International Journal of Physics Research and Applications

ISSN: 2766-2748

Review Article

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

Abdullah Dawar¹, Zahir Shah^2*, 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

Dates: Submitted: 27 July 2018; Approved: 16 August 2018; Published: 17 August 2018

How to cite this article: Dawar A, Shah Z, Islam S, Idress M, Khan W. Magnetohydrodynamic CNTs Casson Nanofluid and Radiative heat transfer in a Rotating Channels. J Phys Res Appl. 2018; 1: 017-032. DOI: 10.29328/journal.ijpra.1001002

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.

Keywords: SWCNTs; MWCNTs; Nanoparticles; Casson fluids; Thermal radiation; MHD; Parallel Plates; Rotating System; HAM

A nanometer-scale tube-like structure is called nanotube. Carbon nanotubes (CNTs) were the first 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, flat 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 fluid. 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 nanofluid past an exponentially stretching sheet has been presented by Nadeem and Lee [3]. Over stretching/shrinking surfaces the mentioned authors [4-7] examined the impacts of nanoparticles for boundary layer flow. Choi [8] was the pioneer who presented the idea of nanofluid by dipping the nanometer-sized particle into the base fluid. For the study of nanofluid flow, the mathematical model was established by Boungiorno [9]. The recent experimental and theoretical study of Sheikholeslami on nanofluids and its applications with different behaviour, properties and effects with uses of different numerical and analytical techniques can be seen in [10^-15]. Sheikholeslami and Rokni [16^-19] have recently investigated Simulation of nanofluids and CuOeH2O nanofluid in a curved porous enclosure by using mesoscopic approach. Elias et al. [20], have examined the influence of shape on heat transmission. Liquid metals, plasmas, slat water, electrolytes are the examples of such magnetofluids. MHD was presented by Hannes Alfven [21]. Cooling of devices, MHD instruments, crystal growth, electromagnetic casting, and magnetic drug targeting are the real-world uses of MHD in the field of engineering and technologies. Recently Shah et al. [22-25], have studied MHD nanofluid with rotating system.

The determination of current research is to study the nanoparticles of SWCNTs and MWCNTs Casson Fluid based on the fluids (water, engine oil, ethylene glycol and kerosene oil) between two rotating parallel plate under the influence of magnetic parameter and thermal radiation. Kerosene oil is taken as base nanofluid. Casson [26] was the first one who proposed the Casson fluid model, this model characterizes a shear retreating fluid which is presumed to obligate infinite viscosity at zero rate of shear stress. Mehmood et al. [27,28], recently investigated Casson micro polar fluid over a stretching sheet with internal heat transmission by using numerical techniques. Singh Megahe et al. [29], inspected the liquid film flow of Casson fluid in the existence of varied heat flux using slip velocity. Abolbashari et al. [30], studied the Casson nanofluid with entropy generation. The other related study about Casson fluid can be seen in [31-36]. The effect of all embedding parameters has been studied graphically. The analytical result for velocities and temperature profiles are obtained using the HAM technique [37-42].

Some recent application of CNTs

CNTS used for

• Cancer treatment

• Cardiac autonomic regulation

• In drug delivery

• Platelet activation

• Tissue regeneration

• In solar cell

Mathematical modeling

Consider the flows of CNTs nanofluid 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 fluid flow and heat transfer is supposed in steady state which is incompressible, laminar and stable. Along y direction, the magnetic field B0 is substituted with which the fluid is rotating as shown in Figure 1. The rheological model that illustrates the Casson fluid is known as [34-36]:

${{\rm T}}_{ij}=2\{\begin{array}{l}\left({\mu }_{{\rm B}}+\frac{{{\rm P}}_y}{\sqrt{2\kappa }}\right)m_{ij},\kappa >{\kappa }_c,\\ \left({\mu }_{{\rm B}}+\frac{{{\rm P}}_y}{\sqrt{2{\kappa }_c}}\right)m_{ij},\kappa >{\kappa }_c.\end{array}\text{ (1)}$

T_ij Denotes Cauchy stress tensor, µB denotes the dynamic viscosity of the Casson fluid, k = m_ij .m_ij is the square of components of strain rate P _y is the yield stress of the fluid and k_c represent critical value of k

The governing equations for the state of problem are [22-24,37]

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0,\text{ (2)}$

${\rho }_{nf}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+2\gamma w\right)=-\frac{\partial P^{\ast }}{\partial x}+\left(1+\frac{1}{\beta }\right){\mu }_{nf}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)-{\sigma }_{nf}{B}_0^{2}u,\text{ (3)}$

${\rho }_{nf}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial P^{\ast }}{\partial y}+{\mu }_{nf}\left(1+\frac{1}{\beta }\right)\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right),\text{ (4)}$

${\rho }_{nf}\left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}-2\gamma u\right)={\mu }_{nf}\left(1+\frac{1}{\beta }\right)\left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}\right)+{\sigma }_{nf}{B}_0^{2}w,\text{ (5)}$

Where $P^{\ast }=P-\frac{{\gamma }^2{x}^2}{2},{\sigma }_{nf},{\mu }_{nf}$ represents the meshes cross flow beside z-axis and β is the parameter of Casson fluid. Mathematically, the phenomenon of heat transmission can be indicated as

$u\frac{\partial {\rm T}}{\partial x}+v\frac{\partial {\rm T}}{\partial y}=\frac{k_{nf}}{{(\rho c)}_{nf}}\left(\frac{{\partial }^2{\rm T}}{\partial x^2}+\frac{{\partial }^2{\rm T}}{\partial y^2}\right)-\frac{\partial Qr}{\partial y}.\text{ (6)}$

${\rm T},{\alpha }_{nf}=\frac{k_{nf}}{{(\rho c)}_{nf}},Qr$ Signifies the fluid temperature, thermal diffusivity and the radiative heat flux respectively. The radiative heat flux is defined as $Qr=-\frac{4{\sigma }^{\ast }}{3k}\frac{\partial {{\rm T}}^4}{\partial y},\text{ (7)}$

Where ${\sigma }^{\ast },k$ indicates the absorption coefficient and Stefan Boltzmann constant respectively.

As ${{\rm T}}^4=4{{\rm T}}_h^3{\rm T}-3{{\rm T}}_h^4$ the equation (6) becomes

$u\frac{\partial {\rm T}}{\partial x}+v\frac{\partial {\rm T}}{\partial y}=\frac{k_{nf}}{{(\rho c)}_{nf}}\left(1+\frac{16{\sigma }^{\ast }{{\rm T}}_h^3}{3k}\right)\frac{{\partial }^2{\rm T}}{\partial y^2}.\text{ (8)}$

The density, heat capacity, and dynamic viscosity of the nanofluid through mathematical equations are [40].

$\begin{array}{l}{(}_{\rho }=\left(1-\phi \right){\rho }_f+\phi {\rho }_{CNT}, {\rho }_{nf}=\left(1-\phi \right){\left(\rho c\right)}_f+\phi {\left(\rho c\right)}_{CNT},\\ {\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\phi \right)}^{2.5}}, k_{nf}=k_f\left(\frac{1-\phi +2\phi \frac{k_{CNT}}{k_{CNT-k_f}}\ln \left(\frac{k_{CNT+k_f}}{2k_f}\right)}{1-\phi +2\phi \frac{k_f}{k_{CNT-k_f}}\ln \left(\frac{k_{CNT+k_f}}{2k_f}\right)}\right).\text{ (9)}\end{array}$

Where, ${\mu }_f,\phi ,k_{nf}$ is the dynamic viscosity of base fluid, nanoparticle volumetric friction, and thermal conductivity respectively. The subscripts CNT, f, nf represent carbon nanotubes, base fluid, and nano fluid respectively. T₀ is the lower temperature at upper wall and T_h is the higher temperature at lower wall. T_h is retained higher than T₀ i.e. T_h > T₀ . The boundary conditions for the system can be defined as [37].

$\begin{array}{l}\overrightarrow{u}=U_w=cx,\overrightarrow{v}=0,\overrightarrow{w}=0,{\rm T}={{\rm T}}_h\text{ at }y=0\text{, (10)}\\ \overrightarrow{u}\text{=0,}\overrightarrow{v}{\text{ =-Z}}_0\text{,}\overrightarrow{w}=0,{\rm T}={{\rm T}}_0\text{ at }y=h.\end{array}$

In equation (9) Z₀ is the uniform suction/injection velocity at the upper wall. If (Z₀ > 0) than it is called uniform suction velocity and if (Z₀ < 0) than it is called uniform injection velocity.

Excluding the pressure gradient and introducing the similarity variables, we have [22-24].

$\overrightarrow{u}=cx{f}^{\prime }\left(\Gamma \right),\overrightarrow{v}=-chf\left(\Gamma \right),\overrightarrow{w}=cxg\left(\Gamma \right),\theta \left(\Gamma \right)=\frac{{\rm T}-{{\rm T}}_0}{{{\rm T}}_h-{{\rm T}}_0},\Gamma =\frac{y}{h}.\text{ (11)}$

$\begin{array}{l}\left(1+\frac{1}{\beta }\right)f^{iv}\left(\Gamma \right)-{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_1\left(f^{\prime }\left(\Gamma \right)f^{″}\left(\Gamma \right)-f\left(\Gamma \right)f^{‴}\left(\Gamma \right)\right)\text{ (12)}\\ -{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_2{g}^{\prime }\left(\Gamma \right)-\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right){\rm M}{f}^{″}\left(\Gamma \right)=0,\end{array}$

$\begin{array}{l}\left(1+\frac{1}{\beta }\right)g^{″}\left(\Gamma \right)-{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_1\left(g\left(\Gamma \right)f^{\prime }\left(\Gamma \right)-f\left(\Gamma \right)g^{\prime }\left(\Gamma \right)\right)\text{ (13)}\\ +2{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_2{f}^{\prime }\left(\Gamma \right)+\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right){\rm M}g\left(\Gamma \right)=0,\end{array}$

$\left(1+Rd\right){\theta }^{″}\left(\Gamma \right)+{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]\frac{\mathrm{Pr}{A}_1}{\left(\frac{k_{nf}}{k_f}\right)}f\left(\Gamma \right){\theta }^{\prime }\left(\Gamma \right)=0,\text{ (14)}$

The relevant boundary conditions are $\begin{array}{l}f\left(0\right)=0,f^{\prime }\left(0\right)=1,g\left(0\right)=0,\theta \left(0\right)=1,\text{ (15)}\\ f\left(1\right)=Q,f^{\prime }\left(1\right)=0,g\left(1\right)=0,\theta \left(1\right)=0.\end{array}$

In equation (14), $Q=\raisebox{1ex}{$Z_0$}\!\left/ \!\raisebox{-1ex}{$ch$}\right.$ is the suction and injection parameter. If $\left(Q>0\right)$ then it is called suction parameter and if $\left(Q<0\right)$ then it is called injection parameter. The above non-dimensional system of equations has the following parameters.

$\begin{array}{l}{A}_1=\frac{c{h}^2}{{\nu }_f}\left(Reynold\text{'}s number\right),A_2=\frac{\gamma h^2}{{\nu }_f}\left(rotation parameter\right),\\ \mathrm{Pr}=\frac{{\mu }_f{c}_p}{k_f}\left(prandtl number\right),Rd=\frac{16{\sigma }^{\ast }{{\rm T}}_h^3}{3k{k}^{*}}\left(radiation parameter\right),\\ {\rm M}=\frac{{\sigma }_f{B}_0^2{h}^2}{{\rho }_f{\nu }_f}\left(magnetic parameter\right),\frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\varphi \left(k_f-k_s\right)}{k_s+2k_f+2\varphi \left(k_f-k_s\right)},\text{ (16)}\\ \frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\varphi }{\left(\frac{{\sigma }_s}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\varphi },\end{array}$

The skin friction coefficient and local Nusselt number are defined as [38]:

${\tilde{C}}_f=\frac{{\mu }_{nf}}{{\mu }_f}{f}^{″}(0)\text{ (17)}$

$N{u}_x=-\frac{k_{nf}}{k_f}\left(1+Rd\right){\theta }^{\prime }\left(0\right).\text{ (18)}$

Solution by HAM

We use HAM [39-42] to solve Eqs. (12, 13, and 14) with boundary conditions (15) by the succeeding process.

The primary suppositions are chosen as follows:

The linear operators are chosen as ${\text{L}}_{\overline{f}}{\text{,L}}_{\overline{g}}{\text{ and L}}_{\overline{\theta }}$

$L_{\overline{f}}(\overline{f})=\overline{f}\text{'}\text{'}\text{'},{\text{ L}}_{\overline{g}}\text{(<mover accent="true"> g <mo>¯</mo> </mover> )=<mover accent="true"> g <mo>¯</mo> </mover> ''},{\text{L}}_{\overline{\theta }}\text{(}\overline{\theta }\text{)=}\overline{{\theta }^{″}},$

Which have the succeeding properties:

$L_{\overline{f}}(b_1+b_2\Gamma +b_3{\Gamma }^2)=0,{\text{ L}}_{\overline{g}}(b_4+b_5\Gamma )=0,{\text{L}}_{\overline{\theta }}(b_6+b_7\Gamma )=0,\text{ (21)}$

Where $b_i(i=1-7)$

The consequence non-linear operators $N_{\overline{f}},N_{\overline{g}}$ and $N_{\overline{\theta }}$ are indicated as:

$\begin{array}{l}{N}_{\overline{f}}\left[\overline{f}(\Gamma ;\psi ),\overline{g}(\Gamma ;\psi )\right]=\left(1+\frac{1}{\beta }\right)\frac{{\partial }^4\overline{f}(\Gamma ;\psi )}{\partial {\Gamma }^4}\\ -{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_1\left\{\begin{array}{l}\frac{\partial \overline{f}(\Gamma ;\psi )}{\partial \Gamma }\frac{{\partial }^2\overline{f}(\Gamma ;\psi )}{\partial {\Gamma }^2}-\\ \overline{f}(\Gamma ;\psi )\frac{{\partial }^3\overline{f}(\Gamma ;\psi )}{\partial {\Gamma }^3}\end{array}\right\}\text{ (22)}\\ -{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_2\frac{\partial \overline{g}(\Gamma ;\psi )}{\partial \Gamma }-\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)M\frac{{\partial }^2\overline{f}(\Gamma ;\psi )}{\partial {\Gamma }^2},\end{array}$

$\begin{array}{l}{N}_{\overline{g}}\left[\overline{f}(\Gamma ;\psi ),\overline{g}(\Gamma ;\psi )\right]=\left(1+\frac{1}{\beta }\right)\frac{{\partial }^2\overline{g}(\Gamma ;\psi )}{\partial {\Gamma }^2}\\ -{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_1\left(\begin{array}{l}\overline{g}(\Gamma ;\psi )\frac{\partial \overline{f}(\Gamma ;\psi )}{\partial \Gamma }\\ -\overline{f}(\Gamma ;\psi )\frac{\partial \overline{g}(\Gamma ;\psi )}{\partial \Gamma }\end{array}\right)\text{ (23)}\\ +2{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_2\frac{\partial \overline{f}(\Gamma ;\psi )}{\partial \Gamma }+\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)M\overline{g}(\Gamma ;\psi ),\end{array}$

$N_{\overline{\theta }}\left[\overline{f}(\Gamma ;\psi ),\overline{\theta }(\Gamma ;\psi )\right]=(1+Rd)\frac{{\partial }^2\overline{\theta }(\Gamma ;\psi )}{\partial {\Gamma }^2}+\left[\left(1-\phi \right)+\phi \frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\right]\frac{\mathrm{Pr}{A}_1}{\left(\frac{k_{nf}}{k_f}\right)}\overline{f}\left(\Gamma ,\psi \right)\frac{\partial \overline{\theta }(\Gamma ;\psi )}{\partial \Gamma }.\text{ (24)}$

The zero^th order problems from In organize to solve Eqs. (12, 13, and 14) are:

$(1-\psi )L_{\overline{f}}\left[\overline{f}(\Gamma ;\psi )-{\overline{f}}_0(\Gamma )\right]=\psi {\hslash }_{\overline{f}}{N}_{\overline{f}}\left[\overline{f}(\Gamma ;\psi ),\overline{g}(\Gamma ;\psi )\right],\text{ (25)}$

$(1-\psi )L_{\overline{g}}\left[\overline{g}(\Gamma ;\psi )-{\overline{g}}_0(\Gamma )\right]=\psi {\hslash }_{\overline{g}}{N}_{\overline{g}}\left[\overline{f}(\Gamma ;\psi ),\overline{g}(\Gamma ;\psi )\right],\text{ (26)}$

$(1-\psi )L_{\overline{\theta }}\left[\overline{\theta }(\Gamma ;\psi )-{\overline{\theta }}_0(\Gamma )\right]=\psi {\hslash }_{\overline{\theta }}{N}_{\overline{\theta }}\left[\overline{f}(\Gamma ;\psi ),\overline{\theta }(\Gamma ;\psi )\right].\text{ (27)}$

The equivalent boundary conditions are:

$\begin{array}{l}{\overline{f}(\Gamma ;\psi )|}_{\Gamma =0}=0,{\frac{\partial \overline{f}(\Gamma ;\psi )}{\partial \Gamma }|}_{\Gamma =0}=1,{\overline{f}(\Gamma ;\psi )|}_{\Gamma =1}=Q,{\frac{\partial \overline{f}(\Gamma ;\psi )}{\partial \Gamma }|}_{\Gamma =1}=0,\\ {\overline{g}(\Gamma ;\psi )|}_{\Gamma =0}=0,{\overline{g}(\Gamma ;\psi )|}_{\Gamma =1}=0,\text{ (28)}\\ {\overline{\theta }(\Gamma ;\psi )|}_{\Gamma =0}=1,{\overline{\theta }(\Gamma ;\psi )|}_{\Gamma =1}=0,\end{array}$

Where $\psi \in [0,1]$ is the emerging parameter? In case of $\psi =0\text{ and }\psi =1$ we have:

Expanding $\overline{f}(\Gamma ;\psi ),\overline{g}(\Gamma ;\psi )\text{ and }\overline{\theta }(\Gamma ;\psi )$ by Taylor’s series

$\begin{array}{l}\overline{f}(\Gamma ;\psi )={\overline{f}}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{f}}_q(\Gamma ){\psi }^q},\\ \overline{g}(\Gamma ;\psi )={\overline{g}}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{g}}_q(\Gamma ){\psi }^q},\text{ (30)}\\ \overline{\theta }(\Gamma ;\psi )={\overline{\theta }}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{\theta }}_q(\Gamma ){\psi }^q}.\end{array}$

Where

As the series (31) converges at, $\psi =1$ changing $\psi =1$ in (31), we get:

$\begin{array}{l}\overline{f}(\Gamma )={\overline{f}}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{f}}_q(\Gamma )},\\ \overline{g}(\Gamma )={\overline{g}}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{g}}_q(\Gamma )},\text{ (32)}\\ \overline{\theta }(\Gamma )={\overline{\theta }}_0(\Gamma )+{\displaystyle \sum _{q=1}^{\infty }{\overline{\theta }}_q(\Gamma )}.\end{array}$

The $q^{th}$ -order problem gratifies the following:

$\begin{array}{l}{L}_{\overline{f}}\left[{\overline{f}}_q(\Gamma )-{\chi }_q{\overline{f}}_{q-1}(\Gamma )\right]={\hslash }_{\overline{f}}{U}_q^{\overline{f}}(\Gamma ),\\ L_{\overline{g}}\left[{\overline{g}}_q(\Gamma )-{\chi }_q{\overline{g}}_{q-1}(\Gamma )\right]={\hslash }_{\overline{g}}{U}_q^{\overline{g}}(\Gamma ),\text{ (33)}\\ L_{\overline{\theta }}\left[{\overline{\theta }}_q(\Gamma )-{\chi }_q{\overline{\theta }}_{q-1}(\Gamma )\right]={\hslash }_{\overline{\theta }}{U}_q^{\overline{\theta }}(\Gamma ).\end{array}$

The equivalent boundary conditions are:

$\begin{array}{l}{\overline{f}}_q(0)={\overline{f^{\prime }}}_q(0)={\overline{f}}_q(1)={\overline{f^{\prime }}}_q(1)=0,\\ {\overline{g}}_q(0)={\overline{g}}_q(1)=0,\text{ (34)}\\ {\overline{\theta }}_q(0)={\overline{\theta }}_q(1)=0.\end{array}$

Here

$\begin{array}{l}{U}_q^{\overline{f}}(\Gamma )=\left(1+\frac{1}{\beta }\right){\overline{f}}_{q-1}^{iv}-A_1{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]\left({\displaystyle \sum _{k=0}^{q-1}{\overline{f^{\prime }}}_{q-1-k}}{\overline{f^{″}}}_k-{\displaystyle \sum _{k=0}^{q-1}{\overline{f}}_{q-1-k}}{\overline{f^{‴}}}_k\right)\\ -A_2{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]{\overline{g}}_{q-1}-\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)M{\overline{f^{″}}}_{q-1},\text{ (35)}\end{array}$

$\begin{array}{l}{U}_q^{\overline{g}}(\Gamma )=\left(1+\frac{1}{\beta }\right){\overline{g^{″}}}_{q-1}-{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_1\left({\displaystyle \sum _{k=0}^{q-1}{\overline{g}}_{q-1-k}}{\overline{f^{\prime }}}_k-{\displaystyle \sum _{k=0}^{q-1}{\overline{f}}_{q-1-k}}{\overline{g^{\prime }}}_k\right)\\ +2{\left(1-\phi \right)}^{2.5}\left[\left(1-\phi \right)+\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right]A_2{\overline{f^{\prime }}}_{q-1}+\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)M{\overline{g}}_{q-1},\text{ (36)}\end{array}$

$U_q^{\overline{\theta }}(\Gamma )=(1+Rd){\overline{{\theta }^{″}}}_{q-1}+\left[\left(1-\phi \right)+\phi \frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\right]\frac{\mathrm{Pr}{A}_1}{\left(\frac{k_{nf}}{k_f}\right)}{\displaystyle \sum _{k=0}^{q-1}{\overline{f}}_{q-1-k}}{\overline{{\theta }^{″}}}_k.\text{ (37)}$

Where

${\chi }_q=\{\begin{array}{l}0,\text{ if }\psi \le \text{1}\\ 1,\text{ if }\psi >\text{1}\end{array}\text{ (38)}$

In order to investigate the flow and heat transfer performance for both SWCNTs and MWCNTs based on kerosene nanoliquids between two rotating parallel plate, (Figure 3,6) are plotted. Figure 1 displays different structure of CNTs and Figure 2 shows physical shape of the flow. Figures 3-7 are plotted to see the impact of various parameters on velocity and temperature profiles for both SWCNTs and MWCNTs-kerosene nanoliquids. These parameters are nanoparticles volume friction φ, Reynolds number (), Rotation parameter (A₂), 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 φ. From Figure 3 we observed that there is petty disparity in $f^{\prime }(\Gamma )$ with the escalating values of φ. We also observed that the MWCNTs has comparatively much greater $f^{\prime }(\Gamma )$ as compared to SWCNTs. Figure 4 is plotted to see the impact of (A₁) on $f^{\prime }(\Gamma )$ with the different escalating values. We observed from the figure, that with the different escalating values of (A₁), the velocity profile changes $f^{\prime }(\Gamma )$ its behavior from rising to reducing at the medium of the two plates $(i.e.\Gamma =0.5)$ $(i.e.\Gamma =0.5)$. This is because of stretching of the lower plates. Figure 5 is plotted to see the impact of (A₂) on $f^{\prime }(\Gamma )$. From figure 5, we observed that the disparity in $f^{\prime }(\Gamma )$ gives dual behavior. Within the region $(0\le \Gamma \le 0.5),$ due to numerous escalating values of (A₂) the velocity profile $f^{\prime }(\Gamma )$ shows declining behavior, however within the region $(0.5<\Gamma \le 1)$, $f^{\prime }(\Gamma )$ gives escalating behavior for various escalating values of (A₂). In addition, at the upper half of the channel (A₂) gives more dominant variation for $f^{\prime }(\Gamma )$. Figure 6 is designed to see the influence of (M) on $f^{\prime }(\Gamma )$. According to Lorentz force theory the magnetic field parameter has reverse effect on $f^{\prime }(\Gamma )$, that is $f^{\prime }(\Gamma )$ 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 $f^{\prime }(\Gamma ).$ Form figure 7, we see that $f^{\prime }(\Gamma ).$ escalates with suction (Q < 0), that is $f^{\prime }(\Gamma ).$ increases with positive the values of (Q), while form Figure 8, it is observed that $f^{\prime }(\Gamma ).$ reduces with injection (Q < 0), that is $f^{\prime }(\Gamma ).$ reduces with the negative (Q). It is fairly obvious that the presence of CNTs nanoparticles has improved the velocity function $f^{\prime }(\Gamma )$, while the velocity function $f^{\prime }(\Gamma )$ increases more when (Q > 0) is present. But the decrement in velocity function $f^{\prime }(\Gamma )$ 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 $f^{\prime }(\Gamma )$. From here we observed that the increasing values of (β) shows reduction in $f^{\prime }(\Gamma )$. Figure 10 is plotted to see the impact of φ on $g(\Gamma )$.

From Figure 10 we observed that there is petty disparity in $g(\Gamma )$ with the escalating values of φ. We also observed that the MWCNTs has comparatively much greater $g(\Gamma )$ as compared to SWCNTs. Figure 11 is plotted to see the impact of on (A₁) $g(\Gamma )$. Figure 11 shows reduction in $g(\Gamma )$ between the two rotating plates with the increasing values of (A₁) and the position of the maximum amount of $g(\Gamma )$ approaches to the stretching sheet. Figure 12 is plotted to see the impact of (A₂) on $g(\Gamma )$. From Figure 12, we observed that the increasing values of (A₂) gives decreasing behavior to $g(\Gamma )$. It is also observed that the disturbance in $g(\Gamma )$ 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 $g(\Gamma )$. According to Lorentz force theory the magnetic field parameter has reverse effect on $g(\Gamma )$, that is $g(\Gamma )$ 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 φ. From figure 14 we observed that there is petty disparity in $\theta (\Gamma )$ with the escalating values of φ.

We also observed that the MWCNTs has comparatively much greater $\theta (\Gamma )$ as compared to SWCNTs. Figure 15 is plotted to see the impact of (A₁) on $\theta (\Gamma )$. From here we observed that the escalating value of (A₁) shows escalation in $\theta (\Gamma )$. As the distance from the surface escalates, $\theta (\Gamma )$ decreases. Figure 16 is plotted to see the impact of (Pr) on $\theta (\Gamma )$. From here we see that the escalating values of (Pr) shows reduction in $\theta (\Gamma )$. Physically, the nanofluids 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 $\theta (\Gamma )$ Thermal radiation has leading rule in heat transmission when the coefficient of convection heat transmission is small. From here we see that the escalating values of (Rd) shows acceleration in $\theta (\Gamma )$.

Tables 1,2 are schemed to see the influences of different embedding parameters on skin fraction coefficient $\left({\tilde{C}}_f\right)$ and local Nusselt number (Nu_x). From Table 1, we see that the effect of A₁ on skin fraction coefficient. It is clear from the table that at 0.1, 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 coefficient 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 A₂ shows reducing behavior. The escalating values of (M) shows decreasing behavior in skin fraction coefficient $\left({\tilde{C}}_f\right)$. 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 $\left({\tilde{C}}_f\right)$. From Table 2, we see that the increasing values of (A₁) and positive Q (i.e. Q > 0) show increasing behavior in local Nusselt number (Nu_x), while (A₂), (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 nanofluids of some base fluids, Thermal conductivity (k_nf) of CNTs with different volume fraction φ respectively.

The investigation for the two-dimensional flow of kerosene oil based nanofluid over an inclined stretching sheet with suction/injection, MHD and radiative heat flux 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 coefficient and local Nusselt number are shown through Table 1 and 2. On the achieved study, the key remarks are listed below.

1. The velocity function $f^{\prime }(\Gamma )$ escalates with the escalation in φ, (β) and suction parameter (i.e. Q > 0), and reduces with the escalation in A₁, (M) and injection parameter (i.e. Q > 0).

2. The velocity function $f^{\prime }(\Gamma )$ shows reducing behavior with escalating values of (A₂) within the region $(0\le \Gamma \le 0.5)$ and shows escalation behavior with escalating values of (A₂) within the region $(0.5<\Gamma \le 1).$

3. The velocity function $g(\Gamma )$ shows escalation behavior with escalation in φ and show reducing behavior with escalation in (A₁), A₂ and (M).

4. The temperature function $\theta (\Gamma )$ show escalating behavior with the escalation in φ and (Rd) shows reducing behavior with the escalation in (A₁) and (Pr).