Magnetohydrodynamic CNTs Casson Nanofluid and Radiative heat transfer in a Rotating Channels

The main purpose of this investigation is to inspect the innovative conception of the magneto hydrodynamic (MHD) nanoparticles of single wall carbon nanotubes base on the fl uids (water, engine oil, and ethylene, glycol and kerosene oil) between two rotating parallel plates. Carbon nanotubes (CNTs) parade sole assets due to their rare structure. Such structure has signifi cant optical and electronics features, wonderful strength and elasticity, and high thermal and chemical permanence. The heat exchange phenomena is deliberated subject to thermal radiation. Kerosene oil is taken as based nano fl uids because of its unique attention due to their advanced thermal conductivities, exclusive features, and applications. The fl uid fl ow is presumed in steady state. With the help of suitable resemblance variables, the fundamental leading equations have been converted to a set of diff erential equations. To obtain the solution of the modeled problem, the homotopic approach has been used. The infl uence of imbedded physical variables upon the velocities and temperature profi les are defi ned and deliberated through graphs. Moreover, for the several values of relevant variables, the skin fraction coeffi cient and local Nusselt number are tabulated. Plots have been presented in order to examine how the velocities and temperature profi le get aff ected by various fl ow parameters. Review Article Magnetohydrodynamic CNTs Casson Nanofluid and Radiative heat transfer in a Rotating Channels Abdullah Dawar1, Zahir Shah2*, Saeed Islam2, Muhammad Idress3 and Waris Khan3 1Department of Mathematics, Qurtuba University of Science and Information Technology, Peshawar 25000, Pakistan 2Department of Mathematics, Abdul Wali Khan University, Mardan 23200, KP, Pakistan 3Department of Mathematics, Islamia College University, Mardan 25000, KP, Pakistan *Address for Correspondence: Zahir Shah, Department of Mathematics, Abdul Wali Khan University, Mardan 23200, KP, Pakistan, Email: zahir1987@yahoo.com Submitted: 27 July 2018 Approved: 16 August 2018 Published: 17 August 2018 Copyright: 2018 Dawar A, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited


Introduction
A nanometer-scale tube-like structure is called nanotube. Carbon nanotubes (CNTs) were the irst to be discovered in 1952. CNTs are commonly divided into two types called Single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The potential of CNTs are catalysts, lat panel display, absorption, and shielding of electro-magnetic waves, nanoelectrodes, sensors, supercapicator and energy conversion. Haq et al. [1], Reported that SWCNTs has higher Nusselt number and skin friction than MWCNTs by considering water as a base luid. Liu et al. [2], Deliberated that ethylene glycol with CNTs have higher thermal conductivities than ethylene glycol suspension without CNTs by studying synthetic engine oil and ethylene glycol in the existence of MWCNTs. The theory of nano luid past an exponentially stretching sheet has been presented by Nadeem and Lee [3]. Over stretching/shrinking surfaces the mentioned authors [4][5][6][7] examined the impacts of nanoparticles for boundary layer low. Choi [8] was the pioneer who presented the idea of nano luid by dipping the nanometer-sized particle into the base luid. For the study of nano luid low, the mathematical model was established by Boungiorno [9]. The recent experimental and theoretical study of Sheikholeslami on nano luids and its applications with different behaviour, properties and effects with uses of different numerical and analytical techniques can be seen in [10][11][12][13][14][15]. Sheikholeslami and Rokni [16][17][18][19] have recently investigated Simulation of nano luids and CuOeH2O nano luid in a curved porous The determination of current research is to study the nanoparticles of SWCNTs and MWCNTs Casson Fluid based on the luids (water, engine oil, ethylene glycol and kerosene oil) between two rotating parallel plate under the in luence of magnetic parameter and thermal radiation. Kerosene oil is taken as base nano luid. Casson [26] was the irst one who proposed the Casson luid model, this model characterizes a shear retreating luid which is presumed to obligate in inite viscosity at zero rate of shear stress. Mehmood et al. [27,28], recently investigated Casson micro polar luid over a stretching sheet with internal heat transmission by using numerical techniques. Singh Megahe et al. [29], inspected the liquid ilm low of Casson luid in the existence of varied heat lux using slip velocity. Abolbashari et al. [30], studied the Casson nano luid with entropy generation. The other related study about Casson luid can be seen in [31][32][33][34][35][36]. The effect of all embedding parameters has been studied graphically. The analytical result for velocities and temperature pro iles are obtained using the HAM technique [37][38][39][40][41][42].

Mathematical modeling
Consider the lows of CNTs nano luid between two parallel plates. The distance between the lower and upper plates is labeled with h. SWCNTs and MWCNTs are used as nano-scale materials where kerosene oil is a base liquid. Through thermal radiation, the heat transportation mechanism is examined. Around the y-axis the plates are rotated with a constant angular velocity γ. It should be noted that γ > 0 indicates that both plates rotate in the same direction, γ > 0 indicates that both plates rotate in the opposite directions, γ > 0 is for the static case. The rotation of the lower plate which is moving with velocity U w = cx (c>0) is quicker than the upper plate. A coordinate system (x, y, z) is chosen in such a way that the x-axis is parallel to the plates, the y-axis is vertical to the plates, and the z-axis is normal to the xy plane. The plates are positioned at y = 0 and y = h. With the help of two forces with same magnitude but opposite direction, the lower plate is being kept stretchable so the position (0, 0, 0) cannot changes. The luid low and heat transfer is supposed in steady state which is incompressible, laminar and stable. Along y direction, the magnetic ield B 0 is substituted with which the luid is rotating as shown in Figure 1. The rheological model that illustrates the Casson luid is known as [34][35][36]: T ij Denotes Cauchy stress tensor,  B denotes the dynamic viscosity of the Casson luid, k = m ij .m ij is the square of components of strain rate P y is the yield stress of the luid and k c represent critical value of k The governing equations for the state of problem are [22][23][24]37]    The density, heat capacity, and dynamic viscosity of the nano luid through mathematical equations are [40]. Where, , , is the dynamic viscosity of base luid, nanoparticle volumetric friction, and thermal conductivity respectively. The subscripts CNT, f, nf represent carbon nanotubes, base luid, and nano luid respectively. T 0 is the lower temperature at upper wall and T h is the higher temperature at lower wall. T h is retained higher than T 0 i.e. T h > T 0 . The boundary conditions for the system can be de ined as [37].
In equation (9) Z 0 is the uniform suction/injection velocity at the upper wall. If (Z 0 > 0) than it is called uniform suction velocity and if (Z 0 < 0) than it is called uniform injection velocity.
The non-dimensional system of equations is The relevant boundary conditions In equation (14), is the suction and injection parameter. If   0 Q  then it is called suction parameter and if   0 Q  then it is called injection parameter. The above non --dimensional system of equations has the following parameters.
The skin friction coef icient and local Nusselt number are de ined as [38]:
The primary suppositions are chosen as follows: The linear operators are chosen as L ,L and L g f Which have the succeeding properties: The consequence non-linear operators , g f N N and N  are indicated as: The zero th order problems from In organize to solve Eqs. (12, 13, and 14) are: The equivalent boundary conditions are: , As the series (31) converges at 1   , changing 1   in (31), we get: The th q -order problem grati ies the following: The equivalent boundary conditions are:  (38)

Results and Discussion
In order to investigate the low and heat transfer performance for both SWCNTs and MWCNTs based on kerosene nanoliquids between two rotating parallel plate, (Figure 3-16) are plotted. Figure 1 displays different structure of CNTs and Figure 2 shows physical shape of the low. Figures 3-7 are plotted to see the impact of various parameters on velocity and temperature pro iles for both SWCNTs and MWCNTs-                kerosene nanoliquids. These parameters are nanoparticles volume friction ( ), Reynolds number (A 1 ), Rotation parameter (A 2 ), Magnetic parameter (M), Suction parameter (Q > 0) and Injection parameter (Q < 0) respectively. Figure 3 is plotted to see the comparison between the SWCNTs and MWCNTs with the escalating values of nanoparticle volume fraction ( ). From Figure 3 we observed that there is petty disparity in ( ) f   with the escalating values of ( ). We also observed that the MWCNTs has comparatively much greater ( ) f   as compared to SWCNTs. Figure 4 is plotted to see the impact of (A 1 ) on ( ) f   with the different escalating values. We observed from the igure, that with the different escalating values of (A 1 ), the velocity pro ile ( ) f   changes its behavior from rising to reducing at the medium of the two plates ( . . 0.5) i e   . This is because of stretching of the lower plates. Figure 5 is plotted to see the impact of (A 2 ) on ( ) f   . From igure 5, we observed that the disparity in ( ) f   gives dual behavior.
Within the region (0 0.5),    due to numerous escalating values of (A 2 ) the velocity pro ile ( ) f   shows declining behavior, however within the region (0.5 1) gives escalating behavior for various escalating values of (A 2 ). In addition, at the upper half of the channel (A 2 ) gives more dominant variation for ( ) f   . Figure 6 is designed to see the in luence of (M) on ( ) f   . According to Lorentz force theory the magnetic ield parameter has reverse effect on ( ) f   , that is ( ) f   reduces with the escalation in (M). Figure 7 and 8 are plotted to see the impact of suction (Q > 0) and injection (Q < 0) parameters at the upper plate for ( ).
f   Form igure 7, we see that ( ) f   escalates with suction (Q < 0), that is ( ) f   increases with positive the values of (Q), while form Figure 8, it is observed that ( ) f   reduces with injection (Q < 0), that is ( ) f   reduces with the negative (Q). It is fairly obvious that the presence of CNTs nanoparticles has improved the velocity function ( ) f   , while the velocity function ( ) f   increases more when (Q > 0) is present. But the decrement in velocity function ( ) f   is due to the fact that (Q < 0) absorbs the internal heat energy from the surface. Figure 9 is plotted to see the impact of (β) on ( ) f   . From here we observed that the increasing values of (β) shows reduction in ( ) f   . Figure 10 is plotted to see the impact of ( ) on ( ) g  .
From Figure 10 we observed that there is petty disparity in ( ) g  with the escalating values of ( ). We also observed that the MWCNTs has comparatively much greater ( ) g  as compared to SWCNTs. Figure 11 is plotted to see the impact of on (A 1 ) ( ) g  . Figure 11 shows reduction in ( ) g  between the two rotating plates with the increasing values of (A 1 ) and the position of the maximum amount of ( ) g  approaches to the stretching sheet. Figure 12 is plotted to see the impact of (A 2 ) on ( ) g  . From Figure 12, we observed that the increasing values of (A 2 ) gives decreasing behavior to ( ) g  . It is also observed that the disturbance in ( ) g  is higher at the medium of the channels as compared to upper and lower surface of the channels. Figure 13 is plotted to see the impact of (M) on ( ) g  . According to Lorentz force theory the magnetic ield parameter has reverse effect on ( ) g  , that is ( ) g  reduces with the escalation in (M). Figure 14 is plotted to see the comparison between the SWCNTs and MWCNTs with the escalating values of nanoparticle volume fraction ( ). From igure 14 we observed that there is petty disparity in ( )   with the escalating values of ( ).
We also observed that the MWCNTs has comparatively much greater ( )   as compared to SWCNTs. Figure 15 is plotted to see the impact of (A 1 ) on ( )   . From here we observed that the escalating value of (A 1 ) shows escalation in ( )   . As the distance from the surface escalates, ( )   decreases. Figure 16 is plotted to see the impact of (Pr) on ( )   . From here we see that the escalating values of (Pr) shows reduction in ( )   . Physically, the nano luids have a large thermal diffusivity with small (Pr), but this effect is revers for higher (Pr), therefore the temperature of liquid shows decreasing behavior. Figure 17 is plotted to see the impact of (Rd) on ( )   . Thermal radiation has leading rule in heat transmission when the coef icient of convection heat transmission is small. From here we see that the escalating values of (Rd) shows acceleration in ( )   .

Discussion
Tables 1,2 are schemed to see the in luences of different embedding parameters on skin fraction coef icient ( ) f   and local Nusselt number (Nu x ). From Table 1, we see that the effect of A 1 on skin fraction coef icient. It is clear from the table that at ( ) f   , the table shows increasing behavior to 0.3 but from 0.3 to 0.5, the table values show decreasing behavior. At the medium of the channel the skin friction coef icient changes its behavior from increasing to decreasing. This is because of stretching of the lower plates. It is also clear form the table that the escalating values of A 2 shows reducing behavior. The escalating values of (M) shows decreasing behavior in skin fraction coef icient ( ) f   . It is due to Lorentz force theory. The increasing and decreasing values of suction/injection parameter (Q) show two different behaviors. The escalating values of injection parameter (Q < 0) show decreasing behavior and also the escalating values of suction parameter (Q > 0) show increasing behavior in skin fraction ( ) Table 2, we see that the increasing values of (A 1 ) and positive Q (i.e. Q > 0) show increasing behavior in local Nusselt number (Nu x ), while (A 2 ), (M) and negative Q (i.e. Q > 0) show decreasing behavior in local Nusselt number (Nu x ). Tables 3-5 are schemed to study the Physical properties of CNTs, thermo physical properties CNTs and nano luids of some base luids, Thermal conductivity (k nf ) of CNTs with different volume fraction ( ) respectively.

Conclusion
The investigation for the two-dimensional low of kerosene oil based nano luid over an inclined stretching sheet with suction/injection, MHD and radiative heat lux effects are examined. SWCNTs and MWCNTs are used in this model. With the help of similarity variables, the system of governing partial differential equations is changed into ordinary differential equations. The impacts of embedded of parameters are shown graphically. The impacts of skin friction coef icient and local Nusselt number are shown through Table 1 and 2. On the achieved study, the key remarks are listed below.

The velocity function ( )
f   shows reducing behavior with escalating values of (A 2 ) within the region (0 0.5)    and shows escalation behavior with escalating values of (A 2 ) within the region (0.