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Far East Journal of Mathematical Sciences (FJMS) © 2016 Pushpa Publishing House, Allahabad, India Published Online: July 2016 http://dx.doi.org/10.17654/MS100020291 Volume 100, Number 2, 2016, Pages 291-299 ISSN: 0972-0871
Received: November 8, 2015; Revised: February 10, 2016; Accepted: February 22, 2016 2010 Mathematics Subject Classification: 65M06, 65Z05, 76W05, 80A20. Keywords and phrases: micropolar fluid, magnetohydrodynamics, boundary layer, Keller-box method.
Communicated by K. K. Azad
UNSTEADY BOUNDARY LAYER MAGNETOHYDRODYNAMICS IN
MICROPOLAR FLUID PAST A SPHERE
Basuki Widodo, Indira Anggriani, Dwi Ariyani Khalimah, Firdha Dwi Shafarina Zainal and Chairul Imron
Mathematics Department Faculty of Mathematics and Natural Sciences Institut Teknologi Sepuluh Nopember Surabaya, Indonesia e-mail: [email protected]
Abstract
We consider heat transfer unsteady boundary layer magnetohydrodynamics in micropolar fluid past a sphere. Dimensional governing equations formed from the physical problem by using the theory of the boundary layer. The dimensional governing equations consist of the continuity equation, momentum equation and energy equation. The dimensional governing equations are further converted into non-dimensional equations by introducing non-dimensional variables. The non-dimensional equations are further transformed into similarity equations and then solved numerically using the Keller-Box method. We obtain that the velocity profiles increase when both the values of magnetic variable, M, and micropolar parameter, K, increase. However, when we increase both these values, we obtain that the value of microrotation decreases.
Basuki Widodo et al. 292
I. Introduction
There are two types of fluids, namely Newtonian fluid and non-Newtonian fluid. There are many researchers who use Newtonian fluid. However, the non-Newtonian fluid is a fluid which has viscosity changes when force is applied on [1, 2]. An example of non-Newtonian fluid is micropolar fluid. The micropolar fluid is a fluid with microstructure. Micropolar fluid consists of rigid, randomly oriented particles with their own and microrotations, suspended in a viscous medium. In the micropolar fluid, rigid particles contained in a small volume element can rotate about the center of the volume element, described by the microrotation vector [3]. We investigate the magnetohydrodynamics effect on micropolar fluid, the influence of magnetic field on the microrotation, magnetohydrodynamics effect on velocity profiles and skin friction coefficient as well.
II. Problem Formulation
We consider boundary layer flow in the magnetohydrodynamic microfluid by including the microrotation of the fluid particle. Figure 1 illustrates 2-D coordinate system and the physical model of the solid sphere. Assume that a laminar flow in an incompressible, electrically-conducting, micropolar fluid past a non-conducting solid sphere. There is also no polarization voltage, which implies that there is no electric field. This causes that the magnetic Reynolds number is also assumed very small and hence induced magnetic field can be neglected.
Unsteady Boundary Layer Magnetohydrodynamics in Micropolar … 293
Figure 1. 2-D physical model and coordinate system.
Further, the dimensional governing equations are obtained from mass, momentum and energy conservation law, as follow [4]:
Continuity equation:
( ) ( ) .0=∂
∂+∂
∂yvr
xur (1)
Momentum equation:
( ) yNk
yu
xukx
pyuvx
uutu
∂∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂+
∂∂+μ+
∂∂−=⎟
⎠⎞⎜
⎝⎛
∂∂+
∂∂+
∂∂ρ 2
2
2
2
( ) ,sin20 xTTguB ∞−βρ+σ−
( ) xNk
yv
xvky
pyvvx
vutv
∂∂+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂+
∂∂+μ+
∂∂−=⎟
⎠⎞⎜
⎝⎛
∂∂+
∂∂+
∂∂ρ 2
2
2
2
( ) .cos20 xTTgvB ∞−βρ−σ− (2)
Basuki Widodo et al. 294
Angular momentum equation:
.22
2
2
2⎟⎠⎞⎜
⎝⎛
∂∂−
∂∂+−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂+
∂∂γ=⎟
⎠⎞
⎜⎝⎛
∂∂+
∂∂+
∂∂ρ x
vyuNk
yN
xN
yNvx
NutNj (3)
We further introduce dimensionless variables, as follow [4]:
( ) ( ) ,,,,, 21axrxra
tUtUuua
yReyaxx ===== ∞
∞
,,, 221
∞∞
∞∞ −−=
ρ== TT
TTTUppU
vRevw
(4)
where Reynolds number .vaURe ∞=
We further substitute (4) into (1) until (3) leads to the following non-dimensional equations, namely:
( ) ( ) ,0=∂
∂+∂
∂yrv
xru (5)
( ) ( ) ,112
2
2
2Muy
NKyuK
xu
ReK
xp
yuvx
uutu −
∂∂+
∂∂++
∂∂++
∂∂−=
∂∂+
∂∂+
∂∂ (6)
( ) ( )2
2
2
2
2111
yv
ReK
xv
ReK
yp
yvvx
vutv
Re ∂∂++
∂∂++
∂∂−=⎟
⎠⎞⎜
⎝⎛
∂∂+
∂∂+
∂∂
yNvx
NutNvRe
MxN
ReK
∂∂+
∂∂+
∂∂−
∂∂−
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂+
∂∂
⎟⎠⎞⎜
⎝⎛ += 2
2
2
2121
yN
xN
ReK
,12 ⎟⎠⎞⎜
⎝⎛
∂∂−
∂∂+− x
vRey
uNK (7)
where micropolar, K, magnetic, M, and microrotation field, N, are dimensionless parameters, respectively.
Unsteady Boundary Layer Magnetohydrodynamics in Micropolar … 295
In order to solve (5), (6) and (7), we apply the same procedure of boundary layer approximation and the stream function, namely [5]:
( ) ( ),,,, yxyxxf θ=θ=ψ (8)
where stream function, ψ, is defined as:
., xvyu∂ψ∂−=
∂ψ∂= (9)
By substituting (8) and (9) into (5), (6) and (7), we obtain:
( ) ⎥⎦
⎤⎢⎣
⎡
∂∂+⎟
⎠⎞⎜
⎝⎛∂∂−+
∂∂+
∂∂+ 2
22
2
2
3
312
321
ηηλη
ηη
ffftffK
tftfMthK∂∂
∂=⎟⎠⎞⎜
⎝⎛
∂∂−+
∂∂+ ληη
21 (10)
and
⎟⎠⎞⎜
⎝⎛
∂∂−
∂∂++
∂∂+
∂∂
⎟⎠⎞⎜
⎝⎛ + ηηλη
ηη
fhhfthhhK23
21
221 2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂++
∂∂= 2
22
ηhhtKt
ht (11)
with respect to the following boundary conditions:
0:0 ==∂∂=< hfft η on the point of ( ),, ηx
,,0:0 2
2
η∂∂−==
η∂∂=≥ fnhfft on ,0=η
0,1 ==η∂∂ hf on .∞→η (12)
We further consider at the lower stagnation point of the solid sphere, when ,0≈x (12), (13) and (14) are reduced to the following ordinary differential equation [4]:
Basuki Widodo et al. 296
( ) [ ( ) ] ( ) tftfMthKffftffK∂′∂=′−+′+′′+′−+′′+′′′+ 112
321 2λη (13)
and
[ ] ( ).223
2221 fhtKthtfhhfthhnhK ′′++∂∂=′−′λ++′+′′⎟
⎠⎞⎜
⎝⎛ + (14)
III. Result and Discussion
Equations (13) and (14) are further solved numerically for some values of the micropolar parameter (K) and magnetic parameter (M) by using Keller-box method [4]. The variation on velocity and microrotation profile at the lower stagnation point, when 0=x with the value of micropolar 1=K and various values of magnetic parameter is illustrated in Figure 2 and Figure 3, respectively. Figure 2 shows that the velocity profile increases when the magnetic parameter increases. We obtain that beyond 3=t the velocity profiles get steady.
In Figure 3 is depicted the microrotation profiles of boundary layer flow in the magnetohydrodynamic micropolar fluid of 0=n at 0=x when
.1=K By increasing M leads to higher microrotation. However, for ,5.0>η the microrotation profiles decrease when the value of magnetic
parameter increases.
Figure 2. Velocity profiles for various value of magnetic parameter M at lower stagnation point 0=x when 1=K and .0=n
Unsteady Boundary Layer Magnetohydrodynamics in Micropolar … 297
Figure 3. Microrotation profiles for various value of M at lower stagnation point 0=x when 1=K and .0=n
Figure 4 depicts the velocity profiles of the boundary layer flow in the magnetohydrodynamic micropolar fluid at various K when 0=n and 1. It is obtained that the velocity profiles of micropolar fluid decrease when we increase the value of micropolar parameter K. It is also evident that the viscous fluid has higher velocity when to be compared towards micropolar fluid.
Figure 4. Velocity profile for various K at lower stagnation point ,0=x 1=M and .0=n
Figures 5 and 6 illustrate the relationship between and the skin friction of
boundary layer flow, ,21ReC f in the magnetohydrodynamic micropolar
fluid past a solid sphere. In Figure 5, the skin friction profile increases when
Basuki Widodo et al. 298
the value of magnetic parameter M increases for .0=n Further, Figure 6
shows skin friction coefficients 21ReC f of a micropolar fluid with ,1=M
0=n vary with x and K. We obtain that for the value of the micropolar parameter K increases then the skin friction profile increases.
Figure 5. Skin friction profile for various M at lower stagnation point ,0=x 1=K and .0=n
Figure 6. Skin friction profile for various K at lower stagnation point ,0=x 1=K and .0=n
IV. Conclusion
This paper considers the boundary layer flow in the magnetohydrodynamic (MHD) microfluid past a solid sphere. From the
Unsteady Boundary Layer Magnetohydrodynamics in Micropolar … 299
analysis and discussion of the result we conclude that the micropolar and magnetic parameter affect the velocity flow and the skin friction of micropolar fluid flow.
Acknowledgement
The authors thank the anonymous referees for their valuable suggestions which let to the improvement of the manuscript.
References
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[2] B. Widodo, Pemodelan Matematika, ITS Press, Surabaya, Indonesia, 2012.
[3] N. F. Mohammad, Unsteady Magnetohydrodynamic Convective Boundary Layer Flow Past a Sphere in Viscous and Micropolar Fluids, Universiti Technology Malaysia, Malaysia, 2014.
[4] B. Widodo, C. Imron, N. Asiyah, G. O. Siswono, T. Rahayuningsih and Purbandini, Viscoelastic fluid flow past a porous circular cylinder when the magnetic field included, Far East J. Math. Sci. (FJMS) 99(2) (2016), 173-186.
[5] A. R. M. Kasim, N. Muhammad and S. Shafie, MHD effect on convective boundary layer flow of a viscoelastic fluid embedded in porous medium with Newtonian heating, Recent Advances in Mathematics (2013), 182-189.
