Unsteady Magnetohydrodynamic Convective Boundary Layer...
Transcript of Unsteady Magnetohydrodynamic Convective Boundary Layer...
Unsteady MagnetohydrodynamicConvective Boundary Layer Flow past aSphere in Viscous and Micropolar Fluids
Nurul Farahain Mohammad
Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia
Name of supervisors:Assoc. Prof. Dr. Sharidan Shafie and Dr. Anati Ali.
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Outline of Presentation
Introduction
Problem 1 / Chapter 4
Problem 2 / Chapter 5
Problem 3 / Chapter 6
Problem 4 / Chapter 7
Problem 5 / Chapter 8
Conclusion
Suggestions for Future Works
Publication / Awards / Attended Conferences
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Introduction
Objectives
• to examine the effects of MHD on unsteady boundary layerflow over a sphere.
• to analyze the behaviour of viscous fluid and micropolarfluid under the influence of MHD.
• to investigate the interaction between MHD flow and heattransfer with or without buoyancy force.
3
Introduction
Scope
• electrically-conducting viscous and micropolar fluids• incompressible unsteady 2D laminar boundary layer flow
past a sphere• uniform magnetic field, transverse of the fluid flow• induced magnetic field is neglected• no polarized or applied voltage enforced on the fluid flow• numerical solution (Keller-Box method)• no real experiments conducted to validate the numerical
results
4
Research Methodology
Mathematical Analysis
• dimensionless variables• stream function• similarity transformation
Keller-Box Method
• finite difference method• Newton’s method• block-tridiagonal factorization scheme
5
Problems Solved
O
a
U∞
x/ayr(x)
x
Tw
T∞
T∞
g
Figure : Physical Coordinate
Viscous fluid Micropolar fluidsBoundary Layer Flow Prob. 1 / Chap. 4 Prob. 4 / Chap. 7Forced Convection Prob. 2 / Chap. 5 -Mixed Convection Prob. 3 / Chap. 6 Prob. 5 / Chap. 8
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Problem 1 / Chapter 4
∂(r (x) u
)∂x
+∂(r (x) v
)∂y
= 0, (1)
∂u∂t
+ u∂u∂x
+ v∂u∂y
= −1ρ
∂p∂x
+µ
ρ
(∂2u∂x2 +
∂2u∂y2
)− σB0
2
ρu, (2)
∂v∂t
+ u∂v∂x
+ v∂v∂y
= −1ρ
∂p∂y
+µ
ρ
(∂2v∂x2 +
∂2v∂y2
)− σB0
2
ρv , (3)
subject to the following initial and boundary conditions:
t < 0 : u = v = 0, for any x , y ,
t ≥ 0 : u = v = 0, at y = 0,u = ue(x), as y →∞.
(4)
7
Dimensional Governing Equations
Non-dimensional Governing Equations
Governing Equations in Stream Function
Non-similar Governing Equations
Dimensionless variables
Stream function
Similarity variables
8
Discretized Governing Equations
Linearized Numerical Scheme
Block Tridiagonal Factorization Scheme
Equations solved
ts, xs, f ′,Cf Re1/2,−h, s,NuRe−1/2
Finite Difference Method
Newton’s Method
LU factorization
Block Elimination Method
Analyse
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Problem 1 / Chapter 4
Table : The separation times of flow past the surface of a sphere.
x M = 0 M = 0 M = 0.1 M = 0.5 M = 1.0 M = 1.3(Ali, 2010) (present)
180◦ 0.3966 0.3960 0.4161 0.5241 0.7963 1.2470171◦ 0.4016 0.4010 0.4217 0.5331 0.8186 1.3103162◦ 0.4177 0.4170 0.4394 0.5623 0.8940 1.5677153◦ 0.4471 0.4463 0.4721 0.6178 1.0627 -144◦ 0.4947 0.4937 0.5257 0.7152 1.4428 -135◦ 0.5709 0.5694 0.6128 0.8937 - -126◦ 0.6987 0.6960 0.7632 1.2953 - -117◦ 0.9442 0.9372 1.0779 - - -108◦ - 1.6751 - - - -
Ali, A. (2010). Unsteady micropolar boundary layer flow and convective heattransfer. Universiti Teknologi Malaysia, Faculty of Science: PhD Thesis.
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Problem 1 / Chapter 4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
f’
M = 0, 0.1, 0.5, 1.0, 1.5
0 10 20 30 40 50 60 70 80 90−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
η
f’
M = 0, 0.1, 0.5, 1.0, 1.5
x = 0◦ x = 180◦
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Problem 1 / Chapter 4
0° 20° 40° 60° 80° 100° 120° 140° 160° 180°−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
x
Cf R
e1/2
t = 0.1, 0.5, 1.0, 1.5, 2.0
0° 20° 40° 60° 80° 100° 120° 140° 160° 180°0
0.5
1
1.5
2
2.5
3
3.5
x
Cf R
e1/2
= 0.1, 0.5, 1.0, 1.5, 2.0 t
Without MHD MHD
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Problem 2 / Chapter 5
∂(r (x) u
)∂x
+∂(r (x) v
)∂y
= 0, (5)
∂u∂t
+ u∂u∂x
+ v∂u∂y
= −1ρ
∂p∂x
+µ
ρ
(∂2u∂x2 +
∂2u∂y2
)− σB0
2
ρu, (6)
∂v∂t
+ u∂v∂x
+ v∂v∂y
= −1ρ
∂p∂y
+µ
ρ
(∂2v∂x2 +
∂2v∂y2
)− σB0
2
ρv , (7)
ρCp
(∂T∂t
+ u∂T∂x
+ v∂T∂y
)= c
(∂2T∂x2 +
∂2T∂y2
), (8)
subject to the following initial and boundary conditions:
t < 0 : u = v = 0,T = T∞ for any x , y ,
t ≥ 0 : u = v = 0,T = Tw at y = 0,
u = ue(x),T = T∞ as y →∞.
(9)
13
Problem 2 / Chapter 5
Ms NuRe−1/2
xCf(x = 0◦) (x = 180◦) (x = 0◦) (x = 180◦)↑ ↓ ↑ ↑ ↓ ↑
Pr ηs NuRe−1/2
↑ ↓ ↑
t NuRe−1/2
↑ ↓
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Problem 3 / Chapter 6∂ (r u)∂x
+∂ (r v)∂y
= 0, (10)
ρ
(∂u∂t
+ u∂u∂x
+ v∂u∂y
)= −∂p
∂x+ µ
(∂2u∂x2 +
∂2u∂y2
)− σB2
0u − ρgβ(T − T∞) sin(
xa
),
(11)
ρ
(∂v∂t
+ u∂v∂x
+ v∂v∂y
)= −∂p
∂y+ µ
(∂2v∂x2 +
∂2v∂y2
)− σB2
0v + ρgβ(T − T∞) cos(
xa
),
(12)
ρCp
(∂T∂t
+ u∂T∂x
+ v∂T∂y
)= c
(∂2T∂x2 +
∂2T∂y2
), (13)
subject to the following initial and boundary conditions:
t < 0 : u = v = 0,T = T∞ for any x , y ,
t ≥ 0 : u = v = 0,T = Tw at y = 0,
u = ue(x),T = T∞ as y →∞.
(14)
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Problem 3 / Chapter 6
M ts xs Cf Re1/2 NuRe−1/2
(x = 0◦) (x = 180◦)↑ ↑ ↓ ↑ ↑ ↓
α ts xs Cf Re1/2 NuRe−1/2
(x = 0◦) (x = 180◦)↑ ↑ ↓ ↑ ↑ ↓
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Problem 4 / Chapter 7∂(r (x) u
)∂x
+∂(r (x) v
)∂y
= 0, (15)
ρ
(∂u∂t
+ u∂u∂x
+ v∂u∂y
)= −∂p
∂x+(µ+κ)
(∂2u∂x2 +
∂2u∂y2
)+κ
∂N∂y−σB2
0u, (16)
ρ
(∂v∂t
+ u∂v∂x
+ v∂v∂y
)= −∂p
∂y+(µ+κ)
(∂2v∂x2 +
∂2v∂y2
)−κ∂N
∂x−σB2
0v , (17)
ρj(∂N∂t
+ u∂N∂x
+ v∂N∂y
)= γ
(∂2N∂x2 +
∂2N∂y2
)− κ
(2N +
∂u∂y− ∂v∂x
), (18)
subject to the following initial and boundary conditions:
t < 0 : u = v = N = 0, for any x , y ,
t ≥ 0 : u = v = 0,N = −n∂u∂y, at y = 0,
u = ue(x),N = 0, as y →∞.
(19)
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Problem 4 / Chapter 7
M ts xs f ′ Cf Re1/2 −h(x = 0◦) (x = 180◦)
↑ ↑ ↓ ↑ ↑ ↑ (η = 0) ↑ (η = 0, η = η∞)↓ (η →∞) ↓ (η →∞)
K ts f ′ −h↑ ↓ ↓ ↓
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Problem 5 / Chapter 8∂(r (x) u
)∂x
+∂(r (x) v
)∂y
= 0, (20)
ρ
(∂u∂t
+ u∂u∂x
+ v∂u∂y
)= −∂p
∂x+ (µ+ κ)
(∂2u∂x2 +
∂2u∂y2
)+ κ
∂N∂y− σB2
0u + ρgβ(
T − T∞)
sin x ,
(21)
ρ
(∂v∂t
+ u∂v∂x
+ v∂v∂y
)= −∂p
∂y+ (µ+ κ)
(∂2v∂x2 +
∂2v∂y2
)− κ∂N
∂x− σB2
0v − ρgβ(
T − T∞)
cos x ,
(22)
ρj(∂N∂t
+ u∂N∂x
+ v∂N∂y
)= γ
(∂2N∂x2 +
∂2N∂y2
)− κ
(2N +
∂u∂y− ∂v∂x
), (23)
ρCp
(∂T∂t
+ u∂T∂x
+ v∂T∂y
)= c
(∂2T∂x2 +
∂2T∂y2
), (24)
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Problem 5 / Chapter 8
subject to the following initial and boundary conditions:
t < 0 : u = v = N = 0,T = T∞ for any x , y ,
t ≥ 0 : u = v = 0,N = −n∂u∂y,T = Tw at y = 0,
u = ue(x),N = 0,T = T∞ as y →∞.
(25)
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Problem 5 / Chapter 8
M ts xs Cf Re1/2 NuRe−1/2
(x = 0◦) (x = 180◦)
↑ ↑ ↓ ↑ ↑ (Pr = 0.7) ↓↓ (Pr = 7)
K = 1,n = 0 α = 1 α = −1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.5
K = 1,n = 0.5 α = 1 α = −1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.7
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Conclusion• 5 different unsteady MHD models in viscous and
micropolar fluids over a sphere.• 3-dimensional numerical schemes.• MATLAB programmings.• Analyses of results obtained in MATLAB.• Results compared with published work are in good
agreement.• MHD is able to resolve issue involving separation of flow.• Given appropriate magnetic strength, separation of flow is
no longer detected.• MHD has potential to increase heat transfer at the surface
of sphere.• Opposing flow requires stronger magnetic field to
encounter separation of flow to be compared to assistingflow.
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Conclusion
M ts xs f ′ Cf Re1/2s NuRe−1/2
(x = 0◦) (x = 180◦) (x = 0◦) (x = 180◦)
↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑ ↓
M ts xs f ′ Cf Re1/2−h NuRe−1/2
(x = 0◦) (x = 180◦) (x = 0◦) (x = 180◦)
↑ ↑ ↓ ↑ ↑↑ (η = 0) ↑ (η = 0, η = η∞) ↑ (Pr = 0.7)
↓↓ (η → ∞) ↓ (η → ∞) ↓ (Pr = 7)
α = 1 α = −1
Pr = 0.7 lowest M highest M
Pr = 7 2nd lowest M 2nd highest M
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Suggestions for Future Works
• induced magnetic field.• electric field.• method to determine appropriate values of M.• other geometries: blunt bodies, circular cylinder, elliptic
cylinder.• other effects: Hall effect, internal heat generation,
Newtonian heating, heat flux, and much more.
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Publication / AwardsISI Indexed Publication :• N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2014). Separation times analysis of
unsteady magnetohydrodynamics mixed convective flow past a sphere. AIP ConferenceProceedings 1605: 349-354.
• N.F. Mohammad, M. Jamaludin, A. Ali, and S. Shafie. (2012). Separation Times Analysis ofUnsteady Boundary Layer Flow Past an Elliptic Cylinder Near Rear Stagnation Point. WorldApplied Sciences Journal 17 (Special Issue of Applied Math): 27-32.
• N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2012). Unsteady mixed convectionboundary layer flow past a sphere in a micropolar fluid. THE 5TH INTERNATIONALCONFERENCE ON RESEARCH AND EDUCATION IN MATHEMATICS: ICREM5. AIPConference Proceedings 1450: 211-217.
Awards :• Skim Saintis Cemerlang (Excellent Scientist Scheme), 2010-2014.
- Research Attachment as Visiting Researcher in Universidade de Coimbra, Portugal (Sept.2012 - May 2013).
• The Abdus Salam ICTP financial support by United Nation (UN) for participation in the 5thWomen in Mathematics Summer School on Mathematics Theories towards EnvironmentalModels at Trieste, Italy (27 May 2013 - 1 June 2013).
• Travel grant for Members of SEAMS for the Asian Mathematical Conference 2013.• Bronze Medal, 15th Industrial Art and Technology Exhibition (INATEX) 2013.
Invention: "Algorithm of the Boundary Layer Flow in Viscoelastic Fluid: Cylinder"• Gold Medal, Malaysia Technology Expo 2014.
Invention: "Algorithm of the Boundary Layer Flow in Viscoelastic Fluid (BLFV)"• Best Award, Malaysia Technology Expo 2014.
Invention: "Algorithm of the Boundary Layer Flow in Viscoelastic Fluid (BLFV)" 25
Attended ConferencesOral presentations:• N.F. Mohammad, A.R.M Kasim, A. Ali, and S. Shafie (2014). Separation Times Analysis of Unsteady
Magnetohydrodynamics Mixed Convective Flow past a Sphere. THE 21ST NATIONAL SYMPOSIUM ONMATHEMATICAL SCIENCES (SKSM21).• N.F. Mohammad, A.R.M Kasim, A. Ali, and S. Shafie (2013). Effect of MHD on Unsteady Boundary LayerFlow past a Sphere. The 3rd Annual International Conference Syiah Kuala University (AIC Unsyiah) 2013 inconjunction with The 2nd International Conference on Multidisciplinary Research (ICMR) 2013. October2-4, 2013, Banda Aceh, Indonesia.• N.F. Mohammad. (2013). Application of Finite Difference Time Domain Method. 3rd PortugueseBioengineering Meeting, 20-22 Feb 2013: Biomechanics and Computational Biology. Page 30. Organizer:University of Minho, Braga.• N.F. Mohammad, A.R.M. Kasim, A. Zaib, A. Ali, and S. Shafie (2012). Separation Time Analysis forUnsteady MHD Mixed Convection Boundary Layer Flow past a Circular Cylinder. 2nd InternationalConference on Mathematical Applications in Engineering (ICMAE2012).• N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2012). Unsteady mixed convection boundary layerflow past a sphere in a micropolar fluid. THE 5TH INTERNATIONAL CONFERENCE ON RESEARCH ANDEDUCATION IN MATHEMATICS: ICREM5.• N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2011). Formulation of UnsteadyMagnetohydrodynamic (MHD) Mixed Convection Flow past a Circular Cylinder in a Micropolar Fluid.NATIONAL SCIENCE POSTGRADUATE CONFERENCE 2011 (NSPC2011): CONFERENCEPROCEEDINGS: 172-181.• N.F. Mohammad, M. Jamaludin, A. Ali, and S. Shafie. (2011). Aliran Olakan Campuran Tak Mantapmelepasi Silinder Berkeratan Rentas Elips di sekitar Titik Genangan Belakang. Simposium KebangsaanSains Matematik ke 19, 9 -11 Nov 2011.
Poster presentation:• N.F. Mohammad, A.R.M Kasim, A. Ali, and S. Shafie (2013). Separation Times Analysis of Unsteady MHD
Forced Convection Flow past a Sphere. Asian Mathematical Conference 2013.26