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

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 x O y , CuO, Al 2 O 3 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 ''smallscale 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 microconduit 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.  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 Graphenebased 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 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 b T 0 and b C 0 denote the temperature and concentration at channel walls respectively. Mathematical modeling for wavy walls is described as: Here, a 1 is amplitude, b t, c is the speed and l be the wavelength of the peristaltic wave respectively. The geometry of the problem is given as:

Electro-magnetohydrodynamics
Generalized Ohmic law prescribed in 31 is: where E = (E x , 0, 0) and B = (0, 0, B 0 ) are applied external electric and magnetic fields respectively. The velocity vector for 2D motion is V = ½Û ,V , 0: Using these parameters, the Lorentz force and Joule heating become: J: The Poisson equation in a symmetric channel according to Tanveer et al. 32 is defined as: hereF indicates electric potential, r e denotes net charged density and e is the electrical permittivity. The density of the charge as per Boltzmann distribution 32 and relationship between cations (n + ) and anions (n À ) are given below: Heren 0 denotes bulk concentration of ions, z charge balance, e electrical charge, K B Boltzmann constant and T av average temperature. By using Debye-Hu¨ckle linearization approach, 33 equation (5) becomes: osmotic parameter in which l D is the Debye length/ thickness of electrical double layer (EDL). Utilizing the following boundary conditions: Solution of the equation (8) is given as: Basic governing equations The basic governing equations with the effects of electromagnetohydrodynamics, viscous dissipation, Brownian motion and thermophoresis and Joule heating according to Refs. 20,28 are modeled as: In above equations,P, r f , T ,Ŝ:L, L,Ŝ 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.
where (rC) 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: h( x where x : = ffiffiffiffiffiffiffiffiffiffiffi ffi 2 tr@ 2 p , @= 1 2Â 1 andÂ 1 =gradV+(gradV) T . Here h ' and h 0 stand for infinite and zero shearingrates of viscosities, x : the shear rate, a material constant, gradV 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: In the light of the above transformation equations (11)-(15) for steady motion in moving frame are as follows: ( Hereŝ:L andŝ ij denote viscous dissipation and extra stress tensor in moving frame. Moreover, the components of extra stress tensor involving parameter of viscosityb( = h ' h 0 ) are given as: Non-dimensionalization Following dimensionless quantities are considered here: Where M, Br, Pr, Ec, and We stand for Hartman, Brinkman, Prandtl, Eckert and Weissenberg numbers respectively, k, S, and U 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: Eliminating pressure between equations (28) and (29) yields: where The boundary conditions dealing with the velocity slip, convective temperature and concentration are: In above equationsÛ w , T w and C w denotes the velocity, temperature and concentration at wall,l andk are heat transfer and mass transfer coefficients.
In which Y is: The dimensionless heat transmission coefficient is given by: Moreover, non-dimensional form of pressure rise per wavelength (DP) is given below: 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.

Temperature variation
Figures 5 to 7 depict the variations of temperature as a function of M, k, and U 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 k: Decrease in thickness of electric double layer (EDL) increases temperature u: Figure 7 depicts the impacts of Helmholtz-Smoluchowski velocity on temperature. Temperature inclines when U hs \0 as E x .0 and favors the flow and promotes the thermal diffusion. If U hs = 0 then there is no effect on thermal diffusion as E x = 0 is non effective. Temperature declines when U hs .0 as E x \0: Heat transfer rate at walls     heat transmission increases for large values of M. As applied magnetic field escalates joule heating. Same growth is noticed for the large values of k. It is due to the bulk layer of EDL which promotes heat exchange. Figure 10 witnesses the decrement in u 0 (h) on increasing U hs : As at higher values of U hs , electric field is acted against the flow and prevent the heat transmission. Figures 11 to 13 indicate the impacts of M, k and U hs on concentration profile. Decay is found in concentration distribution on increasing M, and k 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 hs :

Pressure gradient
Figures 14 to 16 demonstrate the distribution of pressure gradient for different values of M, k, and U hs . All pressure gradient graphs demonstrate an oscillatory behavior. Figure 14 illustrates impact of M on dp dx : It is clear that strong Hartman number decreases pressure gradient. Same behaviors of pressure gradient is observed in Figure 15 for large k. Figure 16 highlights the impact of Helmholtz-Smoluchowski velocity (U hs ) on dp dx : An increase in U 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 19 depicts that pressure rise is increased with higher value of (U hs ):

Comparison with previous results
A comparison study is conducted to check the validity of the numerical technique. Numerical values of DP 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.

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:     Pressure gradient decreases on increasing M and k and increases on increasing U 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 k and increases on changing U hs :

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.