Effect of the magnetic induction on the peristaltic Eyring–Prandtl nanofluid through porous medium

Abstract

The manuscript examines the motion of an Eyring–Prandtl nanofluid inside a horizontal asymmetric peristaltic nonuniform channel. A normal variable magnetic field affects the channel. The slip boundary-conditions are taken into consideration. The boundary-value problem involves the energy as well as the concentration equations. The viscous dissipation of the thermal equation is considered. The flow is immersed in a porous medium. The governing equations of motion are solved by means of the Homotopy perturbation method (HPM). The distributions of the various functions are analytically obtained. The influences of the different parameters are authenticated throughout a set of diagrams. It is found that thermal-diffusion (Soret), diffusion-thermo (Dufour), Lorentz resisting force and induced magnetic have active effects on the distributions of the nano-particle; such as, concentration volume, temperature, and flow characteristics.

Keywords

Eyring–Prandtl model peristaltic transport induced magnetic field porous media heat and mass transfer

1. Introduction

The peristaltic motion supports shape to the moving fluid through the change of its flexibility tube/channel. It has extensive applications in biological systems and engineering. Some of these implementations are; for instance, the movement of lymphatic fluid in lymphatic vessels, urine transport from kidney to bladder via the ureters, bile juice flow from the gallbladder into the duodenum, all these scientists are employed in physiology [1]. Peristalsis has great applications in engineering, including the liquid transport, sewage, and noxious fluid, etc…. Latham [2] was, probably, the first who studied the mechanism of the peristaltic pumping in his Master Thesis. Several researches have been analyzed the phenomenon of peristaltic transport various assumptions. Shapiro et al. [3] used a wavy frame of reference to study the peristaltic motion and contributed to facilitating the study of many researches on the discussion of various physical effects. They debated the peristaltic stream, theoretically and practically, throughout pushing liquid in light of low Reynolds number and long wavelength approximations. They examined the motion of a liquid on a plane with an axi-symmetric stream. Moreover, they explained the reflux and trapping phenomena. Eldabe et al. [4] studied the effects of the induced magnetic field on the peristaltic motion of the Walter’s B fluid with heat transfer throughout a porous medium in symmetric vertical channel in the presence of heat generation. Hayat et al. [5] discussed the influence of variable viscosity on the peristaltic transport of tangent hyperbolic fluid with heat and mass transfer. The viscous dissipation and Joule heating were taken with their account. They observed that the larger values of Soret and Dufour numbers raise the temperature and heat transfer rate. The same result is obtained in our investigation despite the difference of non-Newtonian fluid.

The nonlinear relationship between the shear stress and shear rate may be described by the concept of non-Newtonian fluids. It is currently establishing the fact that the majority of the fluids happening in physiology and in an industry of the non-Newtonian type. Some examples of non-Newtonian fluids Blood are: bile juice, chyme, cosmetic products, mud at low shear rate, etc…The study of the non-Newtonian fluid has been the focus of unlimited research in the past few years. Akbar [6] presented the peristaltic movement of an Eyring–Prandtl fluid in a non-symmetric channel with convected boundary conditions. He found that the pressure rise rises by an upturn in Eyring–Prandtl fluid parameter and Hartman number. Their results are in good agreement with our observed results. Abbasi et al. [7] examined the simultaneous effects of heat and mass transfer on the peristaltic transport of Eyring–Prandtl fluid in an inclined asymmetric channel. They discussed the effects of the inclined magnetic field and Joule heating. They found that the velocity and temperature decrease, whereas the concentration increases by the increasing the non-Newtonian behavior. Their results are in good agreement with our observed results. Iftikhar and Rehman [8] studied the peristaltic flow of an Eyring–Prandtl fluid in a diverging tube. In their work, they observed that the rate of heat transfer and mass transfer decreases, with the increasing of Grashof number. Recently, Eldabe et al. [9] investigated the influence of inconstant electrical conductivity and chemical reaction on the peristaltic motion of non-Newtonian Eyring–Prandtl fluid inside a tapered asymmetric channel. Their governing equations of motion were analytically solved throughout a multi-step differential transform method. They found that the chemical reaction parameter causes a decrease in pressure rise.

The nanotechnology is a helpful topic in science and engineering. It is helpful in numerous applications in medicine [10]; for instance, malignancy treatment, heat transmits including microelectronics, fuel cells, heat exchange and domestic refrigerator. Basically, the nano - fluid is a fluid that is combined by dispersing the nano-particle in the base fluid. Heat transfer fluids, by hanging metallic nano-particles of higher thermic conductivity in the traditional heat transfer fluids, are discussed by Choi and Eastman [11]. Lee et al. [12] explained experimentally the nano-fluids, containing a little measure of nano-particles. They have higher conductivity of the heat than the comparative liquids without nano-particles. Eldabe et al. [13] clarified the peristaltic movement of a non-Newtonian nano-fluid with heat transmits in a permeable medium inside a vertical channel by using a Rung–Kutta–Merson mode and the Newton iteration in a shooting and matching mechanism. Hayat et al. [14] investigated the mixed convective peristaltic flow of the Prandtl fluid under the effects of homogeneous-heterogeneous reactions with Joule heating and thermal radiation in an inclined channel. Throughout their work, they observed that the concentration decreases during both the homogeneous and heterogeneous reactions, where the thermal radiation lessens the fluid temperature and rate of heat transfer. Hina [15] addressed the impacts of heat/mass transfer, wall properties and slip conditions on the peristaltic carriage of the Eyring–Powell model. She found that the temperature is expanded and heat transmits coefficient enlarged when the viscous dissipation impact is intensified. Mehmood et al. [16] contemplated the slip impacts of peristaltic transport in an asymmetric channel within the sight of heat/mass transit and a magnetic field influences.

As of late, the MHD peristaltic flows have taken a lot of trustworthiness because of their applications. The impacts of MHD on the peristaltic stream of the Newtonian and non-Newtonian liquids for various geometries have been examined by many researchers. The influence of the magnetic field is a main topic in medical sciences such as; blood pump machine [17], vascular diseases, cancer tumor treatment [18], hyperthermia, and blood reduction during surgeries. Mekheimer [19] studied the effect of the induced magnetic field on the peristaltic flow of a couple stress fluid. He had been discussing the pressure rise, frictional force per wavelength, the axial induced magnetic field and the distribution of the current density across the channel. Akram and Nadeem [20] deliberated the effects of induced MHD on the peristaltic flow of a Jeffrey fluid in an asymmetric channel in the presence of heat transfer analysis. They discussed the trapping phenomena for different wave shapes. They observed that the pressure rise for sinusoidal wave is less than trapezoidal wave and greater than triangular in a Jeffrey fluid. Hayat et al. [21] investigated the impact of both of the slips and the induced magnetic field on the peristalsis in an asymmetric channel with heat and mass transfer. In their work, they observed that the induced magnetic field in the half regions, in one direction while in the other half it, is in the opposite direction. Moreover, they observed that the magnitude of an induced magnetic field increases when increasing the magnetic Reynolds. These observations are congruent with those observed in our investigation. Sadaf et al. [22] studied the effects of the induced magnetic on the peristaltic flow of Williamson nano-fluid in an annulus. They found that the induced magnetic field and current density enhances by increasing the value of a magnetic Reynolds number. It was also found that the temperature profile upturns with the upturn in the Brownian motion and thermophoresis parameter. Mekheimer et al. [23] showed the slip influence by a sinusoidal peristaltic wavy wall with the Hall current existence on the induced MHD flow in a porous medium. They suggested some results, which service in the mechanical engineering application as the fluid transportation without an external pressure. Mustafa et al. [24] illustrated the effects of an induced magnetic field on the mixed convection peristaltic movement of the nano-fluid in a vertical channel. They found that the peristaltic pumping rate is increased upon the increase of the strengths of the electromagnetic field and the buoyancy force due to the temperature gradient. The contour maps for temperature, vorticity, and streamlines are graphed to obtain a good understanding of the flow behavior.

Typically, several numerical techniques were used to examine enormous domain of linear as well as nonlinear phenomena in mathematical, engineering, and physical problems. On the other hand, many attempts had been performed to treat these phenomena, analytically, ranging from the classical perturbation methods until to the multiple time scale technique. The straightforwardness of the perturbation methods, in it converts the nonlinear equations into linear ones. Indeed, all these perturbation methods depend mainly on a small parameter. Subsequently, the existence of such a parameter is essential to obtain an approximate solution of the given problem. Typically, the solution of a problem may be restricted by the permanence of this parameter. The Homotopy perturbation method (HPM) is one of the momentous techniques, first proposed by Therefore, He [25,26] in 1999 is considered as the first author, who overcomes this difficulty, by introducing the HPM. The main advantage of this method comes from it does not depend on the small parameter in the problem. Furthermore, The approximated solutions are uniformly valid, not only for small parameter, but also at large one. Throughout, this promising method, an artificial embedded parameter 𝜌 ∈ [0,1] is inserted into the problem, to divide it into; two linear and nonlinear parts. It is sometimes called the Homotopy parameter. This technique had been successfully used for solving linear and nonlinear partial differential equations. Throughout this work, He made a modified one of the standard Homotopy and the perturbation techniques by developing the HPM for solving linear, nonlinear, initial, and boundary value problems. This method has an important advantage in that it delivers an analytical approximate solution to the extensive variety of linear and nonlinear problems in the applied sciences. He’s homotopy perturbation technique is demonstrated as an appropriate with the multipurpose nature of the physical problems. It has been in a wide class of functional equations and the references therein.

The present study focuses to incorporate the idea of the induced magnetic field on the peristaltic motion of Eyring–Prandtl nano-fluid. Because of the great importance of the porous media, the current study is performed in a porous medium. The effects of heat generation, nano-particle, and chemical reactions are considered. The problem is modulated through a set of nonlinear partial differential equations. The implications of the slip boundary conditions are converted to a non-dimensional form. By making use of the HPM, the boundary-value problem is analytically solved. The approximations of the low Reynolds number and long wavelength are considered. The influences of the physical factors on the obtained solutions are examined numerically and illustrated graphically through a set of figures. It is found that these factors play a dramatic role in controlling these arrangements. To be precise, the plan of current work is carried out as follows: Section 2 is devoted to formulating the problem together with the non-dimensional analysis. The solution of the boundary-value problem is obtained by the HPM throughout Section 3. Section 4 and its subsections are depicted to discuss, numerically, the outcomes. Finally, the concluding remarks are repeated in Section 5.

Fig. 1.

Geometry of problem.

2. Formulation of the problem and analysis

A peristaltic motion of an incompressible nano-fluid obeying the Eyring–Prandtl model throughout porous media, inside the asymmetric channel with flexible walls is considered. For more convenience, the Cartesian coordinates are considered as the model is sketched in Fig. 1. The system is influenced by a uniform transverse magnetic field of strength H ₀, together with the induced magnetic field is considered, where the total magnetic is defined as follows: $\begin{eqnarray}H=(h_{X}(X,Y,t),h_{Y}(X,Y,t)+H_{0},0).\end{eqnarray}$ The equations of the wave propagations along the channel walls may be written as: $\begin{eqnarray}\displaystyle L_{1}(X,t)=d_{1}+a_{1}∼\cos \left(\frac{2{\pi}}{{\lambda}}(X-c∼t)\right),\quad \text{at the upper wall}, & & \displaystyle\end{eqnarray}$ (1) and $\begin{eqnarray}\displaystyle L_{2}(X,t)=-d_{2}-a_{2}∼\cos \left(\frac{2{\pi}}{{\lambda}}(X-c∼t)+{\Omega}\right),\quad \text{at the lower wall}, & & \displaystyle\end{eqnarray}$ (2) where, a ₁ and b ₁ are the wave amplitudes, 𝜆 is the wavelength, d ₁ + d ₂ is the channel width, c is the velocity of propagation, t is the time.

The phase difference 𝛺 varies in the range 0 ≤ 𝛺 ≤ π. The extreme values of 𝛺 may be described as: at 𝛺 = 0, the peristaltic is considered as a symmetric channel. On the other hand, for 𝛺 = π, the waves are addressed in phase. The various parameters in Eqs (1) and (2) are restricted by the following inequality: $\begin{eqnarray}\displaystyle a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}\cos {\Omega}\leq (d_{1}+d_{2})^{2}. & & \displaystyle\end{eqnarray}$ (3) Let T ₀, T ₁, and H ₀ are the temperature and magnetic field at the channel walls, respectively.

The Cauchy stress tensor S of a non-Newtonian Eyring–Prandtl fluid can be expressed as: $\begin{eqnarray} \displaystyle \text{}\underline{S}=((A^{\ast }∼{\eta}_{0}∼arc\sinh |{\nabla}\text{}\underline{q}|/B^{\ast }))/|{\nabla}\text{}\underline{q}|{\nabla}\text{}\underline{q}, & & \displaystyle \end{eqnarray}$ (4) where A ^∗ and B ^∗ are the material parameters of the fluid model, 𝜂₀ is the coefficient of viscosity, and (U (X, Y, T), V (X, Y, T),0) is the velocity vector where U and V are the velocity components in a horizontal X-axis and vertical Y -axis, respectively.

For an incompressible MHD Eyring–Prandtl model, the governing equations of motion may be listed as given in Ref. [21]:

The incompressibility condition, yields $\begin{eqnarray}\displaystyle \frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0. & & \displaystyle\end{eqnarray}$ (5)

The components of the conservation of momentum, results $\begin{eqnarray}\displaystyle {\rho}_{f}\frac{dU}{dt}=-\frac{\partial P}{\partial X}+\frac{\partial S_{XX}}{\partial X}+\frac{\partial S_{XY}}{\partial Y}-\frac{{\eta}_{0}}{k_{0}}U+{\mu}_{e}\left(h_{X}\frac{\partial }{\partial X}+(H_{0}+h_{Y})\frac{\partial }{\partial Y}\right)h_{X}, & & \displaystyle\end{eqnarray}$ (6) and $\begin{eqnarray}\displaystyle {\rho}_{f}\frac{dV}{dt}=-\frac{\partial P}{\partial Y}+\frac{\partial S_{YX}}{\partial X}+\frac{\partial S_{YY}}{\partial Y}-\frac{{\eta}_{0}}{k_{0}}V+{\mu}_{e}\left(h_{X}\frac{\partial }{\partial X}+(H_{0}+h_{Y})\frac{\partial }{\partial Y}\right)h_{Y}. & & \displaystyle\end{eqnarray}$ (7)

The energy equation may be written as $\begin{eqnarray}\displaystyle ({\rho}c)_{f}\frac{dT}{dt} & = & \displaystyle \bar{k}\left(\frac{\partial ^{2}T}{\partial X^{2}}+\frac{\partial ^{2}T}{\partial Y^{2}}\right)+\frac{D_{B}k_{T}}{C_{s}}\left(\frac{\partial ^{2}C}{\partial X^{2}}+\frac{\partial ^{2}C}{\partial Y^{2}}\right)+Q_{0}(T-T_{0})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{{\sigma}}\left(\frac{\partial h_{Y}}{\partial X}-\frac{\partial h_{X}}{\partial Y}\right)^{2}+S_{XX}\frac{\partial U}{\partial X}+S_{XY}\frac{\partial U}{\partial Y}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼S_{YX}\frac{\partial V}{\partial X}+S_{YY}\frac{\partial V}{\partial Y}+\frac{{\eta}_{0}}{k_{0}}(U+V)\sqrt{U^{2}+V^{2}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼({\rho}c)_{p}\left[D_{B}\left(\frac{\partial C}{\partial X}\frac{\partial T}{\partial X}+\frac{\partial C}{\partial Y}\frac{\partial T}{\partial Y}\right)+\frac{D_{T}}{T_{m}}\left(\left(\frac{\partial T}{\partial X}\right)^{2}+\left(\frac{\partial T}{\partial Y}\right)^{2}\right)\right].\end{eqnarray}$ (8)

The concentration equation may be written as $\begin{eqnarray}\displaystyle \frac{dC}{dt}=D_{B}\left(\frac{\partial ^{2}C}{\partial X^{2}}+\frac{\partial ^{2}C}{\partial Y^{2}}\right)+\frac{D_{m}k_{T}}{T_{m}}\left(\frac{\partial ^{2}T}{\partial X^{2}}+\frac{\partial ^{2}T}{\partial Y^{2}}\right)-k_{1}(C-C_{0}). & & \displaystyle\end{eqnarray}$ (9)

The generalized Ohm’s law defines the total current flow as follows: $\begin{eqnarray}\displaystyle \text{}\underline{J}_{Z}={\sigma}(\text{}\underline{E}+{\mu}_{e}(\text{}\underline{q}\times \text{}\underline{H})),\quad {\nabla}\times \text{}\underline{H}=\text{}\underline{J}_{Z} & & \displaystyle\end{eqnarray}$ (10) where, 𝜌_f, 𝜌_p are the densities of the fluid and the nano-particle, respectively, k ₀ is the permeability of the porous medium, ( B = μ_e H ) represents the magnetic flux density, μ_e is the magnetic permeability, H is the magnetic field, d∕dt represents the material derivative, J _Z is the current density, σ is the electric conductivity, 𝜈_H = 1∕σ μ_e is the magnetic diffusivity, E is the electric field, c _f, c _p are the volumetric volume expansion of the fluid and the nano-particle, respectively, D _B is the Brownian diffusion coefficient, D _T is the thermophoretic diffusion coefficient, T _m is the fluid mean temperature coefficient, C _s is the concentration susceptibility $\bar{k}$ is the thermal conductivity, k _T is the thermal-diffusion ratio, Q ₀ is the heat generation parameter, k ₁ is the chemical reaction of rate constant, T is the temperature, and C is the concentration.

The Maxwell’s equations [19] may be written as $\begin{eqnarray}\displaystyle \frac{\partial \text{}\underline{H}}{\partial t}-{\nabla}\times (\text{}\underline{q}\times \text{}\underline{H})=-{\nabla}\times ({\nu}_{H}{\nabla}\times \text{}\underline{H}), & & \displaystyle\end{eqnarray}$ (11) and $\begin{eqnarray}\displaystyle {\nabla}\times \text{}\underline{E}=-{\mu}_{e}\frac{\partial \text{}\underline{H}}{\partial t},{\nabla}\cdot \text{}\underline{H}=0. & & \displaystyle\end{eqnarray}$ (12) A uniform magnetic field of strength H ₀ is applied in the transverse direction. Additionally, an induced magnetic field H (h _x(X, Y), h _y(X, Y),0) and a total magnetic field H (h _X(X, Y), H ₀ + h _Y(X, Y),0) are taken into consideration.

On presenting a wavy frame (x, y) moving with velocity c away from the fixed frame (X, Y) the transformations are: $\begin{eqnarray}x=X-ct,∼y=Y,∼u=U-c,∼v=V,∼P(x,y)=P(X,Y,t),∼T(x,y)=T(X,Y,t),\end{eqnarray}$ and $\begin{eqnarray}\displaystyle C(x,y)=C(X,Y,t), & & \displaystyle\end{eqnarray}$ (13) where, (u, v) are the velocity components in the moving frame.

The governing equations in the two-dimensional peristaltic movement of an incompressible Eyring–Prandtl nano-fluid through a permeable medium in the wave frame are given as follows: $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}{\rho}_{f}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\frac{\partial S_{xx}}{\partial x}+\frac{\partial S_{xy}}{\partial y}-\frac{{\eta}_{0}}{k_{0}}(u+c)+{\mu}_{e}\left(h_{X}\frac{\partial }{\partial x}+(H_{0}+h_{y})\frac{\partial }{\partial y}\right)h_{X},\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}{\rho}_{f}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\frac{\partial S_{yx}}{\partial x}+\frac{\partial S_{yy}}{\partial y}-\frac{{\eta}_{0}}{k_{0}}v+{\mu}_{e}\left(h_{X}\frac{\partial }{\partial x}+(H_{0}+h_{y})\frac{\partial }{\partial y}\right)h_{y},\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle ({\rho}c)_{f}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right) & = & \displaystyle k\left(\frac{\partial ^{2}T}{\partial x^{2}}+\frac{\partial ^{2}T}{\partial y^{2}}\right)+\frac{D_{B}k_{T}}{C_{s}}\left(\frac{\partial ^{2}C}{\partial x^{2}}+\frac{\partial ^{2}C}{\partial y^{2}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼Q_{0}(T-T_{0})+\frac{1}{{\sigma}}\left(\frac{\partial h_{y}}{\partial x}-\frac{\partial h_{x}}{\partial y}\right)^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼S_{xx}\frac{\partial u}{\partial x}+S_{xy}\frac{\partial u}{\partial y}+S_{yx}\frac{\partial v}{\partial x}+S_{yy}\frac{\partial v}{\partial y}+\frac{{\eta}_{0}}{k_{0}}(u+c+v)\sqrt{(u+c)^{2}+v^{2}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼({\rho}c)_{p}\left[D_{B}\left(\frac{\partial C}{\partial x}\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}\right)+\frac{D_{T}}{T_{m}}\left(\left(\frac{\partial T}{\partial x}\right)^{2}+\left(\frac{\partial T}{\partial y}\right)^{2}\right)\right],\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_{B}\left(\frac{\partial ^{2}C}{\partial x^{2}}+\frac{\partial ^{2}C}{\partial y^{2}}\right)+\frac{D_{B}k_{T}}{T_{m}}\left(\frac{\partial ^{2}T}{\partial x^{2}}+\frac{\partial ^{2}T}{\partial y^{2}}\right)-k_{1}(C-C_{0}) & & \displaystyle\end{eqnarray}$ (18) and $\begin{eqnarray}\displaystyle E=\frac{1}{{\sigma}}\left(\frac{\partial h_{y}}{\partial x}-\frac{\partial h_{x}}{\partial y}\right)+{\mu}_{e}(v∼h_{x}-u(H_{0}+h_{y})). & & \displaystyle\end{eqnarray}$ (19) The slippy conditions for velocity, magnetic, temperature and nanofluid concentration are given as: $\begin{eqnarray}\displaystyle U\pm {\alpha}_{1}∼S_{xy}=0,\quad H=0,\quad T\pm {\alpha}_{2}\frac{\partial T}{\partial y}=\left[\begin{array}{@{}c@{}}T_{1}\\ T_{0}\end{array}\right],\quad C\pm {\alpha}_{3}\frac{\partial C}{\partial y}=\left[\begin{array}{@{}c@{}}C_{1}\\ C_{0}\end{array}\right],\quad \text{at }y=\left[\begin{array}{@{}c@{}}L_{1}\\ L_{2}\end{array}\right], & & \displaystyle\end{eqnarray}$ (20) where, 𝛼₁, 𝛼₂ and 𝛼₃ are the velocity, thermal, and concentration slip coefficients, respectively.

For more conenience, introducing the non-dimensional quantities as the following: $\begin{eqnarray}\displaystyle \hspace{-16.79993pt}\begin{array}{@{}l@{}}\overline{x}={\displaystyle \frac{x}{{\lambda}}},∼\overline{y}={\displaystyle \frac{y}{d}},∼\overline{u}={\displaystyle \frac{u}{c}},∼\overline{v}={\displaystyle \frac{v}{c}},∼l_{1}={\displaystyle \frac{L_{1}}{d_{1}}},l_{2}={\displaystyle \frac{L_{2}}{d_{1}}},∼\overline{t}={\displaystyle \frac{ct}{{\lambda}}},∼\overline{S}={\displaystyle \frac{d_{1}}{{\eta}_{0}c}}S,∼{\delta}={\displaystyle \frac{d_{1}}{{\lambda}}},∼a={\displaystyle \frac{a_{1}}{d_{1}}},∼d={\displaystyle \frac{d_{2}}{d_{1}}},\\[15.0pt] b={\displaystyle \frac{b_{1}}{d_{1}}},∼{\gamma}={\displaystyle \frac{{\rho}k_{1}d_{1}^{2}}{{\eta}_{0}}},∼\overline{h}_{x}={\displaystyle \frac{h_{x}}{H_{0}}},∼\overline{h}_{y}={\displaystyle \frac{h_{y}}{H_{0}}},∼\overline{k}={\displaystyle \frac{k∼c}{{\eta}_{0}d_{1}}},∼M={\displaystyle \frac{{\mu}_{e}H_{0}^{2}d_{1}}{{\eta}_{0}c}},∼D_{a}={\displaystyle \frac{k_{0}}{d_{1}^{2}}},∼\overline{{\psi}}={\displaystyle \frac{{\psi}}{cd_{1}}},\\[15.0pt] {\tau}^{\ast }={\displaystyle \frac{({\rho}c)_{p}}{({\rho}c)_{f}}},∼P_{r}={\displaystyle \frac{{\eta}_{0}c_{f}}{k}},∼E_{c}={\displaystyle \frac{c^{2}}{c_{p}(T_{1}-T_{0})}},∼{\beta}={\displaystyle \frac{Q_{0}d_{1}^{2}}{k}},∼{\theta}={\displaystyle \frac{\bar{T}-T_{0}}{T_{1}-T_{0}}},{\phi}={\displaystyle \frac{\bar{C}-C_{0}}{C_{1}-C_{0}}},∼R_{m}={\sigma}{\mu}_{e}d_{1}c,\\[15.0pt] S_{c}={\displaystyle \frac{{\eta}_{0}}{{\rho}_{f}∼D_{B}}},D_{u}={\displaystyle \frac{k_{T}D_{B}(C_{1}-C_{0})}{c_{f}∼c_{s}∼{\eta}_{0}(T_{1}-T_{0})}},∼S_{t}^{2}={\displaystyle \frac{{\mu}_{e}H_{0}^{2}}{{\rho}∼c^{2}}},∼\overline{P}={\displaystyle \frac{d_{1}^{2}P}{{\lambda}{\eta}_{0}c}},∼\overline{H}={\displaystyle \frac{H}{H_{0}d_{1}}},∼\bar{E}=-{\displaystyle \frac{E}{cH_{0}{\mu}_{e}}},\\[15.0pt] N_{b}={\displaystyle \frac{{\tau}^{\ast }(C_{1}-C_{0})D_{B}∼{\rho}_{f}}{{\eta}_{0}}},∼N_{t}={\displaystyle \frac{{\tau}^{\ast }(T_{1}-T_{0})D_{T}∼{\rho}_{f}}{{\eta}_{0}T_{m}}},∼S_{r}={\displaystyle \frac{{\rho}_{f}k_{T}D_{B}(T_{1}-T_{0})}{T_{m}∼{\eta}_{0}(C_{1}-C_{0})}}\text{ and }R_{e}={\displaystyle \frac{{\rho}_{f}c_{f}d_{1}}{{\eta}_{0}}},\end{array} & & \displaystyle\end{eqnarray}$ (21) where, δ is the wave number, R _e is the Reynolds number, R _m is the Reynolds magnetic number, M denotes the Hartmann number, E _c is the Eckert number, D _a is the Darcy number, D _u is Dufour number, N _t is the thermophoresis parameter, N _b is the Brownian motion parameter, S _t is the Strommer’s number, S _c is the Schmidt’s number, S _r is the Soret number, P _r is the Prandtl number, θ is the temperature, k ₀ is the permeability parameter, ${\hat{H}}$ is the magnetic force function, 𝜓 is the stream function, $\bar{k}$ is the thermal conductivity, c _p is the specific heat, σ is the electrical conductivity, 𝛽 is the non-dimensional heat generation parameter, τ^∗ is the ratio of heat capacity of the nanoparticles and the heat capacity of the fluid, and 𝛾 is the chemical reaction parameter. Since the Soret effect is appeared as a temperature gradient due to the diffusion flux and the Dufour effect is appeared as a chemical potential gradient due to a heat flux.

Typically, in an incompressible two-dimensional flow, one may use a stream function 𝜓 where, $u=\frac{\partial {\psi}}{\partial y}$ , $v=-{\delta}\frac{\partial {\psi}}{\partial x}$ and the induced magnetic field $h_{x}=\frac{\partial {\hat{H}}}{\partial y},h_{y}=-{\delta}\frac{\partial {\hat{H}}}{\partial x}$ .

It follows that Eqs (17)–(20) may be listed, after dropping the bars for simplicity, as: $\begin{eqnarray}\displaystyle R_{e}{\delta}\left(\frac{\partial {\psi}}{\partial y}\frac{\partial }{\partial x}-\frac{\partial {\psi}}{\partial x}\frac{\partial }{\partial y}\right)\frac{\partial {\psi}}{\partial y} & = & \displaystyle -\frac{\partial p}{\partial x}+{\delta}\frac{\partial S_{xx}}{\partial x}+\frac{\partial S_{xy}}{\partial y}-\frac{1}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}+1\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼S_{t}^{2}∼R_{e}\left(\frac{\partial }{\partial y}+{\delta}\left(\frac{\partial {\hat{H}}}{\partial y}\frac{\partial }{\partial x}-\frac{\partial {\hat{H}}}{\partial x}\frac{\partial }{\partial y}\right)\right)\frac{\partial {\hat{H}}}{\partial y},\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle -R_{e}∼{\delta}^{3}\left(\frac{\partial {\psi}}{\partial y}\frac{\partial }{\partial x}-\frac{\partial {\psi}}{\partial x}\frac{\partial }{\partial y}\right)\frac{\partial {\psi}}{\partial x} & = & \displaystyle -\frac{\partial p}{\partial y}+{\delta}^{2}\frac{\partial S_{yx}}{\partial x}+{\delta}\frac{\partial S_{yy}}{\partial y}+\frac{{\delta}^{2}}{D_{a}}\frac{\partial {\psi}}{\partial x}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼S_{t}^{2}∼R_{e}{\delta}\left(\frac{\partial }{\partial y}+{\delta}\left(\frac{\partial {\hat{H}}}{\partial y}\frac{\partial }{\partial x}-\frac{\partial {\hat{H}}}{\partial x}\frac{\partial }{\partial y}\right)\right)\frac{\partial {\hat{H}}}{\partial x},\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle R_{e}∼{\delta}\left(\frac{\partial {\psi}}{\partial y}\frac{\partial {\theta}}{\partial x}-\frac{\partial {\psi}}{\partial x}\frac{\partial {\theta}}{\partial y}\right) & = & \displaystyle \frac{1}{P_{r}}\left({\delta}^{2}\frac{\partial ^{2}{\theta}}{\partial x^{2}}+\frac{\partial ^{2}{\theta}}{\partial y^{2}}\right)+D_{u}\left({\delta}^{2}\frac{\partial ^{2}{\phi}}{\partial x^{2}}+\frac{\partial ^{2}{\phi}}{\partial y^{2}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{{\beta}}{P_{r}}{\theta}+\frac{R_{e}E_{c}S_{t}^{2}}{R_{m}}\left(-{\delta}\frac{\partial ^{2}{\hat{H}}}{\partial x^{2}}-\frac{\partial ^{2}{\hat{H}}}{\partial y^{2}}\right)^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼E_{c}\left({\delta}\left(S_{xx}\frac{\partial ^{2}{\psi}}{\partial x\partial y}-{\delta}∼S_{yx}\frac{\partial ^{2}{\psi}}{\partial x^{2}}\right)+S_{xy}\frac{\partial ^{2}{\psi}}{\partial y^{2}}-{\delta}∼S_{yy}\frac{\partial ^{2}{\psi}}{\partial y\partial x}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{N_{b}}{P_{r}}\left({\delta}^{2}\frac{\partial {\phi}}{\partial x}\frac{\partial {\theta}}{\partial x}+\frac{\partial {\phi}}{\partial y}\frac{\partial {\theta}}{\partial y}\right)+\frac{N_{t}}{P_{r}}\left({\delta}^{2}\left(\frac{\partial {\theta}}{\partial x}\right)^{2}+\left(\frac{\partial {\theta}}{\partial y}\right)^{2}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{E_{c}}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}-{\delta}\frac{\partial {\psi}}{\partial x}+1\right)\sqrt{\left(\frac{\partial {\psi}}{\partial y}+1\right)^{2}+{\delta}^{2}\left(\frac{\partial {\psi}}{\partial x}\right)^{2}},\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle R_{e}S_{c}{\delta}\left(\frac{\partial {\psi}}{\partial y}\frac{\partial {\phi}}{\partial x}-\frac{\partial {\psi}}{\partial x}\frac{\partial {\phi}}{\partial y}\right)=\left({\delta}^{2}\frac{\partial ^{2}{\phi}}{\partial x^{2}}+\frac{\partial ^{2}{\phi}}{\partial y^{2}}\right)+S_{c}S_{r}\left({\delta}^{2}\frac{\partial ^{2}{\theta}}{\partial x^{2}}+\frac{\partial ^{2}{\theta}}{\partial y^{2}}\right)-S_{c}{\gamma}∼{\phi}, & & \displaystyle\end{eqnarray}$ (25) and $\begin{eqnarray}\displaystyle E=\frac{\partial {\psi}}{\partial y}-{\delta}\left(\frac{\partial {\psi}}{\partial y}\frac{\partial {\hat{H}}}{\partial x}-\frac{\partial {\psi}}{\partial x}\frac{\partial {\hat{H}}}{\partial y}\right)+\frac{1}{R_{m}}\left(\frac{\partial ^{2}{\hat{H}}}{\partial y^{2}}+{\delta}^{2}\frac{\partial ^{2}{\hat{H}}}{\partial x^{2}}\right), & & \displaystyle\end{eqnarray}$ (26) where, Eq. (3) is converted to, $\begin{eqnarray}\displaystyle S_{xy}=(A-B({\psi}_{yy})^{2}){\psi}_{yy}, & & \displaystyle\end{eqnarray}$ (27) where, A and B are the non-dimensional forms of the material constants given by $A=\frac{A^{\ast }}{B^{\ast }}$ and $B=\frac{A^{\ast }c^{2}}{3!B^{\ast 2}∼a^{2}}$ ; for instance, see Ref. [7].

Under the assumption of long wavelength and low Reynolds number δ < 1, Eqs (22)–(26) can be written as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial p}{\partial x}=\frac{\partial S_{xy}}{\partial y}-\frac{1}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}+1\right)+S_{t}^{2}∼R_{e}\frac{\partial ^{2}{\hat{H}}}{\partial y^{2}},\end{eqnarray}$ (28) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial p}{\partial y}=0,\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{1}{P_{r}}∼\frac{\partial ^{2}{\theta}}{\partial y^{2}}+D_{u}\frac{\partial ^{2}{\phi}}{\partial y^{2}}+\frac{{\beta}}{P_{r}}{\theta}-\frac{R_{e}E_{c}S_{t}^{2}}{R_{m}}\left(\frac{\partial ^{2}{\hat{H}}}{\partial y^{2}}\right)^{2}+E_{c}\left(S_{xy}\frac{\partial ^{2}{\psi}}{\partial y^{2}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\frac{E_{c}}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}+1\right)^{2}+\frac{N_{b}}{P_{r}}\frac{\partial {\phi}}{\partial y}\frac{\partial {\theta}}{\partial y}+\frac{N_{t}}{P_{r}}\left(\frac{\partial {\theta}}{\partial y}\right)^{2}=0,\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}{\phi}}{\partial y^{2}}+S_{c}S_{r}\frac{\partial ^{2}{\theta}}{\partial y^{2}}-S_{c}{\gamma}∼{\phi}=0.\end{eqnarray}$ (31) and $\begin{eqnarray}\displaystyle E=\frac{\partial {\psi}}{\partial y}+\frac{1}{R_{m}}\frac{\partial ^{2}{\hat{H}}}{\partial y^{2}}. & & \displaystyle\end{eqnarray}$ (32) The corresponding non-dimensional the wall equations are l ₁(x) = 1 + a cos2πx, and l ₂(x) = −d − b cos(2πx + 𝛺).

The non-dimensional slip boundary conditions on the walls are written as: $\begin{eqnarray}\displaystyle {\psi}=\frac{F}{2},∼\frac{\partial {\psi}}{\partial y}+{\alpha}_{1}S_{xy}=-1,∼{\hat{H}}=0,∼{\theta}+{\alpha}_{2}\frac{\partial {\theta}}{\partial y}=1,∼{\phi}+{\alpha}_{3}\frac{\partial {\phi}}{\partial y}=1,\quad \text{at }y=l_{1}, & & \displaystyle\end{eqnarray}$ (33) and $\begin{eqnarray}\displaystyle {\psi}=-\frac{F}{2},∼\frac{\partial {\psi}}{\partial y}-{\alpha}_{1}S_{xy}=-1,∼{\hat{H}}=0,∼{\theta}-{\alpha}_{2}\frac{\partial {\theta}}{\partial y}=0,∼{\phi}-{\alpha}_{3}\frac{\partial {\phi}}{\partial y}=0,\quad \text{at }y=l_{2}, & & \displaystyle\end{eqnarray}$ (34) where, F is the non-dimensional average flux in the wave frame which is related to the non-dimensional time means the flow rate 𝛩 in the laboratory frame by the following expression, see Refs [14,21]. They may be listed as follows: $\begin{eqnarray}\displaystyle {\Theta}=F+d+1,\quad \text{and}\quad F=\int _{l_{2}(x)}^{l_{1}(x)}\frac{\partial {\psi}}{\partial y}∼dy. & & \displaystyle\end{eqnarray}$ (35)

3. Homotopy perturbation method

In what follows, Eqs (28)–(32), with the aid of the boundary conditions Eq. (33) and Eq. (34), are analytically solved by means of a regular perturbation method depends mainly on a small Prandtl parameter. Therefore, any perturbed physical function may be represented as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}(x,y)={\psi}_{0}+B∼{\psi}_{1}+B^{2}∼{\psi}_{2}+O(B^{3})........,\end{eqnarray}$ (36) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\hat{H}}(x,y)={\hat{H}}_{0}+B∼{\hat{H}}_{1}+B^{2}∼{\hat{H}}_{2}+O(B^{3}).........,\end{eqnarray}$ (37) and $\begin{eqnarray}\displaystyle \frac{dP}{dx}(x,y)=\frac{dP_{0}}{dx}+B∼\frac{dP_{1}}{dx}+B^{2}∼\frac{dP_{2}}{dx}+O(B^{3}).......... & & \displaystyle\end{eqnarray}$ (38)

Inserting Eqs (36)–(38) into the governing equations (28)–(32), the solution of these equations may be written in general form as: $\begin{eqnarray}\displaystyle {\psi}(x,y) & = & \displaystyle r_{0}+r_{1}y+r_{2}\cosh (ny)+r_{3}\sinh (ny)+B[r_{4}+r_{5}y+(r_{6}+r_{7}∼y)\cosh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(r_{8}+r_{9}∼y)\sinh (ny)+r_{10}\cosh (3ny)+r_{11}\sinh (3ny)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼B^{2} [r_{12}+r_{13}y+(r_{14}+r_{15}∼y+r_{16}∼y^{2})\cosh (ny)+ (r_{17}+r_{18}∼y+r_{19}y^{2})\sinh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(r_{20}+r_{21}y)\cosh (3ny)+(r_{22}+r_{23}y)\sinh (3ny)+r_{24}\cosh (5ny)+r_{25}\sinh (5ny)],\end{eqnarray}$ (39) $\begin{eqnarray}\displaystyle \frac{dP}{dx} & = & \displaystyle {\Gamma}_{0}+B[{\Gamma}_{1}+{\Gamma}_{2}\cosh (ny)+{\Gamma}_{3}\sinh (ny)+{\Gamma}_{4}\cosh (3ny)+{\Gamma}_{5}\sinh (3ny)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼B^{2} [{\Gamma}_{6}+({\Gamma}_{7}+{\Gamma}_{8}y)\cosh (ny)+({\Gamma}_{9}+{\Gamma}_{10}y)\sinh (ny)+({\Gamma}_{11}+{\Gamma}_{12}y)\cosh (3ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼({\Gamma}_{13}+{\Gamma}_{14}y)\sinh (3ny)+{\Gamma}_{15}\cosh (5ny)+{\Gamma}_{16}\sinh (5ny)],\end{eqnarray}$ (40) and $\begin{eqnarray}\displaystyle {\hat{H}}(x,y) & = & \displaystyle w_{0}+w_{1}y+w_{2}y^{2}+w_{3}\cosh (n∼y)+w_{4}\sinh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼B[w_{5}+w_{6}y+w_{7}y^{2}+(w_{8}+w_{9}y)\cosh (ny)+(w_{10}+w_{11}y)\sinh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼w_{12}\cosh (3ny)+w_{13}\sinh (3ny)]+B^{2} [w_{14}+w_{15}y+w_{16}y^{2}+ (w_{17}+w_{18}y\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼w_{19}y^{2})\cosh (ny)+(w_{20}+w_{21}y+w_{22}y^{2})\sinh (n∼y)+(w_{23}+w_{24}y)\cosh (3ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(w_{25}+w_{26}y)\sinh (3ny)+w_{27}\cosh (5ny)+w_{28}\sinh (5ny)],\end{eqnarray}$ (41) where, $S^{2}=M^{2}+1/D_{a},n=S/\sqrt{A}$ , the coefficients r ₀ − r ₂₅, 𝛤₀ −𝛤₁₆ and w ₀ − w ₂₈ are depended functions on x. For simplicity, the mathematical formulae of the coefficients r ₀ − r ₂₅, 𝛤₀ −𝛤₁₆ and w ₀ − w ₂₈ are not included here. There are available under the request of the referees.

Combining Eqs ((30)) and ((31)), one gets the following form: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-20.39996pt}\frac{\partial ^{2}{\theta}}{\partial y^{2}}+P_{r}D_{u}\frac{\partial ^{2}{\phi}}{\partial y^{2}}+{\beta}{\theta}-P_{r}M^{2}E_{c}\left(E-\frac{\partial {\psi}}{\partial y}\right)^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad +∼P_{r}E_{c}\left(A-B\left(\frac{\partial ^{2}{\psi}}{\partial y^{2}}\right)^{2}\right)\left(\frac{\partial ^{2}{\psi}}{\partial y^{2}}\right)^{2}+\frac{P_{r}E_{c}}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}+1\right)^{2}+N_{b}\frac{\partial {\phi}}{\partial y}\frac{\partial {\theta}}{\partial y}+N_{t}\left(\frac{\partial {\theta}}{\partial y}\right)^{2}=0.\end{eqnarray}$ (42) Equations ((42)) and ((32)) are solved analytically by using the HPM [25] as follows: $\begin{eqnarray}\displaystyle \hspace{-18.0pt}H({\xi},{\theta}) & = & \displaystyle (1-{\xi})[L({\theta})-L({\theta}_{0}^{\ast })]+{\xi}\left[L({\theta})+P_{r}D_{u}\frac{\partial ^{2}{\phi}}{\partial y^{2}}+{\beta}{\theta}+N_{b}\frac{\partial {\phi}}{\partial y}\frac{\partial {\theta}}{\partial y}+N_{t}\left(\frac{\partial {\theta}}{\partial y}\right)^{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+∼P_{r}E_{c}\left(A-B\left(\frac{\partial ^{2}{\psi}}{\partial y^{2}}\right)^{2}\right)\left(\frac{\partial ^{2}{\psi}}{\partial y^{2}}\right)^{2}+\frac{P_{r}E_{c}}{D_{a}}\left(\frac{\partial {\psi}}{\partial y}+1\right)^{2}-P_{r}M^{2}E_{c}\left(E-\frac{\partial {\psi}}{\partial y}\right)^{2}\right]=0,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (43) and $\begin{eqnarray}\displaystyle H({\xi},{\phi})=(1-{\xi})[L({\phi})-L({\phi}_{0}^{\ast })]+{\xi}\left[L({\phi})+S_{c}S_{r}\frac{\partial ^{2}{\theta}}{\partial y^{2}}-S_{c}{\gamma}{\phi}\right]=0,\quad {\xi}\in [0,1] & & \displaystyle\end{eqnarray}$ (44) where, 𝜉 is the artificial Homotopy perturbation parameter, L = ∂ ²∕∂y ² is the linear operator, the initial approximations ${\phi}_{0}^{\ast }$ and ${\theta}_{0}^{\ast }$ can be written as: $\begin{eqnarray}\displaystyle {\theta}_{0}^{\ast }(x,y)=t_{0}+t_{1}y=\frac{{\alpha}_{2}-l_{2}+y}{2{\alpha}_{2}+l_{1}-l_{2}}\quad \text{and}\quad {\phi}_{0}^{\ast }(x,y)=c_{0}+c_{1}y=\frac{{\alpha}_{3}-l_{2}+y}{2{\alpha}_{3}+l_{1}-l_{2}}. & & \displaystyle \nonumber\end{eqnarray}$ One may assume that the solutions of Eqs ((43)) and ((44)) can be written as a power series in 𝜉 as: $\begin{eqnarray}\displaystyle {\theta}(x,y)={\theta}_{0}(x,y)+{\xi}{\theta}_{1}(x,y)+{\xi}^{2}{\theta}_{2}(x,y)+........., & & \displaystyle\end{eqnarray}$ (45) and $\begin{eqnarray}\displaystyle {\phi}(x,y)={\phi}_{0}(x,y)+{\xi}{\phi}_{1}(x,y)+{\xi}^{2}{\phi}_{2}(x,y)+.......... & & \displaystyle\end{eqnarray}$ (46) Substituting from Eqs ((45)) and ((46)) into Eqs ((43)) and ((44)), and then equating the coefficients of like power of 𝜉 on both sides, one finds $\begin{eqnarray}\displaystyle {\theta}(x,y) & = & \displaystyle \mathop{Lim}_{{\xi}\rightarrow 1} [t_{0}+t_{1}y+{\xi}[t_{2}+t_{3}y+t_{4}y^{2}+t_{5}y^{3}+(t_{6}+t_{7}y+t_{8}y^{2})\cosh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{9}+t_{10}y+t_{11}y^{2})\sinh (ny)+(t_{12}+t_{13}y+t_{14}y^{2})\cosh (2ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{15}+t_{16}y+t_{17}y^{2})\sinh (2ny)+(t_{18}+t_{19}y)\cosh (3ny)+(t_{20}+t_{21}y)\sinh (3ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{22}+t_{23}y)\cosh (4ny)+(t_{24}+t_{25}y)\sinh (4ny)+t_{26}\cosh (5ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼t_{27}\sinh (5ny)+t_{28}\cosh (6ny)+t_{29}\sinh (6ny)]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\xi}^{2} [t_{30}+t_{31}y+t_{32}y^{2}+t_{33}y^{3}+t_{34}y^{4}+t_{35}y^{5}+(t_{36}+t_{37}y+t_{38}y^{2})\cosh (ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{39}+t_{40}y+t_{41}y^{2})∼\sinh (ny)+(t_{42}+t_{43}y+t_{44}y^{2})\cosh (2ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{45}+t_{46}y+t_{47}y^{2})\sinh (2ny)+(t_{48}+t_{49}y)\cosh (3ny)+(t_{50}+t_{51}y)\sinh (3ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{52}+t_{53}y)\cosh (4ny)+(t_{54}+t_{55}y)\sinh (4ny)+t_{56}\cosh (5ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼t_{57}\sinh (5ny)+t_{58}\cosh (6ny)+t_{59}\sinh (6ny)]],\end{eqnarray}$ (47) $\begin{eqnarray} \displaystyle {\phi}(x,y) & = & \displaystyle \mathop{Lim}_{{\xi}\rightarrow 1}\left[c_{0}+c_{1}y+{\xi}\left[c_{2}+c_{3}y+\frac{1}{2}{\gamma}S_{c}\left(c_{0}∼y^{2}+\frac{1}{3}c_{1}y^{3}\right)\right]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\xi}^{2}\left[c_{4}+c_{5}y+S_{c}\left(\frac{1}{2}{\gamma}c_{2}-S_{r}t_{4}\right)y^{2}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼S_{c}\left(\frac{1}{6}{\gamma}c_{3}-S_{r}t_{5}\right)y^{3}+\frac{1}{24}{\gamma}^{2}S_{c}^{2}c_{0}y^{4}+\frac{1}{120}{\gamma}^{2}S_{c}^{2}c_{1}y^{5}-S_{c}S_{r} [(t_{6}+t_{7}y\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼t_{8}y^{2})\cosh (ny)+(t_{9}+t_{10}y+t_{11}y^{2})\sinh (ny)+(t_{12}+t_{13}y+t_{14}y^{2})\cosh (2ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{15}+t_{16}y+t_{17}y^{2})\sinh (2ny)+(t_{18}+t_{19}y)\cosh (3ny)+(t_{20}+t_{21}y)\sinh (3ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼(t_{22}+t_{23}y)\cosh (4ny)+(t_{24}+t_{25}y)\sinh (4ny)+t_{26}\cosh (5ny)+t_{27}\sinh (5ny)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.\left.+∼t_{28}\cosh (6ny)+t_{29}\sinh (6ny)\right]\right], \end{eqnarray}$ (48) where the coefficients t ₀ − t ₅₉ and c ₀ − c ₅ are functions of x. Their mathematical formulas are not included in the manuscript to save the space. hey there are available under the request of the referees.

The non-dimensional expression of the pressure difference per wavelength is given as follows: $\begin{eqnarray}\displaystyle {\rm\Delta}P_{{\lambda}}=\int _{0}^{1}\left(\frac{dp}{dx}\right)dx. & & \displaystyle\end{eqnarray}$ (49) The frictional forces at y = l ₁ and y = l ₂ denoted by F _{𝜆_l}. They are given as $\begin{eqnarray}\displaystyle F_{{\lambda}1}=\int _{0}^{1}l_{1}^{2}\left(-\frac{dp}{dx}\right)dx,\quad \text{and}\quad F_{{\lambda}2}=\int _{0}^{1}l_{2}^{2}\left(-\frac{dp}{dx}\right)dx. & & \displaystyle\end{eqnarray}$ (50) The coefficient of the heat transfer Z _l at the walls (y = l ₁ and y = l ₂) is given by $\begin{eqnarray}\displaystyle Z_{l1}=\frac{\partial l_{1}}{\partial x}\frac{\partial {\theta}}{\partial y}.\quad \text{and}\quad Z_{l2}=\frac{\partial l_{2}}{\partial x}\frac{\partial {\theta}}{\partial y}. & & \displaystyle\end{eqnarray}$ (51)

4. Results and discussion

In what follows, the analytical solutions of the considered boundary-value problem have been completed. Therefore, the numerical estimations are done to demonstrate the impact of the physical parameters on the obtained distribution functions. Consequently, the implication of the various parameters of the problem on the axial velocity u, temperature θ, heat transfer coefficient Z _l, pressure gradient dp∕dx, pressure difference Δp _𝜆, frictional force F _𝜆, axial induced magnetic field h _x, total magnetic field ${\hat{H}}$ , the current density J _Z, the concentration of nano-particles 𝜙 are computed. Additionally, the size of the trapped are numerically discussed and illustrated graphically throughout the figures 10–11.

∙ Flow Characteristics

Figures 2(a,b) depict the influence of the velocity versus the y-axis at various values of the Darcy number D _a, and the mean flow rate 𝛩. The effects of D _a on the velocity profile are seen throughout Fig. 2(a). It is observed that the velocity grows with the increasing of the Darcy number D _a in the domain of y ∈ [−0.9,0.9]. Meanwhile, it diminishes as approaches the channel walls and increase at the center of the channel. This indicates that the velocity increases from a porous medium to a non-porous, medium closer to the center of the channel. This observation is physically supported by the fact that the more permeable porous medium will allow less resistance to the fluid flow, therefore, there is an enhanced in the velocity of the fluid. The variation of the velocity with a volume flow rate 𝛩 is depicted in Fig. 2(b). It is showed that the velocity increases with the increasing volume flow rate 𝛩 in all domains of the channel. It is good concurrence with the physical situation. It is occurring due to the volume flow rate, in the high channel, is a larger velocity. This can be seen at lower velocity, the lesser values of volume flow rate.

Fig. 2.

Displays the variation of velocity profiles for different values of D _a and 𝛩, where x = 1, B = 0.001, a = 0.5, b = 1, d = 0.5, 𝛼₁ = 0.1, M = 2, A = 1, 𝛺 = π∕16.

Fig. 3.

Displays the temperature profile for different values of D _a, N _b, D _u, and E _c respectively where x = 1, B = 0.001, a = 0.5, b = 0.5, d = 1, 𝛼₁ = 𝛼₂ = 𝛼₃ = 𝛼₄ = 0.1, M = 2, D _a = 0.3, E _c = 0.4, P _r = 0.7, N _b = 1.5, N _t = 2.5, θ = 2, A = 1, E = 2, D _u = 0.4, 𝛽 = 0.5, R _m = 0.2, 𝛺 = π∕3.

Moreover, the numerical calculations show that the influence of the parameter of the Eyring–Prandtl fluid parameter A behaves like D _a. Meanwhile the influence of Hartman number M is plotted versus the behavior of D _a in the fluid flow. It is observed that the values of M is expanding as the velocity decays near the center of the channel in the domain of y ∈ [−0.8,0.8]. Simultaneously, it increases at the neighborhood of the walls. This appears realistic, because the induced magnetic field acts as occurring a Lorentz force. This force behaves as a resisting force to the flow, therefore, it declines the fluid motion. Furthermore, the influence of 𝛺 of the velocity is studied. It is observed that as the increasing of the phase difference 𝛺, the velocity enlarges near the right wall, and it slows down near the left wall. To avoid the repetition, these figures are excluded. These observations are fully complying with those obtained by Akbar [6] and Eldabe et al. [9].

Fig. 4.

Displays the heat transfer coefficient versus x for the same parameters as consider in Figs 3(a–d).

∙ Heat Characteristics

–

The variation of the temperature θ versus y at different values of the parameters Darcy number D _a, Brownian motion parameter N _b, Dufour number D _u, and Eckert number E _c is explained throughout Figs 3(a–d) as follows:

The influence of D _a is sketched throughout Fig. 3(a). It is found that θ is reduced with the increasing of the value of D _a. Physically, the increasing of D _a reduces the thermal conductivity of the flow. Therefore, the temperature of the fluid is reduced. This is in good agreement with those obtained by Eldabe et al. [4].

The effect of N _b is illustrated throughout Fig. 3(b). It is found that θ increased with the increasing of N _b Physically, the increase of the Brownian motion effect relates to an involved nano-particles movement from the wall of the fluid, which prompts the expansion of the temperature. This is an agreement with those obtained in Refs [4,13–15].

Figure 3(c) displays the impact of D _u of the temperature distribution. One finds that the temperature distribution enlarges with the increase of the diffusion-thermal (Dufour number) D _u. This is in good agreement with those obtained by Abbasi et al. [7].

Figure 3(d) displays the impact of E _c on the temperature. It is shown that θ incremented with the increasing value of E _c. Physically, since the impact of sources and dissipation temperature increases in the rate of energy transport to the fluid. Consequently, it increases the temperature of the fluid. This is an agreement with those obtained in Refs [4,13].

Furthermore, the numerical calculations show that the influence of the parameters P _r, M, and E _c behave like N _b. To avoid the repetition, these figures are excluded. These observations are fully complying with those obtained in Refs [6,7,9].

–

Figures 4(a,b) are sketched to describe the variation of heat transfer coefficient Z _l for various values of heat generation parameter 𝛽 and thermophoresis parameter N _t at y = l ₁. It is observed from these figures that, due to the peristalsis, the heat transfer coefficient has an oscillatory behavior.

It is also indicated from Fig. 4(a,b) that the magnitude of Z _l enlarges with the increasing of 𝛽 and N _t. Physically, this result occurs due to the thermophoretic force that generated by the temperature gradient results in fast and creeping flow away from the stretching surface.

Moreover, the numerical calculations show that the absolute value of the heat transfer coefficient Z _l diminishes with expanding of the Darcy parameter D _a. Meantime, Hartmann number M behaves like N _t. To avoid the repetition, these figures are excluded. These observations are fully congruent with those obtained from the work of Eldabe et al. [4], and Hayat et al. [14].

Fig. 5.

Displays the pressure gradient for different values of E and D _a, where y = l ₁, B = 0.001, A = 1, a = 0.2, b = 0.7, d = 1, 𝛼₁ = 0.1, M = 1.5, D _a = 0.6, E = 2, θ = 1, 𝛺 = π∕16.

∙ Pumping Characteristics

–

Figures 5(a,b) spectacle the behavior of the emerging parameters as; electric field strength E and Darcy number D _a, on the pressure gradient dp∕dx over one wavelength x ∈ [0,1].

The effect of E is illustrated throughout Fig. 5(a). It is displayed that the size of the pressure ramp dp∕dx increases with the increase of E. Physically, these phenomena indicate that, in the presence of the induced magnetic field in the flow field, a higher pressure gradient is necessary to push the flow. This result agrees well with the physical situation, and it is one of the magnetic field applications.

The influence of D _a is illustrated throughout Fig. 5(b). It is found that dp∕dx increases with the increasing of the D _a. This result gives that the pressure gradient is enhanced to push the fluid in the presence of a permeable medium as compared to a clear medium. This result agrees well with the physical situation.

Additionally, the numerical calculations showed that the influence of the Hartmann number M, and amplitude ratio a behave like E. To avoid the repetition, these figures are excluded. These behaviors are fully complying with those obtained in Refs [6,9].

–

Figures 6(a,b) represent the variation of pressure difference ΔP _𝜆 versus the time for mean flow rate 𝛩 for both of D _a and E. Figure 6(a) portrays that the pressure difference ΔP _𝜆 increases by the increasing of the electric field strength E in all the regions of time mean flow rate. Figure 6(b) shows that, with the increasing of the Darcy number D _a, the pumping decreases up to a critical value of volume flow rate 𝛩, and then increases after the critical value. Furthermore, the numerical calculations showed that the influence of the phase difference 𝛺 behaves like D _a, therefore, the impact of Hartmann number M behaves like E. Moreover, the effects of Eyring–Prandtl fluid parameter A and amplitude ratio a behave opposite with D _a effect. To avoid the repetition, these figures are excluded. These behaviors are fully complying with those obtained in Refs [4,6,8,9].

Fig. 6.

Displays the variation of pressure difference versus 𝛩 for the same parameters as consider in Figs 5(a,b).

Fig. 7.

Displays the axial induced magnetic field versus y for different values of R _m and A where x = 1, a = 0.6, b = 0.4, d = 1.1, D _a = 0.3, 𝛼₁ = 0.1, B = 0.001, 𝛺 = π∕6, R _m = 0.75, θ = 5, E = 1.

∙ Magnetic Field Characteristics

–

The expression of the axial induced magnetic field h _x versus the space variable y for different values of the Reynolds magnetic parameter R _m and the Eyring–Prandtl fluid parameter A is depicted in Figs 7(a,b). It is observed that the induced magnetic field h _x increases as R _m increases. This occurs in the upper half of the channel, meanwhile, the opposite effect appears in the lower half. In contrast, the induced magnetic field h _x diminishes with expanding of Eyring–Prandtl fluid parameter A. This appears in the upper half of the channel, but the converse occurs on the other side of the channel. Furthermore, in the half region, the induced magnetic field is in one direction and in the other half. It is in the opposite direction. Moreover, the numerical calculations showed that the influence of the Darcy number D _a, behaves like R _m. On the other hand, the effect of Hartmann number M behaves like A. To avoid the repetition, these figures are excluded. These behaviors are fully complying with those obtained in Refs [4,19–21].

–

Figures 8(a,b) depict the distribution of the current density J _Z versus y for different values of R _m and M. The influence of R _m on J _Z is sketched throughout Fig. 8(a). It is found that the magnitude of J _Z is increased with the increasing of R _m at the center of the channel, but in the neighborhood walls it has an opposite behavior. In contrast, Fig. 8(b) where J _Z decreases with the increasing of M near the center channel and decreases the approach of its sides. These behaviors are fully complying with those obtained in Refs [19–21].

Fig. 8.

Displays the current density J _Z versus y for different values of R _m, M, and for the same parameters as consider in Figs 7(a,b).

Fig. 9.

Displays the concentration versus y for the same parameters as consider in Figs 3(a–d).

∙ The nano-particles Characteristics

Figures 9(a,b) are sketched to discuss the effects for different values of the electric field strength E and the chemical reaction 𝛾 on the concentration of nano-particles 𝜙. It is observed that the viscosity of the fluid decreases with the increasing in the values of E, which leads to a reduction of the nanoparticle volume fraction and then less the density of particles. Otherwise, 𝜙 is increasing with the increasing of the chemical reaction 𝛾 as depicted in Fig. 9(b). This shows the increasing of the values of chemical reaction 𝛾 is corresponding to an increase in density of nano-fluid particles, which causes the nano-particle volume fraction to increase. Moreover, the numerical calculations showed that the influence of the parameters Eckert number E _c, Prandtl number P _r, Schmidt number S _c, and the Soret number S _r behave like E. To avoid the repetition, these figures are excluded. These behaviors are fully complying with those obtained in Refs [7,8].

Fig. 10.

Displays the Streamlines for different values of for a = 0.8, b = 0.5, d = 2, M = 0.3, A = 1, B = 0.001, 𝛼₁ = 0.1, 𝛺 = π∕16 and for (i) D _a = 0.2, (ii) D _a = 0.4, (iii) D _a = 0.6.

Fig. 11.

Displays the streamlines for the same parameters as consider in Fig. 10, except D _a = 1, and a variation (i) a = 0.2, (ii) a = 0.4, (iii) a = 0.6.

∙ Trapping

The streamlines represent the trajectories of fluid-particles in a flow. The trapping concept means a formation of an internally circulating bolus of fluid by the closed streamlines. This trapped bolus is pushed ahead along with the peristaltic wave. Figures 10 and 11 illustrate the streamlines for various values of Darcy number D _a and amplitude ratio a. As seen from these figures, is observed that the sizes of the trapped bolus are increased with the increasing of the Darcy number D _a and amplitude ratio a. In addition, the numerical calculations showed the influence of the variations the Hartmann number M on the trapping. It is clear that the size of trapped bolus is decreased by the increasing of M. To avoid the repetition, these figures are excluded. These behaviors are fully complying with those obtained in Ref. [6].

5. Concluding remarks

The current work examines the influences of the physical parameters on the problem of the peristaltic motion of an Eyring–Prandtl fluid with heat and mass transfer throughout porous media in a horizontal asymmetric channel, taking into account the induced magnetic field. Additionally, the slip effects, heat-generating, and chemical reactions are considered. The governing equations of motion are displayed in a non-dimensional form. The resulted non-linear system is very complicated to analytically solve. Therefore, to relax the complexity of the mathematical procedure, longer wavelength and low Reynolds’s number assumptions is followed by the HPM, are assumed. The effects of the non-dimensional physical parameters of the considered problem are discussed numerically and graphically. The discussion of the results and figures, concluded the following remarks:

The velocity increases with the increasing of both Darcy number D _a and the flow rate 𝛩 in the middle part of the channel. In contrast, this situation is reversed at the boundary of the walls.

The temperature increases with the increasing of the Eckert number E _c, Brownian parameter N _b, Dufour parameter D _u. In contrast, the temperature decreases with the increasing of Darcy number.

The heat transfer coefficient Z _l is increased in accordance with the increasing of the heat generation parameter 𝛽, Hartmann number M, and thermophoresis parameter N _t. However, it decreases with the increase of Darcy number D _a.

The pressure gradient increases with the increasing of both Darcy number D _a and the electric field strength E.

The values of the pressure rise, increases with the increasing of the amplitude ratio a, Hartmann number M, and the electric field strength E, while it decreases with the increasing of the Darcy’s number D _a.

The current density distribution is increased with the increasing of the Reynolds magnetic parameter R _m and Darcy number D _a, meanwhile, it depicted an opposite behavior for Hartmann number M and the electric field strength E.

The nano-particles concentration 𝜙 is reduced with the increase of E, E _c, and P _r, meanwhile, it increases with the increasing of the chemical reaction 𝛾.

The size of the trapped bolus increases with the increase of the Darcy number D _a and amplitude ratio a.

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