INVESTIGATION ON CNTS-WATER AND HUMAN BLOOD BASED CASSON NANOFLUID FLOW OVER A STRETCHING SHEET UNDER IMPACT OF MAGNETIC FIELD

This study aims at considering the properties of heat transfer and magneto-hydrodynamics (MHD) Casson nanofluid at the existence of free convection boundary layer flow with Carbon Nanotubes (CNTs) suspended in human blood/water as based fluid on a stretching sheet. Two types of CNTs nanoparticles, single walled carbon nanotubes (SWCNTs) and multi walled carbon nanotubes (MWCNTs), are taken into account. The governing partial differential equations are transformed to partial differential equations using similar transformation, then solved numerically by an implicit finite difference scheme known as Keller-box method (KBM). The results for physical quantities, the local skin friction, and local Nusselt number, as well as temperature and velocity, are discussed under the magnetic nanofluid Casson parameters. This work is compared with recently published results on the Newtonian fluid as a special case and shows very good agreement.


INTRODUCTION
Casson fluid is a subcategory of Non-Newtonian fluids as toothpaste, food materials, and plastic has several utilizes in medical chemical industries and cosmetics. It is hard to handle this type of fluid, because it behaves as an elastic solid, such that no flow happens with short shear stress. Hence, different fluid models are showed to characterize the effectiveness of the materials. In the current consideration, we investigate the Casson model. This model was first conducted by Casson (1959). After that, many types of researches had been conducted in the same field. Mustafa et al. (2012); Mustafa et al. (2011) investigated the parallel free stream in the presence of the Casson fluid on a semi-infinite flat plate and in the region of stagnation point towards a stretching sheet. Nadeem and UI (2013) studied the boundary layer three-dimensional flow of Casson fluid on a stretching sheet. Another Various researches on Casson fluids can also be found in the literature of Hayat et al. (2012), , Bhattacharyya (2013); Qing et al. (2016), Tahir et al. (2017), Alsagri et al. (2019), Hussanan et al. (2018), andAlwawi et al. (2019).
Recently, several researches have been conducted to develop the heat transfer and reach various levels of thermal competences. This technique can be used through increasing the thermal conductivity. Choi andEastman (1995) were the first who examined the enhancement of heat transfer in convectional fluids with nanoparticles, typically with a size less than 100 nm. After that, many researchers considered it because of its thermal characteristics and its future applications. Moreover, mathematical models have been conducted to investigate in the properties of nanofluids Tiwari and Das (2007). Tiwari-Das model has succeeded in describing the behaviors of nanofluid in boundary layer flow. Besides, many researchers published various articles around the boundary layer flow in a nanofluid, such as Chamkha and Aly (2010); Chamkha et al. (2011), Bhattacharyya (2013; Chamkha (2011), Mukhopadhyay et al. (2013) and Swalmeh et al. (2019 a,b).
It is said that human blood is an example of Casson fluid, and many studies used physical properties of the human blood as based fluid under boundary layer flow, see Khalid et al. (2018), Rashidi et al. (2016), andAlsagri et al. (2019). CNTs (carbon nanotubes) have a long cylindric shape, such as structures of carbon atoms with sizes typically of 0.7-50 nm. In 1991, the concept of CNTs was first investigated by Iijima (1991). Carbon nanotubes are split into two types, namely single wall carbon nanotubes (SWCNTs), and multi wall carbon nanotubes (MWCNTs), relying upon concentric layers of rolled graphene lamina. Moreover, carbon nanotubes are expected to be a creative matter on our time because of their industrial and mechanical applications. There are many studies that take the CNTs nanoparticles into consideration. Haq et al.(2015) considered the effects of the thermo-physical characteristics of CNTs nanoparticles based nanofluids on a stretching surface. The heat transfer flow of nanofluids including CNTs on a flat plate subjected to the Navier slip boundary condition, and the impacts of (MHD) flow on of non-Newtonian as CNTs nanofluid were investigated by Khan et al. (2014), andAman et al. (2017). Several other significant kinds of researches on CNTs based nanofluid have been checked by Khalid et al. (2018); Sallam and Hwang (1984), Nasir et al. (2018), Ding et al. (2006), andAlsagri et al. (2019) Recently, MHD free convection have had great solicitation and importance because of the high wide modern industrial implementations, and thus many articles have been published in this

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org Frontiers in Heat and Mass Transfer (FHMT), 14, 15 (2020) DOI: 10.5098/hmt.14.15 Global Digital Central ISSN: 2151-8629 2 field. MHD free convection with aligned magnetic radiation and variable viscosity on a semi-infinite flat plate were investigated by Seddeek (2002), and Makinde (2012) who considered the collective impacts of Navier slip and Newtonian heat on an unsteady (MHD) magnetic boundary layer flow over a flat plate. The effects of magnetohydrodynamics (MHD) heat transfer and entropy generation flow with the influence of slip-on the flat plate were studied by Ellahi et al. (2018). The (MHD) flow with suitable nanoparticles as Ag suspended in water-based nanofluid over a flat plate in a porous medium,and unsteady magneto-hydrodynamic free convection flow on an infinite vertical flat plate under the characteristics of the uniform transverse magnetic field were considered by Upreti et al. (2018), and Palani and Srikanth (2009) .
The present study has discussed the Heat and mass transfer effect of free convection on MHD Casson nanofluid over a stretching sheet subject to the constant wall temperature. At the same time, it has also used the similar transformations to transform the governing equations in the dimensional form to PDEs. The reduced PDEs are solved numerically by using Keller-box scheme. The numerical calculations for various physical parameters are carried out up to a required level of accuracy (10 −5 ). The results of velocity and temperature profiles are plotted through graphs. The results of the skin friction and Nusselt number are calculated carefully and showed in a form of table.

MATHEMATICAL FORMULATION
Consider the free convection two-dimensional boundary layer flow of incompressible magnetohydrodynamic (MHD), Casson nanofluid over stretching sheet, which the flow begun at 0 y  , and being restricted in 0 y  , as presented in fig. 1, where w u , s u are shrinking/stretching velocity and straining velocity, respectively. Two types of Carbon Nanotubes (CNTs) nanoparticles, such as single walled carbon nanotubes (SWCNTs) and multi walled carbon nanotubes (MWCNTs), in two based fluids namely water and human blood, are investigated. The nanofluid is permeated in a Casson medium and a variable magnetic field strength 0 B and applied in the cross direction of the fluid flow. The flow is supposed to be in the x-direction, which is taken along the stretching sheet in the upward direction and the y-axis is perpendicular to it. Accordingly the same initial temperature is preserved. Instantaneously they are increased to a prescribed wall temperature w TT   the ambient temperature of the fluid that still unchanged. The impacts of nanoparticle volume fraction, Casson, magnetic, and electrical conductivity have merged in the momentum equation. The equation obtained below for determining the rheological property of Casson fluid. Casson (1959) , ij e is the ( , ) ij-th component of the deformation rate, c  , y p , and   respectively are critical values of this product based on the non-Newtonian model, the yield stress of the fluid, and the plastic dynamic viscosity of the non-Newtonian fluid.
In the case of the Casson fluid, c   so: (2), then the kinematic viscosity of the Casson fluid, which B  is purely dependent on plastic dynamic viscosity,  is density and  is parameter of the Casson fluid as Depending on the previous assumptions, the governing equations, the continuity, momentum and energy equations, for Casson nanofluid over a stretching sheet can be presented as follows: Subject to the constant wall temperature boundary condition: where u and v are the velocity components along the x and y directions, respectively. The properties of nanofluid are expressed by Ahmed and Khan (2018) Using the following similarity transformations in the governing equations (4) to (6) subject to the boundary conditions (7) where  is the stream function .
By substituting the equations (6) and (7) we get the following equations and the corresponding boundary condition are Here we have Pr is called the Prandtl number and is magnetic parameter.
For the engineering usefulness, the physical quantities especially, the local skin friction coefficient () Now, the local skin friction coefficient and Nusselt number become as: is denoted the local Reynolds number.

RESULTS AND DISCUSSION
The numerical solutions obtained by using the Keller box method of the above governing equations (10) and (11) subject to the boundary condition (12) are presented in figures and tables, see Keller and Bramble (1970) and Cebeci and Bradshaw (2012). Many convection boundary layer problems use this method to study the numerical solutions and its physical quantities behaviors, this method appears to be the most malleable of the used methods and in spite of novel expansions in other numerical methods, stills a potent and very accurate process for parabolic boundary layer flows. It is also being easily adjustable to solve equations of any order and unconditionally stable on the solutions. The Keller box method procedures are: firstly, reduce PDEs to a first-order system by using a finite difference scheme. Next, Write the difference equations by using central differences, and linearize the resulting differences equations by Newton's method and write them in the matrix-vector form. Finally, solve the linear system by the block tri-diagonal elimination technique, and then writing the steps of Keller-box method in Matlab program see Swalmeh et al (2019 a).  Hassanien et al. (1998) and Salleh et al. (2010). It is noted that the results of the present study are in a good agreement.     Figures 2 and 3 show that both results of the quantities of the temperature and velocity are reduced with an increment of values of the  . Moreover, it is noted that CNTs/human blood has a lower temperature than CNTs/water, but the reverse case happens in the velocity profile with an increase in the value of thermal boundary layer thickness. Also, the MWCENT nanoparticle has a higher temperature opposite to SWCNT nanoparticles, and the reason is that the density of SWCNT is greater than it of MWCNT. Moreover, it can be seen that SWCNT/human blood has a lower velocity than MWCENT/human blood with an increment of the value of thermal boundary layer thickness, but the opposite case happens in MWCENT and SWCNT that are suspending in water.
Figures 4 and 5 exhibit that when the M increase, then the temperature increase, while the velocity profiles are decrement. Practically it has been noticed that the presence of a magnetic field in the flow field region decelerates the fluid motion. The resistive force created is known as the Lorentz force. So, the Lorentz force slows down the velocity of the fluid. In addition, human blood has high temperatures and low velocity compared to water with an increase in the values of the M. Moreover, MWCENT has a high velocity and temperature than SWCENT suspended in human blood Casson nanofluid, meanwhile, when the CNTs in water, we noticed that the SWCENT has a lower temperature than MWCENT nanoparticles, but the reverse case happens in the velocity profile. Figures 6 and 7 display the effects of the  on the temperature and velocity profiles of Casson nanofluid. It is observed that the increasing in  values, leads to increasing in the temperature and decreasing in the velocity profiles, in the presence of CNTs suspended in human blood/water Casson nanofluid. Furthermore, it can be clearly seen that the MWCENT has high temperature than SWCENT nanoparticles. Besides, the MWCENT has less velocity than SWCENT in water, but the opposite case happens when the CENTs nanoparticles suspended in human blood based Casson nanofluid, the MWCENT has high velocity than SWCENT in human blood. On the other hand, human blood has high velocity and less temperature compared with water.

CONCLUSIONS
This paper offers a facilitated mathematical model and numerically studies the heat transfer effects for free convection in MHD Casson nanofluid flow over a stretching sheet. Several properties of the MHD Casson nanofluid parameters on the physical quantities, such as the local skin friction coefficient and the local Nusselt number, as well as temperature and velocity are considered in an attempt to analyse them through various plots and tables. In addition the constant wall temperature boundary condition has been considered in this study. The following results are the findings of the study: 1. The temperature profile of water is more than it for human blood based Casson nanofluid when the values of Casson, magnetic, and nanoparticle volume fraction parameters are increased. 2. When MHD Casson nanofluid parameters increase, local skin friction coefficient in SWCENT/human blood is less than it in MWCENT/human blood based Casson nanofluid . 3. MWCENT human blood /water has a higher temperature profile compared with SWCENT human blood /water with an increase in the MHD Casson nanofluid parameters. 4. SWCENT has lower local skin friction than MWCENT based Human blood Casson nanofluid, but the opposite case happens in water-based Casson nanofluid. Also, SWCENT human blood/water has a high local Nusselt number compared with MWCENT human blood/water.

ACKNOWLEDGEMENTS
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B
Thermal expansion coefficient