Semi-analytical treatment of Hall current effect on peristaltic flow of Jeffery nanofluid

Abstract

In the current paper, a semi-analytical solution is obtained for the problem of the peristaltic motion of Jeffrey nanofluid with heat transfer inside an asymmetric channel. The influences of variable viscosity, thermal conductivity, Hall currents, Joule heat and viscous dissipation are taken into consideration. The problem is modulated mathematically by a system of non-linear partial deferential equations which describe the fluid velocity, temperature and nanoparticles concentration distributions. This system is simplified by considering long wavelength of the peristaltic wave and low Reynolds number. It is solved by using the multi-step deferential transform method to obtain the velocity, temperature and nanoparticles concentration distributions, pressure rise and pressure gradient. These solutions are obtained as functions of the physical parameters of the problem. Tables and figures are discussed to see the effects of these parameters. It is found that the thermal conductivity parameter causes an increase in the pressure gradient, meanwhile it reduces the pressure rise. Furthermore, it is noticed that the behavior of Brownian and thermophoresis parameters on nanoparticles concentration field are quite reverse.

Keywords

Peristaltic flow Jeffrey nanofluid Hall currents Joule heating Mathematica 11

1. Introduction

Peristalsis plays an important role in transporting physiological fluids in the body. It is an involuntary and key mechanism for moving food through the digestive tract, bile movement in a bile duct, transport of spermatozoa in the cervical canal, urine transport from kidney to bladder, the swallowing of food through esophagus and the vasomotion of small blood vessels. It also plays the role of an involuntary key mechanism in the movement of chyme in the gastrointestinal tracts, the movement of ovum in the fallopian tubes, the transport of spermatozoa in the ducts afferent of the male reproductive tract and embryo motion in non-pregnant uterus, etc [1]. Eldabe et al. [2] used homotopy perturbation method (HPM) to solve the system of equations that govern the Magnetohydrodynamic (MHD) peristaltic transport of Eyring–Prandtl nanofluid through a porous medium in the presence of chemical reactions and slip boundary condition. Abouzeid et al. [3] constructed a peristaltic flow with heat transfer through a non-Darcy porous medium on power-law nano-slime, moreover, viscous dispersion effects were taken into consideration.

Jeffrey fluid is one of the rate type materials, Jeffrey model is a relatively simple linear model using time derivatives instead of convected derivatives. Moreover, by choosing Jeffrey model it becomes possible to treat both the eigenvalue problem and the initial value problem analytically [4]. Nadeem et al. [5] scrutinized the influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat transfer in an inclined symmetric or asymmetric channel. In another paper, Hayat et al [6] examined the effects of an endoscope and magnetic field on the peristalsis involving Jeffrey fluid.

Although many studies were available for the peristaltic flow of traditional fluids, only few papers are available for the peristaltic flow of nanofluids. At present, the mathematicians, physiologists, engineers, computer scientists and modelers seem to have interest in the research area of nanofluids concerning peristaltic effectiveness. Nanofluid is a traditional liquid composed of tiny particles of diameter less than 100 nm. It defines a kind of fluids having the distinguishing ability to ameliorate the thermal properties of fluids [7]. Nanofluids have many applications in medical, electrical and engineering processes. Furthermore, some of these applications contributed in hyperthermia, modern drug delivery systems, cryosurgery, batteries of electronic devices, heating and cooling systems … etc. Abouzeid and Mohamed [8] discussed the motion of power-law nanofluid with heat transfer under the effect of viscous dissipation, radiation, and internal heat generation. They solved the governing equations by using HPM and also discussed the effects of the entering physical parameters on the flow’s behavior. Abouzeid [9] analyzed the influence of Cattaneo–Christov heat flux on MHD flow of biviscosity nanofluid between two rotating disks through a porous media.

All the studies mentioned above deal with fluids which have uniform viscosity. The proposition of uniform viscosity fluid fails when peristaltic motion in small lymphatic vessels, intestine and blood vessels are considered. The change in the fluid viscosity has great effects on fluid properties. The heat transfer analysis of such kinds of fluids has many applications in biosciences, lubrications, instruments, viscometry and food processing. Therefore, it is highly eligible to guarantee the effect of temperature dependent viscosity in momentum and thermal transport processes. Mekheimer et al. [10] introduced simultaneous effects of variable viscosity and thermal conductivity on peristaltic flow in a vertical asymmetric channel. It is found that the maximum velocity of the fluid for constant viscosity and thermal conductivity smaller when compared to maximum velocity of a fluid with variable viscosity and variable thermal conductivity.

Elogail and Elshekhipy [11] investigated the influence of temperature dependent viscosity on peristaltic transport of a Newtonian fluid through a porous medium in a vertical asymmetric channel. Moreover, they solved the governing equations by using highly accurate series based method called the multi-step differential transform method (Ms-DTM). There are many studies related to variable viscosity of fluid in Refs. [12–18].

Joule heating occurs when the energy of an electric current is changed into heat as it flows through a resistance. There are many practical applications of Joule heating such as electric stoves, electric heaters, soldering irons, cartridge heaters and electric fuses. The electronic cigarettes usually work by Joule heating, vegetable glycerin and vaporizing propylene glycol. Alvi et al. [19] solved analytically the system of equations that govern MHD peristaltic transport of nonconstant viscosity Jeffrey fluid with suspended nanoparticles under the presence of Brownian motion, thermopherosis, mixed connection, viscous dissipation, and Joule heating. Information on this topic is quite sensible and researchers invoke a few recent representative attempts and several useful references in their investigations [20–23].

The effect of Hall currents was neglected in applying the Ohm’s law as they have no remarkable effect for a weak magnetic field. However, the current trend in the application of MHD flow is towards a strong magnetic field, so that the effect of electromagnetic force could be noticeable. Therefore, Hall currents are paramount and have remarkable influences on both the magnitude and the direction of the current density, and consequently on the magnetic force term. The effects due to the Hall currents become significant when the Hall parameter (the ratio of electron-cyclotron frequency and the electron-atom-collision frequency) is high. This situation occurs as a result of high magnetic field or low collision frequency. The influence of Hall currents with heat transfer is likely to be essential in many states as well as in engineering applications in areas like electric transformers, power generators, Hall accelerators, MHD accelerators, heating elements and refrigeration coils. The works related to this topic are cited in refs [24–28].

In fluid mechanics and in minds of many researchers, investigators, arithmeticians and modelers, peristalsis has acquired the leading place in the due to their great significant of requests in biomedical device and biochemical productions, etc. Generally, peristaltic flow is a form of fluid transport that accomplished via a progressive wave of expansion or contraction, this flow takes place from a region of lower pressure to higher pressure. Peristaltic transport appear in many physiological phenomena for instances, urine transport from kidney to bladder through the urethra, movement of food bolus through esophagus, movement of spermatozoa in the ducts afferents of the male reproductive tract, movement of chyme in the small intestine, movement of eggs in the female fallopian tube, transport of bile in the bile duct, transport of lymph in the lymphatic vessels and circulation of blood in small blood vessels, etc. Abouzeid [29] constructed an undulating surface model for the motility of bacteria gliding on a layer of non-Newtonian nanoslime with heat transfer. Then, various experimental, numerical and theoretical researches have been deliberated to understand peristaltic transport of Newtonian/Non-Newtonian fluids with respect to unlike boundary conditions and geometries [30–33].

The main aim of this work is to investigate the peristaltic flow of Jeffery nanofluid in the presence of variable viscosity, Hall current and other external effects. To the best of our knowledge, this problem has not been investigated yet. Accordingly, we extend the work of Alvi et al. [19] to include the variable thermal conductivity with Hall current effects. Moreover, in order to verify the accuracy of the method of solution a comparison for distributions of temperature and nanoparticles is done between the present work and the work of Alvi et al. [19], It is found the error is too small (≤10⁻⁷).

The paper has been arranged as follows: in Section 2. we formulated the mathematical formulation of our model. In Section 3. we introduced the method of solution, namely differential transform method, and in Section 4. we introduced the concluding remarks and discussions, finally, Section 5. introduce the conclusion of the presented result through the whole manuscript.

2. Mathematical formulations

We consider a peristaltic flow of an incompressible Jeffrey nanofluid in two dimensional asymmetric flexible channel of mean width d ₁ + d ₂. We choose rectangular coordinates ( X , Y ) such that X are along the central line and Y transverse to it. A sinusoidal wave of amplitude b ₁ propagates along the channel walls with uniform speed c. Since we are considering asymmetric channel, the channel flow is generated due to various amplitudes and phases of the peristaltic waves on the channel. A heat transfer along with nanoparticle phenomena has been taken into consideration, (see Fig. 1). The geometry of the wall surface is described by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Y=H_{1}=d_{1}+a_{1}\cos \left[{\displaystyle \frac{2{\pi}}{{\lambda}}}(X-ct)\right]\ldots \text{ Right hand side},\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Y=H_{2}=-d_{2}-b_{1}\cos \left[{\displaystyle \frac{2{\pi}}{{\lambda}}}(X-ct)+{\phi}\right]\ldots \text{Left hand side}.\end{eqnarray}$ (2) The parameters a ₁, b ₁, d ₁ and d ₂ satisfy the condition [19]. $\begin{eqnarray}\displaystyle a_{1}^{2}+b_{1}^{2}+2a_{1}b_{1}\cos {\phi}\leq (d_{1}^{2}+d_{2}^{2}). & & \displaystyle\end{eqnarray}$ (3) The basic equations govern the flow of an incompressible non Newtonian nanofluid in presence of heat and mass transfer with the effects of variable viscosity and thermal conductivity [5,19,24] and [33].

Fig. 1.

Schematic diagram of a two-dimensional axis-symmetric channel.

The continuity equation $\begin{eqnarray}\displaystyle {\nabla}\cdot V=0. & & \displaystyle\end{eqnarray}$ (4) The conversation of momentum yields $\begin{eqnarray}\displaystyle {\rho}_{f}{\displaystyle \frac{dV}{dt}}={\nabla}\cdot {\tau}-J\times B+{\rho}_{f}g{\beta}_{t}(T-T_{0})+{\rho}_{f}g{\beta}_{c}(C-C_{0}). & & \displaystyle\end{eqnarray}$ (5) The energy equation gives $\begin{eqnarray}\displaystyle {\rho}_{f}{\displaystyle \frac{dT}{d\overline{t}}}={\nabla}k\left(T\right)\cdot ({\nabla}T)+\overline{{\tau}}\cdot L+{\tau}_{1}{\rho}_{f}c_{f}\left[D_{B}({\nabla}C\cdot {\nabla}T)+{\displaystyle \frac{D_{T}}{T_{0}}}({\nabla}T\cdot {\nabla}T)\right]+{\displaystyle \frac{1}{{\sigma}}}J^{2}. & & \displaystyle\end{eqnarray}$ (6) The nanoparticles concentration equation gives $\begin{eqnarray}\displaystyle {\displaystyle \frac{dC}{d\overline{t}}}=D_{B}{\nabla}^{2}\overline{C}+{\displaystyle \frac{D_{T}}{T_{0}}}{\nabla}^{2}T & & \displaystyle\end{eqnarray}$ (7) with associated boundary conditions [19] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{1}=d_{1}+a_{1}\cos \left[{\displaystyle \frac{2{\pi}}{{\lambda}}}(X-ct)\right]\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{2}=-d_{2}-b_{1}\cos \left[{\displaystyle \frac{2{\pi}}{{\lambda}}}(X-ct)+{\phi}\right].\end{eqnarray}$ (9)

Introducing a wave frame (x, y) which moving with the velocity c away from the fixed frame (X, Y ) by the transformation x = X − ct, y = Y, u = U − c, v = V and p (x) = P (X, t) in which the boundary shape is stationary. The velocity components in wave are frame and fixed frame of references (u, v), (U, V ), respectively. p and P are pressures in wave and fixed frame of references respectively.

The Cauchy stress S and extra stress τ tensors for Jeffrey fluid are given by [19] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{{\tau}}=-P\overline{I}+\overline{S}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{S}={\displaystyle \frac{\overline{{\mu}}(\overline{T})}{1+{\lambda}_{1}}}\left[\overline{\boldsymbol{A}}_{1}+{\lambda}_{2}\left({\displaystyle \frac{\partial }{\partial \overline{t}}}+\overline{V}\cdot \overline{{\nabla}}\right)\overline{\boldsymbol{A}}_{1}\right]\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{\boldsymbol{A}}_{1}=\overline{{\nabla}}\overline{V}+(\overline{{\nabla}}\overline{V})^{T}.\end{eqnarray}$ (12) The velocity ( V ) and magnetic fields ( B ) are defined as $\begin{eqnarray}\displaystyle \overline{V}=(\overline{U}(\overline{X},\overline{Y},\overline{t}),\overline{V}(\overline{X},\overline{Y},\overline{t}),0),\quad \overline{B}=(0,B_{0},0). & & \displaystyle\end{eqnarray}$ (13) The generalized Ohm’s law can be written as [24] $\begin{eqnarray}\displaystyle \overline{J}={\sigma}(\overline{E}+\overline{V}\times \overline{\boldsymbol{B}}-{\alpha}\overline{J}\times \overline{\boldsymbol{B}}) & & \displaystyle\end{eqnarray}$ (14) where ${\alpha}=\frac{1}{n_{e}e}$ .

Now, consider the following non-dimensional quantities [24]. $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\overline{x}={\displaystyle \frac{x}{{\lambda}}},\quad \overline{y}={\displaystyle \frac{y}{\text{d}_{1}}},\quad \overline{u}={\displaystyle \frac{u}{c}},\quad \overline{v}={\displaystyle \frac{v}{c}},\quad \overline{p}={\displaystyle \frac{d_{1}^{2}}{{\mu}c{\lambda}}},\quad h_{1}={\displaystyle \frac{H_{1}}{d_{2}}},\quad \\ h_{2}={\displaystyle \frac{H_{2}}{d_{1}}},\quad d={\displaystyle \frac{d_{2}}{d_{1}}},\quad a={\displaystyle \frac{a_{1}}{d_{1}}},\quad b={\displaystyle \frac{b_{1}}{d_{1}}},∼R_{e}={\displaystyle \frac{{\rho}_{f}cd_{1}}{{\mu}_{0}}},\\ {\theta}{\displaystyle \frac{T-T_{0}}{T_{1}-T_{0}}},\quad {\varphi}={\displaystyle \frac{C-C_{0}}{C_{1}-C_{0}}},\quad P_{r}={\displaystyle \frac{c_{f}{\mu}_{0}}{k}},\quad \\ G_{t}={\displaystyle \frac{{\rho}_{f}d_{1}^{2}{\rho}g{\beta}_{t}(T-T_{0})}{c{\mu}_{0}}},\quad G_{c}={\displaystyle \frac{{\rho}_{f}d_{1}^{2}{\rho}g{\beta}_{c}(C-C_{0})}{c{\mu}_{0}}},\\ B_{r}={\displaystyle \frac{{\mu}_{0}c^{2}}{(T_{1}-T_{0})k}},\quad M^{2}=\frac{{\sigma}B_{0}^{2}d_{1}^{2}}{{\mu}_{0}},N_{b}={\displaystyle \frac{{\tau}{\rho}_{f}D_{B}(C_{1}-C_{0})}{{\mu}_{0}}},\quad N_{t}={\displaystyle \frac{{\tau}D_{T}{\rho}_{f}(T_{1}-T_{0})}{{\mu}_{0}T_{0}}}.\end{array} & & \displaystyle\end{eqnarray}$ (15) Substituting with Eq. (15) into Eqs (4)–(7) (after dropping bars) yields. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Re∼\left({\delta}u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial x}}+{\delta}{\displaystyle \frac{\partial S_{xx}}{\partial x}}+{\displaystyle \frac{\partial S_{xy}}{\partial y}}-{\displaystyle \frac{M^{2}(u+1)}{1+m^{2}}}+G_{t}{\theta}+G_{c}{\varphi},\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Re∼{\delta}\left({\delta}u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial y}}+{\delta}^{2}{\displaystyle \frac{\partial S_{yx}}{\partial x}}+{\delta}{\displaystyle \frac{\partial S_{yy}}{\partial y}}\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Re∼P_{r}\left({\delta}u{\displaystyle \frac{\partial {\theta}}{\partial x}}-v{\displaystyle \frac{\partial {\theta}}{\partial y}}\right)=({\epsilon}({\theta}_{y})^{2})+((1+{\epsilon}{\theta}){\theta}_{yy})+{\delta}^{2}({\epsilon}({\theta}_{x})^{2})+((1+{\epsilon}{\theta}){\theta}_{xx})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼P_{r}N_{b}\left({\delta}^{2}{\displaystyle \frac{\partial {\varphi}}{\partial x}}{\displaystyle \frac{\partial {\theta}}{\partial x}}+{\displaystyle \frac{\partial {\varphi}}{\partial y}}{\displaystyle \frac{\partial {\theta}}{\partial y}}\right)+P_{r}N_{t}\left({\delta}^{2}\left({\displaystyle \frac{\partial {\theta}}{\partial x}}\right)^{2}+\left({\displaystyle \frac{\partial {\theta}}{\partial y}}\right)^{2}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼B_{r}\left[{\delta}S_{xx}{\displaystyle \frac{\partial u}{\partial x}}+S_{yy}{\displaystyle \frac{\partial v}{\partial y}}+S_{xy}\left({\delta}{\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}}\right)\right]+{\displaystyle \frac{M^{2}B_{r}}{1+m^{2}}}(u+1)^{2}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{cd_{1}}{D_{B}}}\left({\delta}u{\displaystyle \frac{\partial {\varphi}}{\partial x}}+v{\displaystyle \frac{\partial {\varphi}}{\partial y}}\right)=\left({\displaystyle \frac{\partial ^{2}{\varphi}}{\partial y^{2}}}+{\delta}^{2}{\displaystyle \frac{\partial ^{2}{\varphi}}{\partial x^{2}}}\right)+{\delta}^{2}{\displaystyle \frac{N_{t}}{N_{b}}}{\displaystyle \frac{\partial ^{2}{\theta}}{\partial x^{2}}}+{\displaystyle \frac{N_{t}}{N_{b}}}{\displaystyle \frac{\partial ^{2}{\theta}}{\partial y^{2}}}.\end{eqnarray}$ (19) The variable of viscosity and thermal conductivity with dimensionless temperature θ are taken into account of many researchers [10]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\mu}({\theta})={\mu}_{0}e^{-{\beta}{\theta}}\quad \text{or}\quad {\mu}={\mu}_{0}(1-{\beta}{\theta})\quad \text{for}∼{\beta}<1\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle k({\theta})=k_{0}(1+{\epsilon}{\theta})\quad \text{for}∼{\epsilon}<1.\end{eqnarray}$ (21) The extra stress tensor component may be formulated as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S_{xx}={\displaystyle \frac{2{\delta}}{1+{\lambda}_{1}}}\left[1+{\displaystyle \frac{{\lambda}_{2}c{\delta}}{d_{1}}}\left({\psi}_{y}{\displaystyle \frac{\partial }{\partial x}}-{\psi}_{x}{\displaystyle \frac{\partial }{\partial y}}\right)\right]{\psi}_{xy},\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S_{xy}={\displaystyle \frac{1-{\beta}{\theta}}{1+{\lambda}_{1}}}\left[1+{\displaystyle \frac{{\lambda}_{2}c{\delta}}{d_{1}}}\left({\psi}_{y}{\displaystyle \frac{\partial }{\partial x}}-{\psi}_{x}{\displaystyle \frac{\partial }{\partial y}}\right)\right][{\psi}_{yy}-{\delta}^{2}{\psi}_{xx}],\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S_{yy}=-{\displaystyle \frac{{\delta}}{1+{\lambda}_{1}}}\left[1+{\displaystyle \frac{{\lambda}_{2}c{\delta}}{d_{1}}}\left({\psi}_{y}{\displaystyle \frac{\partial }{\partial x}}-{\psi}_{x}{\displaystyle \frac{\partial }{\partial y}}\right)\right]{\psi}_{xy}.\end{eqnarray}$ (24) Defining the stream function 𝜓 = 𝜓(x, y). It is related to u and v by the relations $u=\frac{\partial {\psi}}{\partial y},v=-{\delta}\frac{\partial {\psi}}{\partial x}$ .

Under the assumptions of low Reynolds number and long wavelength, Eqs (16–19) may be written as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle -{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{1-{\beta}{\theta}}{1+{\lambda}_{1}}}{\displaystyle \frac{\partial ^{2}{\psi}}{\partial y^{2}}}\right)-{\displaystyle \frac{M^{2}}{1+m^{2}}}\left({\displaystyle \frac{\partial {\psi}}{\partial y}}+1\right)+G_{t}{\theta}+G_{c}{\varphi}=0\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial p}{\partial y}}=0,\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left({\epsilon}\left({\displaystyle \frac{\partial {\theta}}{\partial y}}\right)^{2}\right)+\left((1+{\epsilon}{\theta}){\displaystyle \frac{\partial ^{2}{\theta}}{\partial y^{2}}}\right)+B_{r}{\displaystyle \frac{1-{\beta}{\theta}}{1+{\lambda}_{1}}}\left({\displaystyle \frac{\partial ^{2}{\psi}}{\partial y^{2}}}\right)^{2}+M^{2}B_{r}\left({\displaystyle \frac{\partial {\psi}}{\partial y}}+1\right)^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼P_{r}N_{b}\left({\displaystyle \frac{\partial {\theta}}{\partial y}}{\displaystyle \frac{\partial {\varphi}}{\partial y}}\right)+P_{r}N_{t}\left({\displaystyle \frac{\partial {\theta}}{\partial y}}\right)^{2}=0\end{eqnarray}$ (27) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial ^{2}{\varphi}}{\partial y^{2}}}+{\displaystyle \frac{N_{t}}{N_{b}}}{\displaystyle \frac{\partial ^{2}{\theta}}{\partial y^{2}}}=0.\end{eqnarray}$ (28) The elimination of the pressure from Eqs (25) and (26), yields $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial ^{2}}{\partial y^{2}}}\left({\displaystyle \frac{1-{\beta}{\theta}}{1+{\lambda}_{1}}}{\displaystyle \frac{\partial ^{2}{\psi}}{\partial y^{2}}}\right)-{\displaystyle \frac{M^{2}}{1+m^{2}}}{\displaystyle \frac{\partial ^{2}{\psi}}{\partial y^{2}}}+G_{t}{\displaystyle \frac{\partial {\theta}}{\partial y}}+G_{c}{\displaystyle \frac{\partial {\varphi}}{\partial y}}=0. & & \displaystyle\end{eqnarray}$ (29) The identical boundary conditions are $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}={\displaystyle \frac{q}{2}},\quad {\theta}=0,\quad {\varphi}=0,\quad {\displaystyle \frac{\partial {\psi}}{\partial y}}=-1,\quad \text{at }∼y=h_{1}=1+a∼cos2{\pi}x\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}={\displaystyle \frac{-q}{2}},\quad {\theta}=1,\quad {\varphi}=1,\quad {\displaystyle \frac{\partial {\psi}}{\partial y}}=-1,\quad \text{at }y=h_{2}=-d-b∼cos(2{\pi}x+).\end{eqnarray}$ (31) The dimensional time mean flow rate Q, in the libratory frame is related to q through the relation [24] $\begin{eqnarray}\displaystyle Q=q+1+d. & & \displaystyle\end{eqnarray}$ (32) In case of a uniform viscosity and thermal conductivity, the parameters 𝛽 and 𝜖 are excluded. Now, the peristaltic flow problem will be numerically solved by means of the Ms-DTM.

3. Method of solution

The governing equations become more complex to handle as supplementary nonlinear terms appear in the equations of motion because of the flow behavior of non-Newtonian nanofluids. So, the exact solutions of such problems are practically impossible, therefor we turn to find one of semi-analytical methods such as the differential transform method (DTM). DTM is proposed by Zhou [34] for solving ordinary, partial differential and integral equations. On designing the DTM technique, the given differential equations and their related boundary/initial conditions are converted into recurrence a relation who leads to the solution of a system of algebraic equations as coefficients of a power series solution. In some situations, in which the governing equations of the system contain highly non-linear terms, the solutions may be diverging. To overcome and remedy that shortcoming, the multi-step differential transform method (Ms-DTM) is applied. This procedure provides the solution in terms of convergent series over a sequence of sub-intervals. Different researches of the Ms-DTM can be found in refs [35–38].

The DTM is unworkable for solving partial differential equations with highly non-linear behavior at infinity. In these cases the series solution does not exhibit the real behaviors of the problem but gives a good approximation to the true solution in a very small region and it has slow convergent rate or completely divergent in the wider region. For this meditate multi-step DTM has been deliberated for the analytical solution of the differential equations along the solicited domain. Multi-step DTM can be defined as the following [35–38].

Let t ∈ [t ₀, t ₀ + L] be the domain of definition of the solution where subdivided into m ∈ Z pieces, L = mh. Here h is chosen sufficiently small so that the series could converge in the subintervals [t _i, t _i+1], t _i = t ₀ + ih, i =1, 2, …, m of equal step size $h=\frac{L}{m}$ by using the nodes t _i = ih with the initial values 𝛼_i. Here 𝛼_i is approximately calculated by the sum of the Taylor series at the second boundary of the preceding subdomain, namely ${\alpha}_{i}=\sum _{k=∼0}^{n}[Y_{k}]_{i-1}h^{k}$ Following this procedure, then eventually one gets, step by step, an approximate solution in the whole domain. The procedure has also been utilized to evaluate boundary value problems. In that case, the initial value 𝛼₀ that corresponds to the first boundary condition is designed systematically via the parameters 𝛼_i, up to the second boundary t ₀ + L where the desired condition is dictated.

The following similar arguments are given by 𝛹(y), 𝛩(y) and Φ(y), the peristaltic flow problem is modulated through the Ms-DTM as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{[k+1][k+2][k+3][k+4]{\Psi}[k+4]}{1+{\lambda}_{1}}}-{\displaystyle \frac{{\beta}}{1+{\lambda}_{1}}}\mathop{\sum }_{r=0}^{k}\left[\begin{array}{@{}c@{}}[r+1][k-r+1][k-r+2]\\ {\Theta}[k-r+2]{\Psi}[k-r+2]\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼{\displaystyle \frac{2{\beta}}{1+{\lambda}_{1}}}\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1][k-r+2][k-r+3]{\Theta}[k-r+1]{\Psi}[k-r+3]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -{\displaystyle \frac{{\beta}}{1+{\lambda}_{1}}}\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1][k-r+2][k-r+3][k-r+4]{\Theta}[r+1]{\Psi}[k-r+4]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼G_{t}[k+1]{\Theta}[k+1]+G_{c}[k+1]{\Phi}[k+1]-{\displaystyle \frac{M^{2}}{1+m^{2}}}[k+1][k+2]{\Psi}[k+2]\end{eqnarray}$ (33) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\epsilon}\left[\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1]{\Theta}[r+1]{\Theta}[k-r+1]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.\quad +∼\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1][k-r+2]{\Theta}[r+1]{\Theta}[k-r+2]\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼[k+1]{\Theta}[k+2]+B_{r}N_{t}\left[\mathop{\sum }_{r=0}^{k}[r+1][k-r+1]{\Theta}[k-r+1]{\Phi}[k-r+2]\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼B_{r}N_{b}\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad {\Theta}[r+1]{\Theta}[k-r+1]-M^{2}B_{r}\left[1+\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1]{\Psi}[k-r+1]+2[k+1]{\Psi}[k+1]\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼{\displaystyle \frac{{\beta}}{1+{\lambda}_{1}}}\left[\displaystyle \mathop{\sum }_{r=0}^{k}[[r+1][k-r+1][k-r+2]{\Psi}[k-r+2]]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.-∼{\beta}\displaystyle \mathop{\sum }_{r=0}^{k}[r+1][k-r+1][k-r+2]{\Theta}[r+1]{\Psi}[k-r+2]\right]\end{eqnarray}$ (34) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle [k+1][k+2]{\Phi}[k+2]+{\displaystyle \frac{N_{t}}{N_{b}}}[k+1][k+2]{\Theta}[k+2]=0.\end{eqnarray}$ (35) with the associated boundary conditions $\begin{eqnarray}\displaystyle {\Psi}_{0}=-{\displaystyle \frac{q}{2}},\quad {\Psi}_{1}=-1,\quad {\Theta}_{0}=1,\quad {\Phi}_{0}=1\quad \text{at}∼y=h_{2}(x)=-d-bCos(2{\pi}x+{\phi}) & & \displaystyle\end{eqnarray}$ (36) $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{k=0}^{n}{\Psi}_{k}[h_{1}(x)-h_{2}(x)]^{k}={\displaystyle \frac{q}{2}},\quad \displaystyle \mathop{\sum }_{k=1}^{n}k{\Psi}_{k}[h_{1}(x)-h_{2}(x)]^{k-1}=-1,\\ \displaystyle \mathop{\sum }_{k=0}^{n}{\Theta}_{k}[h_{1}(x)-h_{2}(x)]^{k}=0,\quad \displaystyle \mathop{\sum }_{k=0}^{n}{\Phi}_{k}[h_{1}(x)-h_{2}(x)]^{k}=0\quad \text{at }y=h_{1}(x)=1+aCos(2{\pi}x)\end{array} & & \displaystyle\end{eqnarray}$ (37) Solving the recurrence relations (33–35) the semi-analytical solutions for 𝜓(y), θ(y) and φ(y) at $Q=-1,{\lambda}_{1}=∼1.5,P_{r}=∼2,{\epsilon}=∼0.1,N_{t}=∼0.4,N_{b}=∼0.5,G_{c}=∼1.1,G_{t}=∼1.2,M=1,B_{r}=∼0.5,a=∼0.4,b=∼0.5,d=∼1.5,m=∼2,{\phi}=\frac{{\pi}}{3}$ are given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}(y)-0.25-1.y+2.5217791993961174y^{2}-1.6758416674056207y^{3}+0.6266368599938942y^{4}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼0.15625251519209107y^{5}+0.0007418763771432943y^{6}+0.03724298836219623y^{7}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -0.03533291225323325y^{8}+0.022276156383795046y^{9}-0.01009828409986365y^{10}.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\theta}(y)=1+0.5020739524699355y-0.3942634846737988y^{2}+0.5995805744516555y^{3}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼0.9290146460089714y^{4}+0.8273067379377093y^{5}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼0.5281636431299402y^{6}+0.2850882758483168y^{7}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼0.15551633640473284y^{8}+0.09849163471767083y^{9}-0.07149257815896616y^{10}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\varphi}(y)=1-1.3009014413766216y+0.4731161816085585y^{2}-0.7194966893419865y^{3}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼1.1148175752107656y^{4}-0.9927680855252511y^{5}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼0.6337963717559283y^{6}-0.34210593101798015y^{7}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼0.1866196036856794y^{8}-0.11818996166120499y^{9}+0.08579109379075939y^{1}0.\nonumber\end{eqnarray}$

4. Numerical results and discussion

Semi-analytical solutions are given in Eqs (33–35) are illustrated graphically through a set of figures. These figures enable us to judge and control the influences of the problem parameters on the various distributions. To do this, we shall consider the following subsections:

Fig. 2.

Pressure rise versus flow rate for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, N _t = 0.4, N _b = 0.5,G _c = 1.1, G _t = 1.2, M = 1, $B_{r}= 0.5,a= 0.4,b= 0.5,d= 1.5,m= 2, \phi=\frac{{\pi}}{3}$ .

Fig. 3.

Pressure rise versus flow rate for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, $B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2, \phi=\frac{{\pi}}{3}$ .

Fig. 4.

Pressure rise versus flow rate for Q = −1, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, $B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

4.1. Pressure rise

Figures (2) and (3) show the variation of the pressure rise ΔP versus the volume flow rate Q for different values of viscosity parameter 𝛽 and Brownian motion parameter N _b, respectively. It is seen from Figs (2) and (3), that the pressure rise decreases, in all pumping regions, with the increasing of 𝛽, while it hardly increases, in all pumping regions, with the increase of N _b, respectively. It is also noted that the pressure rise for different values of 𝛽 and N _b decreases with increasing the volume flow rate, and for small values of both 𝛽 and N _b, the relation between rise ΔP and Q is inversely linear. The behaviors of ΔP with G _c, N _t and 𝜖 are found to be similar to the curves in Fig. (2) with the only difference that the obtained curves are very close to each other near the left side wall. Figure (4) shows the variation of the pressure rise ΔP versus the volume flow rate Q for different values of the ratio of relaxation to retardation times 𝜆₁. It also indicates that the pressure rise decreases in the retrograde pumping (ΔP > 0, Q < 0) and peristaltic pumping (ΔP > 0, Q > 0) regions with an increase of 𝜆₁, while the pressure rise increases in co-pumping region (ΔP < 0, Q > 0) with an increase of 𝜆₁.

Fig. 5.

Pressure rise versus flow rate for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, N _t = 0.6, 𝛽 = 0.1, $G_{t}= 1.2,M= 1,B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}.$

Fig. 6.

Pressure gradient versus x for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, $B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 7.

Pressure gradient versus x for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, $B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}.$

Fig. 8.

Velocity profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, 𝛽 = 0.1, G _t = 1.2, M = 1, B _r = 0.4, $a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 9.

Velocity profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.5, 𝛽 = 0.1, G _t = 1.2, M = 1, B _r = 0.4, $a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 10.

Velocity profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, N _t = 0.6, G _t = 1.2, M = 1, B _r = 0.4, $a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 11.

Temperature profile for Q = −1, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, B _r = 0.4, $a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}.$

Fig. 12.

Temperature profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, B _r = 0.4, $a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 13.

Nanoparticles distribution profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _b = 0.5, 𝛽 = 0.1, G _t = 1.2, $M= 1,B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

Fig. 14.

Nanoparticles distribution profile for Q = −1, 𝜆₁ = 1.5, P _r = 2, 𝜖 = 0.1, G _c = 1.2, N _t = 0.6, 𝛽 = 0.1, G _t = 1.2, M = 1, $B_{r}= 0.4,a= 0.4,b= 0.5,d= 1.5,m= 2,\phi=\frac{{\pi}}{3}$ .

4.2. Pumping characteristics

The variations of the pressure gradient $\frac{dp}{dx}$ versus the space coordinate x for different values of the local nanoparticle Grashof number G _c and Brownian motion parameter N _b are illustrated through Figs (5) and (6), respectively. Figures (5) and (6) indicates that the pressure gradient decreases with the increase of the parameter G _c, i.e., the flow can easily pass without imposition of a large pressure gradient [24]. Contrary while it increases along the channel by increasing N _b, respectively. It is also noted that $\frac{dp}{dx}$ increases with x till a definite value x = x ₀ (represents the maximum value of $\frac{dp}{dx}$ ) and it decreases afterwards. This maximum value of $\frac{dp}{dx}$ increases by increasing N _b, while it decreases by increasing G _c. The effects of both 𝜆₁, 𝜖 and N _t are found to be similar to those obtained in Fig. (5), except that in the left side of the channel, there is no noticeable difference. Figure (7) illustrates the effect of 𝛽 on the pressure gradient $\frac{dp}{dx}$ as a function of the space coordinate x. It is found that in the interval of the space coordinate x ∈ [0.25,0.55], the behavior of $\frac{dp}{dx}$ for various values of 𝛽 is exactly similar to the behavior of $\frac{dp}{dx}$ for various values of N _b given in Fig. (6). It is also noted, from Fig. (7) that in the interval of the space coordinate x ∈ [0,0.25] ∪ [0.55,1], the behavior of $\frac{dp}{dx}$ is an inversed manner of its behavior in the interval x ∈ [0.25, 0.55].

4.3. Velocity profile

Figures (8) and (9) are plotted to indicate the effects of the thermophoresis parameter N _t and Brownian motion parameter N _b on the velocity profile u. It is evident from Fig. (8) that the magnitude of the velocity profile decreases at the central part of the channel, namely in the region y ∈ [−0.75,0.75] with the increase of N _t, while the magnitude value of the velocity profile increases in the left and right hand sides of the channel with the increase of N _t , namely in the regions y ∈ [−1.7, −0.74] ∪ [0.76, 1.4]. Further, we observe from Fig. (9) that the magnitude of the velocity profile increases at the central part of the channel in the region y ∈ [−0.83,0.64] with the increase of N _b, Meanwhile the magnitude value of the velocity profile decreases in the left hand side of that channel with the increase of N _b in the regions y ∈ [−1.7, −0.84] ∪ [0.65,1.4]. Figure (10) shows the variation of the velocity profile u with y for various values of viscosity parameter 𝛽. It indicates that the magnitude of the velocity profile increases in the left hand side of the channel where y ∈ [−1.65, −0.25] with the increase of 𝛽. In the right hand side of the channel, the magnitude value of the velocity profile decreases with the increase of 𝛽, where y ∈ [−0.26,1.4]. These results are in agreement with [19]. The effects of the other parameters are found to be similar to those obtained in Fig. (10); these figures are excluded here to avoid any kind of repetition.

Table 1
Comparison of numerical results for distributions of temperature and nanoparticles

y (x, t) θ(x, y) θ(x, y) Error of solutions φ(x, y) φ(x, y) Error of solutions

Eldabe et al. results Alvi et al. 2016 [19] Eldabe et al. results Alvi et al. 2016 [19]

1. 1 1 0 1 1. 0

0.78 1.00914774689 1.0091477779 3.103 × 10⁻⁸ 0.65805752768 0.658057456905 7.078 × 10⁻⁸

0.56 0.99943658622 0.9994366160 2.982 × 10⁻⁸ 0.38359669620 0.383596622756 7.344 × 10⁻⁸

0.34 0.96951797135 0.9695180025 3.116 × 10⁻⁸ 0.16581874330 0.165818660230 8.307 × 10⁻⁸

0.12 0.91738459278 0.9173846251 3.241 × 10⁻⁸ −0.0020804025 −0.00208049856 9.605 × 10⁻⁸

−0.1 0.84052142436 0.8405214586 3.433 × 10⁻⁸ −0.1235250919 −0.12352520573 1.137 × 10⁻⁷

−0.32 0.73610782824 0.7361078613 3.306 × 10⁻⁸ −0.1991068070 −0.19910693960 1.325 × 10⁻⁷

−0.54 0.60135613968 0.6013561618 2.216 × 10⁻⁸ −0.2272502924 −0.22725043296 1.405 × 10⁻⁷

−0.76 0.43404295908 0.4340429599 8.354 × 10⁻¹⁰ −0.2051372047 −0.20513734060 1.358 × 10⁻⁷

−0.98 0.23322117596 0.2332211544 2.153 × 10⁻⁸ −0.1299096348 −0.12990976743 1.326 × 10⁻⁷

−1.2 5.176 × 10⁻⁸ −2.5 × 10⁻¹⁶ 5.176 × 10⁻⁸ 1.2839 × 10⁻⁷ 1.5866 × 10⁻¹⁶ 1.283 × 10⁻⁷

y (x, t)	θ(x, y)	θ(x, y)	Error of solutions	φ(x, y)	φ(x, y)	Error of solutions
1.	1	1	0	1	1.	0
0.78	1.00914774689	1.0091477779	3.103 × 10⁻⁸	0.65805752768	0.658057456905	7.078 × 10⁻⁸
0.56	0.99943658622	0.9994366160	2.982 × 10⁻⁸	0.38359669620	0.383596622756	7.344 × 10⁻⁸
0.34	0.96951797135	0.9695180025	3.116 × 10⁻⁸	0.16581874330	0.165818660230	8.307 × 10⁻⁸
0.12	0.91738459278	0.9173846251	3.241 × 10⁻⁸	−0.0020804025	−0.00208049856	9.605 × 10⁻⁸
−0.1	0.84052142436	0.8405214586	3.433 × 10⁻⁸	−0.1235250919	−0.12352520573	1.137 × 10⁻⁷
−0.32	0.73610782824	0.7361078613	3.306 × 10⁻⁸	−0.1991068070	−0.19910693960	1.325 × 10⁻⁷
−0.54	0.60135613968	0.6013561618	2.216 × 10⁻⁸	−0.2272502924	−0.22725043296	1.405 × 10⁻⁷
−0.76	0.43404295908	0.4340429599	8.354 × 10⁻¹⁰	−0.2051372047	−0.20513734060	1.358 × 10⁻⁷
−0.98	0.23322117596	0.2332211544	2.153 × 10⁻⁸	−0.1299096348	−0.12990976743	1.326 × 10⁻⁷
−1.2	5.176 × 10⁻⁸	−2.5 × 10⁻¹⁶	5.176 × 10⁻⁸	1.2839 × 10⁻⁷	1.5866 × 10⁻¹⁶	1.283 × 10⁻⁷

4.4. Heat and nanoparticles distribution characteristics

Figures (11) and (12) display the temperature distribution θ versus the coordinate y for different values of the ratio of relaxation to retardation times 𝜆₁ and Brownian motion parameter N _b, respectively. It is clear from these figures that the temperature increases as y increases till a maximum value after which T decreases. Also Fig. (11) depicts that the distribution of temperature decreases with the increase of 𝜆₁, but Fig. (12) shows that the molecules of the liquid increase in the temperature supplies it with more energy. Because of these results the liquid can move more freely. The effects of G _c, N _t and m on the temperature distribution were found to be exactly similar to the effect of N _b given in Fig. (12).

Figures (13) and (14) display the influence of the thermophoresis parameter N _t and Brownian motion parameter N _b on the nanoparticles distribution φ, respectively. Figure (13) shows that the nanoparticles distribution decreases with the increase of N _t, while we notice from Fig. (14) that the nanoparticles distribution increases with the increase of N _b. Also, the nanoparticles distribution decreases with y to a minimum value (at a finite value of y: y = y ₀) after which it increases. The behavior of φ for different values of 𝛽, 𝜆₁ is found to be exactly similar to the curves in Fig. (14).

Table 1 explains a good approval among the Mathematica software package (Parametric NDSolve) and Ms-DTM.

5. Conclusions

This paper studies a peristaltic flow of a Jeffrey nanofluid in an asymmetric channel. The governing equations of temperature and nano particle volume fraction are solved semi-analytically, using the Ms-DTM. The results of our study can be summarized as follows:

Gradual pressure rise decreases in all pumping regions with an increase in 𝜖.

Pressure gradient decreases with an increase in 𝜖 and N _t

The effects of N _t and N _b on pressure gradient are qualitatively opposite.

As seen from Figs 11–12, the distribution of velocity has dual role phenomena.

Brownian motion parameter enhances the temperature.

Brownian motion and thermophoresis parameters have opposite effects on the nanoparticles concentration.

When m = 0 and 𝜖 = 0 the same results as obtained by Alvi et al. [19] are recovered.

Accurate comparisons of the obtained results are available and shown in Table 1.

Footnotes

Acknowledgements

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

Nomenclature

Roman Symbols

Greek symbols

References

Nadeem

and Akram

, Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer, Acta Mechanica Sinica 27 (2011), 237–250.

Eldabe

N.T.

Moatimid

G.M.

and Sayed

, Effect of the magnetic induction on the peristaltic Eyring–Prandtl nanofluid through porous medium, Int J Appl Electromagn Mech 63(2) (2015), 279–298.

Abou-zeid

M.Y.

Shaaban

A.A.

and Alnour

M.Y.

, Numerical treatment and global error estimation of natural convective effects on gliding motion of bacteria on a power-law nanoslime through a non-Darcy porous medium, Journal of Porous Media 18(11) (2015), 1091–1106.

Bergstrom

L.B.

, Transient growth of small disturbances in a Jeffrey fluid flowing through a pipe, Fluid Dynamics Research 32 (2003), 29–44.

Nadeem

and Akram

, Influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric or asymmetric channel, Asia-Pacific Journal of Chemical Engineering 7 (2012), 33–44.

Hayat

Ahmad

and Ali

, Effects of an endoscope and magnetic field on the peristalsis involving Jeffrey fluid, Communications in Nonlinear Science and Numerical Simulation 13 (2008), 1581–1591.

Eldabe

N.T.

Abou-zeid

M.Y.

and Younis

Y.M.

, Magnetohydrodynamic peristaltic flow of Jeffry nanofluid with heat transfer through a porous medium in a vertical tube, Applied Mathematics & Information Sciences 11 (2017), 1097–1103.

Abou-zeid

M.Y.

and Mohamed

M.A.A.

, Homotopy perturbation method for creeping flow of non-Newtonian power-law nanofluid in a non-uniform inclined channel with peristalsis, Z. Naturforsch 72(10) (2017), 899–907.

Abou-zeid

M.Y.

, Implicit homotopy perturabation method for mhd non-newtonian nanofluid flow with Cattaneo–Christov heat flux due to parallel rotating disks, J. Nanofluids 8(8) (2019), 1648–1653.

10.

Mekheimer

Kh.S.

and Abd Elmaboud

, Simultaneous effects of variable viscosity and thermal conductivity on peristaltic flow in a vertical asymmetric channel, Can. J. Phys. 92 (2014), 1541–1555.

11.

Elogail

M.A.

and Elshekhipy

A.A.

, Approximate analytical solutions to non-linear peristaltic flow with temperature-dependent viscosity parameters: Application of multistep differential transform method (MsDTM), Can. J. Phys. 96 (2018), 287–299.

12.

Eldabe

N.T.

El-Sayed

M.F.

Ghaly

A.Y.

and Sayed

H.M.

, Mixed convective heat and mass transfer non Newtonian fluid at a peristaltic surface with temperature dependent viscosity, Arch. Appl. Mech. 78 (2007), 599–624.

13.

Asghar

Hussain

and Hayat

, Peristaltic flow of reactive viscous fluid with temperature dependent viscosity, Math. Comput. Appl. 18 (2013), 198–220.

14.

Nadeem

and Noreen

S.A.

, Peristaltic flow of a Jeffrey fluid with variable viscosity in an asymmetric channel, Z. Naturforsch 64a (2009), 713–722.

15.

Abbasi

F.M.

Hayat

and Ahmed

, Hydromagnetic peristaltic transport of variable viscosity fluid with heat and porous medium, Appl. Math. Inf. Sci. 10 (2016), 2173–2181.

16.

Eldabe

N.T.

Moatimid

G.M.

Abouzeid

M.Y.

Elshekhipy

A.A.

and Abdallah

N.F.

, A semianalytical technique for MHD peristalsis of pseudoplastic nanofluid with temperature-dependent viscosity: Application in drug delivery system, Heat Transfer - Asian Research 49(1) (2020), 424–440.

17.

Manjunatha

Rajashekhar

Vaidya

Prasad

K.V.

and Vajravelu

, Impact of heat and mass transfer on the peristaltic mechanism of Jeffery fluid in a non-uniform porous channel with variable viscosity and thermal conductivity, J Therm Anal Calorim 139 (2020), 1213–1228.

18.

Das

, Influence of thermophoresis and chemical reaction on MHD micropolar fluid flow with variable fluid properties, International Journal of Heat and Mass Transfer 55 (2012), 7166–7174.

19.

Alvi

Latif

Hussain

and Asghar

, Peristalsis of nonconstant viscosity Jeffrey fluid with nanoparticles, Results Phys 6 (2016), 1109–1125.

20.

Eldabe

N.T.M.

Rizkallah

R.R.

Abou-zeid

M.Y.

and Ayad

V.M.

, Thermal diffusion and diffusion thermo effects of Eyring Powell nanofluid flow with gyrotactic microorganisms through the boundary layer, Heat Transfer — Asian Res 49 (2020), 383–405.

21.

Eldabe

N.T.M.

Abou-zeid

M.Y.

Mohamed

M.A.A.

and Abd-Elmoneim

M.M.

, MHD peristaltic flow of non-Newtonian power-law nanofluid through a non-Darcy porous medium inside a non-uniform inclined channel, Arch Appl Mech 91 (2021), 1067–1077.

22.

Eldabe

N.T.M.

Abou-zeid

M.Y.

and Ahmed

O.S.

, Motion of a thin film of a fourth grade nanofluid with heat transfer down a vertical cylinder: Homotopy perturbation method application, J. Adv. Res. Fluid Mech. Thermal Sci. 66(2) (2020), 101–113.

23.

Eldabe

N.T.M.

and Abou-zeid

M.Y.

, Radially varying magnetic field effect on peristaltic motion with heat and mass transfer of a non-Newtonian fluid between two co-axial tubes, Thermal Sci 22(6A) (2018), 2449–2458.

24.

Eldabe

N.T.

Shaaban

A.A.

Abouzeid

M.Y.

and Ali

H.A.

, Magnetohydrodynamic non-Newtonian nanofluid flow over a stretching sheet through a non-Darcy porous medium with radiation and chemical reaction, J. Comput. and Theoret. Nanosci. 12 (2015), 5363–5371.

25.

Abo-Eldahab

E.M.

Barakat

E.I.

and Nowar

K.I.

, Effects of Hall and ion-slip currents on peristaltic transport of a couple stress fluid, Int. J. Appl. Math. Phys. 2 (2010), 145–157.

26.

Mekheimer

K.S.

and El-Kot

M.A.

, Influence of magnetic field and Hall currents on blood flow through a stenotic artery, Appl. Math. Mech.-Engl. Ed. 29(8) (2008), 1093–1104.

27.

Eldabe

N.T.

Moatimid

G.M.

Mohamed

M.A.A.

and Mohamed

Y.M.

, Effects of Hall currents with heat and mass transfer on the peristaltic transport of a Casson fluid through a porous medium in a vertical circular cylinder, Therm. Sci. 24 (2020), 1067–1081.

28.

Gad

N.S.

, Effects of hall currents on peristaltic transport with compliant walls, Appl. Math. Comput. 235 (2014), 546–554.

29.

Abou-zeid

M.Y.

, Homotopy perturbation method to gliding motion of bacteria on a layer of power-law nanoslime with heat transfer, J. Comput. Theor. Nanosci. 12 (2015), 3605–3614.

30.

Eldabe

N.T.M.

Hassan

M.A.

and Abou-zeid

M.Y.

, Wall Properties effect on the peristaltic motion of a coupled stress fluid with heat and mass transfer through a porous media, J. Eng. Mech. 142(3) (2015), 4015102.

31.

Mohamed

M.A.A.

and Abou-zeid

M.Y.

, MHD peristaltic flow of micropolar Casson nanofluid through a porous medium between two co-axial tubes, J Porous Media 22(9) (2019), 1079–1093.

32.

Mansour

H.M.

and Abou-zeid

M.Y.

, Heat and mass transfer effect on non-Newtonian fluid flow in a non-uniform vertical tube with peristalsis, J. Adv. Res. Fluid Mech. Thermal Sci. 61(1) (2019), 44–62.

33.

Abou-zeid

M.Y.

, Homotopy perturbation method for couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid, Microsyst. Technol. 24 (2018), 4839–4849.

34.

Zhou

J.K.

, Differential Transformation Its Applications for Electrical Circuits, Huaz hong University Press, Wuhan, China, 1986.

35.

Momani

Odibat

and Erturk

, Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation, Phys. Lett. A 370 (2007), 379–387.

36.

Erturk

Momani

and Odibat

, Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 1642–1654.

37.

Odibat

Z.M.

Bertelle

Aziz-Alaoui

M.A.

and Duchamp

G.H.E.

, A multistep differential transform method and application to non-chaotic or chaotic systems, Comput. Math. with Appl. 59 (2010), 1462–1472.

38.

Abou-zeid

M.Y.

, Magnetohydrodynamic boundary layer heat transfer to a stretching sheet including viscous dissipation and internal heat generation in a porous medium, J. Porous Media 14(11) (2011), 1007–1018.