Entropy generation and chemical reaction effects on MHD non-Newtonian nanofluid flow in a sinusoidal channel

Abstract

This article discusses the effects of entropy generation as well as slip velocity condition on MHD Jeffery nanofluid flow through a porous medium in a channel with peristalsis. We take the effects of mixed convection, heat source, double diffusion and chemical reaction into consideration. Using the assumption of low-Reynolds number and long-wavelength, series solutions of the governing equations are obtained via homotopy perturbation method. Results will be discussed at various parameters of the problem and drawn graphically. Physically, our model is consistent with the motion of digestive juice in the bowel whenever we are going to insert an endoscopy through it. It is noticed that the axial velocity magnifies with an increase in the values of both first and second slip parameters. Meanwhile, the value of the axial velocity reduces with the elevation in the values of both Grashoff and Darcy numbers. On the other hand, the elevation in the value of thermal radiation leads to a reduction in the value of fluid temperature. Furthermore, increasing in the value of order of chemical reaction parameter makes an enhancement in the value of the solutal concentration. It is noticed also that the entropy generation enhances with the increment in the value of Eckert number. The current study has many accomplishments in several scientific areas like engineering industry, medicine, and others. Therefore, it represents the gastric juice motion depiction in the human body when an endoscope is inserted through it.

Keywords

Entropy generation nanofluid peristaltic flow double diffusive porous medium

1. Introduction

Given the fact that the technology has been improving, it has already been realized that the industrial devices must be cooled into more effective methods [1] as well as the conventional fluids like the water are not suitable anymore, then the idea of adding up particles into a fluid was introduced. Add up nanoparticles for the base fluid affect both the homogeneity of fluids and the uncertainty movement of molecules that it increases.

Such small particles have high heat conductivity, so that blended fluids have a better thermal performance [2–4]. Materials at which of these nano-scale molecules are aluminum oxide (Al₂O₃), copper (Cu), copper-oxide (CuO), gold (Au), silver (Ag), etc., who are suspended in the underlying fluids like the water, oil, acetone, and ethylene glycol. Al₂O₃ and CuO are the most well-known nanoparticles used by various scientists in their research [4–8 ]. They submitted various results due to the shape and the size and so contacting surface of the particles. The increasing popularity of nanofluids can be measured with investigations conducted by researchers for their regular application and one can find writing, for an example [9–17 ].

Adding these nanoparticles up to the fluid makes it base in the fluid inhomogeneous, consequently, thermodynamic is irreversible in flowing increases that will cause more energy and power losses into the system. Saving helpful energy will depend on how to design the effective heat transfer process from a thermodynamic point of view. Energy transformation processes have been led up to the irreversible increase in entropy. Consequently, even if the energy is preserved, the high quality of energy that decreases converting them into a different form of energy at which less work can be obtained. Reducing generated by an entropy will lead to more efficient designs of the energy system. Groundbreaking work at an entropy generation to the fluid flow was made by Bejan [18,19]. Also, he introduced the method called the Entropy Generation Minimization (EGM) to measure and optimization disorder or disorganization generated during a process particularly in the fields of refrigeration (cryogenics), heat transfer, storage, and solar thermal power conversion. There is no question that by “optimize” we mean the stabled process in which the system loses the least energy while still performing its fundamental engineering function. The method is also known as second law analysis and thermodynamic optimization. This field has been developed astoundingly during the 1990s, in both engineering and physics. Good example of such efforts found in references [20–25].

Peristaltic flow has particular concern for various applications inside the industry and physiology. The peristaltic transportation of non-Newtonian fluids is a subject of great interest of research in physiological world. Such an interest is being stimulated because of their phenomenon in several physiological processes such as chyme movement in gastrointestinal tract, urine transportation from the kidneys to the bladder, transport of bile in bile duct, in roller and finger pumps, in vasomotion of small blood vessels. Mekheimer and Arabi [26] have studied nonlinear peristaltic transport of MHD flow through a porous medium. Kumari et al. [27] discussed peristaltic pumping of a conducting Jeffrey fluid in a vertical porous channel with heat transfer. The impact of wall properties on the peristaltic flow of a non-Newtonian fluid is examined by Hayat et al. [28]. Mathematica simulation of peristaltic pumping with double-diffusivity convection in nanofluids a bio-nanoengineering model has been analyzed by Bég and Tripathi [29]. Eldabe and Abou-zeid [30] were reviewed Radially varying magnetic field effect on peristaltic motion with heat and mass transfer of a non-Newtonian fluid between two co-axial tubes. Moreover, Abou-zeid [31] investigated the whole problem of Homotopy perturbation method for a couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid. MHD peristaltic flow of micropolar Casson nanofluid through a porous medium between the two co-axial tubes were studied by Mohamed and Abou-zeid [32]. Heat and mass transfer effect on non-Newtonian fluid flow in a non-uniform vertical tube with peristalsis was discussed by Mansour and Abouzeid [33]. Eldabe et al. [34] considered MHD peristaltic flow of non-Newtonian power-law nanofluid through a non-Darcy porous medium inside a non-uniform inclined channel. Effect of Heat and Mass Transfer on Casson Fluid Flow Between Two Co-Axial Tubes with Peristalsis is reviewed by Eldabe et al. [35]. Also, Eldabe et al. [36] debated the homotopy perturbation approach to the Ohmic dissipation and mixed convection effects on non-Newtonian nanofluid flow between two co-axial tubes with peristalsis. Finally, Peristaltic flow of Herschel Bulkley nanofluid through a non-Darcy porous medium with heat transfer under slip condition is examined by Eldabe et al. [37]. Recently, some important studies on the peristaltic in different geometrical configurations with different flow conditions are mentioned in the references [38–41].

In this problem, we have analyzed the influences of entropy generation and slip velocity on MHD Jeffery nanofluid flow in a peristaltic channel through a porous medium. The fluid flows are in the presence of mixed convection, radiation, heat source, double diffusion medium porosity and chemical effects. The equations of flow are simplified under low-Reynolds number and long-wavelength assumptions. Using homotopy perturbation technique, a semi-analytical solution for the momentum, energy and concentration and nanoparticles volume friction equations have been obtained. Furthermore, the entropy generation is obtained to control problem physical parameters. Results are discussed for some parameters of the problem and sketched graphically. Physically, our model matches the motion of the digestive juice in the intestine when we insert an endoscope through it. Physically, our model corresponds to the transport of the gastric juice in the small intestine when an endoscope is inserted through it.

Fig. 1.

Schematic of the problem

2. Mathematical description

Fluid model is studied in Cartesian coordinate system (x, y), (see Fig. 1). Channel wall has a mathematical description as $\begin{eqnarray}\displaystyle h(X,t)=d_{1}+a_{1}\cos {\displaystyle \frac{2{\pi}}{{\lambda}}}(X-ct). & & \displaystyle\end{eqnarray}$ (1) The equations governing the motion of this model can be represented as [42] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{f}\left(\frac{\partial }{\partial t}+U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y}\right)U=-\frac{\partial P}{\partial X}+\frac{{\mu}_{f}}{1+{\lambda}_{1}}\left(\frac{\partial ^{2}}{\partial X^{2}}+\frac{\partial ^{2}}{\partial Y^{2}}\right)U\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼{\rho}_{f}\text{g}({\beta}_{T}(T-T_{0})+{\beta}_{C}(C-C_{0})-{\beta}_{f}(f-f_{0}))-({\sigma}B_{0}^{2}+k)U,\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{f}\left(\frac{\partial }{\partial t}+U\frac{\partial }{\partial X}+V\frac{\partial }{\partial Y}\right)V=-\frac{\partial P}{\partial Y}+\frac{{\mu}_{f}}{1+{\lambda}_{1}}\left(\frac{\partial ^{2}}{\partial X^{2}}+\frac{\partial ^{2}}{\partial Y^{2}}\right)V-kV,\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\rho}c)_{f}\left(\frac{\partial T}{\partial t}+U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}\right)={\kappa}_{T}\left(\frac{\partial ^{2}T}{\partial X^{2}}+\frac{\partial ^{2}T}{\partial Y^{2}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\frac{{\mu}_{f}}{1+{\lambda}_{1}}\left[2\left(\frac{\partial U}{\partial X}\right)^{2}+\left(\frac{\partial V}{\partial Y}\right)^{2}+\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)^{2}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼({\rho}c)_{p}\left[D_{B}\left(\frac{\partial T}{\partial X}\frac{\partial f}{\partial X}+\frac{\partial T}{\partial X}\frac{\partial f}{\partial Y}\right)+\frac{D_{T}}{T_{m}}\left(\frac{\partial T}{\partial X}+\frac{\partial T}{\partial Y}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼D_{TC}\left(\frac{\partial ^{2}C}{\partial X^{2}}+\frac{\partial ^{2}C}{\partial Y^{2}}\right)+Q_{0}\left(T-T_{0}\right)-\frac{\partial q_{r}}{\partial Y}+{\sigma}B_{0}^{2}U^{2},\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left(\frac{\partial C}{\partial t}+U\frac{\partial C}{\partial X}+V\frac{\partial C}{\partial Y}\right)=D_{s}\left(\frac{\partial ^{2}C}{\partial X^{2}}+\frac{\partial ^{2}C}{\partial Y^{2}}\right)+D_{CT}\left(\frac{\partial ^{2}T}{\partial X^{2}}+\frac{\partial ^{2}T}{\partial Y^{2}}\right)-A\left(C-C_{0}\right),\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left(\frac{\partial f}{\partial T}+U\frac{\partial f}{\partial X}+V\frac{\partial f}{\partial Y}\right)=D_{B}\left(\frac{\partial ^{2}f}{\partial X^{2}}+\frac{\partial ^{2}f}{\partial Y^{2}}\right)+\frac{D_{T}}{T_{m}}\left(\frac{\partial ^{2}T}{\partial X^{2}}+\frac{\partial ^{2}T}{\partial Y^{2}}\right).\end{eqnarray}$ (7)

Consider a wave frame (x, y) which moving with speed c. Coordinates and velocity components in wave frame are related by the following transformations $\begin{eqnarray}\displaystyle x=X-ct,\quad y=Y,\quad u=U-c,\quad v=V,\quad p(x,y)=P(X,Y,t). & & \displaystyle\end{eqnarray}$ (8) Dimensionless quantities can be written as $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle x^{\ast }={\displaystyle \frac{x}{{\lambda}}},\quad y^{\ast }=\frac{y}{d_{1}},\quad t^{\ast }={\displaystyle \frac{ct}{{\lambda}}},\quad u^{\ast }=\frac{u}{c},\quad v^{\ast }={\displaystyle \frac{v}{c{\delta}}},\quad {\delta}_{1}={\displaystyle \frac{d_{1}}{{\lambda}}},\quad h^{\ast }=\frac{h}{d_{1}},a={\displaystyle \frac{a_{1}}{d_{1}}},\quad {\varphi}=\frac{a_{1}}{d_{1}},\\[10.0pt] \displaystyle T^{\ast }=\frac{T-T_{0}}{T_{1}-T_{0}},C^{\ast }=\frac{C-C_{0}}{C_{1}-C_{0}},f^{\ast }=\frac{f-f_{0}}{f_{1}-f_{0}}.\end{array} & & \displaystyle\end{eqnarray}$ (9) According to these transformations, neglecting the star mark and after applying δ₁ ≪ 1, the system of equations can be written as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial p}{\partial x}=\frac{1}{1+{\lambda}_{1}}\frac{\partial ^{2}u}{\partial y^{2}}-\left(M+\frac{1}{Da}\right)u+Gr_{T}T+Gr_{C}C-Gr_{f}f,\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial p}{\partial y}=0,\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left(1+\frac{4}{3}R\right)\frac{\partial ^{2}T}{\partial y^{2}}+\frac{E_{c}P_{r}}{1+{\lambda}_{1}}\left(\frac{\partial u}{\partial y}\right)^{2}+P_{r}N_{t}\left(\frac{\partial T}{\partial y}\right)^{2}+P_{r}N_{b}\left(\frac{\partial T}{\partial y}\right)\left(\frac{\partial f}{\partial y}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼P_{r}N_{TC}\left(\frac{\partial ^{2}C}{\partial y^{2}}\right)+QP_{r}T=0,\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}C}{\partial y^{2}}+N_{CT}\frac{\partial ^{2}T}{\partial y^{2}}-{\delta}C^{m}=0,\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}f}{\partial y^{2}}+\frac{N_{t}}{N_{b}}\frac{\partial ^{2}T}{\partial y^{2}}=0.\end{eqnarray}$ (14) The appropriate boundary conditions are $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle \frac{\partial u}{\partial y}=0,T=0,C=0,f=0\quad \text{at}\quad y=0,\\ \displaystyle u+{\beta}_{1}\frac{\partial u}{\partial y}+{\beta}_{2}\frac{\partial ^{2}u}{\partial y^{2}}=-1,T=1,C=1,f=1\quad \text{at}\quad y=h.\end{array} & & \displaystyle\end{eqnarray}$ (15)

3. Entropy generation analysis

The main aim of the present study is to minify entropy generation to obtain better results by restricting physical parameters of the problem. The dimensionless form of entropy generation can be exhibited mathematically as follows [17–19]: $\begin{eqnarray}\displaystyle \hspace{-18.0pt}Eg=\frac{\partial ^{2}T}{\partial y^{2}}+\frac{3P_{r}}{(3+4R)}\left(\frac{E_{c}}{1+{\lambda}_{1}}\left(\frac{\partial u}{\partial y}\right)^{2}+Mu^{2}+N_{t}\left(\frac{\partial T}{\partial y}\right)^{2}+N_{b}\left(\frac{\partial f}{\partial y}\frac{\partial T}{\partial y}\right)+\right.\left.N_{Tc}\frac{\partial ^{2}C}{\partial y^{2}}+QT\right) & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (16) the proportion between entropy generated by heat transfer to total entropy is defined by Bejan number Bn.

4. Method of solution

The homotopy perturbation technique is a helpful technique which can solve many ordinary or partial differential equations. It entails only a few paces to obtain semi-analytical solutions for non-linear differential equations. furthermore, it is a collection of the perturbation technique and the homotopy analysis technique, which removes the obstacles of the well-known perturbation method, while keeping all their features. First, we construct a suitable homotopy equation which is one of the most serious steps in application. Following [13–15], Eqs (10)–(14) can be formulated as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H(p,u)=L(u)-L(u_{10})+pL(u_{10})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼(1+{\lambda}_{1})p\left(-\frac{\partial P}{\partial x}-\left(\frac{1}{Da}+M^{2}\right)u+Gr_{T}T+Gr_{C}C-Gr_{f}f\right)\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H(p,T)=L(T)-L(T_{10})+pL(T_{10})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\frac{3P_{r}p}{(3+4R)}\left({\displaystyle \frac{E_{c}}{1+{\lambda}_{1}}}\left({\displaystyle \frac{\partial u}{\partial y}}\right)^{2}+N_{t}\left({\displaystyle \frac{\partial T}{\partial y}}\right)^{2}+N_{b}\left({\displaystyle \frac{\partial f}{\partial y}}{\displaystyle \frac{\partial T}{\partial y}}\right)+N_{TC}{\displaystyle \frac{\partial ^{2}C}{\partial y^{2}}}+QT\right),\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H(p,C)=L(C)-L(C_{10})+pL(C)+p\left(N_{CT}{\displaystyle \frac{\partial ^{2}T}{\partial y^{2}}}-{\delta}C^{m}\right),\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H(p,f)=L(f)-L(f_{10})+pL(f)+p{\displaystyle \frac{N_{t}}{N_{b}}}\left({\displaystyle \frac{\partial ^{2}T}{\partial y^{2}}}\right).\end{eqnarray}$ (20) With $L∼=∼\frac{\partial ^{2}}{\partial y^{2}}$ as the linear operator. The initial guess u ₁₀, T ₁₀, C ₁₀ and f ₁₀ can be written as $\begin{eqnarray}\displaystyle u_{10}=-1,\quad T_{10}=C_{10}=f_{10}=\frac{y}{h}. & & \displaystyle\end{eqnarray}$ (21) Now, it is assumed that: $\begin{eqnarray}\displaystyle (u,T,C,f)=(u_{0},T_{0},C_{0},f_{0})+p(u_{1},T_{1},C_{1},f_{1})+p^{2}(C_{2},T_{2},C_{2},f_{2})+\cdots ∼. & & \displaystyle\end{eqnarray}$ (22) The solutions of axial velocity, temperature, solutal concentration and nanoparticle volume friction are: $\begin{eqnarray}\displaystyle u(y)\text{} & =\text{} & \displaystyle -1+a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}+a_{6}y^{4+m},\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle T(y)\text{} & =\text{} & \displaystyle a_{7}+a_{8}y+a_{9}y^{2}+a_{10}y^{3}+a_{11}y^{4}+a_{12}y^{5}+a_{13}y^{2+m},\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle C(y)\text{} & =\text{} & \displaystyle a_{14}+a_{15}y+a_{16}y^{2}+a_{17}y^{3}+a_{18}y^{m}+a_{19}y^{m+1}+a_{20}y^{m+2}+a_{21}y^{2m+2},\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle f(y)\text{} & =\text{} & \displaystyle a_{22}+a_{23}y+a_{24}y^{2}+a_{25}y^{3}.\end{eqnarray}$ (26) The instantaneous volume flow rate, in the fixed frame, is given by $\begin{eqnarray}\displaystyle Q^{\ast }=\int _{0}^{h}u∼dy, & & \displaystyle\end{eqnarray}$ (27) where the dimensionless flux ( $Q∼=∼\frac{Q^{\ast }}{a^{2}c}$ ) is given by $\begin{eqnarray}\displaystyle Q=\int _{0}^{h}u∼dy. & & \displaystyle\end{eqnarray}$ (28)

Now, the skin friction coefficient τ_xy, the heat transfer coefficient (Nusselt number) Nu and the mass transfer coefficient (Sherwood number) Sh at the wavy wall of the outer tube, are defined, respectively, by $\begin{eqnarray}\displaystyle {\tau}_{xy}=\left(\frac{1}{1+{\lambda}_{1}}\right)\frac{\partial u}{\partial y},\quad Nu=\left.\frac{\partial T}{\partial y}\right|_{y=h},\quad Sh=\left.\frac{\partial C}{\partial y}\right|_{y=h}. & & \displaystyle\end{eqnarray}$ (29) The expressions for τ_xy, Nu and Sh have been obtained by substituting from Eqs. (23)–(25) into Eq. (29) respectively, and they have been evaluated numerically for several values of the parameters of the problem, using the software Mathematica package. The obtained results will be discussed in the next section.

Fig. 2.

Graph of three-dimensional axial velocity.

Fig. 3.

Graph of u (y) for different values of 𝛽₂.

Fig. 4.

Graph of u (y) for different values of G _f.

Fig. 5.

Graph of u (y) for different values of 𝜆₁.

Fig. 6.

Graph of u (y) for different values of 𝛽₁.

Fig. 7.

Graph of u (y) for different values of Da.

5. Results and discussion

In our study, we obtained the results of this problem with long wavelength and small Reynolds number, while the wave number is neglected, i.e., we assumed that the parameter δ is very small and more less than unity; moreover, the default values of pertinent parameters are taken as $\frac{\partial p}{\partial x}∼=∼10$ , 𝜆1 = 0.5, M = 1.5, Da = 0.5, G _rt = 1, G _rc = 1, G _rf = 1, R = 0.2, Ec = 3.5, Pr = 1.5, Nt = 1.5, Nb = 0.5, N _TC = 0.1, q1 = 3, N _CT = 0.1, δ = 4.5, 𝜖 = 0.2, m = 1, z = 0.9; 𝛽₂ = 0.05; 𝛽₂ = 0.05.

Fig. 8.

Graph of T (y) for different values of Nt.

Fig. 9.

Graph of T (y) for different values of δ.

Fig. 10.

Graph of T (y) for different values of Nb.

Fig. 11.

Graph of T (y) for different values of R.

Fig. 12.

Graph of T (y) for different values of Q.

Fig. 13.

Graph of entropy generation Eg for different values of Ec.

Our aim of this work is to analyze the emerging parameters effects such as slip parameters 𝛽₁ and 𝛽₂, Darcy number Da, nanoparticles Grashoff number G _rf, the radiation parameter R, Eckert number Ec, Brownian motion parameter Nb, the thermophoresis parameter Nt, Prandtl number Pr and chemical reaction parameter δ on the axial velocity u, temperature T, solute concentration C, the nanoparticles volume friction f.

Fig. 14.

Graph of entropy generation Eg for different values of Da.

Fig. 15.

Graph of entropy generation Eg for different values of N _TC.

Fig. 16.

Graph of entropy generation Eg for different values of R.

Fig. 17.

Graph of entropy generation Eg for different values of N _CT.

In Fig. 2, the effects of transverse coordinate y and axial coordinate x on the axial velocity u are illustrated in a three-dimensional figure. We observed from this figure that the axial velocity u decreases with increasing x, while it increases as y increases till a maximum value, and then it decreases till another minimum value.

Fig. 18.

Graph of C (y) for different values of m.

Fig. 19.

Graph of C (y) for different values of Nb.

Fig. 20.

Graph of C (y) for different values of R.

Fig. 21.

Graph of C (y) for different values of Nt.

Fig. 22.

Graph of f (y) for different values of R.

Fig. 23.

Graph of f (y) for different values of Pr.

Fig. 24.

Graph of f (y) for different values of Nb.

Fig. 25.

Graph of f (y) for different values of Nt.

Table 1

Values of τ_xy for various values of 𝜆₁, R, M, and Gr _T.

𝜆₁	M	Gr _T	τ_xy the present work	τ_xy Eldabe et al. [43 ]
0.1	1.5	1	0.46083	0.2901
0.3	1.5	1	0.42738	0.2198
0.5	1.5	1	0.38621	0.2059
0.5	2.5	1	1.87021	0.9108
0.5	3.5	1	4.29821	2.1506
0.5	3.5	2	2.81242
0.5	3.5	3	1.32663

Table 2

Values of Nu and Sh for various values of Q, R, and δ.

Q	R	δ	Nu	Sh
1	0.2	4.5	−0.00149	3.1045
2	0.2	4.5	−0.20783	3.1361
3	0.2	4.5	−0.68431	3.1677
3	0.5	4.5	−0.55896	3.1094
3	0.8	4.5	−0.40756	3.0737
3	0.8	5.0	−0.43477	3.3421
3	0.8	5.5	−0.46199	3.6177

Figures 3 and 4 give the effects of some parameters such as second order slip parameter 𝛽₂ and nanoparticles Grashoff number Gf on the axial velocity u, respectively, and the effects of the other parameters are found to be similar to them; these figures are excluded here to avoid any kind of repetition. Therefore, in these figures the Eq. (24) is evaluated by setting z = 0.9 and the axial velocity is plotted versus the coordinate y. It is seen from these figures that the axial velocity increases with the increasing of 𝛽₂, whereas it decreases as Gf increases. It is also noted that the axial velocity for different values of 𝛽₂ and Gf becomes greater near the axial of the tube and has a maximum value after which it decreases. Moreover, the obtained curves don’t intersect near the boundary of tube, this is due to the boundary conditions given in (15). The effects of Gt and Gc on the axial velocity are found to be similar to the effect of 𝛽₂ given in Fig. 3, and there are only two differences, the first one is that the obtained curves are very close to those obtained in Fig. 3, and that the second one is that near the boundaries of tube, the effect of Gt and Gc on axial velocity is bounded. Figures 5 and 6 shows the variation of the axial velocity u with y for various values of ratio of relaxation to retardation times 𝜆₁ and first order slip parameter 𝛽₁. From these figures, we observed that the effects of 𝜆₁ and 𝛽₁ on u are found to be similar to the effect of 𝛽₂ on u which is illustrated in Fig. 3. the variation of u with r for different values of Darcy number Da is drawn in Fig. 7, we obtained a figure in which the behavior of the curves are the same as that obtained in Fig. 4. Moreover, the result in Figs 5 and 7 are in agreement with [43].

The effects of the thermophoresis parameter Nt and chemical reaction parameter δ on the temperature T which is a function of the transverse coordinate y are shown in Figs 8 and 9, respectively. It is shown that the temperature T increases as Nt increases, whereas it decreases by increasing δ in the range of y shown in the figure, but near the boundary of tube, namely, 1.1 < r < 1.2, it has an opposite behavior, i.e. in this interval, T increases as Nt decreases and δ increases. Moreover, the temperature increases with y, till a maximum value (at a finite value of y: y = y ₀) after which it decreases. It is clear that the maximum of T increases by increasing Nt and decreasing δ. The result in Fig. 9 agrees with the fact that the reaction rate is the rapidity at which a chemical reaction occurs. It is inversely proportional to the increase in the temperature of a product and to the decrease in the concentration of a reactant per unit time. Moreover, the result agrees with those obtained by [29]. The effect of other parameters is found to be similar to the effect of both Nt and δ on T, but the figures will not be given there to save space. Figure 10 shows the variation of the axial velocity T with y for various values of Brownian motion parameter Nb. From this figure, we observed that the effect of Nb on T is found to be similar to the effect of Nt on T which is illustrated in Fig. 8, with the only difference that the obtained curves are very close to each other, namely near the wall of channel, than those obtained in Fig. 8. Figures 11 and 12 illustrate the change of the temperature T versus the radial coordinate r with several values of Q and R, respectively. It is seen from these figures that the temperature increases with the increasing of Q, whereas it decreases as R increases.

Figures 13 and 14 give the effects of Eckert number Ec, and Darcy number Da on entropy generation, respectively. Therefore, in these figures the Eq. (24) is evaluated by setting x = 0.8 and entropy generation is plotted versus the transverse coordinate y It is seen from these figures that entropy generation increases with the increasing of Ec, whereas it decreases as Da increases. It is also noted that the entropy generation for different values of Ec and Da becomes greater with increasing y and reaches a maximum value (at a finite value of y: y = y ₀) after which it decreases. The following explains the viscous dissipation effect on heat transfer, namely, the result in Fig. 13. It is known that the influence of dissipation generates heat due to traction between the fluid particles, this supplementary heat is a reason to increase of the initial entropying of fluid. This increase of entropy generation causes an additional increment of the force of buoyant. As the buoyant force increases, the fluid velocity increases. So, the bigger traction between the particles of fluid and consequently bigger viscous heating of the fluid. Moreover, the result in Fig. 13 is in agreement with [17]. In Fig. 15, entropy generation distribution Eg is plotted against the transverse coordinate y for various values of N _TC and it is observed that the effect of N _TC on Eg is found to be similar to the effect of Ec given in Fig. 13. The effects of N _CT and R on entropy generation are displayed in Figs. 16 and 17 and are found to be similar to the effect of Da given in Fig. 14, and there are only two differences, the first one is that the obtained curves are very close to those obtained in Fig. 14, and that the second one is that near the channel axis, the effects of N _CT and R on entropy generation are bounded.

Figures 18 and 19 obtain the influence of order of chemical reaction m and Brownian motion parameter Nb on the solutal concentration C, respectively. It is noted that the solutal concentration increases by increasing m, whereas it decreases by increasing values of Nb. Also, it is clear that the solutal concentration is always positive and decreases by increasing y till a minimum value of y, after which it increases. Other parameters affect the solutal concentration, but they are found to be similar to the previous parameters. So, figures are excluded to avoid any kind of repetition. The following clarifies the result in Fig. 19. When the particles suspended in a fluid move in a randomly way, the flow is called Brownian motion. This random motion may decrease or increase the solutal concentration C. The variations of with the transverse coordinate y for various values of the radiation parameter R and the thermophoresis parameter Nt respectively, are displayed in Figs 20 and 21. The graphical results of Figs 20 and 21 indicate that the effect of R and Nt on C are similar to the effects of m and Nb on C, respectively, given in Figs 18 and 19.

Equation (27) evaluates how the nanoparticles volume friction f varies with the transverse coordinate y. The effect of radiation parameter R and Prandtl number Pr on the nanoparticles volume friction f is shown in Figs 22 and 23, respectively. It is found that the nanoparticles volume friction f increases by increasing R, but it decreases by increasing Pr. Furthermore, for small values of R and large values of Pr, it decreases with y till a minimum value of y, after which it increases. Physically, the result in Fig. 22 is due to the following: The radiation effect is a result of heat exchange between enclosure surface and human body, such as building ceiling and wall. It may lead to natural phenomenon such as houses feeling cooler in the winter; this will lead to increase the nanoparticles volume friction. Figures 24 and 25 illustrate the change of the nanoparticles volume friction f versus the transverse coordinate y with several values of Nb and Nt, respectively. It is seen, from these Figs 24 and 25, that the effect of Nb and Nt on f are similar to the effects of R and Pr on f, respectively, given in Figs 22 and 23.

Tables 1 presents numerical results for the skin friction coefficient τ_xy for various values of ratio of relaxation to retardation times 𝜆₁, magnetic parameter M and thermal Grashoff number Gr _T. It is clear from Table 1 that an increase in both 𝜆₁ and Gr _T decreases the values of quantity τ_xy. Furthermore, an increase in the magnetic parameter M gives an opposite behavior to both 𝜆₁ and Gr _T. The numerical results of Nusselt number Nu and Sherwood number Sh for various values of the heat source parameter Q, raddiatio parameter R chemical reaction parameter δ is illustrated in Table 2. It is clear from Table 2 that an increase in Q and δ gives an increase in the values of quantity Sh but decreases the dimensionless quantity Nu. Also, an increase in R gives an opposite behavior to both Q and δ. Moreover, the result in Tables 1 and 2 are in agreement with those obtained by [44,45].

6. Conclusion

In this paper, entropy generation effect on the peristaltic flow of MHD Jeffery nanofluid through a porous media has been analyzed. In our analysis, we are taking into account the effects of radiation, viscous dissipation and chemical reaction. The analytical expressions are constructed for the axial velocity, temperature, nanoparticles volume friction and solute concentration distributions. Also, the skin friction, Nussselt number and Sherwood number at the wavy wall of the channel are obtained. Many physiological flows can be presented by this model. The major findings can be briefed as follows:

The axial velocity u decreases with the increase each of R, Gr _f and Da, whereas it increases as 𝛽₁, 𝛽₂, Nt, Q, $\frac{\partial p}{\partial x}$ and M increase.

The temperature T increases with the increase each of Nb, Nt, Pr and Q whereas it decreases as δ increases. But, near the boundary of channel, the temperature T has an opposite behavior compared to the other range.

The temperature T for different values of Nb, Nt, Pr, Q and δ becomes greater with increasing the transverse coordinate y and reaches maximum value (at a finite value of y: y = y ₀) after which it decreases.

The solutal concentration C has an opposite behavior compared to temperature behavior except that it has the same behavior at the channel boundary.

The nanoparticles volume friction f increases with the increase each of R and Nb whereas it decreases as Pr, Q and Nt increase. The graphs of nanoparticles volume friction distribution f depict that the curve of f has a minimum point.

By increasing each of Da, R and G _f, the entropy generation Eg decreases, while it increases as Nb, 𝛽₁, 𝛽₂, Nt, Q and Ec increase. Furthermore, the obtained curves don’t intersect at the channel boundary, this occurs due to the boundary conditions (15).

7. Applications

The applications of nanofluids are predicted in the nuclear engineering, cooling and mechanical devices, fusion processes, heating devices, extrusion systems, medical applications and many more. The contribution of nanofluids in the energy production becomes more fundamental applications in recent years. The improvement in the mass and heat transportation is predicted when nanoparticles are properly immersed in the base fluid. Some major involvement of nanofluids pronounced in the renewable energy, engineering disciplines, bio-technology, drug delivery etc.

Footnotes

Acknowledgements

The authors thank the reviewers for their valuable remarks which improved and enriched our manuscript.

Nomenclature

Greek symbols

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