Electromagnetohydrodynamic peristaltic flow of a couple stress nanofluid through a vertical annulus in the presence of Hall currents and thermal radiation

Abstract

This paper investigates the electric properties of gold nanoparticles mixed with a convection dielectric couple stress fluid inside a vertical cylindrical tube with moving endoscope in the presence of Hall currents and thermal radiation. Under the long wavelength approximation and the use of appropriate conversion relationships between fixed and moving frame coordinates, the exact solutions have been evaluated for temperature distribution, gold nanoparticles concentration, electrical potential function and nanofluid pressure, while analytical solution is found for the axial velocity using the homotopy analysis method. The results show that the presence of the electric field enhances the effects of Brownian motion parameter, thermophoresis parameter, radiation parameter, Hall currents and wave amplitude ratio on the axial nanofluid velocity, while it was found that its presence reduces the effects of couple stress parameter, thermophoresis diffusion constant and Brownian diffusion constant.

Keywords

Couple stress fluid electromagnetic field gold nanoparticles Hall currents vertical oscillatory tube thermal radiation

1. Introduction

Gold nanoparticles are small parts of gold with diameters less than 100 nm. All nanoparticles have different properties from those of conventional particles. The mechanical, thermal, optical, magnetic, and electrical properties of nanoparticles are much greater than those in conventional particles. Gold nanoparticles are characterized by their ability to spread throughout the body and can pass through cell membranes easily and without causing them any damage. Therefore, they are used to deliver the drug to the affected cells and are also used in the diagnosis of some diseases. Gold nanoparticles are also distinguished by their high atomic number, which enables them to generate energy and deliver enormous heat. There are many successful experiments in which the gold nanoparticles have been used to treat animals with different types of cancerous tumors [1]. In these experiments, the body is injected with gold nanoparticles, and after these particles penetrate the cancer cells, a laser is projected onto them, so the gold nanoparticles convert the light energy into thermal energy and burn the cancer cells. Then within a week, the body can get rid of the remaining gold nanoparticles without damaging living cells. The many uses of nanoparticles have caused a huge technological revolution in all aspects of life and some of these applications have been mentioned in the list of references [2–6]. In 1995 Choi [7] invented new types of heat transfer fluids by suspending metallic nanoparticles in conventional heat transfer fluids and the resulting mixture was named “nanofluids”. It has been found that the resulting nanofluids have a much higher thermal conductivity compared to conventional heat transfer fluids. The problem of the mechanism of peristaltic transport of nanofluids in the human body has attracted the attention of many investigators. Mekheimer et al. [8] have studied the effect of adding gold nanoparticles to third grade non-Newtonian fluid flowing through a flexible heating tube and in the presence of a catheter. They found that the gold nanoparticles increased the temperature distribution. Hussain et al. [9] have studied the peristaltic transport of couple stress fluid conveying gold nanoparticles through coaxial tubes. They have indicated that the nanofluid velocity has been supported by all parameters except for thermophoresis parameter. Additional articles on the nanofluids flow have also been cited [10–13]. At the present time, the magnetic field is used in various fields in medicine. It is used in magnetic resonance imaging (MRI), in the dissolution of stones in the kidneys and gallbladder, in the treatment of bed sore, and in reducing bleeding during surgical procedures. Therefore, several studies have emerged to examine the effect of the magnetic field on the movement of vital fluids within the human body. Some of these studies considered vital fluids to behave the same as non-Newtonian fluids [14–18], and others considered them as couple stress fluids [19–23]. Stokes [24] was the first to develop a mathematical model for the movement of couple stress fluids and since that time many researchers used this model to study the effect of couple stress on the movement of vital fluids within the human body in various forms. A number of studies containing couple stress fluids have been investigated in [25–31]. A few papers have studied the effect of the electric field on the flow of vital fluids through a flexible tube. Some of these papers had to use the perturbation method twice [32–35]. The first time in terms of the wave amplitude ratio, and in the second time in terms of the wave number. Thus, the method of solution was very difficult and stressful. In some other papers, the duct was considered micro-duct [36–38]. Recently, Haroun [39] investigated the effect of an electric field on Jeffrey fluid flow through a heating vibrating tube with endoscope. In this study, the researcher presented a new technique that enabled him to obtain analytical solutions using the perturbation method only once. In general, the numerical techniques do not give us the form of solutions as functions in coordinates and time but only give us the values of these functions at some points and in a specific domain. Thus, it is difficult to deeply understand the nature of the problem and to know its solutions in general. It is also well known that perturbation techniques are based primarily on the presence of a physical parameter in the mathematical model that can be considered as a small quantity (which is a constraint on the problem). In other words, if the small parameter is not present in the problem then we cannot solve the problem by perturbation method. Unlike the other perturbation techniques, the homotopy perturbation method has the advantage that it does not require small physical parameters. Furthermore, the main advantage of the homotopy perturbation method is that the number of terms in the series solution can be greatly reduced; meanwhile the accuracy of the solution can be well retained. In fact, the homotopy perturbation method is considered as a special case of the homotopy analysis method. Both methods are based on the Taylor series with respect to an embedding parameter. The difference is that the homotopy perturbation method should use a good enough initial guess, but this is not necessary for the homotopy analysis method. This is because the homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of series solution [40]. The current study generalizes the paper of Hussain et al. [9] to include electromagnetic force, moving endoscope, Hall currents and thermal radiation. As far as we know, our current research is the first study examining the effect of a variable AC electric field on the movement of gold nanoparticles mixed with a couple stress fluid through vital vessels as a simulation of treatment processes with gold nanoparticles. We also succeeded in obtaining the nanofluid pressure, unlike previous studies, which were not able to obtain it and were only able to obtain a pressure gradient. Furthermore, exact solutions of all dependent variables in the problem were obtained except the axial velocity for which an analytical expression was obtained using the homotopy analysis method. The effects of various physical parameters on these expressions were discussed and illustrated graphically.

2. Formulation of the problem

We consider the unsteady flow of an incompressible electrically conducting couple stress nanofluid through coaxial infinite tubes such that the inner tube is an endoscope which is moving with constant velocity V ₀ and the outer tube has a sinusoidal wave travelling down on its surface. Cylindrical polar coordinates (R, θ, Z) are used. The Z-axis is taken along the common axis of the two cylinders and R is in the radial direction. The outer tube is maintained at temperature θ₁ and concentration 𝛾₁ while the inner tube is maintained at temperature θ₀ and concentration 𝛾₀. A variable electric field E is imposed across the tubes. The outer tube is kept at the electrical potential V ₁ and the inner tube is grounded. Also, the system is stressed by a constant magnetic field B = (0, 0, B ₀) perpendicular to the electric field (see Fig. 1). Denoting the nanofluid density by 𝜌, the viscosity by μ and the pressure by P, the equation of continuity and the equation of momentum are [9,24] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}.\text{}\underline{\mathbf{V }}=0.\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}\left(\frac{\partial \text{}\underline{\mathbf{V }}}{\partial t}+\text{}\underline{\mathbf{V }}.{\nabla}\text{}\underline{\mathbf{V }}\right)={\nabla}.∼{\mathcal{S}}+\frac{1}{2}∼{\nabla}\times ({\nabla}.∼{\mathcal{M}})+\frac{1}{2}∼{\nabla}\times ({\rho}\text{}\underline{{\Sigma}})+{\rho}\text{}\underline{\mathbf{f}},\end{eqnarray}$ (2) where, 𝛴, f are the body couple and the external body force per unit mass. The constitutive equations for the force stress tensor ${\mathcal{S}}$ and the couple stress tensor ${\mathcal{M}}$ are given by: $\begin{eqnarray}\displaystyle {\mathcal{S}}_{ij} & = & \displaystyle -\text{P}{\delta}_{ij}+2{\mu}d_{ij}-\frac{1}{2}{\epsilon}_{ijk}({\mathcal{M}}_{ll,k}+4{\xi}_{1}{\mathcal{W}}_{k,ll}+{\rho}{\Sigma}_{k}),\nonumber\\ \displaystyle {\mathcal{M}}_{ij} & = & \displaystyle \frac{1}{3}{\mathcal{M}}_{kk}{\delta}_{ij}+4{\xi}_{1}{\mathcal{W}}_{j,i}+4{\xi}_{2}{\mathcal{W}}_{i,j}.\nonumber\end{eqnarray}$ Here, comma in the suffixes denotes covariant differentiation, $\text{}\underline{{\mathcal{W}}}=∼\frac{1}{2}{\nabla}\times \text{}\underline{\mathbf{V }}$ is the spin vector, d _ij is the rate of deformation tensor and the material constants 𝜉₁, 𝜉₂ are the couple stress coefficients. For an incompressible couple stress fluid and in the absence of the body couples, equation ((2)) reduces to $\begin{eqnarray}\displaystyle {\rho}\left(\frac{\partial \text{}\underline{\mathbf{V }}}{\partial t}+\text{}\underline{\mathbf{V }}.{\nabla}\text{}\underline{\mathbf{V }}\right)=-{\nabla}\text{P}+{\mu}{\nabla}^{2}\text{}\underline{\mathbf{V }}-{\xi}_{1}{\nabla}^{4}\text{}\underline{\mathbf{V }}+\text{}\underline{\mathbf{f}}_{e}+\text{}\underline{\mathbf{J}}\times \text{}\underline{\mathbf{B}}+{\rho}\text{}\underline{\mathbf{g}}, & & \displaystyle\end{eqnarray}$ (3) where $\text{}\underline{\mathbf{J}}=∼{\sigma}(\text{}\underline{\mathbf{V }}\times \text{}\underline{\mathbf{B}}-\frac{1}{e_{1}n_{e}}\text{}\underline{\mathbf{J}}\times \text{}\underline{\mathbf{B}})$ is the current density (generalized Ohm’s law for Hall effects), g is the acceleration due to gravity. In the above equations V = (U, 0, W) is the velocity vector, σ electrical conductivity, e ₁ electric charge, n _e number density of electrons, 𝜖 dielectric constant, 𝜌_e free charge density and f _e is the electrical body force, also known as Korteweg-Helmholtz body force density, which expressed as [41,42 ] $\begin{eqnarray}\displaystyle \text{}\underline{\mathbf{f}}_{e}={\rho}_{e}\text{}\underline{\mathbf{E}}-\frac{1}{2}\text{E}^{2}{\nabla}{\epsilon}+\frac{1}{2}{\nabla}\left({\rho}\frac{\partial {\epsilon}}{\partial {\rho}}\text{E}^{2}\right). & & \displaystyle\end{eqnarray}$ (4) The first term in Eq. ((4)) is the Coulombic force exerted on free charges. The second term is the polarization force exerted on the dipoles. The last term is the electrostriction force density associated with volumetric change in the material; See Melcher [42] for a detailed derivation.

Fig. 1.

Geometry of the problem.

Using the Oberbeck-Boussinesq approximation, the last term in Eq. (3) can be written as [43–45] $\begin{eqnarray}{\rho}\text{}\underline{\mathbf{g}}=(({\rho}_{p}-{\rho}_{f})({\Omega}-{\gamma}_{1})+(1-{\gamma}_{1}){\rho}_{f}{\beta}_{T}({\theta}-{\theta}_{0}))\text{}\underline{\mathbf{g}}.\end{eqnarray}$ The energy equation is [44,46] $\begin{eqnarray}\displaystyle ({\rho}c_{0})_{f}\left(\frac{\partial {\theta}}{\partial t}+\text{}\underline{\mathbf{V }}.{\nabla}{\theta}\right)=K{\nabla}^{2}{\theta}+({\rho}c_{0})_{p}\left(D_{b}{\nabla}{\Omega}.{\nabla}{\theta}+\frac{D_{T}}{{\theta}_{1}}{\nabla}{\theta}.{\nabla}{\theta}\right)-{\nabla}.\text{}\underline{\mathbf{q}}, & & \displaystyle\end{eqnarray}$ (5) where θ, 𝛺, K, 𝛽_T, D _b, D _T, c _0f and c _0p denote the temperature, nanoparticles concentration, thermal conductivity, volumetric coefficient of expansion, Brownian motion coefficient, thermophoretic diffusion coefficient, specific heat of base fluid and specific heat of gold nanoparticles, respectively.

Fig. 2.

Change in W with z when A = 0.3, a ₀ = 0.05, m = 0.01, M = 0.001, 𝜉 = 3, V ₀ = 25, N _T = 1.5, N _b = 2, R _N = 1, τ = 0.3 and r = 0.5. (a) Effect of G _r for B _r = 3, L ₁ = 0. (b) Effect of G _r for B _r = 3, L ₁ = 4. (c) Effect of B _r for G _r = 1, L ₁ = 0. (d) Effect of B _r for G _r = 1, L ₁ = 4.

Fig. 3.

Change in W with r when A = 0.1, a ₀ = 0.3, m = 0.1, M = 2, V ₀ = 1, B _r = 3, G _r = 1, R _N = 1, N _T = 1.5, N _b = 2 and z = 0.4. (a) Effect of 𝜉 for R _N = 1, L ₁ = 0, τ = 0.3. (b) Effect of 𝜉 for R _N = 1, L ₁ = 15, τ = 0.3. (c) Effect of R _N for 𝜉 = 2.2, L ₁ = 0, τ = 0.4. (d) Effect of R _N for 𝜉 = 2.2, L ₁ = 15, τ = 0.4.

Fig. 4.

Change in W with r when A = 0.1, a ₀ = 0.2, m = 0.1, M = 2, 𝜉 = 4.2, V ₀ = 1, R _N = 1, G _r = 1, τ = 0.4 and z = 0.4. (a) Effect of N _b for N _T = 1.5, L ₁ = 0. (b) Effect of N _b for N _T = 1.5, L ₁ = 5. (c) Effect of N _T for N _b = 2, L ₁ = 0. (d) Effect of N _T for N _b = 2, L ₁ = 5.

Fig. 5.

Change in W with r when A = 0.1, N _b = 2, N _T = 1.5, M = 2, 𝜉 = 4.2, V ₀ = 1, R _N = 1, G _r = 1, τ = 0.3 and z = 0.4. (a) Effect of m for a ₀ = 0.2, L ₁ = 0. (b) Effect of m for a ₀ = 0.2, L ₁ = 15. (c) Effect of a ₀ for m = 0.1, L ₁ = 0. (d) Effect of a ₀ for m = 0.1, L ₁ = 15.

Fig. 6.

Distribution of P on r when A = 0.15, a ₀ = 0.3, L ₂ = 1.1 and z = 0.4. (a) Effect of N _T for L ₁ = 5, R _N = 2, N _b = 4, τ = 0.4. (b) Effect of N _b for L ₁ = 5, R _N = 2, N _T = 4, τ = 0.4. (c) Effect of R _N for L ₁ = 5, N _b = 2, N _T = 2, τ = 0.3. (d) Effect of L ₁ for N _b = 2.5, R _N = 1, N _T = 4, τ = 0.25.

Fig. 7.

Distribution of θ on r when A = 0.15, τ = 0.4 and z = 0.4. (a) Effect of N _T for R _N = 1, N _b = 1, a ₀ = 0.01. (b) Effect of N _b for R _N = 1, N _T = 1, a ₀ = 0.01. (c) Effect of R _N for N _b = 3, N _T = 3, a ₀ = 0.01. (d) Effect of a ₀ for N _b = 3, N _T = 3, R _N = 1.

In the above equation q = (q _r, 0, 0) is the heat flux. Based on the Rosseland approximation, the radiative heat flux can be written as [31,51] $\begin{eqnarray}\displaystyle q_{r}=\frac{-16{\sigma}^{\ast }{\theta}_{1}^{3}}{3k_{R}}\frac{\partial {\theta}}{\partial r}, & & \displaystyle\end{eqnarray}$ (6) where σ^∗ and k _R denote the Stefan-Boltzmann constant and the Roseland mean absorption coefficient. The equation of gold nanoparticle volume fraction in the nanofluid [9,45] is given by $\begin{eqnarray}\displaystyle \frac{\partial {\Omega}}{\partial t}+\text{}\underline{\mathbf{V }}.{\nabla}{\Omega}=D_{b}{\nabla}^{2}{\Omega}+\frac{D_{T}}{{\theta}_{1}}{\nabla}^{2}{\theta}. & & \displaystyle\end{eqnarray}$ (7) The dielectric constant 𝜖 is defined by [48]: $\begin{eqnarray}\displaystyle {\epsilon}={\epsilon}_{0}[1-e({\theta}-{\theta}_{1})]. & & \displaystyle\end{eqnarray}$ (8) The geometry of the inner and the outer tubes are $\begin{eqnarray}\displaystyle R=a_{1}\quad \text{and}\quad R=a_{2}+{\eta}(Z,t)\quad \text{in which }{\eta}(Z,t)=a∼\text{Cos}\left[\frac{2{\pi}}{{\lambda}}(Z-Ct)\right], & & \displaystyle\end{eqnarray}$ (9) where a ₁, a ₂, a, 𝜆, C, e and 𝜖₀ are apparent radius of the inner tube, radius of the outer tube, wave amplitude, wave length, wave speed, thermal expansion coefficient and permittivity at vacuum. We shall carry out this investigation in a coordinate system moving with the wave speed, in which the boundary shape is stationary. We employ these coordinates in the governing equations and then introduce the following non-dimensional variables and parameters: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\overline{z}=\frac{z}{{\lambda}},\quad \overline{r}=\frac{r}{a_{2}},\quad \overline{u}=\frac{{\lambda}u}{Ca_{2}},\quad \overline{w}=\frac{w}{C},\quad \overline{{\eta}}=\frac{{\eta}}{a_{2}},\quad \overline{t}=\frac{Ct}{{\lambda}},\quad \overline{\text{P}^{\ast }}=\frac{a_{2}^{2}∼\text{P}^{\ast }}{C{\lambda}{\mu}},\quad \overline{{\theta}}=\frac{{\theta}-{\theta}_{1}}{{\theta}_{0}-{\theta}_{1}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\overline{{\Omega}}=\frac{{\Omega}-{\gamma}_{1}}{{\gamma}_{0}-{\gamma}_{1}},\quad \overline{{\Phi}}=\frac{{\Phi}}{E_{0}\sqrt{a_{2}{\lambda}}},\quad \overline{{\psi}}=\frac{{\psi}}{C{a_{2}}^{2}},\quad {\gamma}=\frac{L_{1}}{L_{2}RR},\quad L_{2}=e({\theta}_{0}-{\theta}_{1}),\quad {\tau}=\frac{({\rho}c_{0})_{p}}{({\rho}c_{0})_{f}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}{\xi}^{2}=\frac{{\xi}_{1}}{a_{2}^{2}∼{\mu}},\quad \text{Reynolds number RR}=\frac{Ca_{2}{\rho}_{f}}{{\mu}},\quad \text{electrical Rayleigh number }L_{1}=\frac{{\epsilon}_{0}a_{2}E_{0}^{2}e^{2}({\theta}_{0}-{\theta}_{1})^{2}}{C∼{\mu}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{wave number }{\delta}=\frac{a_{2}}{{\lambda}},\quad \text{wave amplitude ratio }a_{0}=\frac{a}{a_{2}},\quad \text{Prandtl number Pr}=\frac{{\mu}∼c_{0f}}{K},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{Schmidt number SC}=\frac{{\mu}}{{\rho}_{f}D_{b}},\quad \text{radiation parameter }\text{R}_{\text{N}}=\frac{16∼{\sigma}^{\ast }{\theta}_{1}^{3}}{3∼k_{R}∼K},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{Brownian motion parameter }\text{N}_{\text{b}}=\frac{D_{b}({\gamma}_{0}-{\gamma}_{1})({\rho}c_{0})_{f}}{K},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{thermophoresis parameter }\text{N}_{\text{T}}=\frac{D_{T}({\theta}_{0}-{\theta}_{1})({\rho}c_{0})_{f}}{K{\theta}_{1}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{thermophoresis diffusion constant }G_{r}=\frac{g∼a_{2}^{2}(1-{\gamma}_{1})({\theta}_{0}-{\theta}_{1}){\rho}_{f}{\beta}_{T}}{C∼{\mu}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{Brownian diffusion constant }B_{r}=\frac{g∼a_{2}^{2}({\rho}_{p}-{\rho}_{f})({\gamma}_{0}-{\gamma}_{1})}{C∼{\mu}},\quad \text{Hartman number }M=\frac{B_{0}∼a_{2}\sqrt{{\sigma}}}{\sqrt{{\mu}}},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-20.0pt}\text{Hall parameter }m=\frac{B_{0}∼{\sigma}}{e_{1}∼n_{e}}.\nonumber\end{eqnarray}$ Dropping the pare over the symbols and then using the long wavelength assumption, the equations of motion will reduce to: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\text{RR}∼{\gamma}}{2}\left(\frac{\partial {\Phi}}{\partial r}\right)^{2}\left(\frac{\partial {\theta}}{\partial r}\right)-\frac{\partial \text{P}^{\ast }}{\partial r}=0,\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}-{\xi}^{2}\left(\frac{\partial ^{4}w}{\partial r^{4}}+\frac{2}{r}\frac{\partial ^{3}w}{\partial r^{3}}-\frac{1}{r^{2}}\frac{\partial ^{2}w}{\partial r^{2}}+\frac{1}{r^{3}}\frac{\partial w}{\partial r}\right)+\frac{\text{RR}∼{\gamma}}{2}\left(\frac{\partial {\Phi}}{\partial r}\right)^{2}\left(\frac{\partial {\theta}}{\partial z}\right)-\frac{\partial \text{P}^{\ast }}{\partial z}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -\frac{M^{2}(w+1)}{(1+m^{2})}+\text{B}_{r}∼{\Omega}+\text{G}_{r}∼{\theta}=0,\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\tau}∼\text{N}_{\text{b}}\frac{\partial {\Omega}}{\partial r}\frac{\partial {\theta}}{\partial r}+{\tau}∼\text{N}_{\text{T}}\left(\frac{\partial {\theta}}{\partial r}\right)^{2}+(1+\text{R}_{\text{N}})\left(\frac{\partial ^{2}{\theta}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\theta}}{\partial r}\right)=0,\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{N}_{\text{b}}\left(\frac{\partial ^{2}{\Omega}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\Omega}}{\partial r}\right)+\text{N}_{\text{T}}\left(\frac{\partial ^{2}{\theta}}{\partial r^{2}}+\frac{1}{r}\frac{\partial {\theta}}{\partial r}\right)=0,\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial }{\partial r}\left(r(1-L_{2}∼{\theta})\frac{\partial {\Phi}}{\partial r}\right)=0.\end{eqnarray}$ (15) The corresponding boundary conditions are $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle w=V_{0}-1,\quad u=0,\quad {\theta}=1,\quad {\Omega}=1,\quad {\Phi}=0\quad \text{at }r=A,\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle w=-1,\quad u=-\frac{\partial h(z)}{\partial z},\quad {\theta}=0,\quad {\Omega}=0,\quad {\Phi}=V_{1}\quad \text{at }r=1+h(z),\end{eqnarray}$ (17) where h (z) = a ₀ Cos(2πz), $A=∼\frac{a_{1}}{a_{2}}$ , $V_{0}=∼\frac{\text{v}_{0}}{C}$ , $a_{0}=∼\frac{a}{a_{2}}$ , $V_{1}=∼\frac{\text{v}_{1}}{E_{0}\sqrt{a_{2}{\lambda}}}$ and P* is the modified pressure.

3. Method of solution

Solving Eqs (11) and (13)–(15) subject to the boundary conditions (16)–(17), we obtain the closed solutions for temperature θ and gold nanoparticles concentration 𝛺 [8,49], electrical potential function Φ and nanofluid pressure P^∗ as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}(\text{r},\text{z})=\text{G}_{1}(z)r^{-\text{G}_{2}(z)}+\text{G}_{3}(z),\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Omega}(\text{r},\text{z})=\text{G}_{4}(z)r^{-\text{G}_{2}(z)}+\text{G}_{5}(z)\log (r)+\text{G}_{6}(z),\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Phi}(\text{r},\text{z})=\text{G}_{7}(z)\log (r)+\text{G}_{8}(z)\log (\text{G}_{10}(z)-r^{-\text{G}_{2}(z)})+\text{G}_{9}(z),\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{P}^{\ast }(\text{r},\text{z})=\frac{\text{G}_{11}(z)∼\text{}_{2}F_{1}\left(1,{\displaystyle \frac{2}{\text{G}_{2}(z)}};1+{\displaystyle \frac{2}{\text{G}_{2}(z)}};{\displaystyle \frac{r^{-\text{G}_{2}(z)}}{\text{G}_{10}(z)}}\right)}{r^{2}}-\frac{\text{G}_{12}(z)}{r^{2}(r^{-\text{G}_{2}(z)}-\text{G}_{10}(z))}+\text{G}_{13}(z),\end{eqnarray}$ (21) where, ₂ F ₁(−, −; −; −) is the hypergeometric function.

To get the solution of Eq. (12), we employ the homotopy analysis method [9,50] as follows:

Regarding the solution, the linear and nonlinear operators are chosen as $\begin{eqnarray}\displaystyle L\{w(r,{\Gamma})\} & = & \displaystyle \frac{d^{2}}{dr^{2}}\{w(r,{\Gamma})\},\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle N\{w(r,{\Gamma})\} & = & \displaystyle \left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-{\xi}^{2}\left(\frac{d^{4}}{dr^{4}}+\frac{2}{r}\frac{d^{3}}{dr^{3}}-\frac{1}{r^{2}}\frac{d^{2}}{dr^{2}}+\frac{1}{r^{3}}\frac{d}{dr}\right)-\frac{M^{2}}{(1+m^{2})}\right)\{w(r,{\Gamma})\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼F(r,z).\end{eqnarray}$ (23) The deformation equation for the equation (12) is defined as [50] $\begin{eqnarray}\displaystyle H(w,{\Gamma})=(1-{\Gamma})L\{w(r,{\Gamma})-W_{00}(r)\}-h_{w}{\Gamma}N\{w(r,{\Gamma})\}=0. & & \displaystyle\end{eqnarray}$ (24) Here, W ₀₀(r) is initial guess and 𝛤 is embedding parameter which has the range 0 ≤ 𝛤≤ 1; under the condition that for 𝛤 = 0; we get the initial solution and for 𝛤 = 1; we seek the final solution. Let us expand w (r, 𝛤) in Taylor’s series as:. $\begin{eqnarray}\displaystyle w(r,{\Gamma})=\mathop{\sum }_{n=0}^{\infty }{\Gamma}^{n}w_{n}(r). & & \displaystyle\end{eqnarray}$ (25) Here, w _n(r) are unknown functions that can be determined from the following formula: $\begin{eqnarray}\displaystyle L\{w_{n+1}(r)-{\chi}_{n}∼w_{n}(r)\}=\frac{h_{w}}{(n)!}\left(\frac{\partial ^{n}N\{w(r,{\Gamma})\}}{\partial {\Gamma}^{n}}\right)_{|_{{\Gamma}=0}}. & & \displaystyle\end{eqnarray}$ (26) In the above, the non-zero auxiliary parameter h _w controls the convergence of the series solution [51,52]. Substituting Eq. (25) into Eq. (26) and solving the resulting equations, the analytical solution for the axial velocity up-to second iterations (when 𝛤 → 1) can be obtained as: $\begin{eqnarray}\displaystyle \hspace{-10.0pt}w(r,z) & = & \displaystyle \log (r)∼\text{F1}(r,z)+\text{F2}(r,z)+\text{h}_{w}∼r^{4}∼\text{G}_{14}(z)+\text{h}_{w}∼r^{5}∼\text{G}_{15}(z)+r∼\text{G}_{16}(z)+r^{2}∼\text{G}_{17}(z)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼r^{3}∼\text{G}_{18}(z)+\frac{\text{B}_{r}∼\text{h}_{w}\text{G}_{5}(z)}{2}r^{2}\log (r)+\frac{\text{h}_{w}^{2}\text{G}_{19}(z)}{2}r\log ^{2}(r)+\text{G}_{19}(z)\text{h}_{w}r\log (r)+\text{G}_{20}(z),\end{eqnarray}$ (27) where $\begin{eqnarray}\displaystyle \text{F1}(r,z) & = & \displaystyle \text{h}_{w}\text{G}_{21}(z)r-\text{h}_{w}\text{G}_{22}(z)r^{3}-\text{h}_{w}\text{G}_{23}(z)r^{4}+\text{h}_{w}^{2}\text{G}_{24}(z)r+\text{B}_{r}\text{h}_{w}^{2}\text{G}_{5}(z)r^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\frac{1}{2}\text{B}_{r}\text{h}_{w}\text{G}_{5}(z)r^{2}-\frac{{\xi}^{2}∼\text{G}_{19}(z)∼\text{h}_{w}^{2}}{2∼r},\quad \nonumber\\ \displaystyle \text{F2}(r,z) & = & \displaystyle \frac{\text{h}_{w}\text{F3}(r,z)}{r}+\text{F4}(r,z)-\frac{\text{h}_{w}\text{G}_{25}(z)}{r}-\text{h}_{w}\text{G}_{26}(z)r^{2-\text{G}_{2}(z)}.\nonumber\end{eqnarray}$ The remaining functions G₁(z) − G₂₆(z), F3(r, z) and F4(r, z) were obtained and are not considered here.

Table 1

Physical properties of couple stress fluid (base fluid) and gold nanoparticles

Material	Density (kg/m ³)	Specific heat capacity (J/kg k)	Thermal conductivity (W/m k)
Gold	19300	129	318
Blood (considerd as couple stress fluid)	1050	3617	0.25

Special cases and validation of the results

We will present some previous studies as special cases of the current study:

* When 𝛾 → 0, M → 0, R_N → 0 and V₀ → 0, the current study reduce to the previous article of Hussain et al. [9].

* When 𝛾 → 0, M → 0, V₀ → 0, 𝜉 → 0, R_N → 0 and third grade parameter → 0, the results are the same as obtained by Mekheimer et al. [8].

* In the absence of the electric field (As 𝛾 → 0) and when V₀ → 0, m → 0, B_r → 0, G_r → 0, 𝜆₁ → 0 and 𝜆₂ → 0, the current study is reduced to the previous article of Abou-zeid [31].

To validate the results of the current study, a comparison was made between the current results of the axial velocity W for various values of r and z and the previous results obtained by Mekheimer et al. [8] and Hussain et al. [9]. As illustrated in Table 2, our results are in good agreement with those of Mekheimer et al. [8] and Hussain et al. [9].

Fig. 8.

Change in 𝛺 with r when A = 0.15 and z = 0.4. (a) Effect of N _T for R _N = 1, N _b = 2, a ₀ = 0.01, τ = 0.25. (b) Effect of N _b for R _N = 1, N _T = 3.5, a ₀ = 0.01, τ = 0.25. (c) Effect of R _N for N _b = 0.1, N _T = 1, a ₀ = 0.01, τ = 0.3. (d) Effect of a ₀ for N _b = 0.15, N _T = 1, R _N = 1, τ = 0.35.

Table 2

Comparison of results for the axial velocity W when 𝛾 → 0, V ₀ → 0, M → 0, R_N → 0, third grade parameter → 0, $\frac{dp}{dz}=∼1.5$ and r ₁ = 0.1

r	Our results, W (r)	Hussain et al. [9 ], W (r)	z	Our results, W (z)	Mekheimer et al. [8 ], W (z)
0.20	−0.98862	−0.98620	0.30	−1.81929	−1.81344
0.25	−0.99204	−0.99493	0.35	−1.77194	−1.77960
0.30	−1.00576	−1.00562	0.40	−1.70074	−1.70422
0.35	−1.01290	−1.01826	0.45	−1.64994	−1.64975
0.40	−1.03885	−1.03284	0.50	−1.63712	−1.63064
0.45	−1.04267	−1.04934	0.55	−1.64994	−1.64975
0.50	−1.06287	−1.06771	0.60	−1.70074	−1.70422
0.55	−1.08292	−1.08786	0.65	−1.77194	−1.77960
0.60	−1.10101	−1.10964	0.70	−1.81929	−1.81344

Fig. 9.

Variation of 𝜙 with r when A = 0.15, V ₁ = 1.5, L ₂ = 0.95, τ = 0.3 and z = 0.4. (a) Effect of N _T for R _N = 0.1, N _b = 2, a ₀ = 0.3. (b) Effect of N _b for R _N = 0.1, N _T = 1, a ₀ = 0.3. (c) Effect of R _N for N _b = 5, N _T = 5, a ₀ = 0.4. (d) Effect of a ₀ for N _b = 5, N _T = 5, R _N = 5.

4. Results and discussion

The thermophysical quantities for density, specific heat capacity and thermal conductivity for gold and couple stress fluid are listed in Table 1 [8]. To illustrate the effects of the physical parameters on the behavior of the resulting solutions, we have used the Mathematica program to study the influence of each of these parameters graphically. In this study, we use the following data p _r = 0.7, RR = 50, V ₁ = 1.3, L ₂ = 0.09 and Sc = 0.5. The effects of thermophoresis diffusion constant G _r and Brownian diffusion constant B _r on the axial nanofluid velocity w are shown in Fig. 2. Intensive observation proved that the axial nanofluid velocity increases with the increase of G _r and B _r and the graphics also show that the effects of these parameters in the presence of electric field are less than those in the absence of electric field. As illustrated in Figs 3 and 4, the axial nanofluid velocity decreases near the endoscope but increases as we move away from the endoscope toward the outer flexible tube with an increase in couple stress parameter 𝜉 and thermophoresis parameter N _T while it has a reverse behavior with an increase in Brownian motion parameter N _b and radiation parameter R _N. Also, the effects of N _b, N _T and R _N on the nanofluid axial velocity in the presence of the electric field are more obvious than those in the absence of the electric field while the presence of the electric field reduces the effect of the couple stress parameter 𝜉. As shown in Fig. 5, the presence of the electric field causes deformation and acceleration of the flow in the radial direction and it follows that the curves shift to the right and upward. The axial nanofluid velocity decreases with an increase in Hall parameter m and that is because, an increase in the magnetic field increases the Lorentz force which acts as a force resisting the movement of the fluid. Also, the axial nanofluid velocity decreases with an increase in wave amplitude ratio a ₀. The reason is that increasing the amplitude of the wave on the wall of the tube reduces the speed of this wave and thus the speed of the nanofluid layers close to the wall of the tube decreases. In Figure 6, thermophoresis parameter N _T and Brownian motion parameter N _b result in the decline in the nanofluid pressure distribution P. On the contrary, the pressure distribution is an increasing function with the radiation parameter R _N and electrical Rayleigh number L₁. Physically, an increase in the thermophoresis parameter and Brownian motion parameter increases the temperature of the nanofluid and thus increases its velocity and it follows that, based on Bernoulli’s principle, the pressure of the nanofluid will decrease. In the same way, the increase in the radiation parameter leads to an increase in the nanofluid pressure. As shown in Fig. 7, the temperature increases with the increase of both N _T and N _b, but it decreases with the increase of R _N and a ₀. The changes in the curves in Fig. 7 are due to the following reasons: First, the thermophoresis phenomenon accelerates particles from the hotter region to the cooler region and thus heat travels rapidly from the hotter surface to the fluid and consequently this leads to an increase in temperature. Second, the increment in Brownian motion parameter accelerates the random movement of nanoparticles in fluid and this random movement accelerates the nanoparticles collision with fluid molecules and as a consequent, molecules kinetic energy is converted into thermal energy and hence temperature rises. Finally, from the definition of the radiation parameter in the current research, as its increase leads to an increase in temperature of the outer tube wall θ₁ and then the difference between the temperature of the nanofluid and the temperature of the outer tube wall (θ − θ₁) decreases and consequently the value of the non-dimensional temperature of the nanofluid decreases. It is observed through Fig. 8 that the gold nanoparticles concentration profile is decreasing with increasing N _T and a ₀, but a reverse effect is observed by increasing N _b and R _N. Figure 9 depicts the variation of N _T, N _b, R _N and a ₀ on the electrical potential function 𝜙. It is noticed that the electrical potential function decreases near the endoscope but increases as we move away from the endoscope toward the outer flexible tube with an increase in R _N and a ₀ while it has a reverse behavior with an increase in N _T and N _b. Finally, Figs 10 and 11 illustrate the streamlines for different values of N _T and N _b. It is observed that the size of trapped bolus decreases with the increase in the N _T, whereas it has an opposite effect with an increase in N _b. Furthermore, the size of trapped bolus in the presence of electric field are greater than those in the absence of electric field.

Fig. 10.

Streamlines at 𝜉 = 4, A = 0.1, a ₀ = 0.2, V₀ = 1, M = 2, m = 0.1, B _r = 3, G _r = 1, N _b = 1. (a) N _T = 4, L₁ = 0; (b) N _T = 5, L₁ = 0; (c) N _T = 4, L₁ = 20; (d) N _T = 5, L₁ = 20.

Fig. 11.

Streamlines at 𝜉 = 4, A = 0.1, a ₀ = 0.2, V₀ = 1, M = 2, m = 0.1, B _r = 3, G _r = 1, N _T = 4. (a) N _b = 1, L₁ = 0; (b) N _b = 2, L₁ = 0; (c) N _b = 1, L₁ = 30; (d) N _b = 2, L₁ = 30.

5. Conclusions

This study clarifies the theoretical framework for examining the effect of the presence of an electromagnetic field on the flow of couple stress fluid mixed with gold nanoparticles through two coaxial vertical tubes. The inner tube was considered a moving medical endoscope while the outer tube was considered as a flexible tube that oscillates in the form of a sinusoidal wave as a simulation of the movement of vital vessels within the human body. Under the long wavelength approximation, the exact solutions have been evaluated for temperature distribution, nanoparticles concentration, electrical potential function and pressure, while an analytical solution is found for the axial velocity of the nanofluid with the help of perturbation method. The most important results of this study have been listed below:

The effects of Brownian motion parameter, thermophoresis parameter, radiation parameter, amplitude ratio and Hall currents on the axial nanofluid velocity in the presence of the electric field are more obvious than those in the absence of the electric field, while the effect of couple stress parameter is less than that in the absence of the electric field.

The nanofluid pressure decreases with the increase in Brownian motion parameter and thermophoresis parameter, whereas it increases with the increase in the radiation parameter and electrical Rayleigh number.

The temperature increases with the increase of Brownian motion parameter and thermophoresis parameter, whereas it decreases as radiation parameter and amplitude ratio increase.

The concentration of gold nanoparticles increases with the increase of both Brownian motion parameter and radiation parameter, whereas it decreases as thermophoresis parameter and amplitude ratio increase.

The electrical potential function increases near the endoscope by increasing thermophoresis parameter and Brownian motion parameter, whereas it decreases as we move away from the endoscope towards the outer flexible tube.

The effects of thermophoresis and Brownian motion parameters on the volume of trapped bolus in the presence of electric field are greater than those in the absence of electric field.

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