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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.

1

Outline of Presentation

Introduction

Problem 1 / Chapter 4





Conclusion

Suggestions for Future Works

Publication / Awards / Attended Conferences

2

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

6


∂(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

9


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.

10


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◦

11


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

12


∂(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)


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


Ms NuRe−1/2

xCf(x = 0◦) (x = 180◦) (x = 0◦) (x = 180◦)↑ ↓ ↑ ↑ ↓ ↑

Pr ηs NuRe−1/2

↑ ↓ ↑

t NuRe−1/2

↑ ↓

14

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)


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)

15


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◦)↑ ↑ ↓ ↑ ↑ ↓

16

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)


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)

17


M ts xs f ′ Cf Re1/2 −h(x = 0◦) (x = 180◦)

↑ ↑ ↓ ↑ ↑ ↑ (η = 0) ↑ (η = 0, η = η∞)↓ (η →∞) ↓ (η →∞)

K ts f ′ −h↑ ↓ ↓ ↓

18

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)

19



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)

20


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

21

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.

22

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

23

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.

24

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