An electroosmotic peristaltic flow of graphene-lubrication oil nanofluid through a symmetric channel

Abstract

The purpose of this manuscript is to analyze the electroosmotic peristaltic motion of graphene-lubrication oil nanofluid. Rheological characteristics of such fluids are predicted by using the Carreau’s model. Effects of magnetic field, electric field, viscous dissipation, Joule heating, Brownian motion and thermophoresis are also reckoned. Debye-Hückel linearization and lubrication approach are employed in mathematical modeling. Obtained non-linear system of equations are analytically solved by using the builtin command NdSolve and parametric study is carried out to visualize the variation in temperature, velocity, heat transfer rate, concentration profiles, pressure gradient and pressure rise per wavelength. Results unveil that strong electroosmotic parameter enhances both velocity and temperature. Temperature and velocity decline on the enlargement of the Helmholtz-Smoluchowski velocity. Variation in electroosmotic parameter decreases and Helmholtz-Smoluchowski velocity increases mass transfer rate. Pumping region can also be maintained by thickening and thinning the Electric Double Layer (EDL). Present study has useful applications in industries, defect sensor, energy saving, domestic air conditioning, cooling power electronic components and heat extraction and heat transportation.

Introduction

Nanofluids (NF) are studied because of their incredible thermal and rheological attributes. Nanofluids are colloidal suspensions of nanoscale metallic particles into conventional base fluids. Naoparticles are made of metals (Ag, Cu etc.), metallic oxides (Fe_xO_y, CuO, Al₂O₃ etc.), carbons (graphite, diamond, carbon nanotubes etc.), metal carbides (SiC etc) and metal nitrides (ZrN, TaN, AIN etc). Water, oil and ethylene-glycol are commonly used as base fluids. Nanofluids have tremendous applications in biological, pharmaceutical and technological sciences. Novel idea of Nanofluids was introduced by Choi and Eastman ¹ and addition of solid nanoparticles augments the thermal features of working fluid. Tiwari and Das ² unveiled two-phase model to analyze the nanoliquids by considering effective viscosity, density and thermal conductivity. Buongiorno ³ suggested a model to scrutinize the nanofluid motion by using thermophoresis and Brownian motion effects. Recently, Izadi et al.^4,5 reported useful analysis over natural convection of a micropolar and hybrid nanofluid inside a porous cavity with different configurations. Amelioration of pool boiling thermal performance by using the graphene-silver and new hybrid was discussed by Xu et al. ⁶ and Izadi et al. ⁷ respectively. Study of MHD thermogravitational convection of a micropolar nanoliquids in a porous chamber was also presented by Izadi et al. ⁸

Peristalsis phenomena is a mode of fluid motion inside a flexible channel or tube due to expansion and compression of continuous waves. Many physiological applications of this phenomena are found in daily life like urine movement from kidneys to bladder, food delivery via esophagus, chyme movement via gastrointestinal and vasomotion of small vessels etc. Industrial applications of peristalsis include movement of corrosive and sensitive fluids in finger and roller pumps etc. Initially, Latham ⁹ investigated the peristaltic motion of Newtonian fluids. Later, Shapiro et al. ¹⁰ theoretically investigated the peristaltic flow by using the “small-scale Reynold number and long wave length” approach. Ebaid ¹¹ reported the impacts of magnetohydrodynamics over peristaltic movement of Newtonian fluid inside asymmetric duct along with wall slip conditions. Many studies are conducted to analyze the attributes of non-Newtonian fluids driven by peristaltic as well. Tiripathi et al. ¹² studied peristaltic flow to see the slip effects of viscoelastic fluid by using fractional Burger’s model. Hayat et al. ¹³ explained the Jeffery fluids motion driven by peristalsis. Zahir et al. ¹⁴ illustrated the entropy generation in peristalsis of Casson fluid. Abbasi et al. ¹⁵ analyzed the magnetohydrodynamics peristaltic motion of Carreau-Yasuda (CY) fluid with Hall effects through curved channel.

Electroosmosis phenomena is defined as the ionized fluid flow in any capillary tube or microchannel under the effect of applied electric field. When a solid charged surface is attached with ionized solution or water, the negative charges gather on the surface. The cations from the liquid are attracted toward surface and anions are repelled from it. The remaining unbalanced charges then generate a thin layer is called electric double layer (EDL). While an electric field becomes parallel to this solid surface, the EDL moves in the same direction of electric field. Its applications are found in industrial, biomedical, oil and gas fields, mechanical and chemical engineering and so on as it works in micro channels. Chakraborty ¹⁶ explained the electroosmotically dragged capillary transport of biofluid through microchannel. Electroosmotic motion of power-law fluid in a micro-conduit was explained by Zhao et al. ¹⁷ Tang et al. ¹⁸ analyzed non-Newtonian fluids inside microchannel. The electroosmotic effect of power-law fluid for distinct zeta potential is discussed by Vasu and De. ¹⁹ Tripathi et al. ²⁰ analyzed the EDL impacts on peristaltic transport. Magnetohydrodynamics (MHD) is an investigation of fluid flow subjected to the external magnetic field. It has applications in MRI, detection of tumors, petroleum industries and many more. Noreen ²¹ and Tanveer et al. ²² discussed peristaltic movement of MHD Carreau fluid in a curved conduit. Peristaltic movement of Carreau nanofluid in an asymmetric tapered conduit under the effect of magnetic field is reported by Kothandapani and Prakash. ²³

Rheology is a study of materials flow behavior under applied stresses. The flow behavior of Newtonian fluids (e.g. urine) are studied through constitutive equations but rheological attributes of non-Newtonian fluids (shampoo, ketchup, drill mud, blood, and polymers) cannot be studied through a single constitutive relation. Therefore some interesting models are developed to study specific rheological attribute of non-Newtonian fluids. For example, Burger’s model, Maxwell model and Jeffery’s model are used to study the viscoelastic fluids. Casson fluid model is characterized for shear thinning behavior of non-Newtonian fluid. Carreau model, Power-law model, Second order fluid model, Eyring-Powell, Sisko, Hershel-Bulkley and Ellis model are used for Generalized Newtonian Fluids (GNF). Carreau fluid model illustrates the rheological attributes of non-Newtonian liquids and lubricants. BN-EG and Silicon oil are examples of Carreau nanofluids. Sobh, ²⁴ Ali and Hayat ²⁵ reported the peristalsis phenomena of Carreau fluid (CF) through asymmetric channel. Hayat et al. ²⁶ discussed the peristaltic motion of CF subjected to the induced magnetic field. Carreau fluid motion through a regular duct is explained by Nadeem et al. ²⁷ Abbasi et al. ²⁸ unveiled the Hall effects over peristaltic motion of BN-EG nanofluid along with temperature depending on thermal conductivity. Bakak et al. ²⁹ have proved during their experimental study that rheological attributes of nanofluids (suspension of Graphene nanoparticles into ethylene glycol and lubrication oil) are well computed by Carreau-Yasuda (CY) model. Moreover, the gathered results unveiled that CY is more suitable for GN-LO for the entire range of shear rates as compared to the GN-EG. The Graphene-based nanofluids have several applications in defect sensor, heat transfer, anti-infection therapy, cancer treatment, biomedical and energy saving system. ³⁰

Rheological properties of graphene nanoparticles dispersed into lubrication oil have not got much attention so far. Present study is an attempt to reduce this space. This article explores the electroosmotic flow of GN-LO nanofluid driven by peristaltic in a symmetric channel. Effects of magnetic field, electric field, Joule heating, Brownian motion, thermophoresis and viscous dissipation are reckoned. Parametric study is performed for velocity, temperature, Nusselt number and concentration profiles, pressure gradient and pressure rise per wavelength. GN-LO has useful applications in industries, defect sensor, energy saving, domestic air conditioning, cooling power electronic components, heat extraction and heat transportation.

Problem construction

Problem Statement

Assume a $2D$ electroosmotic flow of graphene-Lubrication oil (GN-LO) nanofluid passing through a symmetric channel having width $2d.$ Fluid motion is generated through sinusoidal waves traveling along $X$ -axis. The channel’s length is placed along $X$ -axis while $Y$ -axis is perpendicular to it (see Figure 1). Electromagnetic forces are generated in the flow direction by applied electric field. Let ${\hat{T}}_0$ and ${\hat{C}}_0$ denote the temperature and concentration at channel walls respectively. Mathematical modeling for wavy walls is described as:

$$\pm \hat{W}(\hat{X},\hat{t})=\pm d\pm a_1\cos (\frac{2\pi }{\lambda }(\hat{X}-c\hat{t})).$$

(1)

Here, $a_1$ is amplitude, $\hat{t}$ , $c$ is the speed and $\lambda$ be the wavelength of the peristaltic wave respectively. The geometry of the problem is given as:

Figure 1.

Geometry of physical model.

Electro-magnetohydrodynamics

Generalized Ohmic law prescribed in ³¹ is:

$$\mathbf{J}={\sigma }_f[\mathbf{V}\times \mathbf{B}+\mathbf{E}],$$

(2)

where $\mathbf{E}=(E_x,0,0)$ and $\mathbf{B}=(0,0,B_0)$ are applied external electric and magnetic fields respectively. The velocity vector for $2D$ motion is $\mathbf{V}=[\hat{U},\hat{V},0].$ Using these parameters, the Lorentz force and Joule heating become:

$$\mathbf{F}=\mathbf{J}\times \mathbf{B}={\sigma }_f[-B_0^2\hat{U},-(B_0{E}_x+B_0^2\hat{V}),0],$$

(3)

$$\frac{\mathbf{J}.\mathbf{J}}{{\sigma }_f}={\sigma }_f[E_x^2+2B_0{E}_x\hat{V}+B_0^2({\hat{V}}^2+{\hat{U}}^2)].$$

(4)

The Poisson equation in a symmetric channel according to Tanveer et al. ³² is defined as:

$${\nabla }^2\hat{\mathrm{\Phi }}=-\frac{{\rho }_e}{\epsilon },$$

(5)

here $\hat{\mathrm{\Phi }}$ indicates electric potential, ${\rho }_e$ denotes net charged density and $\epsilon$ is the electrical permittivity. The density of the charge as per Boltzmann distribution ³² and relationship between cations $({\hat{n}}^{+})$ and anions ( ${\hat{n}}^{-}$ ) are given below:

$${\rho }_e=-\mathrm{ze}({\hat{n}}^{-}-{\hat{n}}^{+}),$$

(6)

$${\hat{n}}^{\pm }={\hat{n}}_{0}e(\pm \frac{\mathrm{ez}}{T_{\mathrm{av}}{K}_B}\hat{\mathrm{\Phi }}),$$

(7)

Here ${\hat{n}}_0$ denotes bulk concentration of ions, $z$ charge balance, $e$ electrical charge, $K_B$ Boltzmann constant and $T_{\mathrm{av}}$ average temperature. By using Debye-Hückle linearization approach, ³³ equation (5) becomes:

$$\frac{d^2\mathrm{\Phi }}{d{y}^2}={\kappa }^2\mathrm{\Phi }$$

(8)

Where $\kappa =d\left(\mathrm{ez}\sqrt{\frac{2n_0}{\epsilon {\hat{T}}_{\mathrm{av}}{K}_B}}\right)=\left(\frac{d}{{\lambda }_D}\right)$ stands for electro-osmotic parameter in which ${\lambda }_D$ is the Debye length/thickness of electrical double layer (EDL). Utilizing the following boundary conditions:

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d\mathrm{\Phi }}{\mathrm{dy}}=0\mathrm{when}}& y=0,\hfill \\ \multicolumn{1}{c}{\mathrm{\Phi }=1\mathrm{when}}& y=w.\hfill \end{array}$$

(9)

Solution of the equation (8) is given as:

$$\Phi (y)=\frac{\cosh (\kappa y)}{\cosh (\kappa w)}.$$

(10)

Basic governing equations

The basic governing equations with the effects of electro-magnetohydrodynamics, viscous dissipation, Brownian motion and thermophoresis and Joule heating according to Refs.^20,28 are modeled as:

$$\frac{\partial (\hat{U})}{\partial \hat{X}}+\frac{\partial (\hat{V})}{\partial \hat{Y}}=0,$$

(11)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial \hat{t}})\hat{U}=-\frac{\partial \hat{P}}{\partial \hat{X}}+(\frac{\partial {\hat{S}}_{\hat{X}\hat{X}}}{\partial \hat{X}}+\frac{\partial {\hat{S}}_{\hat{X}\hat{Y}}}{\partial \hat{Y}})}\\ +{\rho }_e{\hat{E}}_x-{\sigma }_f{B}_0^2\hat{U},\hfill \end{array}$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial \hat{t}})\hat{V}=-\frac{\partial \hat{P}}{\partial \hat{Y}}+(\frac{\partial {\hat{S}}_{\hat{Y}\hat{X}}}{\partial \hat{X}}+\frac{\partial {\hat{S}}_{\hat{Y}\hat{Y}}}{\partial \hat{Y}})}\\ -{\sigma }_f(B_0{E}_x+B_0^2\hat{V}),\hfill \end{array}$$

(13)

$$\begin{array}{c}(\rho C{)}_f(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial \hat{t}})T=K_f(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})T\hfill \\ +\varsigma (\rho C{)}_f\left[\begin{array}{c}{D}_B(\frac{\partial C}{\partial \hat{X}}\frac{\partial T}{\partial \hat{X}}+\frac{\partial C}{\partial \hat{Y}}\frac{\partial T}{\partial \hat{Y}})\hfill \\ \multicolumn{1}{c}{+\frac{D_T}{T_0}{(\frac{\partial T}{\partial \hat{X}}+\frac{\partial T}{\partial \hat{Y}})}^2}\end{array}\right]\hfill \\ +\hat{S}.L+{\sigma }_f[E_x^2+2B_0{E}_x\hat{V}+B_0^2({\hat{V}}^2+{\hat{U}}^2)],\hfill \end{array}$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial \hat{t}})C=D_B(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})C}\\ +\frac{D_T}{T_0}(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})T.\hfill \end{array}$$

(15)

In above equations, $\hat{P},$ ${\rho }_f,$ $T$ , $\hat{S}.L$ , $L$ , ${\hat{S}}_{\mathrm{ij}}$ , and $C$ denote pressure, density, temperature, viscous dissipation term, velocity gradient, extra stress tensor components and concentration respectively. $C_f$ , $K_f$ , $D_B$ , and $D_T$ stand for specific heat, thermal conductivity, mass and thermal diffusivity of the nanofluid. Moreover $\varsigma (=\frac{{(\rho C)}_p}{{(\rho C)}_f})$ , where $(\rho C{)}_p$ denotes effective heat capacity. Rheological characteristics of graphene-lubrication oil nanofluid can be correctly computed by using the Carreau’s model. ²⁸ For Carreau’s model, the extra stress tensor is:

$$\hat{S}={\hat{A}}_1\eta ,$$

(16)

$$\eta ({\chi }^{.})=[{\eta }_{\infty }+\frac{{\eta }_0-{\eta }_{\infty }}{{(1+{(\alpha {\chi }^{.})}^2)}^2}]$$

(17)

where ${\chi }^{.}=\sqrt{2\mathrm{tr}{\aleph }^2}$ , $\aleph =\frac{1}{2}{\hat{A}}_1$ and ${\hat{A}}_1=\mathrm{grad}\mathbf{V}+(\mathrm{grad}\mathbf{V}{)}^T$ . Here ${\eta }_{\infty }$ and ${\eta }_0$ stand for infinite and zero shearing-rates of viscosities, ${\chi }^{.}$ the shear rate, $\alpha$ material constant, $\mathrm{grad}\mathbf{V}$ the velocity gradient and $n$ non-dimensional form of power-law index.

The transformation of coordinates and velocities from fixed to moving frames are defined as:

$$\begin{array}{c}\hat{p}(\hat{x},\hat{y})=\hat{P}(\hat{X},\hat{Y},\hat{t}),\hat{u}=\overline{U}-c,\hat{x}=\hat{X}-c\overline{t},\hfill \\ \hat{y}=\hat{Y},\hat{v}=\overline{V}.\hfill \end{array}$$

(18)

In the light of the above transformation equations (11)–(15) for steady motion in moving frame are as follows:

$$\frac{\partial (\hat{u})}{\partial \hat{x}}+\frac{\partial (\hat{v})}{\partial \hat{y}}=0,$$

(19)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})(\hat{u}+c)=-\frac{\partial \hat{p}}{\partial \hat{x}}+(\frac{\partial {\hat{S}}_{\hat{x}\hat{y}}}{\partial \hat{x}}+\frac{\partial {\hat{S}}_{\hat{x}\hat{y}}}{\partial \hat{y}})}\\ +{\rho }_e{\hat{E}}_x-{\sigma }_f{B}_0^2(\hat{u}+c),\hfill \end{array}$$

(20)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})\hat{v}=-\frac{\partial \hat{p}}{\partial \hat{y}}+(\frac{\partial {\hat{S}}_{\hat{y}\hat{x}}}{\partial \hat{x}}+\frac{\partial {\hat{S}}_{\hat{y}\hat{y}}}{\partial \hat{y}})}\\ -{\sigma }_f(B_0{E}_x+B_0^2\hat{v}),\hfill \end{array}$$

(21)

$$\begin{array}{c}(\rho C{)}_f((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})T=K_f(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})T\hfill \\ +\varsigma (\rho C{)}_f[D_B(\frac{\partial C}{\partial \hat{x}}\frac{\partial T}{\partial \hat{x}}+\frac{\partial C}{\partial \hat{y}}\frac{\partial T}{\partial \hat{y}})+\frac{D_T}{T_0}{(\frac{\partial T}{\partial \hat{x}}+\frac{\partial T}{\partial \hat{Y}})}^2]\hfill \\ +\hat{s}.L+{\sigma }_f[E_x^2+2B_0{E}_x\hat{v}+B_0^2({\hat{v}}^2+{(\hat{u}+c)}^2)],\hfill \end{array}$$

(22)

$$((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})C=D_B(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})C+\frac{D_T}{T_0}(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})T.$$

(23)

Here $\hat{s}.L$ and ${\hat{s}}_{\mathrm{ij}}$ denote viscous dissipation and extra stress tensor in moving frame. Moreover, the components of extra stress tensor involving parameter of viscosity $\hat{\beta }(=\frac{{\eta }_{\infty }}{{\eta }_0})$ are given as:

$$\begin{array}{c}{\hat{S}}_{\hat{x}\hat{x}}=2{\eta }_0[1+\frac{{\alpha }^{2}n}{2}(\hat{\beta }-1)\{2({\hat{u}}_{\hat{x}}^2+{\hat{v}}_{\hat{y}}^2)+{({\hat{u}}_{\hat{y}}+{\hat{v}}_{\hat{x}})}^2\}]{\hat{u}}_{\hat{x}},\hfill \\ {\hat{S}}_{\hat{x}\hat{y}}={\eta }_0[1+\frac{{\alpha }^{2}n}{2}(\hat{\beta }-1)\{2({\hat{u}}_{\hat{x}}^2+{\hat{v}}_{\hat{y}}^2)+{({\hat{u}}_{\hat{y}}+{\hat{v}}_{\hat{x}})}^2\}]({\hat{u}}_{\hat{y}}+{\hat{v}}_{\hat{x}}),\hfill \\ {\hat{S}}_{\hat{y}\hat{y}}=2{\eta }_0[1+\frac{{\alpha }^{2}n}{2}(\hat{\beta }-1)\{2({\hat{u}}_{\hat{x}}^2+{\hat{v}}_{\hat{y}}^2)+{({\hat{u}}_{\hat{y}}+{\hat{v}}_{\hat{x}})}^2\}]{\hat{v}}_{\hat{y}}.\hfill \end{array}$$

(24)

Non-dimensionalization

Following dimensionless quantities are considered here:

$$\begin{array}{c}x=\frac{\hat{x}}{\lambda },y=\frac{\hat{y}}{d},u=\frac{\hat{u}}{c},v=\frac{\hat{\nu }}{c\delta },\delta =\frac{d}{\lambda },w=\frac{\hat{W}}{d},\hfill \\ a=\frac{a_1}{d}p=\frac{d^2\hat{p}}{c\lambda {\eta }_0},\nu =\frac{{\eta }_0}{{\rho }_f},S_{\mathrm{xy}}=\frac{d}{{\eta }_{0}c}{\hat{s}}_{\hat{x}\hat{y}},M^2=\frac{{\sigma }_f}{{\eta }_0}{B}_0^2{d}^2,\hfill \\ \mathrm{Re}=\frac{{\rho }_f\mathrm{cd}}{{\eta }_0},\mathrm{Pr}=\frac{{\eta }_0{C}_f}{K_f},\mathrm{Ec}=\frac{c^2}{C_f{\hat{T}}_0},\mathrm{Br}=\mathrm{Pr}.\mathrm{Ec},\hfill \\ \theta =\frac{T-T_0}{T_0},\varphi =\frac{C-C_0}{C_0},N_t=\frac{\varsigma D_T}{\nu },N_b=\frac{\varsigma D_B{C}_0}{\nu },\hfill \\ W_e=-\frac{\mathrm{ac}}{d},\kappa =\frac{d}{{\lambda }_D},U_{\mathrm{hs}}=-\frac{{\hat{E}}_x\epsilon }{c{\eta }_0},\mathrm{\Phi }=\frac{\mathrm{ez}}{T_{\mathrm{av}}{K}_B}\hat{\mathrm{\Phi }},\hfill \\ S=\frac{{\sigma }_f{d}^2{E}_x^2}{K_f{T}_0},u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}.\hfill \end{array}$$

(25)

Where $M,$ $\mathrm{Br},$ $\mathrm{Pr},$ $\mathrm{Ec}$ , and $\mathrm{We}$ stand for Hartman, Brinkman, Prandtl, Eckert and Weissenberg numbers respectively, $\kappa ,$ $S$ , and $U_{\mathrm{hs}}$ are electroosmotic, Joule heating parameters and Helmholtz-Smoluchowski velocity, $N_b$ and $N_t$ stand for Brownian motion and thermophoresis parameters, respectively.

Utilizing the above dimensionless quantities and employing the “small-scale Reynolds number and long wavelength” approximation, equations (19)–(23) are reduced and given below:

$$\begin{array}{c}\multicolumn{1}{c}{0=\frac{\partial p}{\partial x}+\frac{\partial }{\partial y}[(1+\frac{n}{2})(\hat{\beta }-1)W_e^2{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2]\frac{{\partial }^2\psi }{\partial y^2}}\\ +{\kappa }^2{U}_{\mathrm{hs}}\Phi -M^2(\frac{\partial \psi }{\partial y}+1),\hfill \end{array}$$

(26)

$$0=\frac{\partial p}{\partial y},$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{0=\frac{{\partial }^2\theta }{\partial y^2}+\mathrm{Br}\mathrm{\Omega }+\mathrm{Pr}{N}_b\frac{\partial \varphi }{\partial y}\frac{\partial \theta }{\partial y}+\mathrm{Pr}{N}_t{\left(\frac{\partial \theta }{\partial y}\right)}^2}\\ +\mathrm{Br}{M}^2{(1+\frac{{\partial }^2\psi }{\partial y^2})}^2+S,\hfill \end{array}$$

(28)

$$0=\frac{\partial \varphi }{\partial y}+\frac{N_b}{N_t}\frac{\partial \theta }{\partial y}.$$

(29)

Eliminating pressure between equations (28) and (29) yields:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2}{\partial y^2}[(1+\frac{n}{2})(\hat{\beta }-1)W_e^2{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2]\frac{{\partial }^2\psi }{\partial y^2}}\\ \multicolumn{1}{c}{+{\kappa }^2{U}_{\mathrm{hs}}\frac{\partial \Phi }{\partial y}-M^2\frac{{\partial }^2\psi }{\partial y^2}=0,}\end{array}$$

(30)

where

$$\mathrm{\Omega }=[(1+\frac{n}{2})(\hat{\beta }-1)W_e^2{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2]\frac{{\partial }^2\psi }{\partial y^2}.$$

(31)

The boundary conditions dealing with the velocity slip, convective temperature and concentration are:

$$\hat{U}-{\hat{U}}_w=\hat{\beta }{\hat{S}}_{\hat{x}\hat{y}},-K_f\frac{\partial T}{\partial Y}=\hat{l}(T-T_w),-D\frac{\partial C}{\partial Y}=\hat{k}(C-C_w).$$

(32)

In above equations ${\hat{U}}_w,$ $T_w$ and $C_w$ denotes the velocity, temperature and concentration at wall, $\hat{l}$ and $\hat{k}$ are heat transfer and mass transfer coefficients. The stream function satisfies the continuity equation identically. The non-dimensional slip boundary conditions to compute the stream function, temperature and concentration are:

$$\begin{array}{c}\psi =0,{\psi }_{\mathrm{yy}}=0,{\theta }_y=0,{\varphi }_y=0\mathrm{at}y=0,\hfill \\ \psi =\mathrm{\Theta },{\psi }_y+{\beta }_1{S}_{\mathrm{xy}}=-1,{\theta }_y+\mathrm{Bi}\theta =0,{\varphi }_y+\mathrm{Mi}\hfill \\ \varphi =0\mathrm{at}y=w,\hfill \end{array}$$

(33)

where

$$w(x)=1+a\cos (2\pi x).$$

(34)

Here, ${\beta }_1=\left(\frac{\hat{\beta }}{d}\right),$ $\mathrm{Bi}=\left(\frac{\hat{l}}{d}\right)$ and $\mathrm{Mi}=\left(\frac{\hat{k}}{d}\right)$ are dimensionless velocity slip parameter, heat and mass transmission Biot numbers respectively.

The correlation between the dimensionless flow rates $\mu =\left(\frac{\hat{Q}}{\mathrm{cd}}\right)$ (in fixed frame) and $\mathrm{\Theta }=\left(\frac{\hat{q}}{\mathrm{cd}}\right)$ (in moving frame) is:

$$\mu =\mathrm{\Theta }+1.$$

(35)

In which $\mathrm{\Theta }$ is:

$$\mathrm{\Theta }=\underset{0}{\overset{w}{\int }}\frac{\partial \psi }{\partial y}\mathrm{dy}.$$

(36)

The dimensionless heat transmission coefficient is given by:

$$\chi =w_x|{\theta }_y{|}_{y=w}.$$

(37)

Moreover, non-dimensional form of pressure rise per wavelength $(\mathrm{\Delta }P)$ is given below:

$$\mathrm{\Delta }P=\underset{0}{\overset{1}{\int }}\frac{\mathrm{dp}}{\mathrm{dx}}\mathrm{dx}.$$

(38)

The system of dimensionless equations (30), (31), and (32) having boundary conditions (35) is solved numerically by using builtin command NDSolve in Mathematica. Outcomes are compiled graphically to facilitate physical analysis.

Outcomes and related discussion

The aim of this portion is to examine the consequences of Hartman number $(M),$ electro osmotic parameter $(\kappa )$ and Helmholtz-Smoluchowski velocity $(U_{\mathrm{hs}})$ on the velocity, temperature, heat transmission rate, concentration characteristics, pressure gradient and pressure rise per wavelength via graphs. While sundry parameters according to Refs.^12,28,34 are $N_b=0.5,$ $N_t=0.5,$ $\mathrm{Br}=0.3,$ $\mathrm{Pr}=0.5,$ $a=0.2,$ $\eta =0.8,$ $x=0,$ $\hat{\beta }=0.9,$ ${\beta }_1=0.3,$ $\mathrm{Bi}=0.5,$ $\mathrm{Mi}=0.5,$ $n=2.5,$ $\mathrm{We}=0.1,$ $\kappa =2,$ $U_{\mathrm{hs}}=-1$ and $S=1$ respectively and are fixed unless stated alter-wise.

Velocity variation

Figures 2 to 4 portray the impacts of different values of $M,$ $\kappa$ , and $U_{\mathrm{hs}}$ on velocity profile. It is evident from Figure 2 that for the large values of $M,$ velocity decreases. As magnetic field facilitates the drag forces (Lorentz forces) against the flow and causes velocity decrement. It is observed via Figure 2 that axial velocity greatly depends on the $\kappa .$ Here, $\kappa$ is inversely proportional to the Debye thickness ${\lambda }_D$ of electric double layer (EDL) and directly proportional to the channel width. Velocity increases significantly in the upper part of the channel (for larger width and smaller thickness of EDL) on the enhancement of $\kappa .$ $U_{\mathrm{hs}}=-\frac{{\hat{E}}_x\epsilon }{c{\eta }_0}$ defines that Helmholtz-Smoluchowski velocity is directly proportional to ${\hat{E}}_x$ the electric field. In Figure 4, the fluid velocity is analyzed for three cases $(U_{\mathrm{hs}}<0),$ $(U_{\mathrm{hs}}>0)$ and $(U_{\mathrm{hs}}=0).$ For all cases of $U_{\mathrm{hs}},$ velocity diminishes.

Figure 2.

Velocity variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5.$

Figure 3.

Velocity variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5.$

Figure 4.

Velocity variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5$ .

Temperature variation

Figures 5 to 7 depict the variations of temperature as a function of $M,$ $\kappa$ , and $U_{\mathrm{hs}}$ respectively. Figure 5 reveals the impact of $M$ on temperature. Significant rise in temperature is recorded for large value of Hartman number owing to the Joule heating triggered by the constant effect of applied magnetic field. Figure 6 portrays the enhancement in temperature for large values of electro osmotic parameter $\kappa .$ Decrease in thickness of electric double layer (EDL) increases temperature $\theta .$ Figure 7 depicts the impacts of Helmholtz-Smoluchowski velocity on temperature. Temperature inclines when $U_{\mathrm{hs}}<0$ as $E_x>0$ and favors the flow and promotes the thermal diffusion. If $U_{\mathrm{hs}}=0$ then there is no effect on thermal diffusion as $E_x=0$ is non effective. Temperature declines when $U_{\mathrm{hs}}>0$ as $E_x<0.$

Figure 5.

Temperature variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5$ .

Figure 6.

Temperature variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5$ .

Figure 7.

Temperature variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ and $N_t=0.5$ .

Heat transfer rate at walls

Figures 8 to 10 illustrate the consequences of M, κ and $U_{\mathrm{hs}}$ on heat transmission rate. Results highlight that heat transmission increases for large values of M. As applied magnetic field escalates joule heating. Same growth is noticed for the large values of κ. It is due to the bulk layer of EDL which promotes heat exchange. Figure 10 witnesses the decrement in $\theta \prime (h)$ on increasing $U_{\mathrm{hs}}.$ As at higher values of $U_{\mathrm{hs}},$ electric field is acted against the flow and prevent the heat transmission.

Figure 8.

Heat transfer rate variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 9.

Heat transfer rate variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 10.

Heat transfer rate variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Concentration variation

Figures 11 to 13 indicate the impacts of M, κ and $U_{\mathrm{hs}}$ on concentration profile. Decay is found in concentration distribution on increasing M, and κ and has good relevance with previous results provided by Gul et al. ²⁸ and Yasmin and Iqbal, ³⁵ respectively. The growth is observed in the Figure 13 on increasing $U_{\mathrm{hs}}.$

Figure 11.

Concentration variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 12.

Concentration variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 13.

Concentration variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Pressure gradient

Figures 14 to 16 demonstrate the distribution of pressure gradient for different values of M, κ, and $U_{\mathrm{hs}}$ . All pressure gradient graphs demonstrate an oscillatory behavior. Figure 14 illustrates impact of M on $\frac{\mathrm{dp}}{\mathrm{dx}}.$ It is clear that strong Hartman number decreases pressure gradient. Same behaviors of pressure gradient is observed in Figure 15 for large κ. Figure 16 highlights the impact of Helmholtz-Smoluchowski velocity $(U_{\mathrm{hs}})$ on $\frac{\mathrm{dp}}{\mathrm{dx}}.$ An increase in $U_{\mathrm{hs}}$ augments pressure gradient. It is owing to the EDL existence in charged surface which actually provides the resistance against the flux. So pumping region can also be maintained by thickening and thinning the Electric Double Layer (EDL).

Figure 14.

Pressure gradient variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 15.

Pressure gradient variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 16.

Pressure gradient variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Pressure rise per wavelength

Figures 17 to 19 outline the average pressure rise $(\mathrm{\Delta }p)$ versus mean flow rate $(\mathrm{\Theta })$ for distinct parameters $M$ , $\kappa$ , and $U_{\mathrm{hs}}.$ Figure 17 declares that on increasing $M,$ pressure rise increases in retrograde $(\mathrm{\Theta }<0,\mathrm{\Delta }p>0)$ and reverse $(\mathrm{\Theta }<0,\mathrm{\Delta }p>0)$ pumping region while decreases in augmented flow region $(\mathrm{\Theta }>0,\mathrm{\Delta }p<0).$ It is observed in the Figure 18 that pressure rise drops at large value of $\kappa$ in all pumping regions. Figure 19 depicts that pressure rise is increased with higher value of $(U_{\mathrm{hs}}).$

Figure 17.

Pressure rise per wavelength variance with $\mathrm{Br}=0.3,$ $K=1,$ $a=0.2,$ Θ = 0.8, x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 18.

Pressure rise per wavelength variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi = 0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $U_{\mathrm{hs}}=-1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Figure 19.

Pressure rise per wavelength variance with $\mathrm{Br}=0.3,$ $M=1,$ $a=0.2,$ $\eta =0.8,$ x = 0, $\hat{\beta }=0.5,$ ${\beta }_1=0.5,$ Bi =0.5, Mi = 0.5, n = 2.5, $W_e=0.1,$ Pr = 0.5, $K=1,$ κ = 2, S = 1, $N_b=0.5$ , and $N_t=0.5$ .

Comparison with previous results

A comparison study is conducted to check the validity of the numerical technique. Numerical values of ΔP from exact solution calculated by Ali et al. ³⁶ and from particular case of the present study are provided in Table 1. Both results have accuracy up to three decimal places which is a good agreement with previous study.

Table 1.

Comparison table for critical values of Θ above which ΔP < 0 and below which ΔP > 0 calculated by Ali et al. ³⁶ with present study.

${\beta }_1$	Ali et al. 36	Present outcomes
0.0	0.3683	0.367888
0.03	0.3533	0.354241
0.06	0.3411	0.340593

Conclusions

Electroosmotic peristaltic flow graphene-lubrication oil nanofluid through a symmetric conduit is examined. Effects of Brownian motion, viscous dissipation, thermophoresis and Joule heating are also reckoned. Key features are summarized below:

• Temperature rapidly rises at higher Hartman number.

• Strong electroosmotic parameter enhances velocity and temperature.

• Velocity and temperature declines on the enlargement of the Helmholtz-Smoluchowski velocity.

• Concentration decreases for higher values of $M$ and $\kappa$ but increases for large value of $U_{\mathrm{hs}}.$

• Pressure gradient decreases on increasing $M$ and $\kappa$ and increases on increasing $U_{\mathrm{hs}},$ respectively.

• Enhancement in pressure rise is noticed in retrograde pumping and co-pumping region while decrement in augment region for strong $M.$

• The pumping rate decreases on enhancement of $\kappa$ and increases on changing $U_{\mathrm{hs}}.$