Electromagnetohydrodynamic Effects with Single-Walled Carbon Nanotubes Particles in a Corrugated Microchannel

. One of the advanced products of nanotechnology having notable mechanical and physical properties are the Single-Walled Carbon Nanotubes (SWCNTs). This motivated us to investigate the effect of single-walled carbon nanotubes (SWCNTs) suspended in a microchannel with corrugated walls under the effect of electromagnetic hydrodynamic flow (EMHD). The corrugation of the wavy walls are described by periodic sinusoidal waves of small amplitudes either in phase or out of phase. The problem simulated with a system of governing equations such as potential, momentum, and heat equations were solved analytically using the perturbation method. The behavior of nanofluid velocity, temperature, volumetric flow rate, and average velocity investigated under the multi-conditions of using three models for the sizes and shapes of carbon nanoparticles tubes. By addition of SWCNTs, the fluid velocity is reduced at the center of the channel by providing resistance to the fluid motion. The size and concentration of SWCNTs influence enhance the rate of heat transfer. Also, the Xue's model has the highest heat transfer rate when compared to the Maxwell and Hamilton-Crosser's (H-C) models. Finally, the obtained results for the flow rate were compared with previously published data and appeared in good agreement.


Introduction
Some years ago, lot of researchers have interested to study the impact of nanoparticles due to its new physical and Thermophysical properties, and how these new feature affect in the industry and biomedical fields.Choi [1] was the first one that studied the efficacy of heat transfer using nanoparticles in his work , since this time the concept of nanofluid starts in spread.Khanafer et al. [2] investigated the heat transfer enhancement due to using of the Carbon nanotubes in the base fluid inside an enclosure, According to the obtained results that they deduced the mechanical and thermodynamic properties have improved.Kakac et al. [3] present a review of how the heat transfer improved by using a metal nanoparticles in a base fluid such as water, Furthermore, the review shows the investigation of variables based on the development and presentation of Thermophysical properties of a Nano fluid.Turkylmazoglu [4] investigated the flow and heat transfer of nanofluid over a rotating disk, the study considered five different types of nanoparticles, they concluded that the nanofluids due to using Cu, CuO and Ag give low mount of axial fluid comparing with Al2O3 and TiO2 , Moreover, a high heat transfer rate occur in the presence of Cu -nanoparticles and low rate for TiO2 .Abbasi et al. [5] studied the Brownian motion and the thermophoresis of the non-Newtonian nanofluid runs within peristaltic channel.Hayat et al. [6] Investigated the mixed convection of peristaltic nanofluid flow runs within a channel, they considered two types of nanotubes single and double walls.According to the results obtained, adding of SWCNTs leads to decrease in fluid velocity near the center of the channel.Moreover, when the size of the SWCNTs fraction increases, the heat transfer rate in the boundary increase, also they concluded that the pressure in the closed section of the pressure also leads to a decrease in the pressure rise per wavelength when suspended carbon nanotubes (SWCNTs) are inserted.
In recent years, many microfluidic devices have been invented There has recently been in analytical and numerical models of EMHD flow in microchannel.Chakraborty et al. [7] analyzed the combined effect of magnetic and electric on the fluid flow between two parallel plates, A significant rise in volumetric flow rates happens with the help of a relatively modest magnitude magnetic field, according to the study.They are, however, highly unreliable in high-strength magnetic fields.Duwairi and Abdullah [8] Analyzed analytically and numerically the flow and temperature distribution of hydrodynamic magnetohydrodynamic fluid under the presence of Lorentz force in a micro-pump.Yoon et al. [9] they discussed the effects of wavy surface form, magnetic field, and heat flux from the disc surface of magnetohydrodynamic (MHD) flow across a fast rotating axisymmetric wavy disc.The influence of the wavy surface's amplitude on the heat transfer properties of MHD flow across a wavy surface was also explored.Buren et al. [10] studied the combined effect of magnetic and electric on Jeffrey fluids moving through a microchannel with corrugated walls.Andreozzi et al. [11] they investigated the effects of heat transport on nanofluid and ribs in a channel.According to this study Triangular ribs have the best thermal performance.to them.Rashid et al. [12] Analyzed the second grade fluid with a combined effect of magnetic and electric runs between two corrugated walls in microchannel.Abo-Elkhair et al. [13] they Examined using of electric and magnetic fields to control flow rates of a fluid with variable viscosity moving inside a peristaltic micro channel.Liu et al. [14] investigated the entropy generation rate of electro-hydro magnetic flow of Newtonian fluid moving through a curved rectangular microchannel, they concluded that the entropy rate increases with increase of magnetic field parameter Ha until a certain value of it then remain constant as Ha increases.As well as the entropy rate reaches to its maximum value at the walls.Liu et al. [15] discussed electro-viscous effects on the EMHD flow of Maxwell fluid in microchannel were.They reached to an analytical solution and analyzed the flow transport properties in relation to the streaming potential and electro viscous flow.Xue [16] He discussed the Maxwell's theorem and the methods for determining effective heat, as well as the thermal conductivity of carbon nanotube-based compounds.Si and Jian [17] studied the effect of both magnetic and electric field of the viscous Jeffrey fluid which moving in two parallel corrugated walls under the driven of Lorentz force.Rashid and Nadeem [18] explored analytically the flow of a corrugated microchannel wall with a porous media under EMHD effects.They used the perturbation technique to analyze the flow of the second-grade fluid, which they regarded a working fluid.They explored the flow transport phenomenon in this work without describe the temperature distribution associated with such flow.Reza el al. [19] explained analytically the effect of hydrodynamic electromagnetic flow supported by nanoparticles in a tiny channel with uniform undulating rough wall in the presence of an applied magnetic field and a transverse electric field.
The purpose of this work is to look at the EMHD flow of Nano fluid through a corrugated channel.The thermal conductivity of carbon nanotubes immersed in water has been studied using a variety of models.The system is affected by the Lorenz force, which is created by the interaction of electric and magnetic fields.The perturbation strategy has used to address the problem in the microchannel.And the investigation of the features of fluid flow speed and temperature in a tiny channel with corrugated walls, where the two walls are small-amplitude periodic sinusoidal waves that are either in phase or half-period out of phase.

Essential equations as well as the mathematical conceptualization:
Consider a laminar, Newtonian, incompressible EMHD flow between two immovable corrugated walls with height2, the origin of the coordinate system is believed to be fixed in the middle of the channel.The layer thickness,,  ≫ 2, is substantially more than the width of the channel along the -axis and the length along the -axis (see figure 1).The top and lower wavy walls are situated at where  is the wave number and  is small amplitude Figure 1.Geometry of the problem

Electric Potential distribution
To calculate the net charge density   in the EDL in a corrugated wall channel, first calculate the EDL potential Φ * = Φ * (x , y) There exists an electric potential where ion density, electronic charge, valence, and Boltzmann constant are denoted by  0 ,  0 ,  0 and   respectively.  and  0 denote the absolute temperature and the solution's permittivity constant, respectively.Let Φ * is Very small, then ) ≪ 1 so the term ( 0 Φ/    ) can be used to approximate the term (sinh  0 Φ/    ) [21].Debye-Hückle linearization is the name for this principle.Finally, the linearized Poisson equation is obtained in the following from Where  =  0 √ 2 0  0     is Debye-Hückle parameter Reza el al. [19], then ).
The non-dimensional variables are introduced Assume that the  potential is constant.To have the linearized Poisson equation dimensionless and thus becomes to make dimensionless the linearized Poisson equation by substitution in eq. ( 4) where  =  is the EDL's normalised reciprocal thickness, which denotes the ratio of the microchannel's half height to Debye length (1/).
The dimensionaless boundary conditions for the electrical potential are defined as: The Electric Potential function can be described by the regular perturbation expansion in small values of , [10]- [19].
We get Differential equations for powers of: are obtained by substituting equation ( 8) into (5).
In a Taylor series concerning the mean wall positions  = 1 and  = −1, the boundary constraints ( 6) and ( 7) can be expanded we get.
We get for the boundary condition equations ( 9), (10) and (11), respectively, by collecting terms of equal powers.

Velocity distribution
The dimensional form of the continuity and momentum balance equations for a fluid is J. H. Masliyah and S. Bhattacharjee [20]. Where The net electric charge density and applied electric field are represented by   and  ⃑⃑⃑ , respectively.The Lorentz force, which is formed by the interaction of fluid flow and applied magnetic field, is denoted by .where the magnetic field is y-Direction  ⃑ =  0   and the uniform electric field is  ⃑ =     +     and the electric current density  =   [ ⃑ ×  ⃑ ] +    ⃑ caused by an x-directional electric field J. C. Maxwell [21] Therefore, The momentum equation in z-direction takes the form.
Dimensionless variables are created using the non-dimensional variables in Eq. (22). Where Where the ratio of Lorentz force to viscous force is described by the Hartmann number () and the transverse electric field is defined by , With the following Boundary Conditions Using the following transformation in equation ( 23) One can deduce the following form of the velocity equation Where With the boundary conditions The fluid velocity can be described by a regular perturbation expansion in small values of  [18]- [20].
We get Differential equations for powers of : are obtained by substituting equation ( 30) into (27).
In a Taylor series concerning the mean wall positions  = 1 and  = −1, the boundary constraints ( 28) and (29) can be expanded as.
We get the non-dimensional boundary conditions for equations (31), ( 32) and (33), respectively, by collecting terms of equal powers of .
From equations (31) and (36), we can get u 0 ± ( , ) =  −√ ( √ −  √ −  √(2+) +  √  +  √(1+2) ( + )) From equations (32) and (37) then From equations ( 33) and (38) then The volume flow rate per unit channel width can be written , where in the last two integrals the function w is expanded in a Taylor series about the mean wall positions.We get the result by averaging over one wavelength of corrugations The mean velocity (45)

Temperature distribution
In the presence of nanoparticles, we have to investigate the thermal characteristics and its effect on the flow.Carbon nanotube particles suspended in water make up a nanofluid (SWCNTs).The following models were used in this study: Maxwell's model J. C. Maxwell [21].
Hamilton-Crosser's (H-C) model R. L. ) With Boundary Conditions By using the following transformation into the equation ( 55) We deduced that Then, the boundary conditions take the form The Temperature function can be described by a regular perturbation expansion in small values of  [18]- [20].
We get Differential equations for powers of  are obtained by substituting equation ( 62) into (59) We get the non-dimensional boundary conditions for equations (63), ( 64) and (65), respectively, by collecting terms of equal powers of .
From equations ( 64) and (69) then From equations ( 65) and (70) then , ,( where ,( The Nusselt number, which represents the strength of convective heat exchange, can be calculated as follow where Nusselt number is can be written as

Discussion and Outcomes
In this section, the behavior of the fluid supported by (SWCNTs) nanoparticles through corrugated walls is explored graphically.The problem simulated by a set of partial differential equations and solved analytically using the perturbation method.The approximate analytical solutions were extracted using Mathematica software.The flow rate and heat transfer rate at different points in the microchannel have been studied.In order to check the accuracy of the obtained results a comparison with the previous published data under the same conditions is shown in Tables 1 and 2, at  = 0,  = 0 ,  = 0,   = 0.5,  = 0.25,  = 0.025.
It is worth mentioning that, the channel's half height is considered to be ~40 for microfluidic inquiry.The following main parametric values are used to investigate and analyze the effects of wavy roughness and porous of micro-channel on electromagnetically driven flow and temperature Buren et al. [10] and Reza et al. [19].The values of non-dimensional parametric variables have been calculated according to Table 3, [3], [8], [10], [19].

Effect of nano-particles on the flow velocity
The impact of magnetic field (Hartmann Number) on the volumetric flow rate depicted in Table 2 for different vlaue of nanoparticles concentration φ.It is clear that flow rate decreases as the Hartmann number raises under an concentration of the nanoparticles this due to the double effect of the body force resulting from the applied electromagnetic forces.The first component is the 'flow assist' (∼       ) which is formed between the electric and magnetic fields.The second component is the "opposite flux" (∼     2 ) which the Lorenz electromagnetic force is controlled by the presence of a magnetic field thses forces generate a normal component of velocity and decreses the main velocity of the flow.On the other hand by investigation the effect of nanoparticles concentraion on the flow rate, one can found that by increase the nanoparticle concentration the flow rate decreases due to the raises of fluid viscocity.

Values [units]
Characteristic   By increase the electric field and the magnetic field to  = 50 ,  = 1, as shown in Figure 7 the flow increases in the middle region and decreases when we approach to the wall's region, and an increase in the velocity occur also with the presence of particles.Layers of liquids are greatly affected by the Lorentz force, which increases when the Hartmann number (Ha) increases and in the presence of a tangential electric field S. The impact of this force is observed and is clearly shown in the previous figures.Figure 8 shows the average velocity   with increasing of Hartman Number  if the phase difference between the two walls is 0° where it is represented by the solid lines and if the phase difference between the two walls is 180° (represented by the dashed lines).It is clear that the mean velocity   reaches gradually to maximum value as the Hartman goes between 1.0 to 1.5 and decay again to minimum value.Also one can observe that the presence of nanoparticles reduces the mean velocity of the flow especially for the opposite phase channel.

Effect of wall roughness on flow temperatue in the presence of nanofluidic particles.
After deriving heat equation (55) and using Maxwell's model with n as the structural coordination of nanoparticles is given by 3/ where  is the spherically of nanoparticles.For spherical nanoparticles  = 1 or  = 3 but the Hamilton-Crosser's (H-C) model reduces nanoparticles to cylindrical shapes  = 0.5 or  = 6.The objective of the study of temperature is to examine the effect of the different models of heat conduction mentioned.Respectively, Figure 9 shows the behavior of both transverse magnetic and electric fields.It plays a role in controlling the temperatures of the flowing liquid.Thermal distribution shows an increase in temperatures with the increase of nanoparticles.When the transverse magnetic and electric fields are increased  = 25 ,  = 0.5, the degree increases when the nanoparticles are increased, as shown in Figure 10.When studying the effect of nanoparticles particles on heat transfer, it is noted that the rate of increase of the Nusselt number  increases when the concentration of nanoparticles increases in Figure 11 shows the Nusselt number with increasing  of Hartmann Number  if the phase difference between the two walls is 0° is represented by the solid lines and if the phase difference between the two walls is 180°, The electric and magnetic fields control the rate of heat transfer.It is observed that from the energy equation, the joule temperature increases with the increase of the transverse magnetic and electric fields.Therefore, the temperature spread is further increased by increasing the transverse magnetic and electric fields.
In Figure 12, studying the effect of nanoparticles on heat generation/absorption , if the phase difference between the two walls is 0° is represented by the solid lines and if the phase difference between the two walls is 180° it is noted that the rate of increase of the Nusselt number  increases when the concentration of nanoparticles increases.Tables 4 and 5 Whereas, the H-C model has the lowest heat transfer rate of 5.19% at 0.2 concentration.For Maxwell model, it is higher than the previous model in heat transfer by 15.01% at 0.2 concentration.However, the Xue model increases the heat transfer rate by 23.33% at 0.2 concentration.We also note an increase in the heat transfer rate by increasing the fracture size in the nanotubes in the three models.When comparing the Xue model with the other models, we note that the heat transfer rate predicted by the Xue model is higher than the other models.These results are in agreement with previous findings by Hayat et al. [6].Thermal expansion coefficient (1/)10 −6 210 15

Conclusion
The flow and heat rate effect of SWCNTs and wall surface roughness were studied through a fine corrugated channel in the presence of magnetic and electric fields at constant pressure gradient and the following points were concluded.
-Due to the phase difference between the two walls, the speed and temperature distribution depend on the shape of the channel wall.-When the corrugation increases, the phenomenon of ripple speed and temperature distribution becomes apparent.-One can observe that the velocity distribution decreases due to the increasing of the concentration due to a rising in fluid viscosity.-The impact of transverse electric field () on the velocity contour appears it is observed that the raising of transverse electric field improves the fluid velocity and the wavy phenomenon of the flow be clearer.-By addition of SWCNTs, the fluid velocity is reduced at the center of the channel by providing resistance to the fluid motion.-When the transverse electric field is increased, there is a spread and a rise in temperature.
-The size and concentration of SWCNTs influence enhance the rate of heat transfer.
-Heat generation/absorption increases when the transverse magnetic and electric fields increase with the change in size and concentration of SWCNTs.-The Xue's model has the highest heat transfer rate when compared to the Maxwell and H-C models.

Figure 9 . 1 𝜑Figure 11 .
Figure 9.Contour plot for Temperature distribution(Same Phase)  = 0.1 ,  = 0. 2 ,  = 10 ,  = 25 ,  = 0.5 ,  = 1,  = 1 . shows the numerical values of heat transfer rates for changes in the fraction size and nanoparticle shapes.A comparison of the Maxwell model, the H-C model, and the Xue model is made through this table.

Figure 10 .Figure 12 .
Figure 10.Contour plot for Temperature distribution(Same Phase)  = 0.1 ,  = 0. 2 ,  = 10,  = 50 ,  = 1 ,  = 1,  = 1 (b)  = 0.1  = 0.2 [22]lton and O. K. Crosser[22]In the above equations,   indicates to the base fluid's thermal conductivity,   the nanofluids actual thermal conductivity,   the thermal conductivity of nanoparticles,   the thermal conductivity of SWCNTs and  the nanotubes volume fraction.now invariant of the axial coordinate for thermally fully developed flow,   is the channel wall temperature, which is constant across the channel's cross section,   is the channel's average inward wall heat flow, for thermally fully developed flow with constant heat flux, we may write: * +  + .(49)Where *  the temperature of the fluid,  is the dimensional heat generation/absorption, J is the Joule is Dimensionless variables are created using the non-dimensional variables in Eq. (51).

Table 1 :
Comparison of numerical values of volumetric flow rate for different Hartmann number ()

Table 2 :
The volumetric flow rate for different concentration of nanoparticles  on corrugated walls

Table 3 :
Typical values of the physical variables

Table 4 :
Numerical values of the Nusselt number rate at the upper wall for different values of concentration