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1 Copyright © 2008 by ASME

Proceedings of IMECE2008

2008 ASME International Mechanical Engineering Congress and Exposition

October 31- November 06, 2008, Boston, Massachusetts

IMECE2008- 66629

MAGNETOHYDRODYNAMIC FLOW IN TUBES OF ARBITRARY CONTOUR

Mario F. Letelier Christopher P. Moraga

Juan S. Stockle

Dennis A. Siginer

Department of Mechanical Engineering

University of Santiago of Chile

mletelier@usach.cl; cmoraga@usach.cl

jstockle@usach.cl

Department of Mechanical Engineering

Wichita State University

dennis.siginer@wichita.edu

ABSTRACT

In this paper it is explored the effects of cross-section shape on the steady flow of a Newtonian fluid in long tubes affected by an

electromagnetic force. A general method of analysis is presented, which draws on previous results of the authors in the study of flow

in tubes of arbitrary contours.

The velocity field is determined and analyzed for a variety of cross-sectional shapes that include the triangle, square and others.

These results aim, in part, at providing a basis for the investigation of more complex flows, such as magnetorheological flows, in tubes

of non-circular shapes. Such flows are important in many applications as related to damping devices and others where controlled

modulation of resistance forces is desired.

INTRODUCTION

Electric and magnetic fields are presently used in many applications in which a fluid is a means for transmitting and regulating

forces or for transporting determined amounts of mass. In particular magnetorheological fluids (MRF) are increasingly being used

self-powered magnetorheological dampers [1], stay cable vibration control [2] and wheeled vehicle dampers [3]. In all cases, a

magnetic field is applied to a magnetic fluid so that its properties change according to the desired response to active loads. Such loads

can be very big, especially in the case of seismic-dampers, and also in heavy vehicles and aircraft landing gear [4].

Complementarily, other uses of magnetic fields relate to chemical and metallurgy industries [5], to flows in porous media [6],

and chaotic stirrers [7], among other. Also ferrofluids are used in inkjet printing, sink float separation, sealing, loud-speakers and

domain observation.

In many cases the fluid must flow through passages of shape different from circular. So far the analysis, as shown in the

technical literature, has been restricted to simple shapes.

2 Copyright © 2008 by ASME

In this paper it is analyzed the effect of cross-section geometry on the flow velocity for non-zero values of the Hartman number.

The method of analysis is an extension of a former method used by the authors for Newtonian and non-Newtonian pipe flows [8, 9,

10].

ANALYSIS

The equation of motion is

[

]

ρθ

ν

ρ

kBJw

rr

w

r

rrz

Pˆ

111

02

2

2

⋅×

−+

∂

∂

+

∂

∂

∂

∂

+

∂

∂

−= (1)

Where w = axial velocity, P = pressure, J= current density,

B

= magnetic field,

ρ

=density,

ν

= kinematic viscosity, and

r

= radial

coordinate.

The pressure gradient is a function of time defined, in general, as

φ

=

∂

∂

−

z

P (2)

The magnetic force becomes

[

]

2

0

ˆwBkBJ

σ

−=⋅× (3)

The following dimensionless terms are defined, viz.

0

*

w

w

w= (4)

a

r

r=

* (5)

φ

µ

φ

0

2

*

w

a

= (6)

µ

σ

22

02 aB

Ha = (7)

In this a = pipe radius,

µ

= dynamic viscosity, and Ha is the Hartmann number.

Equation (1) is conveniently rendered dimensionless as follows

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( )

**

2

2

*2

2

*

*

*

*

2

*

*2 11

φ

θ

=+

∂

∂

+

∂

∂

+

∂

∂

−wHa

w

r

r

w

r

r

w (8)

In the following two complementary albeit different analytical approaches are applied, in order to extend the study to a geater

manifold of tube shapes.

First approach

In this first approach, a perturbation of the tube contour is exploited. The resulting shapes in this case are not pre-determined, but they

come out as a direct result of the method.

Dropping asterics, the particular solution of (8) is

( )

⋅

−= )(

)(

1);(

0

0

2HaI

rHaI

Ha

Harw p

φ

(9)

where I0 is the modified Bersel fuction of order zero.

The homogeneous solution is found by separating variables, i.e.,

)()(),(

θθ

TrRrwh⋅= (10)

The solutions for R and T are found through equations (11) and (12), as follows.

0

2

2

2

=+

∂

∂

λ

θ

T

T (11)

( )

0

22

2

2

2

2=−−

∂

∂

+

∂

∂

λ

RrHaR

r

R

r

r

R

r (12)

The corresponding solutions, in terms of the separation constant λ, are

)cos(

λθ

CT

=

(13)

(14)

where C is an arbitrary constant

4 Copyright © 2008 by ASME

The homogeneous velocity is

(15)

Where K is an arbitrary constant

The full velocity is

(16)

The tube contour`s shapes are defined by putting w=o in (16).

Second approach

In the second approach, the velocity is given a prescribed for, i.e.,

!"#$ !#%&&& (17)

Where the first factor determines the tube Shapes when the boundary condition is imposed. Thus w (r,

θ

)=0 lead to

1-r2+

ε

r

λ

cos

λθ

=0 (18)

Expression (18) constitutes a “shape factor”, which has been exploited previously by the authors [8]. In (17) the functions ƒ0, ƒ1,…are

determined by substituting (17) in (16). The resulting expressions are

#$'(

%)* +,*

+, (19)

#%'+-* ./01

%)*'(*-./01

%)* +,*

+, (20)

Where

2%

345

6 (21)

And

2

+-+,

%7 8

79:

;97;< (22)

5 Copyright © 2008 by ASME

The expression for constant C2 is found by imposing the continuity condition in the function ƒ2 for r=1. This leads to equate the

numerator in (20) to zero, wherefrom C2 is determined. The final, continuous forms for ƒ0 and ƒ1, are

=$>

?@%,?@?@

%7?@8%AB"

7?@:%ABAB8"

;97;< (23)

=%>BC./0DE

?FG,?FGC?F H?F8IC

%J CD7 ?F:IC AB"

KLJ CD7 M?F8IC

CDA%7 M?F:IC

%J CDA%7 M?FNIC%MB"

KLJ CDA%7M?F:IC%AB"

7 CDA 7 M?FNIC%A B"

%J 7 CDA 7 M?F(,IC%A B"

KLJ 7 CDA 7 <O

(24)

RESULTS

In this paper are shown results for a variety of values of λ and ε. The contour and isovelocity plots are shown in the following

figures and the respective velocity profile for maximum and minimum contour radius and for three different Hartmann numbers.

The selected figures depict the base circular contour followed by shapes that approach the ellipse, equilateral triangle, square and

others.

These figures are grouped according to the respective analytical approach.

First approach

Fig.1 Contour and isovelocity curves for

6 Copyright © 2008 by ASME

Fig.2 Velocity profile for

Fig.3. Contour and isovelocity curves for P and &QP

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Fig.4 Velocity profile for máximum contour radius, &PPP, P &RP, Q &SS

Fig.5 Velocity profile for minimum contour radius, (1) &PPP, (2) &RP, (3) &SS

8 Copyright © 2008 by ASME

Second approach

Fig.6 Contour and isovelocity curves for λ=2 and ε=0.

Fig.7 Velocity profile for contour in fig.6.

Fig.8 Contour and isovelocity curves for λ=2 and ε=0,8.

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Fig.9 Velocity profile for maximum contour radius in fig.8.

Fig.10. Velocity profile for minimum contour radius in fig.8.

Fig.11 Contour and isovelocity curves for λ=3 and ε=0,37.

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Fig.12 Velocity profile for maximum contour radius in fig.11.

Fig.13 Velocity profile for minimum contour radius in fig.11.

Fig.14 Contour and isovelocity curves for λ=4 and ε=0,24.

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Fig.15Velocity profile for maximum contour radius in fig.14.

Fig.16 Velocity profile for minimum contour radius in fig.14.

All results here shown are analytically exact, since they come from exact solutions of the momentum equation. On the other hand, the

tube shapes that can be studied result by combining the parameters λ and ε. The parameter λ determines a family of continuous shapes

that start with a circle and can reach involved forms such as shown in figures 8, 11 and 14. These continuous shapes have a common

number of maximum and minimum contour radio, that are numerically equal to λ, which in this paper is always an integer number.

The evolution of the shapes is governed by the parameter ε, small variation of which leads to considerable shape deformation for a

given value of λ. The maximum allowable value of ε for a given family of shapes depends on λ.

The effect on the Hartmann number is always a velocity decrease. This effect is shown in the velocity profiles associated to each given

contour. The isolvelocity curves are very similar in form for different values of the Hartmann number, so that they are depicted only

once for a given shape.

CONCLUSIONS

The analytical model here presented for determining the velocity field in pipe flow of non-circular cross-section under the effect

of an electro-magnetic force yields results for a wide variety of pipe geometries.

The method determines an exact perturbation component to the velocity in circular pipes.

12 Copyright © 2008 by ASME

In this way the exact velocity field is found for many pipe contours, some of which may be of interest in technological

applications such as those cited in the Introduction.

In all cases it is found that the electro-magnetic force diminishes the velocity, according to the value of the Hartmann number.

The effect of the Hartmann number increases as the cross-section deviates from the circular shape.

One limiting aspect of the method is the difficulty that appears for generating sharp triangular and square shapes.

On the other hand, for the cases here analyzed, the cross-sectional shapes that can be obtained by changing the perturbation

parameter are different and more varied that in the case of Newtonian flow.

ACKNOWLEDGMENTS

The author gratefully acknowledges the financial support of DICYT- Chile, of the University of Santiago of Chile, and Wichita

State University.

REFERENCES

[1] Letelier M. F, Siginer D. A, Rouliez J.P and Corral O. F. (2005): Mathematical modelling of magneto-hydrodynamic dampers with

time-varying fluid properties, ASME International Mechanical Engineering Congress and Exposition, paper 81172.

[2]Lee D.Y and Wereley N. M. (1999): Quasi-steady Herschel-bulkley analysis of electro and magneto-rheological flow mode

dampers-journal of intell. Mar. Syst. And Struc. Vol.10, pp 761-769.

[3] Hong S. B., Choi S. B., Choi Y. T and Wereley N. M (2003): Non-dimensional analysis for effective design of semi-active

electrorheological damping control systems. - Proc. Instn. Mech. Engrs. Vol.217, pp. 1095-1106.

[4] Droguer U., Gordaninejad F. and Evrensen C. A. (2003): A new magneto- rheological fluid damper for high-mobility multi-

purpose wheeled vehicle (HMMMWV)- Smart Structure and materials 2003, Proceedings of SPIE, vol 5052. pp. 198-206.

[5]Liao Shi-Jun (2003): On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet 2003

Cambridge University Press, vol .488, pp.189-212.

[6]Geindreau.C and Auriault, J.L.(2002): Magnetohydrodynamic flows in porous media. 2002 Cambridge University Press, vol .466, pp.343-

363.

[7]Mingoiano Y.I, Shizhi. Qian and Haim H. Bau (2002): A magnetohydrodynamic chaotic stirrer. 2002 Cambridge University Press,

vol .468, pp.153-177.

[8]Letelier.M.F. D.S. Siginer, Cáceres. C., “Pulsating Flow of Viscoelastic Fluids in Straight Tubes of Arbitrary Cross-Section, Part I:

Longitudinal Field”. International Journal of Non-Linear Mechanical N°37, pp369-393, 2002.

[9]D.A.Siginer, Letelier, M.F., “Pulsating Flow of Viscoelastic Fluids in Straight Tubes of Arbitrary Cross-Section, Part II: Secondary

Flows”, International Journal of Non-Linear Mechanical N°37, pp395-407, 2002.

[10]Letelier. M.F., D.A. Siginer, “Secondary Flows of Viscoelastic Liquids in Straight Tubes”, International Journal of Solids an

Structures N°40, pp5081-5095, 2003.

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