MIXTURE OF Al2O3-Cu/H2O-(CH2OH)2 MHD HYBRID NANOFLUID FLOW DUE TO A STRETCHABLE ROTATING DISKS SYSTEM UNDER THE INFLUENCE OF NON-UNIFORM HEAT SOURCE OR SINK AND THERMAL RADIATION

This paper investigates the theoretical analysis of hybrid nanofluid flow due to stretchable rotating disks system under the influence of non-uniform heat source or sink and thermal radiation. Two types of nanoparticles Copper (Cu) and Alumina (Al2O3) mixed with the base fluid (ethylene glycol and water) in a ratio 50:50 were considered. The governing partial differential equations were considered in a cylindrical coordinate and Von Karman transformations were rendered into the system to obtain equivalent Ordinary differential equations. The resulting non-linear Ordinary differential equations together with their initial and boundary conditions were solved using finite differences method (FDM) with the aid of maple 18.0 software. The numerical result obtained shows the effect of Reynolds number, Radiation parameter, magnetic parameter and volume fraction of hybrid nanoparticles on the total Entropy generation and Bejan number. Also, the Skin friction Coefficient and Nusselt number at the lower and upper rotating disk were examined for different parameters and the effects of various parameters on the Axial, radial and tangential velocities and the thermal field presented graphically.


INTRODUCTION
The use of copper as nanoparticle suspended in the base fluid have received a lot of attention in the recent decades due to it extraordinary performance as antibacterial and antimicrobial agents as well as its heat conduction activities. Majzlik et al. [1]. Its application in industries such as pharmaceuticals, food, aeronautic, cosmetics, electronics and environmental science cannot be over-emphasized.
Rotating disk with hydrodynamic flow was first investigated in the year 1921 by Von Kanman [2], in his work which introduced a similarity transformation that was later adopted by many researchers. Magnetohydrodynamic (MHD) flow of Cu-water nanofluid due to a rotating disk with partial slip was studied by Hayat et al. [3]. Zangooee et al. [4] considered hydrothermal analysis of MHD nanofluid (TiO2-GO) flow between two radiative stretchable rotating disks using AGM. Chaoli Zhang et al. [5] studied MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Nanofluid flow and heat transfer due to a rotating disk was studied by Mustafa Turkyilmazoglu [6]. Mushtaq and Mustafa [7] studied the computations for nanofluid flow near a stretchable rotating disk with axial magnetic field and convective conditions. Recently, Akindele and Ogunsola [8] analyzed the study of non-isothermal permeable flow of nano-fluids in a stretchable rotating disk system. Vimal Kumar Joshi et al. [9] analytically discussed the numerical investigation of magnetic nanofluids flow over rotating disk embedded in a porous medium. MHD fluid flow and heat transfer due to stretchable rotating disk was investigated by Mustafa [10]. The researchers [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] presented their valuable contributions on the flow of nanofluids in a rotating disks system. Recent researches on nanotechnology as proven that hybrid nanoparticles are more effective compare to ordinary nanoparticles. Hybrid nanoparticles are prepared by suspending different types of nanoparticules (more than one) in the base fluid.
Scholars that have worked on hybrid nanoparticles are as follows: Mabood et al. [30] investigated Cu-Al2O3-H2O hybrid nanofluid flow with melting heat transfer, irreversibility analysis and nonlinear thermal radiation. Siddiqui et al. [31] carried out on trade-off for dispersion stability and thermal transport of Cu-Al2O3 hybrid nanofluid for various mixing ratios. 3 MIXTURE OF Al2O3-Cu/H2O-(CH2OH)2 MHD HYBRID NANOFLUID Investigation on thermophysical properties of Tio2-Cu/H2O hybrid nanofluid transport dependent on shape factor in MHD stagnation point flow was considered by Ghadikolaei et al. [32]. Flow of hybrid nanofluid across a permeable longitudinal moving fin along with thermal radiation and natural convection was studied by Gireesha et al. [33]. Waini et al. [34] reviewed MHD flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge.
Khan et al. [35] discussed Darcy-Forchheimer hybrid (MoS2, SiO2) nanofluid flow with entropy generation. The Study of Two-Phase Newtonian Nanofluid Flow Hybrid with Hafnium Particles under the Effects of slip was explored by Ellahi et al. [36]. Waini et al. [37] performed Hybrid nanofluid flow past a permeable moving thin needle. Aladdin et al. [38] carried out Cu-Al2O3/water hybrid nanofluid flow over a permeable moving surface in presence of hydromagnetic and suction effects. Tayebi and Chamkha [39] studied the natural convection enhancement in an eccentric horizontal cylindrical annulus using hybrid nanofluids. Free convection enhancement in an annulus between horizontal confocal elliptical cylinders using hybrid nanofluids was performed by Tayebi and Chamkha [40]. Gorla et al. [41] discussed heat source/sink effects on a hybrid nanofluid-filled porous cavity. Chamkha et al. [42] analyzed the numerical analysis of unsteady conjugate natural convection of hybrid water-based nanofluid in a semicircular cavity. Transpiration and Viscous Dissipation Effects on Entropy Generation in Hybrid Nanofluid Flow over a Nonlinear Radially Stretching Disk was examined by Farooq et al. [43]. Afridi et al. [44] carried out the Entropy Generation in Cu-Al2O3-H2O Hybrid Nanofluid Flow over a Curved Surface with Thermal Dissipation. Khan et al. [45] carried out Computational analysis of nanofluid and hybrid nanofluid in Darcy's squeezing flow with entropy optimization. Magneto rotating flow of hybrid nanofluid with entropy generation was discussed by Khan et al. [46]. Sadaf and Abdelsalam [47] worked on the adverse effects of a hybrid nanofluid in a wavy non-uniform annulus with convective boundary conditions. Mixed convective slip flow of hybrid nanofluid (MWCNTs + Cu + Water), nanofluid (MWCNTs + Water) and base fluid (Water) was comparatively investigated by Mahammad et al. [48].
Zainala et al. [49] worked on MHD mixed convection stagnation point flow of a hybrid nanofluid past a vertical flat plate with convective boundary condition. Venkateswarlu and Narayana [50] investigated Cu2Al2O3/H2O hybrid nanofluid flow past a porous stretching sheet 4 OGUNSOLA AMOS WALE, AKINDELE AKINTAYO OLADIMEJI due to temperature dependent viscosity and viscous dissipation. Roy et al. [51] considered Heat transfer of a hybrid nanofluid past a circular cylinder in the presence of thermal radiation and viscous dissipation. Heat transfer analysis of Cu-Al2O3 hybrid nanofluid with heat flux and viscous dissipation was investigated by Ali et al. [52]. Aly and Pop [53] explored MHD flow and heat transfer near stagnation point over a stretching/shrinking surface with partial slip and viscous dissipation: Hybrid nanofluid versus nanofluid. Acharya et al. [54] worked on the hydrothermal features of magnetized TiO2-CoFe2O4 water-based steady hybrid nanofluid flow over a radiative revolving disk. Cu-Al2O3/Water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM was examined by Usman et al. [55]. There are some latest explorations about the hybrid nanofluid as presented in the references . Sachin Shaw [94] explained the Impact of Cattaneo-Christov heat flux on Al2O3-Cu/H2O-(CH2OH)2 hybrid nanofluid flow between two stretchable rotating disks.
The objective of the present study is to analyze the development of the Mixture of Al2O3-Cu/H2O-(CH2OH)2 MHD hydrid nanofluid flow due to a stretchable rotating disks system under the influence of non-uniform heat source or sink and thermal radiation in a cylindrical coordinate.
The velocities profiles, pressure profile, thermal field, skin friction coefficient, Nusselt number, total entropy generation and Bejan number are taken into consideration. The system of governing non-linear partial differential equations was transformed into their ordinary differential equations equivalents and the resulting equations were solved numerically by Newton's Finite difference technique subjected to appropriate boundary conditions.

MATHEMATICAL FORMULATION
Following [8], the flow in the gap between two stretchable rotating disks in cylindrical coordinate systems (r, θ, z) was considered. The gap contains a fluid that is considered as a hybrid nanofluid with basefluid as a mixture of ethylene glycol ( 2 6 2) and water ( 2 ) in a ratio 50:50. The upper and the lower disks are separated by distance h. The two disks rotate in an anticlockwise direction with rotational velocities Ω1 and Ω2. The disks are deformable and a1 (lower) and a2 (upper) are their stretching rates. Magnetic force takes place between the two disks and the upper disk is maintained at temperature T2 and the lower disk at lesser temperature of T1. The flow geometry is as shown on figure   1. The conservation equation for mass (continuity), radial, tangential and axial momentum and energy conservations equation models are as follows:

SIMILARITY TRANSFORMATIONS
Following [8], the Von Karman similarity transformation is invoked using: where, u, v, w are the radial, tangential, and axial velocity components in the (r, θ, z) directions respectively. T is the temperature, p is hydrodynamic pressure of the fluid and ρhnf is the density of the hybrid nanofluid, hnf and hnf are the dynamic viscosity and thermal diffusivity of the hybrid nanofluid.
The mass conservation law equation (1) is identically satisfied. However, the radial, tangential and axial momentum equations (replaced with the pressure equation), the energy and the concentration equations are reduced to the equivalent nonlinear coupled system of ordinary differential equations: The corresponding boundary conditions (7) at lower and upper disk transform to: Upper disk: is the radiative parameter, For making simpler form of Equation (9) and removing , it can be differentiated with respect toand then we have:

SKIN FRICTION AND NUSSELT NUMBER
The radial and tangential shear stresses are given as The total shear stresses at the surface of the lower and upper rotating disks, respectively, are given as: Therefore the skin friction coefficients Cf(0) and Cf(1) at the lower and upper disks, respectively, are as follows: The heat flux qw at the lower and upper rotating disks, respectively, are as follows: (26) hence the Nusselt numbers at the lower and upper rotating disks surfaces, respectively, are written as:

ENTROPY GENERATION AND BEJAN NUMBER
Entropy generation rate is the determination of the irreversibilities encountered in a specific process. The entropy generation is written as: The entropy generation is the combination of three terms as shown below: Substituting the similarity transformation of equation (7)
In this method, the system of ODEs is converted to a first-order system by introducing a new dependent variable then the derivatives are replaced by finite-difference approximations, to (A2) respectively. The effects of parameters were discussed and graphical results were presented to explore stimulating aspects of the parameters on related profiles.

RESULTS AND DISCUSSION
In this paper, the theoretical analysis of hybrid nanofluid flow due to stretchable rotating disks system under the influence of non-uniform heat source or sink and thermal radiation. Two  The effect of the stretching parameter at the lower disk (A1) is observed in Figure.3. As shown in Figure.3a and b, the axial and the radial velocities increase respectively with increment in the stretching rate parameter at the lower disk. In Figure.3c and d, the tangential and the temperature profiles decays respectively near the lower disc with increasing (A1).  The total entropy generation and Bejan number at the lower disk were discussed in Figure.9 and 10 respectively with increasing Reynolds number, the entropy generation decreases which is obvious as shown in equation (32) where the non-dimensional entropy generation is 14 OGUNSOLA AMOS WALE, AKINDELE AKINTAYO OLADIMEJI inversely proportional to the Reynolds number (see Figure.9). Also the Bejan number reduces with increasing Reynolds number (see Figure.10).The effect of (Q) and (Q1) are demonstrated on Fig.11 and Fig.12 respectively. It was found that both (Q) and (Q1) enhance the thermal field. Fig.13, it was found that an increase in radiation parameter (Rd) enhance the absorbing rate (k*) which decrease the temperature profile. Fig.14 shows the impact of increasing magnetic field parameter (M) on the thermal field; (M) is noticeably increasing.
The comparison results of skin friction coefficient and Nusselt number of some existing literature: Turkyilmazoglu et al [28], Hosseinzadeh et al [29] and Sachin Shaw [94], under some certain conditions (absent of hybrid nanoparticles: φ1=φ2=0) were tabulated on      Figure.14. Effect of M on Temperature profile (θ)

CONCLUSION
In this paper, the effect of Reynolds number, stretching rate parameter at the lower and upper disk, skin friction coefficient at the lower and upper disk, Nusselt number at the lower and upper disk, the total entropy generation, Bejan number, radiation parameter, temperature dependent heat source parameter, surface dependent heat source parameter and magnetic parameter on the Mixture of Al2O3-Cu/H2O-(CH2OH)2 MHD hybrid nanofluid flow due to a stretchable rotating disks system have been investigated using the Newton's finite differential method. It is confirmed in this work that the radiation parameter enhance the adsorption rate which decreases the temperature profile. Base on the work the following remarks were made. 1) As the Reynolds number increase, the axial, radial and thermal field increases significantly while tangential velocities decrease.
2) Increasing the stretching rate parameter at the lower disk (A1), the axial and the radial velocities enhances near the lower disk while the tangential velocity and the temperature profile decreases near the lower disk but the impact of increasing stretching rate parameter at the upper disk (A2) is the opposite of (A1) on all the profiles.
3) Increasing the temperature dependent heat source parameter and the surface dependent heat source parameter evokes a corresponding increase in the thermal field for both parameters. 4) By increasing the Reynolds number, the total entropy generation and Bejan number at the lower disk decreases, which are obvious as shown in equation (32) where the nondimensional entropy generation is inversely proportional to the Reynolds number.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.