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Magnetohydrodynamic Flow in Tubes of Arbitrary Contour

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The effect of the cross-sectional shape on the steady flow of a Newtonian fluid in long tubes under an electromagnetic force is explored. 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.
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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
=
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
3 Copyright © 2008 by ASME
( )
**
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
#$'(
%)* +,*
+, (19)
#%'+-* ./01
%)*'(*-./01
%)* +,*
+, (20)
Where
2%
345
6 (21)
And
2
+-+,
%7 8
79:
;97;< (22)
5 Copyright © 2008 by ASME
The expression for constant C2 is found by imposing the continuity condition in the function ƒ2 for r=1. This leads to equate the
numerator in (20) to zero, wherefrom C2 is determined. The final, continuous forms for ƒ0 and ƒ1, are
=$>
?@%,?@?@
%7?@8%AB"
7?@:%ABAB8"
;97;< (23)
=%>BC./0DE
?FG,?FGC?F H?F8IC
%J CD7 ?F:IC AB"
KLJ CD7 M?F8IC
 CDA%7 M?F:IC
%J CDA%7 M?FNIC%MB"
KLJ CDA%7M?F:IC%AB"
 7 CDA 7 M?FNIC%A B"
%J 7 CDA 7 M?F(,IC%A B"
KLJ 7 CDA 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
7 Copyright © 2008 by ASME
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.
9 Copyright © 2008 by ASME
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.
10 Copyright © 2008 by ASME
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.
11 Copyright © 2008 by ASME
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.
13 Copyright © 2008 by ASME
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