MHD Newtonian and non-Newtonian Nano Fluid Flow Passing On A Magnetic Sphere with Mixed Convection Effect

International Journal of Innovation Engineering and Science Research
www.ijiesr.com
Volume 2 Issue 6 November-December 2018 26|P a g e
ABSTRACT
MHD Newtonian and non-Newtonian Nano Fluid
Flow Passing On A Magnetic Sphere with Mixed
Convection Effect
Basuki Widodo, Yolanda Norasia, Dieky Adzkiya
Department of Mathematics, Faculty of Mathematics, Computing, and Data Sciences
Institut Teknologi Sepuluh Nopember
Indonesia
This paper considers the problem of magneto-hydrodynamics (MHD) Newtonian and non-Newtonian nano fluid
flow passing on a magnetic sphere with mixed convection effect. Nano Fluid is a combination of liquid fluid as a
base fluid with small solid nano particles. Water is chosen as Newtonian base fluid and oil is chosen as non-
Newtonian base fluid. Then, Alumina and Copper are chosen as solid particle in nano fluid. We further construct
governing equation by applying continuity equation, momentum equation, and energy equation to obtain
dimensional governing equations. The dimensional governing equations that have been obtained are converted
into non-dimensional governing equations by substituting non-dimensional variables. The non-dimensional
governing equations are further transformed into similarity equations using stream function and solved
numerically using Euler Implicit Finite Difference method. We further analyse the effect of magnetic parameter
towards velocity and temperature in MHD nano fluid flow. The results show that the increases of magnetic
parameter impacts to the decrease of velocity and temperature. Then, the velocity and temperature of Newtonian
nano fluid are higher than the velocity and temperature of non-Newtonian nano fluid. Also, the velocity and
temperature of copper-water are higher than the velocity and temperature of Alumina-water.
Keywords—Newtonian and non-Newtonian nano fluid; MHD; Sphere; Euler Implicit Finite Difference.
I. INTRODUCTION
Nano fluid is a combination of liquid fluid as a base fluid with small solid nano particles [1]. Nano fluid is
divided into two types, i.e. Newtonian nano fluid and non-Newtonian nano fluid. Newtonian nano fluid is
a base fluid in nano fluid which has a linear relationship between viscosity and shear stress. However,
non-Newtonian nano fluid is the opposite of Newtonian nano fluid. In this paper, water is chosen as
Newtonian base fluid and oil is chosen as non-Newtonian base fluid. Then, Alumina (𝐴𝑙2 𝑂3) and
Copper (𝐶𝑢) are chosen as solid particle in nano fluid. Alumina (𝐴𝑙2 𝑂3) contains metal oxide and
copper (𝐶𝑢) contains metal. These types of fluids are used in industrial area that needs for heating and
cooling based on heat transfer [2].
Because of those, we conduct a research how to analyse MHD nano fluid flow problem using
numerical simulation based on mathematical modelling. Putra et al [3] have illustrated the natural
convection of nano-fluids. Their investigations stated that the thermal conductivity of solid nano
particles can be increased when mixed with base fluid. Wen and Ding [4] have discussed about
experimental investigation into convection heat transfer of nano fluids at the entrance region under
laminar flow conditions. Akbar et al [5] have investigated unsteady MHD nano fluid flow through a
channel with moving porous walls and medium by using Runge Kutta. The results show that the heat
transfer rate increases and mass transfer rate decreases with the increase of Reynolds number. Mahat
et al [6] also have observed mixed convection boundary layer flow past a horizontal circular cylinder in
visco-elastic nano fluid with constant wall temperature and solved numerically by using the Keller-Box

Basuki Widodo et al. “International Journal of Innovation Engineering and Science Research”
method. The results indicate that the velocity and temperature are increased by increasing the values
of nano particles volume fraction and mixed convection parameter. Juliyanto et al [7] also have solved
the problem of the effect of heat generation on mixed convection in nano fluids over a horizontal
circular cylinder numerically by using Keller-Box method. The result of their investigations show that
the velocity increase and temperature decrease when mixed convection parameter increases. In the
present paper, we are interested to develop mathematical modelling of the problem of MHD newtonian
and non-newtonian nano fluid flow passing on a magnetic sphere with mixed convection effect. The
influence of magnetic parameter ( 𝑀), mixed convection parameter ( 𝜆), and volume fraction (𝜒) towards
velocity and temperature in Newtonian and non-Newtonian nano fluid are investigated.
II. MATHEMATICAL FORMULATION
The unsteady MHD Newtonian and non-Newtonian nano fluid flow passing on a magnetic sphere with
mixed convection effect is considered. Fig. 1illustrates the physical model of the problem and the
coordinate system used to develop the mathematical model. The fluid used is Newtonian nano fluid
and non-Newtonian nano fluid. The bluff body used is a magnetic sphere with radius a. The flow of
nano fluid is assumed laminar flow and incompressible. The magnetic Reynolds number is assumed to
be very small. Therefore, there is no electrical voltage which makes electric field. With potential theory,
where the velocity potential is perpendicular with stream function, so the 3D dimensional governing
equations can be transform into 2D dimensional governing equations.
Fig. 1 Physical model and coordinate system
Based on the physical model and coordinate system, unsteady MHD Newtonian and non-Newtonian
nano fluid flow passing on a magnetic sphere is illustrated in Fig. 1. The 2D dimensional governing
equations are developed from the law of conservation mass, the second law of Newton, and the first
law of Thermodynamics. We further obtain continuity equation, momentum equation, and energy
equation, which can be written as follows:
Continuity Equation:
𝜕𝑟 𝑢
𝜕𝑥
+
𝜕𝑟 𝑣
𝜕𝑦
= 0 (1)
Momentum Equation :
at x axis
𝜌𝑓𝑛
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= −
𝜕𝜌
𝜕 𝑥
+ 𝜇 𝑓𝑛
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2 + 𝜎𝐵0
2
𝑢 − 𝜌𝑓𝑛 𝛽 𝑇 − 𝑇∞ 𝑔 𝑥 (2)

aty axis
𝜌𝑓𝑛
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= −
𝜕𝜌
𝜕 𝑥
+ 𝜇 𝑓𝑛
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2 + 𝜎𝐵0
2
𝑢 − 𝜌𝑓𝑛 𝛽 𝑇 − 𝑇∞ 𝑔 𝑦 (3)
Energy Equation :
𝜕𝑇
𝜕𝑡
+ 𝑢
𝜕𝑇
𝜕𝑥
+ 𝑣
𝜕𝑇
𝜕𝑦
= 𝛼𝑓𝑛
𝜕2 𝑇
𝜕𝑥2 +
𝜕2 𝑇
𝜕𝑦2 (4)
With the initial and boundary condition as follows :
𝑡 = 0: 𝑢 = 𝑣 = 0, 𝑇 = 𝑇∞, for every𝑥, 𝑦
𝑡 > 0: 𝑢 = 𝑣 = 0, 𝑇 = 𝑇𝑤 ,for𝑦 = 0
𝑢 = 𝑢 𝑒 𝑥 , 𝑢 = 𝑣 = 0, 𝑇 = 𝑇∞as𝑦 → ∞
where𝜌𝑓𝑛 is density of nano fluid, 𝜇 𝑓𝑛 is dynamic viscosity of nano fluid, 𝑔is the gravitational
acceleration, and 𝛼𝑓𝑛 is thermal diffusivity of nano fluid. In addition, the value of 𝑟is defined as
𝑟 𝑥 = 𝑎 sin(𝑥 /𝑎).
Further, the 2D dimensional governing equations (1)-(4) are transformed into non-dimensional
equations by using both non-dimensional parameters and variables. In this problem, the non-
dimensional variables are given as in [7], i.e.:
𝑥 =
𝑥
𝑎
; 𝑦 = 𝑅𝑒1/2
𝑦
𝑎
; 𝑡 =
𝑈∞ 𝑡
𝑎
; 𝑢 =
𝑢
𝑈∞
𝑣 = 𝑅𝑒1/2
𝑢
𝑈∞
; 𝑟(𝑥) =
𝑟(𝑥)
𝑎
where𝑔 𝑥and 𝑔 𝑦are defined as in [7]
𝑔 𝑥 = −𝑔 𝑠𝑖𝑛
𝑥
𝑎
𝑔 𝑦 = 𝑔 𝑐𝑜𝑠
𝑥
𝑎
Boundary layer theory [8] is applied to non-dimensional governing equation. We obtain the following
results
Continuity Equation
𝜕(𝑟𝑢 )
𝜕𝑥
+
𝜕(𝑟𝑣)
𝜕𝑦
= 0 (5)
Momentum Equation
atx axis
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= −
𝜕𝑝
𝜕𝑥
+
𝑣 𝑛𝑓
𝑣 𝑓
∂2 𝑢
∂𝑦2 + 𝑀𝑢 + 𝜆𝑇𝑠𝑖𝑛 𝑥 (6)
aty axis
−
𝜕𝑝
𝜕𝑦
= 0 (7)
Energy Equation
𝜕𝑇
𝜕𝑡
+ 𝑢
𝜕𝑇
𝜕𝑥
+ 𝑣
𝜕𝑇
𝜕𝑦
=
1
𝑃𝑟
𝛼 𝑓𝑛
𝛼 𝑓
𝜕2 𝑇
𝜕𝑦2 (8)

where these nano fluid constants are defined as [9], i.e. :
Density of nanofluid :
𝜌𝑓𝑛 = 1 − 𝜒 𝜌𝑓 + 𝜒𝜌𝑠
Dynamic viscosity :
𝜇 𝑛𝑓 = 𝜇 𝑓
1
(1 − 𝜒)2.5
Specific heat :
(𝜌𝐶𝑝) 𝑛𝑓 = 1 − 𝜒 (𝜌𝐶𝑝) 𝑓 + 𝜒(𝜌𝐶𝑝)𝑠
Heat conductivity :
𝑘 𝑛𝑓 =
𝑘 𝑠 + 2𝑘𝑓 − 2𝜒(𝑘𝑓 − 𝑘 𝑠)
𝑘 𝑠 + 2𝑘𝑓 + 𝜒(𝑘𝑓 − 𝑘 𝑠)
𝑘𝑓
The thermo-physical properties of nano particles and base fluid is given in Table 1 [10].
TABLE I. THERMO-PHYSICAL PROPERTIES
Properties Water Oil Cu 𝑨𝒍 𝟐 𝑶 𝟑
density 997.1 884 8933 3970
specific heat of constant
pressure
4179 1900 385 765
thermal conductivity 0.613 0.145 400 40
We substitute those nano fluid constants into (6) and (8). We obtain
Momentum Equation :
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= −
𝜕𝑝
𝜕𝑥
+
1
1−𝜒 2.5
1
1−𝜒 +𝜒
𝜌 𝑠
𝜌 𝑓
∂2 𝑢
∂𝑦2 + 𝑀𝑢 + 𝜆𝑇𝑠𝑖𝑛 𝑥 (9)
And
Energy equation :
𝜕𝑇
𝜕𝑡
+ 𝑢
𝜕𝑇
𝜕𝑥
+ 𝑣
𝜕𝑇
𝜕𝑦
=
1
𝑃𝑟
𝑘 𝑠+2𝑘 𝑓−2𝜒(𝑘 𝑠−𝑘 𝑓)
𝑘 𝑠+2𝑘 𝑓+𝜒(𝑘 𝑠−𝑘 𝑓)
1
1−𝜒 +𝜒
(𝜌𝐶 𝑝 ) 𝑠
(𝜌𝐶 𝑝 ) 𝑓
𝜕2 𝑇
𝜕𝑦2 (10)
Further, by converting (9) and (10) into non-similarity equations using stream function, which is given
as follows [11]
𝑢 =
1
𝑟
𝜕𝜓
𝜕𝑦
𝑣 = −
1
𝑟
𝜕𝜓
𝜕𝑥
Where
𝜓 = 𝑡
1
2 𝑢 𝑒 𝑥 𝑟 𝑥 𝑓 𝑥, 𝜂, 𝑡 ,
𝜂 =
𝑦
𝑡
1
2
,

𝑇 = 𝑠 𝑥, 𝜂, 𝑡
The equation (9) and (10) are modified by substituting stream function as follows:
Momentum Equation :
1
1−𝜒 2.5 1−𝜒 +
𝜌 𝑠
𝜌 𝑓
𝜕3 𝑓
𝜕𝜂3 +
𝜂 𝜕2 𝑓
2𝜕𝜂2 + 𝑡
𝜕𝑢 𝑒
𝜕𝑥
1 −
𝜕𝑓
𝜕𝜂
2
+ 𝑓
𝜕2 𝑓
𝜕𝜂2) = 𝑡
𝜕2 𝑓
𝜕𝜂𝜕𝑡
+ 𝑡𝑢 𝑒
𝜕𝑓
𝜕𝜂
𝜕2 𝑓
𝜕𝑥𝜕𝜂
−
𝜕𝑓
𝜕𝑥
𝜕2 𝑓
𝜕𝜂2 −
1
𝑟
𝜕𝑟
𝜕𝑥
𝑓
𝜕2 𝑓
𝜕𝜂2 +
𝑀𝑡 1 −
𝜕𝑓
𝜕𝜂
−
𝜆𝑠𝑡
𝑢 𝑒
sin 𝑥 (11)
Energy Equation :
𝑘 𝑠+2𝑘 𝑓−2𝜒 𝑘 𝑠−𝑘 𝑓
𝑘 𝑠+2𝑘 𝑓+𝜒 𝑘 𝑠−𝑘 𝑓
1
1−𝜒 +𝜒
𝜌𝐶 𝑝 𝑓
𝜕2 𝑠
𝜕𝜂2 + 𝑃𝑟
𝜂
2
𝜕𝑠
𝜕𝜂
+ Pr 𝑡
𝜕𝑢 𝑒
𝜕𝑥
𝑓
𝜕𝑠
𝜕𝜂
= 𝑃𝑟 𝑡
𝜕𝑠
𝜕𝜂
+ 𝑢 𝑒
𝜕𝑓
𝜕𝜂
𝜕𝑠
𝜕𝑥
−
𝜕𝑓
𝜕𝑥
𝜕𝑠
𝜕𝜂
−
1
𝑟
𝜕𝑟
𝜕𝑥
𝑓
𝜕𝑠
𝜕𝜂
(12)With the initial and boundary condition are as follows :
𝑡 = 0 ∶ 𝑓 =
𝜕𝑓
𝜕𝜂
= 𝑠 = 0untuksetiap𝑥, 𝜂
𝑡 > 0 ∶ 𝑓 =
𝜕𝑓
𝜕𝜂
= 0 , 𝑠 = 1ketika𝜂 = 0
𝜕𝑓
𝜕𝑦
= 1 , 𝑠 = 0ketika𝜂 → ∞
By substituting local free stream for sphere case [12],𝑢 𝑒 =
3
2
sin 𝑥 into (11) and (12) respectively, we
obtain
Momentum Equation :
1
1−𝜒 2.5 1−𝜒 +
𝜌 𝑠
𝜌 𝑓
𝜕3 𝑓
𝜕𝜂3 +
𝜂 𝜕2 𝑓
2𝜕𝜂2 +
3
2
𝑡 cos 𝑥 1 −
𝜕𝑓
𝜕𝜂
2
+ 2𝑓
𝜕2 𝑓
𝜕𝜂2) = 𝑡
𝜕2 𝑓
𝜕𝜂𝜕𝑡
+
3
2
𝑡 sin 𝑥
𝜕𝑓
𝜕𝜂
𝜕2 𝑓
𝜕𝑥𝜕𝜂
−
𝜕𝑓
𝜕𝑥
𝜕2 𝑓
𝜕𝜂2 +
𝑀𝑡 1 −
𝜕𝑓
𝜕𝜂
−
2
3
𝜆𝑠𝑡 (13)
Energy Equation :
𝑘 𝑠+2𝑘 𝑓−2𝜒 𝑘 𝑠−𝑘 𝑓
𝑘 𝑠+2𝑘 𝑓+𝜒 𝑘 𝑠−𝑘 𝑓
1
1−𝜒 +𝜒
𝜌𝐶 𝑝 𝑓
𝜕2 𝑠
𝜕𝜂2 + 𝑃𝑟
𝜂
2
𝜕𝑠
𝜕𝜂
+ 3 𝑐𝑜𝑠 𝑥𝑃𝑟 𝑡 𝑓
𝜕𝑠
𝜕𝜂
= 𝑃𝑟 𝑡
𝜕𝑠
𝜕𝜂
+ Pr 𝑡
3
2
sin 𝑥
𝜕𝑓
𝜕𝜂
𝜕𝑠
𝜕𝑥
−
𝜕𝑓
𝜕𝑥
𝜕𝑠
𝜕𝜂
(14)
III. NUMERICAL PROCEDURES
MHD Newtonian and non-Newtonian nano fluid flow passing on a magnetic sphere with mixed
convection effect have been investigated numerically by using Euler Implicit Finite Difference method.
he set of similarity equation and boundary condition are discretized by a second order central
difference method and solved by a computer program which has been developed.
Momentum Equation :
1
1 − 𝜒 2.5( 1 − 𝜒 + 𝜒
𝜌 𝑠
𝜌 𝑓
)
𝜕2
𝑢
𝜕𝜂2
+
𝜂
2
𝜕𝑢
𝜕𝜂
+
3
2
𝑡 1 − 𝑢 2
+ 𝑓
𝜕𝑢
𝜕𝜂
= 𝑡
𝜕𝑢
𝜕𝑡
+ 𝑀𝑡 1 − 𝑢 −
2
3
𝜆𝑠𝑡

by using Euler implicit finite difference method we obtain
1
1 − 𝜒 2.5( 1 − 𝜒 + 𝜒
𝜌 𝑠
𝜌 𝑓
)
1
∆𝜂2
𝑢𝑖+1
𝑛+1
− 2𝑢𝑖
𝑛+1
+ 𝑢𝑖−1
𝑛+1
+
𝜂𝑖
2
3𝑢𝑖+1
𝑛+1
− 4𝑢𝑖
𝑛+1
+ 𝑢𝑖−1
𝑛+1
2∆𝜂
+
3
2
𝑡 𝑛+1
1 − 𝑢𝑖
𝑛+1 2
+ 2
1
2∆𝜂
𝑓𝑖
𝑛
3𝑢𝑖+1
𝑛+1
− 4𝑢𝑖
𝑛+1
+ 𝑢𝑖−1
𝑛+1
= 𝑡 𝑛+1
1
2∆𝑡
3𝑢𝑖
𝑛+1
− 4𝑢𝑖
𝑛
+ 𝑢𝑖
𝑛−1
+ 𝑀𝑡 𝑛+1
1 − 𝑢𝑖
𝑛+1
−
2
3
𝜆𝑠𝑖
𝑛
𝑡 𝑛+1
where𝐾𝑖
𝐾𝑖 =
1
1 − 𝜒 2.5( 1 − 𝜒 + 𝜒
𝜌 𝑠
𝜌 𝑓
)
1
∆𝜂2
𝑢𝑖+1
𝑛
− 2𝑢𝑖
𝑛
+ 𝑢𝑖−1
𝑛
+
𝜂𝑖
4
1
∆𝜂
3𝑢𝑖+1
𝑛
− 4𝑢𝑖
𝑛
+ 𝑢𝑖−1
𝑛
+
3
2
𝑡 𝑛+1
1 − 𝑢𝑖
𝑛 2
+
𝑓𝑖
𝑛
∆𝜂
3𝑢𝑖+1
𝑛
− 4𝑢𝑖
𝑛
+ 𝑢𝑖−1
𝑛
− 𝑡 𝑛+1
𝑀 1 − 𝑢𝑖
𝑛
+
2
3
𝜆𝑠𝑖
𝑛
𝑡 𝑛+1
and for
𝐴0 =
1
4
𝜂𝑖
∆𝜂
+
3
2
𝑡 𝑛+1
𝑓𝑖
𝑛
∆𝜂
𝐴1 =
1
1−𝜒 2.5( 1−𝜒 +𝜒
𝜌 𝑠
𝜌 𝑓
)
∆𝜂2
+ 𝐴0
𝐴2 =
3
2
𝑡 𝑛+1
∆𝑡
+ 2
1
1−𝜒 2.5( 1−𝜒 +𝜒
𝜌 𝑠
𝜌 𝑓
)
∆𝜂2
− 𝑡 𝑛+1
𝑀 + 3 𝑡 𝑛+1
𝑢𝑖
𝑛
+ 4𝐴0
𝐴3 =
1
1−𝜒 2.5( 1−𝜒 +𝜒
𝜌 𝑠
𝜌 𝑓
)
∆𝜂2
+ 3𝐴0
Energy Equation :
Pr 𝑡
𝜕𝑠
𝜕𝑡
=
𝑘 𝑠 + 2𝑘𝑓 − 2𝜒(𝑘𝑓 − 𝑘 𝑠)
( 𝑘 𝑠 + 2𝑘𝑓 + 𝜒 𝑘𝑓 − 𝑘 𝑠 )( 1 − 𝜒 +
𝜒 𝜌(𝐶𝑝) 𝑠
𝜌(𝐶𝑝) 𝑓
𝜕2
𝑠
𝜕𝜂2
+ 𝑃𝑟
𝜂
2
𝜕𝑠
𝜕𝜂
+ 3 Pr 𝑡 𝑓
𝜕𝑠
𝜕𝜂
by using implicit finite difference method we get
Pr 𝑡 𝑛+1
1
2∆𝑡
3𝑠𝑖
𝑛+1
− 4𝑠𝑖
𝑛
+ 𝑠𝑖
𝑛−1
=
𝑘 𝑠 + 2𝑘𝑓 − 2𝜒(𝑘𝑓 − 𝑘 𝑠)
( 𝑘 𝑠 + 2𝑘𝑓 + 𝜒 𝑘𝑓 − 𝑘 𝑠 )( 1 − 𝜒 +
𝜌(𝐶𝑝) 𝑓
1
∆𝜂2
𝑠𝑖+1
𝑛+1
− 2𝑠𝑖
𝑛+1
+ 𝑠𝑖−1
𝑛+1
+ 𝑃𝑟
𝜂𝑖
∆𝜂
1
2
3𝑠𝑖+1
𝑛+1
− 4𝑠𝑖
𝑛+1
+ 𝑠𝑖−1
𝑛+1
+ 3 Pr 𝑡 𝑛+1
𝑓𝑖
𝑛
1
2∆𝜂
3𝑠𝑖+1
𝑛+1
− 4𝑠𝑖
𝑛+1
+ 𝑠𝑖−1
𝑛+1
Where𝐿𝑖

𝐿𝑖 =
𝑘 𝑠 + 2𝑘𝑓 − 2𝜒(𝑘𝑓 − 𝑘 𝑠)
( 𝑘 𝑠 + 2𝑘𝑓 + 𝜒 𝑘𝑓 − 𝑘 𝑠 )( 1 − 𝜒 +
𝜌(𝐶𝑝) 𝑓
1
∆𝜂2
𝑠𝑖+1
𝑛
− 2𝑠𝑖
𝑛
+ 𝑠𝑖−1
𝑛
+
1
4
𝑃𝑟
𝜂𝑖
∆𝜂
3𝑠𝑖+1
𝑛
− 4𝑠𝑖
𝑛
+ 𝑠𝑖−1
𝑛
+ 3 Pr
𝑡 𝑛+1
2∆𝜂
𝑓𝑖
𝑛
3𝑠𝑖+1
𝑛
− 4𝑠𝑖
𝑛
+ 𝑠𝑖−1
𝑛
and for
𝐵0 =
1
4
𝜂𝑖
∆𝜂
+
3
2
𝑃𝑟 𝑡 𝑛+1
𝑓𝑖
𝑛
∆𝜂
𝐵1 =
𝑘 𝑠+2𝑘 𝑓 −2𝜒(𝑘 𝑓−𝑘 𝑠)
( 𝑘 𝑠+2𝑘 𝑓 +𝜒 𝑘 𝑓−𝑘 𝑠 )( 1−𝜒 +
𝜒 𝜌 (𝐶𝑝 ) 𝑠
𝜌 (𝐶𝑝 ) 𝑓
∆𝜂2
+ 𝐵0
𝐵2 =
3
2
𝑃𝑟
𝑡 𝑛+1
∆𝑡
+ 2
( 𝑘 𝑠+2𝑘 𝑓 +𝜒 𝑘 𝑓−𝑘 𝑠 )( 1−𝜒 +
∆𝜂2
+4𝐵0
𝐵3 =
( 𝑘 𝑠+2𝑘 𝑓 +𝜒 𝑘 𝑓−𝑘 𝑠 )( 1−𝜒 +
∆𝜂2
+ 3𝐵0
IV. RESULTS AND DISCUSSION
In this research, the effect of magnetic parameter (M) to velocity and temperature in Newtonian and
non-Newtonian nano fluid are analyzed.
Fig. 2 Velocity Profile of Cu-Oil and Cu-Water with Magnetic Influence
Water is chosen as Newtonian base fluid and oil is chosen as non-Newtonian base fluid.
Then, Alumina (𝐴𝑙2 𝑂3) and Copper (Cu) are chosen as solid particle in nano fluid. The numerical
results of the velocity and temperature with respect to the position in front of the lower stagnation at

the point 𝑥 = 0°
with value of magnetic parameters 𝑀 = 1 are depicted in Fig. 2 and Fig. 3
respectively.
Fig. 3 Temperature Profile of Cu-Oil and Cu-Water with Magnetic Influence
Fig. 2 shows the velocity profiles of the MHD Newtonian and non-Newtonian nano fluid flow
passing on a magnetic sphere with mixed convection effect. Cu-oil is used as non-Newtonian nano
fluid and Cu-water is used as Newtonian nao fluid. The results show that the velocity of Newtonian
nano fluid is higher than the velocity of non-Newtonian nano fluid. Also, Fig. 3 shows that the
temperature of Newtonian nano fluid is higher than the temperature of non-Newtonian nano fluid.
Fig. 4 Velocity Profile of Cu-Water and 𝐴𝑙2 𝑂3-Water with Magnetic Influence
Fig. 4 and Fig. 5 show the velocity profiles and temperature profiles of Cu-Water and 𝐴𝑙2 𝑂3-
water respectively. Alumina (𝐴𝑙2 𝑂3) contains metal oxide and copper (Cu) contains metal. The results
show that the velocity of Cu-Water is higher than the velocity of 𝐴𝑙2 𝑂3-water. Fig. 5 also shows that
the temperature of Cu-water is higher than the temperature of 𝐴𝑙2 𝑂3-water.

Fig. 5 Temperature Profile of Cu-Water and 𝐴𝑙2 𝑂3-Water with Magnetic Influence
The numerical results of the velocity and temperature in Newtonian nano fluid Cu-Water with
respect to the position in front of the lower stagnation at the point 𝑥 = 0°
with various value of
magnetic parameters 𝑀 = 0, 1, 3, and 5are illustrated in Fig. 6 and Fig. 7 respectively.
Fig. 6 shows the velocity profiles of the MHD Newtonian nano fluid Cu-Water flow passing on
a magnetic sphere at various M when mixed convection parameter 𝜆 = 1and volume fraction 𝜒 = 0.1.
The results show that velocity and temperature of Newtonian nano fluid Cu-Water in Fig. 6 and Fig. 7
decrease when magnetic parameter increases. The magnetic parameter represents the presence of
Lorentz force in a magnetic sphere. Therefore, when magnetic parameter increases, then the Lorentz
force also increases. It impacts to decrease of the velocity and temperature in Newtonian nano fluid
Cu-Water.
Fig. 6 Velocity Profile for various M of Cu-Water
The numerical results of the velocity and temperature in non-Newtonian nano fluid Cu-Oil with respect
to the position in front of the lower stagnation at the point 𝜒 = 0.1with various value of magnetic
parameters 𝑀 = 0, 1, 3, and 5 are illustrated in Fig. 8 and Fig. 9 respectively.

Fig. 7 Temperature Profile for various M of Cu-Water
Fig. 8 Velocity Profile for various M of Cu-O
Fig. 8 shows the velocity of the MHD non Newtonian nano fluid Cu-Oil flow passing on a magnetic
sphere at various M when mixed convection parameter 𝜆 = 1 and volume fraction 𝜒 = 0.1. Fig. 9
shows the temperature of the MHD non Newtonian nano fluid Cu-Oil flow passing on a magnetic
sphere at various M when mixed convection parameter 𝜆 = 1and volume fraction 𝜒 = 0.1.The results in
Fig. 8 and Fig. 9 show that velocity profiles and temperature profiles of non-Newtonian nano fluid Cu-
Oil decrease when magnetic parameter increases.

Fig. 9 Temperature Profile for various M of Cu-Oil
V. CONCLUSIONS
MHD Newtonian and non-Newtonian nano fluid flow passing on a magnetic sphere with mixed
convection effect have been investigated numerically by using Euler Implicit Finite Difference method.
We have considered water as Newtonian base fluid and oil is chosen as non-Newtonian base fluid.
Further, Alumina (𝐴𝑙2 𝑂3) and Copper (Cu) are chosen as solid particle in nano fluid. We further obtain
numerical results that when effects of magnetic parameter, mixed convection parameter, and volume
fraction are included, the velocity and temperature profiles change. It is concluded that the velocity
and temperature of Newtonian nano fluid Cu-water are higher than the velocity and temperature of
non-Newtonian nanofluid Cu-Oil. The velocity and temperature of copper-water Cu-Water also are
higher than the velocity and temperature of Alumina-water 𝐴𝑙2 𝑂3-Water. Further, the velocity profiles
and temperature profiles of Newtonian Cu-Water and non-Newtonian nano fluid Cu-Oil decrease
when magnetic parameter increases.
ACKNOWLEDGMENT
This research is supported by the Institute for Research and Community Services,
InstitutTeknologiSepuluhNopember (ITS) Surabaya, Indonesia with Funding Agreement Letter number
970/PKS/ITS/2018. We further are very grateful to LPPM-ITS for giving us a chance to submit this
paper in an International Journal.
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MHD Newtonian and non-Newtonian Nano Fluid Flow Passing On A Magnetic Sphere with Mixed Convection Effect

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