Homotopy perturbation approach for Ohmic dissipation and mixed convection effects on non-Newtonian nanofluid flow between two co-axial tubes with peristalsis

Abstract

In this study, the effects of internal heat generation and mixed convection with thermal radiation on peristaltic motion of a non-Newtonian fluid are investigated. The fluid used is third-grade model. The flow is through the gap between two co-axial vertical tubes under the effect of radially varying magnetic field. The outer tube is flexible with sinusoidal deformations. The problem is modulated mathematically by a system of partial differential equations which describes the equations of momentum, heat transfer and nanoparticles concentration which are simplified by using long wave length and low-Reynolds number assumptions. The closed solutions of fluid temperature and nanoparticle concentration are obtained, and the solution of velocity is obtained by using the homotopy perturbation method (HPM). The radially varying magnetic field effect on the axial velocity is discussed and it is shown that the increase of magnetic field parameter tends to reduce the fluid flow.

Keywords

Non-Newtonian nanofluid peristaltic flow varying magnetic field heat transfer

1. Introduction

A nanofluid is a liquid consisted of very small particles of diameter less than 100 nm. Although, the study of peristaltic flow of traditional fluids has been attracted by many researchers, only a few of them tends to the study of peristaltic flow of nanofluids. Recently, the research area of nanofluids has gained a big importance because it has applications in many fields such as physiology, engineering, computer science and medicine. Abouzeid and Mohamed [1] discussed the peristaltic transport of power-law nanofluid under the effects of thermal radiation, viscous dissipation and internal heat generation. The problem of peristaltic flow of non-Newtonian nanofluid in the presence of mixed convective effects through asymmetric channel was studied by Hayat et al. [2]. Mekheimer et al. [3] analyzed the flow of third-grade nanofluid between two co-axial pipes under the effect of mixed convection. There are many studies which represent peristaltic transport of nanofluid under different external forces [4–7].

The study of non-Newtonian fluids are differ from Newtonian fluids due to its applications in industry, engineering and medicine. Due to the sub-stances contained like fibrinogen, protein and the blood cell’s chain structure, the human blood can classified as third-grade fluid. A third-grade fluid is a subclass of non-Newtonian fluids which has attracted many researchers particularly in describing the blood properties. The inclined magnetic field effect on peristaltic flow of non-Newtonian fluid with heat transfer in asymmetric channel is scrutinized by Nadeem et al. [8]. In another paper, Hayat et al. [9] examined the problem of peristaltic motion of Jeffrey fluid under the effect of an endoscope and magnetic field. MHD peristaltic flow of biviscosity nanofluid through a porous medium in eccentric annuli is studied by Abouzeid [10]. Eldabe et al. [11] studied MHD Eyring powell nanofluid transport over a stretching sheet with non-Darcy porous medium under the effects of chemical reaction and radiation. Bhatti et al. [12] investigated the entropy analysis and mass transfer process on asymmetric peristaltic propulsion of Williamson nanofluid under convective and magnetic forces. Khan et al. [13] examined the flow of Oldroyd-B nanofluid containing the gyrotactic microorganisms in presence of partial slip effects. Shahid et al [14] analyzed the dissipative influence into an unsteady MHD fluid flow embedded in a porous medium over a shrinkable sheet. Some others studies regarding peristaltic flow of non-Newtonian fluids are presented in Refs [15–19].

Physicists, mathematicians and engineers turned their attention towards the radially MHD flow due its applications such as optimization, the process solidification, MHD power generators and reduction of bleeding during surgeries. Information on this topic is quite sensible and several useful references in their investigations [20–26].

The present study extends the work of Mekheimer et al. [3] to cover the effects of internal heat generation, radially varying magnetic field and thermal radiation. The governing equations are simplified using long wavelength and low-Reynolds number assumptions. The analytical solution of axial velocity is obtained using HPM [27–30], while the temperature and nanoparticles concentration have an exact solution. Effects of the physical parameters of problem on these solutions are discussed numerically and depicted graphically. No doubt that this study has practical significance in many scientific fields such as medical, physiological and industrial fields. Physically, our model matches to the gastric juice transport in the small intestine when an endoscope is put in it.

2. Mathematical model

Consider a third-grade fluid saturated through the gap between two co-axial tubes with internal radius R ₁ and external radius R ₂. The inner tube is uniform and fixed, while the outer one has a sinusoidal wave. A varying magnetic field B (R) stressed the system. Consider a cylindrical coordinates system (R, θ, Z); the radii equations are: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{1}=b_{1},\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{2}=b_{2}+b∼\cos \frac{2{\pi}}{{\lambda}}(Z-ct).\end{eqnarray}$ (2) The constitutive relations for an incompressible third-grade fluid are given by [3]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{{\tau}}=-PI+\text{}\underline{S},\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S={\mu}∼A_{1}+{\alpha}_{1}A_{2}+{\alpha}_{2}{A_{1}}^{2}+{\beta}_{1}A_{3}+{\beta}_{2}(A_{1}A_{2}+A_{2}A_{1})+{\beta}_{3}(tr∼{A_{1}}^{2})A_{1},\end{eqnarray}$ (4) where 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ and 𝛽₃are the constants of material and A ₁, A ₂ and A ₃ are Rivlin–Ericksen tensors and can be defined as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{1}={\nabla}\text{}\underline{v}+({\nabla}\text{}\underline{v})^{t},\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{n+1}={\displaystyle \frac{dA_{n}}{dt}}+A_{n}{\nabla}\text{}\underline{v}+({\nabla}\text{}\underline{v})^{t}A_{n},\quad (n=1,2,\ldots ).\end{eqnarray}$ (6) The flow parameters are not depend on the azimuthal coordinate θ, so, the velocity is taken in the form v = (U,0, W). The electric current J generated magnetic field B which is defined by Biot–Savart law as follows: $\begin{eqnarray}\displaystyle B=B_{0}{\displaystyle \frac{R_{1}}{R}},\quad B_{0}={\displaystyle \frac{{\mu}_{0}J}{2{\pi}R_{1}}}. & & \displaystyle\end{eqnarray}$ (7) The governing equations for an incompressible flow in the fixed frame are given as [26] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial U}{\partial R}}+{\displaystyle \frac{U}{R}}+{\displaystyle \frac{\partial W}{\partial Z}}=0,\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{f}∼\left(U{\displaystyle \frac{\partial U}{\partial R}}+W{\displaystyle \frac{\partial U}{\partial Z}}\right)=-{\displaystyle \frac{\partial P}{\partial R}}+{\displaystyle \frac{\partial {\tau}_{RR}}{\partial R}}+{\displaystyle \frac{\partial {\tau}_{RZ}}{\partial Z}}-{\displaystyle \frac{{\sigma}B_{0}^{2}}{R^{2}}}U,\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle {\rho}_{f}∼\left(U{\displaystyle \frac{\partial W}{\partial R}}+W{\displaystyle \frac{\partial W}{\partial Z}}\right)\text{} & =\text{} & \displaystyle -{\displaystyle \frac{\partial P}{\partial Z}}+{\displaystyle \frac{\partial {\tau}_{RZ}}{\partial R}}+{\displaystyle \frac{\partial {\tau}_{ZZ}}{\partial Z}}-{\displaystyle \frac{{\sigma}B_{0}^{2}}{R^{2}}}W\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\rho}_{f}∼g{\alpha}(T-T_{1})+{\rho}_{f}∼g{\alpha}(f-f_{1}),\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\rho}∼c)_{f}\left(U{\displaystyle \frac{\partial T}{\partial R}}+W{\displaystyle \frac{\partial T}{\partial Z}}\right)=k∼\left({\displaystyle \frac{\partial ^{2}T}{\partial R^{2}}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial T}{\partial R}}+{\displaystyle \frac{\partial ^{2}T}{\partial Z^{2}}}\right)-{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial (Rq_{r})}{\partial R}}+{\sigma}B_{0}^{2}(U^{2}+W^{2})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad +∼Q_{0}(T-T_{2})+({\rho}c)_{p}∼\left[D_{B}∼\left({\displaystyle \frac{\partial T}{\partial R}}{\displaystyle \frac{\partial f}{\partial R}}+{\displaystyle \frac{\partial T}{\partial Z}}{\displaystyle \frac{\partial f}{\partial Z}}\right)+\frac{D_{T}}{T_{2}}\left(\left({\displaystyle \frac{\partial T}{\partial R}}\right)^{2}+\left({\displaystyle \frac{\partial T}{\partial Z}}\right)^{2}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad +∼{\displaystyle \frac{{\rho}_{f}D_{B}K_{T}}{C_{s}}}\left({\displaystyle \frac{\partial ^{2}f}{\partial R^{2}}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial f}{\partial R}}+{\displaystyle \frac{\partial ^{2}f}{\partial Z^{2}}}\right),\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle U{\displaystyle \frac{\partial f}{\partial R}}+W{\displaystyle \frac{\partial f}{\partial Z}}=D_{B}\left({\displaystyle \frac{\partial ^{2}f}{\partial R^{2}}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial f}{\partial R}}+{\displaystyle \frac{\partial ^{2}f}{\partial Z^{2}}}\right)+{\displaystyle \frac{D_{T}K_{T}}{T_{2}}}\left({\displaystyle \frac{\partial ^{2}T}{\partial R^{2}}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial T}{\partial R}}+{\displaystyle \frac{\partial ^{2}T}{\partial Z^{2}}}\right). & & \displaystyle\end{eqnarray}$ (12) By using Rosseland approximation [22], the radiative heat flux is given by $\begin{eqnarray}\displaystyle q_{r}={\displaystyle \frac{-4{\sigma}^{\ast }}{3k_{R}}}{\displaystyle \frac{\partial T^{4}}{\partial r}}. & & \displaystyle\end{eqnarray}$ (13) Then, T ⁴ can be expanded in a Taylor series about T ₂ with neglecting higher-order terms, $\begin{eqnarray}\displaystyle T^{4}\approx 4T_{2}^{3}T-3T_{2}^{4}. & & \displaystyle\end{eqnarray}$ (14) Introducing a wave frame (r, z) which moving with the velocity c away from the fixed frame (R, Z) by the transformations: r = R, z = Z − ct, u = U − c, w = W.

We used the following non-dimensional variables to simplify the above equations. $\begin{eqnarray} \displaystyle \begin{array}{@{} l@{}}r^{\ast }={\displaystyle \frac{r}{b_{2}}},\quad z^{\ast }={\displaystyle \frac{z}{{\lambda}}},\quad u^{\ast }={\displaystyle \frac{{\lambda}}{cb_{2}}}u,\quad w^{\ast }={\displaystyle \frac{w}{c}},\quad P^{\ast }={\displaystyle \frac{{b_{2}}^{2}}{{\lambda}c{\mu}}}P,\quad T^{\ast }={\displaystyle \frac{T-T_{2}}{T_{1}-T_{2}}},\\[0.5pc] {\delta}={\displaystyle \frac{b_{2}}{{\lambda}}},\quad {Q_{0}}^{\ast }={\displaystyle \frac{Q_{0}b_{2}}{{\rho}c_{p}c}},\quad f^{\ast }={\displaystyle \frac{f-f_{2}}{f_{1}-f_{2}}},\quad t^{\ast }={\displaystyle \frac{c}{{\lambda}}}t,\quad h={\displaystyle \frac{H}{b_{2}}},\quad {\varepsilon}={\displaystyle \frac{b_{1}}{b_{2}}},\quad {\varphi}={\displaystyle \frac{b}{b_{2}}} \end{array} . & & \displaystyle \end{eqnarray}$ (15) Equations ((8))–((12)) become, after dropping star mark and assuming long wavelength and low-Reynolds number approximation: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial P}{\partial r}}=0,\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial P}{\partial z}}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial w}{\partial r}}\right)+{\gamma}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r\left({\displaystyle \frac{\partial w}{\partial r}}\right)^{3}\right)-{\displaystyle \frac{M^{2}}{r^{2}}}w+Gr∼T+Br∼f,\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle 0\text{} & =\text{} & \displaystyle {\displaystyle \frac{1}{\Pr }}\left(1+{\displaystyle \frac{4}{3}}R_{1}\right)\left(\frac{\partial ^{2}T}{\partial r^{2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}\right)+Nt∼\left({\displaystyle \frac{\partial T}{\partial r}}\right)^{2}+Nb∼\left({\displaystyle \frac{\partial f}{\partial r}}\frac{\partial T}{\partial r}\right)+{\displaystyle \frac{1}{Sh}}\left({\displaystyle \frac{\partial ^{2}f}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial f}{\partial r}}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\displaystyle \frac{M^{2}}{r^{2}}}w^{2}+Q_{0}T,\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial ^{2}f}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial f}{\partial r}}+{\displaystyle \frac{Nt}{Nb}}\left({\displaystyle \frac{\partial ^{2}T}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}\right)=0. & & \displaystyle\end{eqnarray}$ (19) The boundary conditions for the wave frame in dimensionless are $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}w=-1,\quad T=f=1\quad \text{at }r=r_{1},\\ w=-1,\quad T=f=0\quad \text{at }r=r_{2}.\end{array} & & \displaystyle\end{eqnarray}$ (20)

3. Exact solution form

This section presents the solutions of the system of Eqs ((17))–((19)) with boundary conditions ((20)), the temperature and nanoparticles concentration can be written as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle T(r,∼z)=r^{-a_{1}}(C_{1}J_{a_{1}}(a_{2}r)+C_{2}Y_{a_{1}}(a_{2}r)),\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f(r,∼z)=-{\displaystyle \frac{Nt}{Nb}}T(r,∼z)+C_{3}\ln ∼r+C_{4}.\end{eqnarray}$ (22) Here J _a1 and Y _a1, are Bessel functions of the first and second kind of order a ₁, respectively. A combination of homotopy method and perturbation method is called homotopy perturbation method which removes the drawbacks of the classical perturbation methods. Now, we construct a suitable homotopy equation [27–30]. Equation ((17)) can be written as follows: $\begin{eqnarray}\displaystyle H(p,∼w)\text{} & =\text{} & \displaystyle (1-p)[L_{1}(w)-L_{1}(w_{10})]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼p∼\left(L_{1}(w)-{\displaystyle \frac{\partial P}{\partial z}}+{\gamma}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r\left({\displaystyle \frac{\partial w}{\partial r}}\right)^{3}\right)+Gr∼T+Br∼f\right),\end{eqnarray}$ (23) with $L_{1}=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\right)-\frac{M^{2}}{r^{2}}$ as the linear operator. The initial guess w ₁₀ can be written as $\begin{eqnarray}\displaystyle w_{10}(r,∼z)=C_{5}r^{-a_{3}}+C_{6}r^{a_{3}}, & & \displaystyle\end{eqnarray}$ (24) Assuming that w = w ₀ + pw ₁ + p ² w ₂ + ⋯ ,.

The solution of axial velocity (for p = 1) are constructed as follows $\begin{eqnarray} \displaystyle w(r,∼z)\text{} & =\text{} & \displaystyle C_{7}(r^{-a_{3}}+r^{a_{3}})+C_{8}(r^{-a_{4}}+r^{a_{4}})+C_{9}(r^{-a_{5}}+r^{a_{5}})+C_{10}(r^{-a_{6}}+r^{a_{6}})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad C_{11}(r^{-a_{7}}+r^{a_{7}})+C_{12}(r^{-a_{8}}+r^{a_{8}})+C_{13}(r^{-a_{9}}+r^{a_{9}})+C_{14}(r^{-a_{10}}+r^{a_{10}})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad+∼C_{15}r^{a_{15}}∼\text{}_{1}F_{2}∼(a_{11};∼∼a_{12},a_{13};∼a_{14}r^{2})+C_{16}r^{-a_{18}}∼\text{}_{1}F_{2}∼(a_{16};∼∼a_{12},a_{17};∼a_{14}r^{2})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼C_{17}r^{a_{22}}∼\text{}_{1}F_{2}∼(a_{19};∼∼a_{20},a_{21};∼a_{14}r^{2})+C_{18}r^{-a_{26}}∼\text{}_{1}F_{2}∼(a_{23};∼∼a_{20},a_{24};∼a_{14}r^{2})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\left[C_{19}(r^{-a_{25}}+r^{a_{25}})+C_{20}(r^{-a_{26}}+r^{a_{26}})+C_{21}(r^{-a_{27}}+r^{a_{27}})+C_{22}(r^{-a_{28}}+r^{a_{28}})\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \left.+∼C_{23}(r^{-a_{29}}+r^{a_{29}})+C_{24}(r^{-a_{30}}+r^{a_{30}})+C_{25}(r^{-a_{31}}+r^{a_{31}})\right]\ln ∼r. \end{eqnarray}$ (25) In Eq. ((25)), ₁ F ₂ is the regularized generalized-hypergeometric function. The mathematical formulas of the constants a ₁–a ₃₁ and C ₁–C ₂₅ are not included here. However, they are available upon request from the author.

The instantaneous dimensionless volume flow rate, in the fixed frame, is given by $\begin{eqnarray}\displaystyle Q(z,t)=2\int _{r_{1}}^{r_{2}}rw∼dr. & & \displaystyle\end{eqnarray}$ (26) The coefficient of heat transfer Nu and nanoparticles transfer coefficient Sh _r at the outer tube, are represented by the following equations $\begin{eqnarray}\displaystyle Nu=\left.{\displaystyle \frac{\partial T}{\partial r}}\right|_{r=r_{2}},\quad Sh_{r}=\left.{\displaystyle \frac{\partial f}{\partial r}}\right|_{r=r_{2}}. & & \displaystyle\end{eqnarray}$ (27) The expressions for Nu and Sh _r were calculated from Eqs ((21)) and ((22)) into Eq. ((27)), and they were evaluated and tabulated numerically for different values of the problem parameters.

Table 1

Nb	Nt	Nu (our work)	Nu Mekheimer et al. [3]	Sh _r(our work)	Sh _r Mekheimer et al. [3]
3.5	4.5	−2.7233	−2.1247	2.1274	1.3578
4.5	4.5	−2.9624	−2.3625	1.7602	1.1602
5.5	4.5	−3.2016	−2.6052	1.5265	1.0386
5.5	5.5	−3.4398	−2.8517	2.2376	1.6494
5.5	6.5	−3.6766	−3.1011	3.0335	2.3534

The acquired results will be discussed in the following section.

4. Results and discussion

In this problem, the semi-analytical solutions of the axial velocity, temperature and nanoparticles concentration are obtained, as shown in the previous section. Moreover, we discussed the impact of some entering parameters on the obtained solutions. We used Mathematica package Ver. 10, to calculate numerical results for the heat transfer coefficient Nu and the nanoparticles transfer coefficient Sh_r .

It is clear that we have to long wave length and small Reynolds number, while the wave number is abandoned. Moreover, these approximations restricted us to choose the following values: 𝜆 = 8.2 cm, c = 3 cm/sec and d = 2.3 cm.

Figures 1 and 2 reveal the variation of the axial velocity w versus the radial coordinate r for several values of the magnetic field parameter M and the radiation parameter R ₁, respectively. It is noticed, from Figs 1 and 2, that the axial velocity decreases in response to the increase of M, whereas it increases as R ₁ increases, respectively. Moreover, it is recorded that for each value of both M and R ₁, w is always negative, and all obtained curves intersect at the boundary of tubes. It is also seen that the change in the axial velocity for different rates of M and R ₁ becomes greater with the increase in the radial coordinate r and achieves minimum value, after which it rises. The results in Fig. 1, agrees qualitatively with physical and medical expectations. The change in MHD flow of fluid is caused by the Lorentz force which retards the flow. The effects of the other parameters are recorded to be similar to them; these figures are omitted here to avoid repetition.

Fig. 1.

The axial velocity is plotted versus r for different values of M for a system have the particulars R ₁ = 1, Sh = 1.5, Q ₀ = 10, 𝛾 = 0.5, Br = 01, Gr = 1.5, $\frac{dP}{dz}=1$ , Nt = 4.5, Nb = 3.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

Fig. 2.

The axial velocity is plotted versus r for different values of R ₁ for a system have the particulars Sh = 1.5, Q ₀ = 10, M = 0.8, 𝛾 = 0.5, Br = 01, Gr = 1.5, $\frac{dP}{dz}=1$ , Nt = 4.5, Nb = 3.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

The variations of the temperature distribution T with the dimensionless radial coordinate r for various values of the thermophoresis parameter Nt and the radiation parameter R ₁ are shown in Figs 3 and 4, respectively. The graphical results of Figs 3 and 4 reveal that the temperature distribution T arises with the increase in the parameter Nt, while it decreases by increasing the parameter R ₁. In addition, it is noticed from these figures that for large values of R ₁ and small values of Nt, T increases with r till a definite value r = r ₀ (represents the maximum value of T) and it decreases later. This highest value of T increases by increasing R ₁, while it decreases by increasing Nt. Moreover, it is found that for each value of both Nt and R1, T is always positive. We believe that an upturn in thermophoresis parameter Nt enhances the fluid temperature. This arises on account of gradual enhancement in nanoparticles percentage with Nt. Meanwhile, we discern an opposite result in concentration profiles. The result in Fig. 4 is due to the fact that thermal radiation is electromagnetic radiation which is generated by the particles thermal motion. The fluid loses about 76% of its heat through radiation, heat is lost in air temperatures which is lower than 78 °F. The fluid loses about 3% of its heat through air conduction. Other parameters have an effect on the temperature, but they are recorded to be similar to those obtained in Figs 3 and 4. So, figures are excluded here to save space.

Fig. 3.

The temperature is plotted versus r for different values of Nt for a system have the particulars R ₁ = 1, Sh = 1.5, Q ₀ = 10, M = 0.8, 𝛾 = 0.5, Br = 01, Gr =1.5, $\frac{dP}{dz}=1$ , Nb = 3.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

Fig. 4.

The temperature is plotted versus r for different values of R ₁ for a system have the particulars Sh = 1.5, Q ₀ = 10, M = 0.8, 𝛾 = 0.5, Br = 01, Gr = 1.5, $\frac{dP}{dz}=1$ , Nt = 4.5, Nb = 3.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

Figures 5 and 6 delineate the behaviors of the nanoparticles concentration distribution f with the dimensionless radial coordinate r for different values of Brownian motion parameter Nb and the heat source parameter Q ₀, respectively. Brownian motion is the random movement of particles suspended in a fluid as a result of their collision with the quick atoms or molecules in the fluid. Einstein, who studied Brownian collision of the suspended particle with the molecules of the fluid, presented a physical explanation for Brownian motion. It is indicated from Figs 5 and 6, that the nanoparticles concentration distribution arises in response to the increase of Nb, whereas it decreases as Q ₀ increases, respectively. It is also found that the difference of the nanoparticles concentration distribution for different values of Nb and Q ₀ becomes lower with increasing the radial coordinate r and reaches minimum value, after which it increases. The effects of other parameters (figures are removed) are found to be exactly similar to the effect of Q ₀ on f given in Fig. 6, with the only difference that the resulting curves are very close to each other than those noted in Fig. 6.

Fig. 5.

The nanoparticles is plotted versus r for different values of Nb for a system have the particulars R ₁ = 1, Sh = 1.5, Q ₀ = 10, M = 0.8, 𝛾 = 0.5, Br = 01, Gr = 1.5, $\frac{dP}{dz}=1$ , Nt = 4.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

Fig. 6.

The nanoparticles is plotted versus r for different values of Q for a system have the particulars R ₁ = 1, Sh = 1.5, M = 0.8, 𝛾 = 0.5, Br = 01, Gr = 1.5, $\frac{dP}{dz}=1$ , Nt = 4.5, Nb = 3.5, r ₁ = 0.3, z = 0.3 and ϵ = 0.25.

Table 1 presents a comparison between the numerical results of present study and those obtained by Mekheimer et al. [3] for Nusselt number Nu and nanoparticles transfer coefficient Sh _r for various values of both the Brownian motion parameter Nb and the thermophoresis parameter Nt. It can be concluded from Table 1 that the present results are in a good agreement with those obtained by Mekheimer et al. [3].

5. Conclusion

Peristaltic pumping through an annulus (coaxial tubes) is significant in real many problems in human organs such as, the endoscope problem and the catheterized artery. Nevertheless, from a fluid dynamic point of view, there is no variation between an endoscope and a catheter. This paper generalizes the problem of mixed convection and diffusion thermo effects on peristaltic motion of a third grade non-Newtonian nanofluid via porous medium in the presence of radially varying magnetic field. The expressions of both temperature and nanoparticles are obtained in a closed form, while the solution of velocity is obtained using the homotopy perturbation method. These expressions have been studied graphically. The following observations have been noted.

The axial velocity w increases with the increase R ₁ whereas it decreases as other parameters increase.

The axial velocity w for different values of all parameters becomes lower with increasing the radial coordinate r and reaches minimum value (at a finite value of r: r = r ₀) after which it increases.

The temperature decreases with the increase of R ₁, whereas it increases as other parameters increase.

The temperature for various values of Sh, Q ₀, R ₁, Nb, and Nt becomes higher with increasing the radial coordinates r and reaches the maximum value at r = 0.6.

The nanoparticles concentration has an opposite behavior compared to the temperature behavior.

Footnotes

Acknowledgements

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

Conflict of interest

The authors declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Nomenclature

$\begin{eqnarray} \displaystyle \begin{array}{@{} clcl@{}}b_{1},b_{2} & \text{The radius of inner and outer tubes} & R_{1} & \text{Radiation parameter}\\ b & \text{The wave amplitude} & \mathit{Re} & \text{Reynolds number}\\ B_{0} & \text{The uniform magnetic field} & \text{}\underline{S} & \text{The extra stress tensor}\\ c & \text{The propagation velocity along } Z \text { direction} & \mathit{Sh} & \text{Sherwood number}\\ C_{s} & \text{The nanoparticle susceptibility} & T & \text{The fluid temperature}\\ D_{\text{B}} & \text{Brownian diffusion coefficient} & T_{1} & \text{Temperature at }r=r_{1}\\ D_{\text{T}} & \text{Thermophoretic diffusion coefficient} & T_{2} & \text{Temperature at }r=r_{2}\\ f & \text{Nanoparticles concentration} & t & \text{The time}\\ f_{1} & \text{Nanoparticles concentration at }r=r_{1} & U & \text{Radial velocity}\\ f_{2} & \text{Nanoparticles concentration at }r=r_{2} & W & \text{Axial velocity}\\ I & \text{The identity tensor} & Z & \text{Axial coordinate}\\ J & \text{The electric current} & & \mathbf{Greek∼symbols}\\ k_{R} & \text{The mean absorption coefficient} & {\sigma} & \text{The electrical conductivity}\\ K_{T} & \text{The thermal diffusion ratio} & {\sigma}^{\ast } & \text{Stefan Boltzmann constant}\\ k & \text{Thermal conductivity} & {\varphi} & \text{The amplitude ratio}\\ M & \text{The magnetic field parameter} & {\gamma} & \text{Third-grade parameter}\\ \mathit{Nb} & \text{The Brownian motion parameter} & {\lambda} & \text{The wavelength}\\ \mathit{Nt} & \text{The thermophoresis parameter} & {\mu} & \text{The coefficient of viscosity}\\ P & \text{The fluid pressure} & {\mu}_{0} & \text{The magnetic permeability}\\ \mathit{Pr} & \text{Prandtl number} & {\rho}_{f} & \text{The density of the fluid}\\ q_{r} & \text{The radiative heat flux} & ({\rho}c)_{f} & \text{heat capacity of the fluid}\\ Q_{0} & \text{The volumetric rate of heat generation} & ({\rho}c)_{p} & \text{heat capacity of nanoparticle material}\\ R & \text{Radial coordinate} & \text{}\underline{{\tau}} & \text{Stress vector} \end{array} & & \displaystyle \nonumber \end{eqnarray}$

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