-
Electromagnetic drag on a magnetic dipole near a translating
conducting bar
Maksims Kirpo, Saskia Tympel, Thomas Boeck, Dmitry Krasnov, and
André Thessa)
Institute of Thermodynamics and Fluid Mechanics, Ilmenau
University of Technology, P.O. Box 100565,98684 Ilmenau,
Germany
(Received 27 January 2011; accepted 4 April 2011; published
online 13 June 2011)
The electromagnetic drag force and torque acting on a magnetic
dipole due to the translatory
motion of an electrically conducting bar with square cross
section and infinite length is computed
by numerical analysis for different orientations and locations
of the dipole. The study is motivated
by the novel techniques termed Lorentz force velocimetry and
Lorentz force eddy current testing
for noncontact measurements of the velocity of a conducting
liquid and for detection of defects in
the interior of solid bodies, respectively. The present,
simplified configuration provides and
explains important scaling laws and reference results that can
be used for verification of future
complete numerical simulations of more realistic problems and
complex geometries. The results of
computations are also compared with existing analytical
solutions for an infinite plate and with a
newly developed asymptotic theory for large distances between
the bar and the magnetic dipole.
We finally discuss the optimization problem of finding the
orientation of the dipole relative to the
bar that produces the maximum force in the direction of motion.
VC 2011 American Institute ofPhysics. [doi:10.1063/1.3587182]
I. INTRODUCTION
The present work is devoted to the theoretical investiga-
tion of a conceptually simple prototype problem for Lorentz
force velocimetry (LFV) and Lorentz force eddy current test-
ing (LET). LFV is a modern, contactless technique for meas-
uring flow rates and velocities of moving conducting liquids.
It
can be used in situations where mechanical contact of a
sensor
with the flowing medium must be avoided due to environmen-
tal conditions (high temperatures, radioactivity) and
chemical
reactions.1 Possible applications include flow measurement
during the continuous casting of steel, in ducts and open
chan-
nel flows of liquid aluminum alloys in aluminum production,2
and in other metallurgical processes where hot liquid metal
or
glass flows are involved. LET can serve as a basic tool for
detecting subsurface defects such as cracks in metallic con-
structions where these defects are critical for safety, e. g.
air
and railroad transport, engines, bridges, etc.
LFV is not the only one known technique for flow rate
measurements in opaque conducting liquids,3 however none of
these known techniques have found commercial realtime appli-
cation in metallurgy. Invasive probes, such as the Vives’
probes5 or mechanical reaction probes,4 are not very
suitable
for flow rate measurements at high temperatures because they
require direct contact between the sensor and the often
aggres-
sive liquid metal. Ultrasound sensors6 have similar
problems,
but can be used for hot melts with temperatures up to 800
�C.7
Commercially available electromagnetic flowmeters8,9 are
often not usable either, since heavy working conditions are
typ-
ical for metallurgical applications. Inductive flow
tomography10
can sometimes be used for reconstruction of the melt flow
structure in closed ducts. This technique, however, is too
complex to be applied for simple flow metering and requires
solution of inverse problems. So its adaption to industrial
proc-
esses seems to be overly complicated.
At the origin of LFV and LET is Lenz’ rule of magnetic
induction. Its application for flow rate measurement was al-
ready proposed by Shercliff.8 Eddy currents are induced in a
conductor, which is moving in an external (primary) mag-
netic field. The interaction of these eddy currents with the
primary magnetic field depicted in Fig. 1(a) creates a force
that opposes the motion according to Lenz’ rule. The mag-
netic system, which creates the primary magnetic field,
expe-
riences a drag force acting along the direction of the
conductor motion. Simple estimations show that this force is
F � rvB2, where r is the electrical conductivity of the mov-ing
conductor, v is the magnitude of the velocity and B is themagnitude
of the magnetic induction. Measuring this force,
acting on the magnetic system, allows us to measure the ve-
locity of the moving conductor with high accuracy.
Since the drag force is proportional to the square of the
magnetic induction, it is possible to improve the sensitivity
of
the measurement technique by increasing the magnetic field
intensity in the case when the velocity is independent of
the
magnetic field, i. e., for solid bodies and duct flows with
small
Hartmann number. The method, therefore, can be applied to
poorly conducting substances like electrolytes, ceramics or
glass melts. However, this still requires further research on
the
proper magnetic system design and optimization as well as an
accurate and advanced force measurement system.
In general, one cannot find an analytical solution for the
force acting on a realistic magnet system even when the
motion of the conducting body is very simple. Only a few
cases, which replace the real magnet system by a magnetic
dipole or a simple coil, are known to have analytical solu-
tions.11–15 However, these simplified problems are of great
importance for the theories of LFV because they allow a
deeper understanding of the involved processes and provide
reference data for complex numerical simulations.
a)Author to whom correspondence should be addressed. Electronic
mail:
[email protected].
0021-8979/2011/109(11)/113921/11/$30.00 VC 2011 American
Institute of Physics109, 113921-1
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http://dx.doi.org/10.1063/1.3587182http://dx.doi.org/10.1063/1.3587182
-
In the present paper, we consider a moving electrically
conducting bar with infinite length and square cross
section,
subjected to the field produced by a magnetic dipole. This
problem represents a canonical problem for LFV and LET
theory. It generalizes the case of an infinite conducting
plate
which can be treated analytically.13,15 Its solution can be
directly compared with results obtained from LVF and LET
applications for duct flows and solids without defects,
respectively.
The paper is organized as follows: The next section will
provide a brief theoretical description of the problem and
explain approaches used for the numerical solution; Sec. III
will give an overview and analysis of the obtained numerical
results and discuss the discovered dependencies in detail.
Section IV will introduce the asymptotic theory which
explains the behavior of the Lorentz force when the dipole
is
far away from the bar. Finally, conclusions and further
steps
of research will be discussed.
II. FORMULATION AND NUMERICAL SOLUTION OFTHE PROBLEM
We consider an electrically conducting nonmagnetic in-
finite solid bar with a square cross section d� d [Fig.
1(b)]which is moving with a constant velocity v ¼ v � ex in
xdirection in the field B of a magnetic dipole with themagnetic
dipole moment m ¼ mðkxex þ kyey þ kzezÞ andk2x þ k2y þ k2z ¼ 1. Our
ultimate goal is to compute the elec-tromagnetic drag force and
torque acting on the dipole for ar-
bitrary dipole location and orientation. The bar translating
in
the direction orthogonal to its cross section is of primary
in-
terest for basic LET configurations. Besides that, it can
also
be viewed as the first approximation of the mean velocity
distribution in a turbulent duct flow with the average
velocity
v under constant pressure gradient in x direction. The
latter,i.e., determination of the average duct flow velocity, is
the
main task of LFV. Hence, the solution of the moving bar
problem is aimed toward a better understanding of LFV prin-
ciples for fluid flows. With this particular motivation in
mind, we do not consider other possible translatory motions
of the bar in the present work.
The magnetic field of the dipole at the point r isgiven by16
BðrÞ ¼ l04p
3ðm � rÞ rr5�m
r3
h i(1)
assuming that the origin of the coordinate system corre-
sponds to the dipole location.
Eddy currents are induced in the bar when it crosses the
magnetic field lines. They create a secondary magnetic field
bwhich in this work is assumed to be just a very small
perturba-
tion of the external field. This is satisfied in the
quasistatic
approximation,17 i.e., when the magnetic diffusion time is
small compared to the advection time L/v by the velocity v.The
length scale L corresponds to the variation scale of theexternal
field, i. e., distance between the dipole and the bar h.The
estimation of the magnetic diffusion time is less obvious
but its upper limit should be based on the same length scale
L¼ h.18 The assumptions of the quasistatic approximationsare
then satisfied when the magnetic Reynolds number
Rm ¼ l0rvL is less than unity, where r denotes the
electricconductivity. Additionally we consider the so-called
kinematic
problem where the motion of the bar (velocity v) is
prescribed.For further formulation of the problem and
presentation
of the numerical results we will use nondimensional units
based on the characteristic length L0 ¼ d, characteristic
ve-locity equal to the bar velocity V0 ¼ v and the
characteristicmagnetic field intensity B0 ¼ l0md�3. This choice of
thecharacteristic parameters leads to the following expressions
for the current density j ¼ rvl0md�3j� and Lorentz forceF ¼
rvl20m2d�3F� where “star” symbol represents nondi-mensional
quantities. This presentation of the force F allowsus to express F�
as a function of the dipole orientationsk ¼ m� ¼ ðkx; ky; kzÞ,
distance between the dipole and thebar h� ¼ h=d and the dipole
displacement in y directionF� ¼ F�ðkx; ky; kz; h�; y�Þ. Similarly,
the torque is defined asT ¼ rvl20m2d�2T� and T� ¼ T�ðkx; ky; kz;
h�; y�Þ. From nowonly nondimensional variables will be used and the
“star’’
symbol will be omitted below.
In the quasistatic approximation, the electric field can be
represented as the gradient of the electrical potential /.
Theinduced current density can be expressed by Ohm’s law for a
moving conductor as:
j ¼ �r/þ v� B: (2)
The moving bar is electrically neutral and according to the
conservation of electric charge, the induced currents should
be divergence-free. Hence, the Poisson equation for electri-
cal potential can be obtained taking the divergence of Ohm’s
law (2). Since the magnetic field of the dipole (1) is
irrota-
tional (r� B ¼ 0) and the velocity distribution is uniform,the
electrical potential satisfies the Laplace equation:
r2/ ¼ 0: (3)
A solution of this equation is required to obtain the eddy
cur-
rents using Eq. (2). Appropriate boundary conditions (BC)
require zero normal currents on all side surfaces of the bar,
i.
e., j � n ¼ 0. We also require that the electrical potential
andcurrents should vanish at the remote ends, i. e., at x! 61.
The braking Lorentz force and torque acting on the bar
are given by the volume integrals
FIG. 1. Sketch of the studied problem (left) and parameters of
the geometry
(right).
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Fbar ¼ð
V
j� B dV and
Tbar ¼ð
V
r� ðj� BÞdV: (4)
By virtue of Newton’s third law, an opposite force and tor-
que act on the dipole. This force and torque on the dipole
are
of equal magnitude as Fbar and Tbar. They are given by
theformulas F ¼ ðk�rÞb and T¼ k�b. The secondary mag-netic field b
in these expressions is produced by the inducedcurrents in the
moving bar. It can be computed at the loca-
tion of the dipole using the Biot-Savart law.
Our attempt to find a general analytical solution of the
Laplace equation with the described BC was not successful.
For this reason an automated MatlabTM
script coupled with the
ComsolTM
FEM Laplace “pardiso” solver19 was used to solve
the problem numerically for the electrical potential using
sec-
ond-order Lagrangian elements.20 The elementary force and
torque from every current carrying element of the bar were
evaluated and the total force and torque values were
obtained
taking the volume integral. All integration procedures were
implemented using built-in ComsolTM
functions.
Preliminary computations showed that the accurate solu-
tion of the problem requires a very fine grid in the zone of
large magnetic field gradients if the dipole is very close
to
the bar, i. e., h� 1 and hmin ¼ 0:01. Therefore, a refinedgrid
was used for simulations as shown in Fig. 2. We have
verified by a grid convergence study that computational ac-
curacy is within 5% if the distance between the dipole and
the top surface of the moving bar h equals the doubled
char-acteristic size of the element. The computational grid was
further refined for very small h 8� 10�2. The maximalnumber of
elements in the grid was around 105.
A second numerical approach has been used to verify
the asymptotic theory presented later in section IV for
large distances between the dipole and the bar. This
approach is based on an in-house finite-difference code
for the direct numerical simulation of turbulent magneto-
hydrodynamic flows.21 The numerical scheme is of second
order with collocated grid arrangement. It has very good
conservation properties for mass, momentum and electric
charge thanks to the particular formulations of the discrete
equations proposed in references.22,23 The Poisson equa-
tion is solved with the Poisson solver FishPack. Verifica-
tions of this code versus a spectral code and details on the
algorithms can be found in Krasnov et al.21 The codecomputes
force and torque by the volume integrals (4)
with the trapezoidal rule.
This in-house code was adapted to simulate a solid bar
because it can use much larger structured grids than Comsol,
which is important for h 1 cases. To resolve all the effectsin
the case of large distances h the length of the solid barwas chosen
to be proportional to the distance to the dipole, i.
e., it was 7.5ph. The code used periodic boundary conditionsat
x¼6 7.5ph/2 for electrical potential. They did not influ-ence the
obtained solution due to sufficient length of the
computational domain. The numerical resolution of the uni-
form mesh was 8192� 256� 256 points in x, y, z in all
simu-lations for h 1.
III. NUMERICAL RESULTS
A. Magnetic dipole in the plane y 5 0
We begin the discussion of the numerical results for the
solid bar with a reference case for which an analytical
result
is available. This reference case is a translating infinite
plate
of unit thickness, whereas the dipole is vertically
oriented.13
The analytical formulas for the x component of the force andthe
y component of the torque are
F0x ¼1
128ph3� 1� h
3
ð1þ hÞ3
" #; (5)
T0y ¼ �1
128ph2� 1� h
2
ð1þ hÞ2
" #: (6)
For this case, the components of the magnetic moment are
k¼ (0, 0, 1) in our nondimensional representation, and wecan
compare our simulation results directly by evaluating the
ratios Fx=F0x and Ty=T
0y for different h, as shown in Fig. 3.
As expected, the force and torque ratios decrease monot-
onously as h increases. When h tends to zero, i. e., the
mag-netic dipole approaches the top surface of the bar, the
curves
reach unity. This behavior agrees with our expectations
FIG. 2. Examples of the refined grid used for numerical
simulations with
Comsol (the full bar size is 7:5� 1� 1) (a) is in a plane z =
const. and (b) ina plane x = const.
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because the rectangular bar acts on the dipole exactly as
the
infinite plate if the dipole is very close to its surface. If
the
dipole is moved to a new position away from the bar, then
the force ratio starts to decay until it reaches very small
val-
ues. This is because the “useful” volume, which the magnetic
dipole interacts with, decreases, too. This is, clearly, a
dis-
tinct feature of the rectangular bar compared to the
infinite
plate. The data for h> 2 indicate a transition to
scalingbehavior for large distances, and the ratios Fx=F
0x and Ty=T
0y
are related by a constant factor in this range. It can also
be
noticed that the two curves start to decay at somewhat
differ-
ent values of h with the torque ratio Ty=T0y decaying
earlier.
The critical distance when the influence of the translating
bar
can be approximated by the infinite plate Eqs. (6) and (7)
within 1% error is hC 0:1.We consider now the influence of
dipole orientation. As
the first step, we focus on the cases when the dipole is
aligned with one of the coordinate axes (main orientations),
i. e. (1, 0, 0), (0, 1, 0), and (0, 0, 1). These results
(integral
force and torque) are shown in Figs. 4 and 5. They are tabu-
lated also for other selected ðkx; ky; kzÞ in Tables I and II.
Itcan be seen that the other orientations of the magnetic
dipole
provide smaller values of the Lorentz forces compared with
the vertically oriented dipole if it is placed in the lateral
mid-
plane of the bar y¼ 0. For example, the Lorentz force on adipole
oriented in y direction is approximately equal to 25%of the force
for a vertically oriented dipole. One might guess
that this is due to the lower external field intensity at
the
base point on the bar beneath the dipole, which is indeed
twice lower for the y orientation (0, 1, 0) than for the z
orien-tation. However, this argument seems misleading because
the same reduction in the field strength for the x
orientationreduces the force only to about 75% of the value
obtained
for vertically oriented dipole. The spatial organization of
the
induced currents is therefore decisive for the actual force.
We illustrate the influence of the dipole orientation on the
current density distribution on the surface of the bar in
Fig.
6. It can be noticed that the current density magnitude for
(1,
0, 0)-oriented dipole is higher than for (0, 1, 0)-oriented
dipole and the current density maximum is better localized
in the region of the highest magnetic field intensity for
the
dipole with (1, 0, 0)-orientation. An analytical solution by
Priede et al.15 for Lorentz force in a layer of infinite
horizon-tal extent and arbitrary depth produced by a dipole of
FIG. 3. Ratios of nondimensional forces Fx=F0x and torques
T=T
0y acting on
a magnetic dipole located at x¼ 0, y¼ 0 for different h and
oriented perpen-dicular to the closest surface, i. e., k¼ (0, 0,
1).
FIG. 4. Nondimensional force Fx acting on the magnetic dipole
located atx¼ 0, y¼ 0 for different h and dipole orientations.
FIG. 5. Absolute values of nondimensional torque Ty acting on
the magneticdipole with any k ¼ ðkx; 0; kzÞ, k2x þ k2z ¼ 1 located
at x¼ 0, y¼ 0 for differ-ent h.
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arbitrary orientation shows the same Lorentz force depend-
ence on dipole orientation.24
Different orientations of the dipole can break the symme-
try in current density distributions as shown in Figs. 6(d)
and
6(e). In such cases not only the main Fx force componentappears.
The simulations have shown, that for ð
ffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
; 0Þthe force component Fy is present as well. Although it has
asmaller absolute value than Fx, the order of magnitude is thesame.
If the dipole is oriented as ð
ffiffiffiffiffiffiffi0:5p
; 0;ffiffiffiffiffiffiffi0:5p
Þ, then theforce component Fy vanishes. Instead a smaller
attractive forceFz between the dipole and the bar appears. It is at
least oneorder of magnitude smaller than Fx.
It is also interesting to note that the maximal values of
the torque component Ty are obtained when the magneticdipole has
ky ¼ 0, i. e., it is oriented in xz-plane only as it isshown in
Table II. Moreover, these Ty values are equalwithin the accuracy of
numerical results for different dipole
orientations with ky ¼ 0 and k2x þ k2z ¼ 1 in the xz-plane.
Thetranslational motion of the conducting bar tries to rotate
the
dipole located in xz-plane around the y axis with a
constanttorque for given h. This result can be used for rotary
flowmeters where flow rate is determined by measuring the rota-
tion frequency of a freely rotating magnet placed near the
channel. A constant torque has also been noted by Priede
et al.,14 where an analytical solution for the angular
velocityof a long rotating cylindrical magnet above a translating
con-
ducting layer is obtained in two-dimensional approximation.
A partial explanation of these results follows simply
from inspection of the formula T¼ k�b for the torque onthe
dipole. We can see that Ty ¼ kzbx � kxbz vanishes if thedipole is
oriented in y direction, i. e., k¼ (0, 1, 0). However,the fact that
the torque stays the same for any orientation
with ky ¼ 0 and k2x þ k2z ¼ 1 is not obvious. Equation
(4)gives
Ty ¼ð
V
ðrzfx � rxfzÞdV; (7)
where r is taken in the coordinate system whose origin
corre-sponds to the dipole. The integrated torque Ty thus
containscontributions from spatial distributions of the Lorentz
force
densities fx and fz in the bar. Figure 7 shows the force
densityintegrated in every cross section
vi ¼ð ð
fi dy dz; i ¼ x; y; z (8)
depending on the coordinate x for the magnetic dipole withk ¼
ð1; 0; 0Þ. It can be clearly seen that the integrals vx andvz are
approximately of same order of magnitude and the in-tegral vy
vanishes. The total force in vertical direction
Fz ¼ð
vzdx (9)
vanishes because vz is an antisymmetric function. But theproduct
rxfz always has a positive value which contributes tothe torque Ty.
Both products rzfx and rxfz balance in a waythat the integrated
torque Ty remains constant for any dipoleorientation with ðkx; 0;
kzÞ, k2x þ k2z ¼ 1. The same applies forthe infinite plate as shown
by Priede et al.15 Their analyticalresult for Ty depends on the sum
ðk2x þ k2z Þ only.
After examination of the orientation, we finally com-
ment on the asymptotic behavior of force and torque with the
distance h. The double logarithmic representations in Figs. 4and
5 reveal different power law approximations for differ-
ent distances h. The power law fitting provides Fx � h�3,Ty �
h�2 for small h and Fx � h�7, Ty � h�6 for large h. Itcan be
noticed that these estimations exactly correspond to
Eqs. (5) and (6) for the infinite plate in small h region.
TABLE I. Values of the nondimensional force Fx acting on the
magnetic dipole located at x¼ 0, y¼ 0 for selected h and dipole
orientations.
h (0, 0, 1) (0, 1, 0) (1, 0, 0) ð0;ffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
Þ ðffiffiffiffiffiffiffi0:5p
; 0;ffiffiffiffiffiffiffi0:5p
Þ ðffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
; 0Þ ðffiffiffiffiffiffiffiffi1=3
p;ffiffiffiffiffiffiffiffi1=3
p;ffiffiffiffiffiffiffiffi1=3
pÞ
0.02 3:11� 102 7:83� 101 2:34� 102 1:95� 102 2:73� 102 1:56� 102
2:08� 1020.10 2:46� 100 5:98� 10�1 1:85� 100 1:53� 100 2:16� 100
1:22� 100 1:64� 1000.20 2:80� 10�1 5:75� 10�2 2:02� 10�1 1:69� 10�1
2:41� 10�1 1:30� 10�1 1:80� 10�10.50 1:01� 10�2 1:20� 10�3 6:56�
10�3 5:63� 10�3 8:31� 10�3 3:88� 10�3 5:94� 10�31.00 4:12� 10�4
3:91� 10�5 2:59� 10�4 2:26� 10�4 3:35� 10�4 1:49� 10�4 2:37�
10�42.00 9:53� 10�6 9:64� 10�7 5:99� 10�6 5:25� 10�6 7:76� 10�6
3:48� 10�6 5:50� 10�65.00 3:51� 10�8 3:83� 10�9 2:22� 10�8 1:95�
10�8 2:87� 10�8 1:30� 10�8 2:04� 10�810.0 3:74� 10�10 4:15� 10�11
2:37� 10�10 2:08� 10�10 3:05� 10�10 1:39� 10�10 2:17� 10�10
TABLE II. Values of the nondimensional torque Ty acting on the
magnetic dipole located at x¼ 0, y¼ 0 for selected h and dipole
orientations.
h (0, 0, 1) (0, 1, 0) (1, 0, 0) ð0;ffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
Þ ðffiffiffiffiffiffiffi0:5p
; 0;ffiffiffiffiffiffiffi0:5p
Þ ðffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
; 0Þ ðffiffiffiffiffiffiffiffi1=3
p;ffiffiffiffiffiffiffiffi1=3
p;ffiffiffiffiffiffiffiffi1=3
pÞ
0.02 �6:22� 100 0.0 �6:20� 100 �3:11� 100 �6:21� 100 �3:10� 100
�4:14� 1000.10 �2:38� 10�1 0.0 �2:30� 10�1 �1:19� 10�1 �2:34� 10�1
�1:15� 10�1 �1:56� 10�10.20 �4:52� 10�2 0.0 �4:62� 10�2 �2:26� 10�2
�4:57� 10�2 �2:31� 10�2 �3:05� 10�20.50 �2:99� 10�3 0.0 �3:01� 10�3
�1:50� 10�3 �3:00� 10�3 �1:50� 10�3 �2:00� 10�31.00 �2:01� 10�4 0.0
�2:01� 10�4 �1:00� 10�4 �2:01� 10�4 �1:01� 10�4 �1:34� 10�42.00
�8:10� 10�6 0.0 �8:11� 10�6 �4:05� 10�6 �8:11� 10�6 �4:06� 10�6
�5:41� 10�65.00 �6:70� 10�8 0.0 �6:71� 10�8 �3:35� 10�8 �6:71� 10�8
�3:36� 10�8 �4:47� 10�810.0 �1:37� 10�9 0.0 �1:37� 10�9 �6:84�
10�10 �1:37� 10�9 �6:87� 10�10 �9:13� 10�10
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However, the Lorentz force and torque decay faster for mov-
ing solid bar than for infinite plate for large h and theFx �
h�7 dependence for large h is not so obvious.
B. Magnetic dipole in cross-sectional plane x 5 0
Since the conducting bar has infinite length, the results
are independent of the positions of the dipole along the
length of the bar. However, the Lorentz force Fx shouldchange if
the dipole is shifted from the symmetry plane
y¼ 0. The presented numerical approach allows us to studythis
dependence of Fx on the coordinates y and z of thedipole. Pairs of
y> 0 and z> 0 values are selected in thex¼ 0 plane, and the
Lorentz force is computed for threemain orientations of the dipole.
The decay of the Lorentz
force is fast and the distribution of the decimal logarithm
of
the force magnitude allows better resolution of the Fxðy;
zÞdistributions. It can be seen that for the dipole with k¼ (1,
0,0) the distribution of Fxðy; zÞ is completely symmetric aboutthe
diagonal y¼ z as shown in Fig. 8. For the dipole withk¼ (0, 0, 1)
the force distribution Fxðy; zÞ keeps the symme-try with respect to
the y¼ 0 plane only as it is shown onFig. 9. These distributions of
Fxðy; zÞ are compatible with thepreviously described results: the
Lorentz force has the small-
est value if the dipole is oriented in y direction, i. e., the
iso-lines of the same force magnitude are closer to the surface
of
the bar. The force distribution Fxðy; zÞ for the dipole withk¼
(0, 0, 1) also represents the results for the dipole withk¼ (0, 1,
0) upon reflection around the diagonal.
The nondimensional Lorentz force depends on position
of the dipole and on its orientation. It is therefore natural
to
inquire about the optimal orientation of the dipole for the
given positions ðy; hÞ, which gives the largest force compo-nent
Fx. In the present mathematical model, the induced cur-rents are a
linear functional of the applied magnetic field,
and the induced magnetic field is neglected in the Lorentz
force. For these reasons, the Lorentz force depends
quadrati-
cally on the applied field. The integrated Lorentz force
com-
ponents are therefore quadratic forms of the dipole
orientation vector ðkx; ky; kzÞ. In particular,
Fx ¼ kTi Aijkj; (10)
where summation on the repeated indices is understood.
If the eigenvalue problem Aa¼ ka is solved,25 then
theeigenvector a which corresponds to the largest eigenvaluemax(k)
of the matrix Aij provides orientation of the dipolewith the
largest force component Fx. The matrix Aij is sym-metric and its
six independent elements can be computed
using the data from force computations for six different
dipole
orientations and solving the obtained linear equation
system.
We have verified that the dipole orientation ðkx;ky; kzÞmax,
which provides the maximum force Fx, has kx ¼ 0within the accuracy
of numerical results. For different ðy; hÞpairs the components ky
and kz, providing the maximum Lor-entz force, are different,
requiring tilting of the dipole by a
certain angle h between the z axis and direction of the
dipolewhich lies in x¼ 0 plane. This definition of h is sketched
inFig. 10.
FIG. 6. Contours of the nondimensional current density
magnitudes and current density vectors on the top surface (z¼ 1/2)
of the bar for h¼ 0.2 and differentorientations of the magnetic
dipole. Only the central part of the surface is shown. The
different dipole orientations are (1,0,0) for case (a), (0,1,0) for
case (b),
(0,0,1) for case (c), (1,1,0)/ffiffiffi2p
for case (d), (1,0,1)/ffiffiffi2p
for case (e), and (0,1,1)/ffiffiffi2p
for case (f).
113921-6 Kirpo et al. J. Appl. Phys. 109, 113921 (2011)
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The computed eigenvectors with orientations ð0; ky;kzÞmax are
shown in Fig. 11 where every arrow representsdirection of the
optimal orientation for the dipole located at
certain y and h. The angle h between z axis and direction ofthe
eigenvector is shown in Fig. 12 for all studied ðy; hÞ val-ues and
in Fig. 13 for two selected distances h. The angle hvaries from
almost zero for vertical dipole orientation k¼ (0,0, 1) at y¼ 0 and
any h to more than 50 � for large y andsmall h. If the dipole is
placed very close to the surface ofthe bar (h! 0) then the angle h
remains small with increas-ing y until the edge of the bar, i. e.,
while y< 0.5, andchanges rapidly to 50 � at y> 0.5 as it is
shown in Fig. 13. Ifthe dipole is located at larger h, then the
angle h increasesmonotonously with y. It can be seen that if y or h
becomeslarge, i. e., the dipole is placed very far away from the
bar,
then its orientation providing the maximum Lorentz force
points to the location of the bar.
Several numerical simulations were performed for the
selected pairs of points (y, h) to verify that the
obtaineddipole orientation provides the largest Lorentz force Fx.
Theresults of force comparison are combined into Table III and
clearly show that the Lorentz force for the optimal orienta-
tion of the dipole is always a little greater than the force
computed for other selected direction of the dipoles in this
point, e. g. the difference is almost 12% at point (y,h)¼ (0.4,
0.4), which can be significant for LFV applicationfor very small
velocity measurements or cases with small
electrical conductivity. However, in LFV applications it
could be difficult to realize different orientations of
magnet-
ization for the real magnets due to their finite size.
IV. ASYMPTOTIC THEORY FOR LARGE h
The decay of the Lorentz force with the power h�7 atlarge
distances is more rapid than one would expect from a
simple estimate. The straightforward estimation would
involve Fx � B20V based on a characteristic value B0 of
themagnetic field and an effective volume V of the bar affectedby
the magnetic field. With the dipole field decaying accord-
ing to B0 � h�3 and V ¼ hd2 one would obtain Fx � h�5.The decay
with h�7 is therefore not obvious. It can be
explained with the help of asymptotic expansions in the
small parameter e:¼ 1/h. Alternatively, the asymptoticapproach
can be regarded as a long-wave expansion along
the length of the bar.
The goal of our asymptotic approach is to estimate the
Lorenz force Fx acting on the dipole for h 1. We assumethat the
dipole is placed in the symmetry plane y¼ 0 butallow for arbitrary
dipole orientation.
The asymptotic solution is based on the rescaled coordi-
nates x ¼ hx̂, y ¼ ŷ, z ¼ ẑ, whereby one can exploit the
slow
FIG. 7. Force density integrals vi ¼ÐÐ
fi dy dz, i¼ x, y, z depending on thecoordinate x for the dipole
with k¼ (1, 0, 0) and h¼ 0.2.
FIG. 8. Decimal logarithm of the force magnitude distribution
for nondi-
mensional force Fxðy; zÞ, k¼ (1, 0, 0). The gap in the isoline
closest to thebar is due to computational reasons.
FIG. 9. Decimal logarithm of the force magnitude distribution
for nondi-
mensional force Fxðy; zÞ, k¼ (0, 0, 1). The gap in the isoline
closest to thebar is due to computational reasons.
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variation in x when the parameter e tends to zero. The
quanti-ties of interest (i. e., B, /, j, F) are then represented as
regu-lar perturbation expansions in the small parameter e, e.
g.,for the magnetic field:
Bðx̂; ŷ; ẑÞ ¼ B0ðx̂Þ þ eB1ðx̂; ŷ; ẑÞ þ e2B2ðx̂; ŷ; ẑÞ þ…:
(11)
The superscripts denote the order of approximation for every
term. The expressions for B0 are given in Appendix A.
Thevelocity field is constant and therefore independent of e.
We would like to limit ourselves by three leading terms
of the Lorentz force Fx series expansion:
F0x ¼ð
j0 � B0� �
xdV; (12)
F1x ¼ð
j1 � B0 þ j0 � B1� �
xdV; (13)
F2x ¼ð
j2 � B0 þ j1 � B1 þ j0 � B2� �
xdV: (14)
Then the evaluation of Fx requires computation of these
sixintegrals. Each of them and details of the calculation are
con-
sidered in the appendix. We only present the key steps of
the
procedure at this point.
The Laplace equation for the leading term /0 of theelectrical
potential is easily solved by /0 ¼ zB0y � yB0zþconst. Therefore j0
¼ 0 and all integrals containing the cur-rent j0 vanish. The
first-order term j1 of the current does notvanish. We can determine
its components in the yz plane bya stream function representation.
There is no contribution to
the Lorentz force from such a planar current distribution
interacting with a field B0 that is constant on each yz plane,i.
e.
Ððj1 � B0ÞxdV ¼ 0. However, there is a contribution
from the interaction with B1:
e2ð
j1 � B1� �
xdV ¼ � 15
2 � 2:253220ph7
ð5k2x þ 7k2z Þ: (15)
FIG. 10. Definition of the angle h between the z axis and
direction of thedipole.
FIG. 11. The eigenvectors show dipole orientations which provide
the high-
est Lorentz force for different (y, h) pairs.
FIG. 12. Angle h in degrees between the z axis and direction of
theeigenvector.
FIG. 13. Angle h in degrees between the z axis and direction of
the eigen-vector for two selected values of the distance h.
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For the last remaining integral we use the continuity
equation
and the result for /0 to get
e2ð
j2 � B0� �
xdV ¼ � 15
216ph7ð35k2x þ 8k2y þ 57k2z Þ: (16)
By combining all evaluated integrals we see that the leading
term of the Lorentz force is given by
Fx ¼ �15
216ph745:561k2x þ 8k2y þ 71:785k2z� �
: (17)
We have thereby demonstrated that Fx � h�7 when h 1.The
dependence of the force on the orientation differs from
the limit h! 0 as can be seen from the coefficients multiply-ing
the components ki. In particular, the y orientationbecomes even
less effective than in the case h! 0.
The in-house finite differences code described in Sec. II
is capable of calculating the total Lorentz force for large
dis-
tances exceeding h > 103, while the commercial code usedin
Sec. III can resolve the effects at small distances. In the
region where both codes could be used (h between 2 and 80)the
largest relative error between the results obtained with
the two different codes was not greater than 2%.
The asymptotic theory agrees with the values obtained
by in-house solver for large h as it is shown in Fig. 14.
Theobserved differences are less than 2% for all orientations
of
the dipole.
V. CONCLUSIONS
The electromagnetic drag force and torque acting on a
magnetic dipole due to the motion of an electrically
conducting
bar with square cross section and infinite length have been
computed for different orientations and locations of the
dipole.
The results show that the largest magnitude of the Lorentz
force can be obtained for the magnetic dipole oriented in
the
vertical direction and located in the symmetry plane y¼ 0 ofthe
bar. The force dependence on the distance h between thedipole and
the bar is governed by power laws when the dis-
tance h is either small or large relative to the width of the
bar.For small distances, the power law is identical to the case of
an
infinite plate. At large distance, the power law for the bar
shows a more rapid decay, which is proportional to h�7, thanfor
the infinite plate. The asymptotic theory that explains the
slope of the force decay for large h 1 was developed too.The
force magnitude for really large distance h is very
small and the obtained analytical result cannot find a
practical
application for flow rate measurements in liquid metal
flows.
From the other side, there is a substantial demand for LFV
application for electrolyte flows. In these applications the
electrical conductivity is several orders of magnitude
smaller
r 10� 100 S/m and, therefore, measured Lorentz force isin a
range of 10�5 N. The developed asymptotic theory can beused to
build LFV prototypes for low-conducting liquids. In
particular, to investigate them within the model environment
where LFV device is located at some distance away from the
experimental liquid-metal channel.
The slope of the Lorentz force decay for h 1, whichis
proportional to h�7, is expected to be present in other simi-lar
problems involving laminar or turbulent flows in ducts or
pipes. Derivation of the asymptotic theory for the transla-
tional motion of a solid cylindrical body of round cross
sec-
tion is straightforward and gives a different prefactor but
almost the same dependencies on k.It also was found that the
optimal orientation of the
dipole that produces the maximum Lorentz force strongly
depends on its position (y, h).The torque dependence on the
distance h is also given by
power laws. The torque found to be constant within the accu-
racy of numerical results for y¼ 0, ky ¼ 0 and k2x þ k2z ¼ 1.The
discussed results allow us to evaluate the Lorentz
force magnitude and torque for specified h and m. If we takean
aluminum bar with v¼ 1 m/s, d ¼ 5� 10�2 m,r ¼ 3:54� 107 (X m)�1, h
¼ 10�2 m, and dipole momentm¼ (0, 0, 1) Am2, then the corresponding
nondimensionalforce value taken from Table I is F�x ¼ 0:28 andFx ¼
rvl20m2d�3F�x ¼ 0:125 N. The analogous computationfor Ty with T
�y ¼ �4:52� 10�2 gives Ty ¼ �1:01� 10�3N
m. Such values are easily measured with typical laboratory
equipment. They show that the LFV approach is quite practi-
cal for the envisaged applications.
TABLE III. Nondimensional force Fx for different dipole
orientations whenthe dipole is located at selected points ðy; hÞ.
Three last rows show theLorentz force magnitudes computed for the
dipoles which are oriented to get
the highest force.
Dipole orientation Fx at (y, h)
ðkx; ky; kzÞ (0.56, 0.04) (0.8, 0.8) (0.4, 0.4)
(1, 0, 0) 0.678 2:10� 10�4 0:88� 10�2(0, 1, 0) 0.671 1:38� 10�4
0:38� 10�2(0, 0, 1) 0.633 2:39� 10�4 1:17�
10�2ðffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
; 0Þ 0.678 1:74� 10�4 0:63� 10�2ðffiffiffiffiffiffiffi0:5p
; 0;ffiffiffiffiffiffiffi0:5p
Þ 0.659 2:24� 10�4 1:03� 10�2ð0;
ffiffiffiffiffiffiffi0:5p
;ffiffiffiffiffiffiffi0:5p
Þ 1.007 3:18� 10�4 1:20� 10�2(0.136, 0.725, 0.688) 1.010
(0, 0.563, 0.826) 3:28� 10�4(0, 0.400, 0.916) 1:36� 10�2
FIG. 14. A comparison between numerically obtained values and
large h as-ymptotic theory for two different dipole
orientations.
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Future numerical work will focus on cylindrical geome-
tries and more realistic velocity distributions resembling
actual
pipe or duct flows. Investigations of the coupled problem
where
the flow is modified by the Lorentz force will be performed
with the finite-difference method presented by Krasnov et
al.21
Experimental verification of the obtained results will be
per-
formed for different dipole orientations and (y, h) pairs
bymembers of our Research Training Group “Lorentz Force
Velocimetry and Lorentz Force Eddy Current Testing’’
project.
ACKNOWLEDGMENTS
The authors gratefully acknowledge financial support
from the Deutsche Forschungsgemeinschaft in the frame-
work of the Research Training Group “Lorentz Force Veloc-
imetry and Lorentz Force Eddy Current Testing” (grant
GRK 1567/1). Computer resources were provided by the
computing center of Ilmenau University of Technology.
APPENDIX A: DETAILS OF THE ASYMPTOTICANALYSIS
We assume that the dipole is placed in the symmetry
plane y¼ 0 but allow arbitrary dipole orientation. In this
sit-uation, the leading order terms of the nondimensional mag-
netic field components are
B0xðx̂Þ ¼1
4ph3kxð2x̂2 � 1Þ þ 3kzx̂ðx̂2 þ 1Þ5=2
;
B0yðx̂Þ ¼1
4ph3�ky
ðx̂2 þ 1Þ3=2;
B0z ðx̂Þ ¼1
4ph3�3kxx̂þ kzð2� x̂2Þðx̂2 þ 1Þ5=2
: (A1)
They depend only on x̂. The y component of the magnetic
fieldvanishes to leading order for dipoles with orientations k¼
(0,0, 1) and k¼ (1, 0, 0), while B0z vanishes for k¼ (0, 1, 0).
To obtain the currents j0, j1 and j2 one has to solve theLaplace
Eq. (3) for the electrical potential up to the required
order of approximation. We find the corresponding equations
and boundary conditions for the electric potential at the
dif-
ferent orders by substitution of the expansions and grouping
terms of different orders. For the expansion of derivatives
we note that the Nabla operator takes the form
r ¼ e @@x̂;@
@ŷ;@
@ẑ
� �: (A2)
For the zero-order approximation the appropriate Laplace
equation becomes
@2/0
@ŷ2þ @
2/0
@ẑ2¼ 0: (A3)
There is no normal current at insulating walls and
@/0
@ŷ
ŷ¼60:5
¼ �B0z ;
@/0
@ẑ
ẑ¼60:5
¼ B0y :(A4)
The solution for the potential is then given by
/0 ¼ ẑB0y � ŷB0z þ const; (A5)
which gives the currents j0 � 0. Hence, the integral (12)
van-ishes and F0x � 0. Also, all the integrands involving j0 inEqs.
(13) and (14) are zero.
The currents in the bar have to fulfill the continuity
equation $ � j ¼ 0. Because of Eq. (A2) we have
@j0x@x̂þ@j1y@ŷþ @j
1z
@ẑ¼ 0: (A6)
This equation can be automatically satisfied for j0x ¼ 0 if
astream function w ¼ wðx̂; ŷ; ẑÞ is introduced, i. e.,j1y ¼
@w=@ẑ and j1z ¼ �@w=@ŷ. We choose w to vanish at thewalls and
automatically satisfy boundary conditions for cur-
rents j � n¼ 0 because the boundary is a streamline of
electriccurrent.
To obtain an equation for the stream function w we con-sider the
x component of the current curl,
ðr � j1Þx ¼@j1z@ŷ�@j1y@ẑ¼ � @
2w@ŷ2� @
2w@ẑ2
: (A7)
Ohm’s law (2) gives
ðr � j1Þx ¼@
@ŷ� @/
1
@ẑþ B1y
� �� @@ẑ� @/
1
@ŷ� B1z
� �
¼ � @B0x
@x̂: (A8)
The above result is obtained remembering that the magnetic
field of the dipole is solenoidal and therefore
@B0x@x̂þ@B1y@ŷþ @B
1z
@ẑ¼ 0: (A9)
Hence, we have to solve the following Poisson equation for
the stream function
@2w@ŷ2þ @
2w@ẑ2¼ @B
0x
@x̂; (A10)
where wjwall ¼ 0. Equation (A10) also governs the laminarflow
profile in a rectangular duct.26 It can be solved using an
infinite series expansion, whereby one finds:ð0:5�0:5
ð0:5�0:5
wdŷdẑ ¼ 2:25364
@B0x@x̂
: (A11)
This identity is used to calculate the x component of the
firstterm from integral (13) which also vanishes:
ððj1 � B0ÞxdV ¼ �
2:253
64
ð1�1
B0x@B0x@x̂
hdx̂ � 0: (A12)
The second-order terms Eq. (14) should be evaluated to
obtain
a nonvanishing Fx. There are two such terms:Ððj2 � B0ÞxdV
andÐðj1 � B1ÞxdV. The latter integral can be solved using
the
stream function w by taking into account that
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ðj1 � B1Þx ¼@
@ŷðwB1yÞ þ
@
@ẑðwB1z Þ þ w
@B0x@x̂
: (A13)
By using Stokes’ theorem we see that the first two terms do
not contribute:
ð0:5�0:5
ð0:5�0:5
@
@ŷðwB1yÞ þ
@
@ẑðwB1z Þ
� �dŷdẑ
¼þ
w|{z}¼0
B1ydẑ� w|{z}¼0
B1z dŷ
0@
1A ¼ 0: (A14)
Then the integral can be transformed as
ððj1 � B1ÞxdV ¼
ð ð ðwðx̂; ŷ; ẑÞdŷdẑ @B
0x
@x̂hdx̂: (A15)
Using Eq. (A11), it integrates to
ððj1 � B1ÞxdV ¼ �
152 � 2:253220ph5
5k2x þ 7k2z� �
: (A16)
The integralÐðj2 � B0ÞxdV contains j2. To compute the cur-
rent at second order we again use the continuity equation in
the following form
@j2y@ŷþ @j
2z
@ẑ¼ � @j
1x
@x̂¼ � @
2/0
@x̂2: (A17)
We can use this equation to obtain
j1x ¼ ẑ@B0y@x̂� ŷ @B
0z
@x̂;
j2y ¼1
2ŷ2 � 1
4
� �@2B0z@x̂2
;
j2z ¼ �1
2ẑ2 � 1
4
� �@2B0y@x̂2
; (A18)
which satisfy the boundary conditions for the current. The
xcomponent of the first term of F2x in Eq. (14) is then given
by
ððj2 � B0ÞxdV ¼
ðj2yB
0z � j2z B0y
� �dV; (A19)
which can be computed analytically by integration by parts
and variable substitution. We thereby obtain the expressionððj2
� B0ÞxdV ¼ �
15
216ph535k2x þ 8k2y þ 57k2z� �
: (A20)
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bar the relevant diffusion time could
be based on the width of the bar d assuming that the dominant
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situation would
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and weaker
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it is actually preferable to work
with the induction equation in the quasi-static limit in order
to estimate the
magnitude of the induced currents. In this formulation, the
source term for
the induced field is the x derivative of the applied magnetic
field. We havecomputed the modulus of this quantity for different
dipole orientations,
and find that it accounts for the different magnitudes of the
Lorentz force
in a straightforward manner. We shall not discuss this issue
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113921-11 Kirpo et al. J. Appl. Phys. 109, 113921 (2011)
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