^{2}Q{y}^2\sin \alpha A_1{A}_2{A}_4{h}_2-3Gr{Ha}^{2}Q{y}^2\sin \alpha A_2{A}_4^2{h}_2\\ \tanh \frac{1}{2}m{h}_1-h_2{h_1}^{\prime }-{h_2}^{\prime }-12\cosh \frac{\sqrt{Ha}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}sinh\frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}\\ GrQsin\alpha A_1{A}_2{m}^2{A}_1-Ha{A}_4+HaRd+A_3{A}_4-m^{2}Px{A}_1+HaPx+m^2\omega A_4\tanh \frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}{h_1}^{\prime }-{h_2}^{\prime }\\ +12\cosh \frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}sinh\frac{\sqrt{Ha}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\\ GrQsin\alpha A_1{A}_2{m}^2{A}_1-Ha{A}_4+HaRd+A_3{A}_4-m^{2}Px{A}_1+HaPx+m^2\omega A_4\tanh \frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}{h_1}^{\prime }-{h_2}^{\prime }\\ +6{Ha}^2{m}^2\omega \cosh my-sinhmyRd+A_3{A}_4^2{h_1}^{\prime }+{h_2}^{\prime }+6{Ha}^2{m}^2\omega \cosh my+sinhmyRd+A_3{A}_4^2{h_1}^{\prime }+{h_2}^{\prime }\\ +6{Ha}^2{m}^2\omega \mathrm{sech}\frac{1}{2}m{h}_1-h_2\cosh my-\frac{1}{2}m{h}_1+h_2-sinhmy-\frac{1}{2}m{h}_1+h_{2}Rd+A_3{A}_4^2{h_1}^{\prime }+{h_2}^{\prime }\\ +6{Ha}^2{m}^2\omega \mathrm{sech}\frac{1}{2}m{h}_1-h_2\cosh my-\frac{1}{2}m{h}_1+h_2+sinhmy-\frac{1}{2}m{h}_1+h_{2}Rd+A_3{A}_4^2{h_1}^{\prime }+{h_2}^{\prime }\\ -6GrHaQ{y}^2\sin \alpha A_2{A}_4-m^2{A}_1+Ha{A}_4{h_1}^{\prime }+{h_2}^{\prime }\\ -12\cosh \frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}GrQsin\alpha A_1{A}_2{m}^2{A}_1-Ha{A}_4+HaRd+A_3{A}_4-m^{2}Px{A}_1+HaPx+m^2\omega A_4{h_1}^{\prime }+{h_2}^{\prime }\\ -12\cosh \frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}\cosh \frac{\sqrt{Ha}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\mathrm{sech}\frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}\\ GrQsin\alpha A_1{A}_2{m}^2{A}_1-Ha{A}_4+HaRd+A_3{A}_4-m^{2}Px{A}_1+HaPx+m^2\omega A_4{h_1}^{\prime }+{h_2}^{\prime }\\ +12\mathrm{sech}\frac{\sqrt{Ha}\sqrt{A_4}{h}_1-h_2}{2\sqrt{A_1}}sinh\frac{\sqrt{Ha}y\sqrt{A_4}}{\sqrt{A_1}}sinh\frac{\sqrt{Ha}\sqrt{A_4}{h}_1+h_2}{2\sqrt{A_1}}\\ GrQsin\alpha A_1{A}_2{m}^2{A}_1-Ha{A}_4+HaRd+A_3{A}_4-m^{2}Px{A}_1+HaPx+m^2\omega A_4{h_1}^{\prime }+{h_2}^{\prime }\\ -3GrHaQ{y}^2\sin \alpha A_2{A}_4-m^2{A}_1+Ha{A}_4\\ 2+m{h}_1\tanh \frac{1}{2}m{h}_1-h_2+m{h}_2\tanh \frac{1}{2}m{h}_1-h_2{h_1}^{\prime }--2+m{h}_1\tanh \frac{1}{2}m{h}_1-h_2+m{h}_2\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_2\\ +6y-m^2{A}_1+Ha{A}_4-22HaRd{A}_4{Px}^{\prime }+2Ha{A}_3{A}_4{Px}^{\prime }-GrmQsin\alpha A_1{A}_2\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_1\\ +HamPxRd+A_3{A}_4\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_1-{h^{\prime }}_2-GrHaQsin\alpha A_2{A}_4{h}_1{h_2}^{\prime }+GrmQsin\alpha A_1{A}_2\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_2\\ +GrHaQsin\alpha A_2{A}_4{h}_{2}2+m{h}_1\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_1-m{h}_1\tanh \frac{1}{2}m{h}_1-h_2{h^{\prime }}_2.\end{array}\end{array}$$

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