ANALYTICAL STUDY OF UNSTEADY SQUEEZED FLOW OF WATER BASE CNTS NANOFLUID WITH MAGNETIC FIELD AND VARIABLE THERMAL CONDUCTIVITY OVER A STRETCHING SURFACE

This research paper explains the analytical solution unsteady squeezing flow of water based CNTs for both MWCNT and SWCNT in the presence of magnetic field and variable thermal conductivity. The given partial differential equation is converted to nonlinear ordinary differential equation by using the similarity transformation and solve by analytical method namely optimal homotopy asymptotic method (OHAM) to obtain analytical solution of the nonlinear problem which analyze the problem. The result of important parameter for both velocity and temperature profiles are plotted and discussed. The BVPh 2.0 package is used to obtain the convergence of the problem up to 25 iteration. The skin friction coefficient and Nusselt number is explained in table form.


INTRODUCTION
Due to the progress in fluid mechanics and its application in industrial, squeezing flow is interesting area for the researcher in the present time. The application of squeezing flow is polymer processing, food engineering, physical comparison, injection molding and biophysical. Kuzma et al. (1968) used squeezing films to discuss fluid inertia effect. Gupta and Gupta (1977) used parallel plate to discuss squeezing flow, the researchers (Wang and Watson, 1979;Bhattacharyya and Pal, 1997;Xu et al., 2010;Ganji et al., 2014) discussed heat transfer involving squeezing flow. There are some important applications of the analysis of heat transfer, for example, energy production, heat conduction in tissues, astro physical flow, missiles physical flow and electrical power generation. Grubka and Bobba (1985) used stretching surface to discuss heat transfer phenomena. Dutta (1988) used stretching surface to investigate the hydro magnetic flow and heat transfer. The researcher Rudraswamy and Gireesha, 2014;Kuiry and Bahadur, 2015;Rudraswamy et al., 2016;Ganesh et al., 2017) discussed the effect of thermal radiation and heat transfer. Chaim (1998) used stretching sheet to discuss the impact of variable thermal conductivity and heat transfer. Sharma and Singh (2009) used linearly stretching sheet to discus MHD flow with variable thermal conductivity.  used Keller box method to investigate the effect of thermal conductivity and MHD flow, for this they use stretching cylinder. Malik et al. (2016) used stretching cylinder to investigate MHD flow of Sisko fluid.
The achievement of energy is not enough, but also to adjust the consumptions of energy and this is possible only to approve the development heat transmission liquids to mechanism the expenditures of energy and to improvement the most heat transmission which is the demand of the industry and other related scientific fields. Former to the application of nanotechnology, analysts and engineers have challenged such huge numbers of questions, recognizing with heat transmission fluids, still, with the development of nanometer sized particles and its uses in the heat transfer fluids have overall improved thermal conductivity. Rehman et al. (2019c) used stretching sheet to discuss the viscous dissipation of thin film unsteady flow of GO-EG/GO-W nanofluids. The most common lecture of carbon family is carbon nanotube (CNT). Carbon nanotube is used as a nanomaterial to rise heat transmission. Single wall carbon nanotube (SWCNT) and multi wall carbon nanotube (MWCNT) are two subclass of carbon nanotube (CNT).
The key use of carbon nanotube (CNT) in engineering is for fluidization and heat conversation. Iijima was the first one to present carbon nanotube (CNT) in 1991 and also the first time present the multi wall carbon nanotube (MWCNT) by using Krastschmer and Huddman method, see (Iijima, 1991;Ajayan and Iijima, 1993). Donald Bethune in 1993 studied multiwall carbon nanotube (MWCNT). They discuss the diameter range of multiwall carbon nanotube and show that the diameter range of multiwall carbon nanotube is 0.4 × 10 −9 to 3 × 10 −9 (To, 2006). In multiwall carbon nanotube there are 2 to 50 coaxial nanotube with diameter range from 3 × 10 −9 to 30 × 10 −9 (Dresselhaus et al., 1995). Hone (2004) investigated that at room temperature the thermal conductivity of MWCNT is 3000 −1 −1 and for SWCNT is 6600 −1 −1 . Haq et al. (2015) discussed the compression of engine base (CNT) and kerosene oil base (CNT) and show that the skin friction of water base (CNT) is greater than the kerosene oil base (CNT). Khan et al. (2014) used stretching plate to discuss the thermal conductivity and navier slip boundary condition for CNT. Kamali and Binesh (2010) used CNT base nanofluid to discuss the model of power law. Rehman et al. (2019a) used stretching cylinder to discuss Marangoni convection of thin film spray of CNT nanofluid. Rehman et al. (2019b) discussed the impact

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org 2 of magnetic field and viscous dissipation on the thin film unsteady flow of GO-EG/GO-W nanofluids. Atashafrooz et al. (2020) investigated interacting infancies of Lorentz force and bleeding on the hydrothermal behaviors of nanofluid flow in a trapezoidal recess with help of second law of thermodynamics. Sheikholeslami (2019) used Al2O3 nanoparticles to discuss magnetic force and radiation influences by using permeable media. Sheikholeslami et al. (2014) investigated the influence of radiative heat transfer on thermal characteristics of nanofluid flow in the presence of magnetic field. Atashafrooz (2019)  In this research paper, we explain the analytical solution of unsteady squeezing flow of water based CNTs for both MWCNT and SWCNT in the presence of magnetic field and variable thermal conductivity. The given partial differential equation is converted to nonlinear ordinary differential equation by using the similarity transformation and solve by analytical method, i.e., optimal homotopy asymptotic method (OHAM). Liao (2010) used this method for the solution of nonlinear problem and shows that this method is quickly convergent to the approximate solution. This method gives us series solution in the form of functions and all the physical parameters of the problem involved in this method. The result of important parameter for both velocity and temperature profiles are plotted and discussed. Consider the unsteady boundary layer flow of water base CNTs. Figure 1 shows the closed squeezed channel with height ℎ( ), supposed that ℎ( ) is greater than that boundary layer thickness. represent micro cantilever of length which is enclosed inside the channel. I t is supposed that the squeezing stream is to be occurred from the tip to the surface. Moreover it is assumed that lower plat is in rest. The continuity and momentum equations are settled, see :

MATHEMATICAL FORMULATION
(2) Here and are the velocity component with and direction. Kinematic viscosity is denoted by , the stream velocity is denoted by , the variable thermal conductivity is denoted by , the power law index is denoted by , the time constant is denoted by , the electric charge density is denoted by , the fluid pressure is denoted by , and the temperature is denoted by . By eliminating the pressure term from equation (2) and equation (3), we obtain with boundary condition The free stream temperature wall heat flux and free stream velocity is denoted by ∞ , ( , ) and ( ), = ∞ (1 + ) represent the thermal conductivity. The similarity transformation is defined as where is the squeezed flow index, is arbitrary constant, show strength squeezing flow parameter, heat flux is represented by 0 and is the thermal conductivity. Using the similarity transformation in equation (4) and equation (5) the partial differential equation is transform to ordinarily differential equation: where = is Prandtl number, 0 = √ is permeable velocity, = √2 Γ √ and = 0 2 is magnetic parameter and is the power law index.
The coefficient of skin friction and Nusselt number is defined as

Method of Solution
The analytical method (OHAM) is used to find the approximate analytical solution of the nonlinear ordinary differential equation. The BVPh 2.0 package function of Mathematica is used to obtain the numerical results, the result of important parameters. The given equations (7) and (8) are solved analytically by analytical method, i.e., optimal homotopy asymptotic method (OHAM) which is given below where is linear operator, is independent variable, ( ) is the unknown function, is the nonlinear operator and ( ) is a boundary operator. By using this method, we first find a family of equations Let the solution of equation (14) be When = 1, equation (14) become as which has a solution, let the solution is ( ).
Thus by increasing from 0 to 1, the solution change form 0 ( ) to ( ). Choose the supplementary function as where 1 , 2 , 3 , … are constants to be determined. Expanding ( , ) in a series with respect to , we have ∞ =1 for = 1,2,3 … . Put equation (19) into equation (11) and comparing the coefficient of different power of , we get differential equations with boundary condition. By solving this equation, we find a series solution in the form of functions 0 ( ), 1 ( , 1 ), 2 ( , 2 ), … and the solution of equation (10) can be written as The residual is as follow If in equation (21), ( , ) = 0, then equation (20) has exact solution but for nonlinear differential equation is not possible. We try to minimize the ( , ) to get the best approximate analytical solution of equation (10).  3.133 × 10 −5 8.006632 × 10 −9 Fig. 2 Effect of Weissenberg number on velocity profile.       The main objective of this section is to study the effect of various model factors like , 0 , , , , , (magnetic parameter, permeable velocity, Prandtl number, Weissenberg number, small quantity, power law index and squeezed flow index on velocity and temperature distribution). In Tables 1 and 2, the numeric results illustrate the impacts of different model factors on the skin friction coefficient and Nusselt number of both MWCNT and SWCNT. The effect of different parameters on local skin friction coefficient is shown in Table 1. The Table 1 depicts that skin friction coefficient decreasing in both MWCNT and SWCNT for the increasing value of squeezed flow index and magnetic field. Whereas, Table 2 shows the Nusselt number coefficient decreasing in both MWCNT and SWCNT for the increasing value of Prandtl number and Weissenberg number. The convergence for both MWCNT and SWCNT nanofluid has been obtained up to the 25 th iteration for the SWCNT and MWCNT nanofluid in Tables 3 and 4 oneto-one. The Tables 3 and 4 show that by increasing the number of iteration reduces the order of residual error and strong convergence attained. Fig. 2 shows the effect of Weissenberg number on velocity profile. The relation between Weissenberg number and velocity is inverse relation, by increasing Weissenberg number the velocity profile decreasing. By increasing the Weissenberg number, increase the relaxation time of fluid due to the resistance produced in the flow direction and as a result velocity profile is decreasing as shown in Fig. 2.  Fig. 3 shows the effect of squeezed flow index on velocity profile. The relation between squeezed flow index and velocity is inversely relation. By increasing squeezed flow index, the velocity profile decreasing because by increasing squeezing phenomena, the kinetic energy of the fluid particle increase but there is inverse relation between strength of squeeze flow and squeezed flow index. So as a result velocity profile decrease as shown in Fig. 3. Fig. 4 shows the effect of permeable velocity on velocity profile. The relation between permeable velocity and velocity is inversely relation. The velocity profile decreasing by increasing the permeable velocity because by increasing permeable velocity suction is produced and due to this fluid to be more attached to surface. Fig. 5 shows the effect of magnetic parameter on velocity profile. The relation between magnetic parameter and velocity is inversely relation. By increasing magnetic parameter the velocity profile decreasing because by increasing the magnetic parameter, resistance in the flow direction increasing due to which velocity profile decreasing as shown in Fig. 5. Fig. 6 shows the effect of power law index on velocity profile, the relation between power law index and velocity is inversely relation. By increasing power law index the velocity profile decreasing because by increasing the power law index will reduce the fluid motion. So as a result velocity profile decreasing as shown in Fig. 6. Fig. 7 shows the effect of the against the ( ) temperature field. The relation between temperature field and is inverse relation, so by enhance the decreasing the temperature profile as displayed in Fig. 7. In fact, the thickness of the momentum boundary layer to be larger than that of the thermal boundary layer, or that the viscous diffusion is larger than the thermal diffusion and therefore, the larger amount of the reduces the thermal boundary layer. Fig. 8 shows the effect of Weissenberg number on temperature profile. The relation between Weissenberg number and temperature is direct relation. By increasing Weissenberg number the temperature profile increasing. By increasing Weissenberg number the thermal boundary layer decreasing, so as a result the temperature profile increasing. Fig. 9 shows the effect of squeezed flow index on temperature profile. The relation between squeezed flow index and temperature is inversely relation. By increasing squeezed flow index the temperature profile decreasing because by increasing squeezed flow index the thickness of the momentum boundary layer to be larger than that of the thermal boundary layer, so as result temperature profile decreasing. Fig.  10 shows the effect of on temperature profile. The relation between and temperature is direct relation. By increasing the temperature profile increasing because by increasing the small partial, increase the kinetic energy of the fluid particle which increase the variation of thermal characteristics, so as a result the temperature profile increase.

CONCLUSIONS
This paper explained the unsteady squeezing flow of water based nanofluid CNTs for both MWCNT and SWCNT in the presence of magnetic field and variable thermal conductivity. The model is solved by analytical method namely optimal homotopy asymptotic method (OHAM) to obtain analytical solution of the nonlinear problem which analyze the problem. The result of important parameter for both velocity and temperature profiles are plotted and discussed. The skin friction coefficient and Nusselt number is explained in table form. The obtained outputs are deliberated as follows. It is observed that:  Increasing Weissenberg number, decreasing the velocity field.  Increasing the squeezed flow index, decreasing the velocity profile.  Increasing the values of the magnetic field, decreasing the velocity profile.  By increasing the Prandtl number , decreasing the temperature profile.  By increasing the , increasing the temperature field.