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Electromagnetic drag on a magnetic dipole near a translating conducting
bar
Maksims Kirpo, Saskia Tympel, Thomas Boeck, Dmitry Krasnov, and André Thess
Citation: J. Appl. Phys. 109, 113921 (2011); doi: 10.1063/1.3587182
View online: http://dx.doi.org/10.1063/1.3587182
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i11
Published by the American Institute of Physics.
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JOURNAL OF APPLIED PHYSICS 109, 113921 (2011)
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
C 2011 American Institute of
bar that produces the maximum force in the direction of motion. V
Physics. [doi:10.1063/1.3587182]
I. INTRODUCTION
The present work is devoted to the theoretical investigation of a conceptually simple prototype problem for Lorentz
force velocimetry (LFV) and Lorentz force eddy current testing (LET). LFV is a modern, contactless technique for measuring 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 environmental conditions (high temperatures, radioactivity) and chemical
reactions.1 Possible applications include flow measurement
during the continuous casting of steel, in ducts and open channel 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 constructions 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 application 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 aggressive 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 typical 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
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-8979/2011/109(11)/113921/11/$30.00
solution of inverse problems. So its adaption to industrial processes 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 already proposed by Shercliff.8 Eddy currents are induced in a
conductor, which is moving in an external (primary) magnetic 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 magnetic system, which creates the primary magnetic field, experiences 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 moving conductor, v is the magnitude of the velocity and B is the
magnitude of the magnetic induction. Measuring this force,
acting on the magnetic system, allows us to measure the velocity 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 solutions.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.
109, 113921-1
C 2011 American Institute of Physics
V
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113921-2
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
BðrÞ ¼
FIG. 1. Sketch of the studied problem (left) and parameters of the geometry
(right).
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 OF
THE PROBLEM
We consider an electrically conducting nonmagnetic infinite solid bar with a square cross section d d [Fig. 1(b)]
which is moving with a constant velocity v ¼ v ex in x
direction in the field B of a magnetic dipole with the
magnetic dipole moment m ¼ mðkx ex þ ky ey þ kz ez Þ and
kx2 þ ky2 þ kz2 ¼ 1. Our ultimate goal is to compute the electromagnetic drag force and torque acting on the dipole for arbitrary dipole location and orientation. The bar translating in
the direction orthogonal to its cross section is of primary interest 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 principles 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 is
given by16
l0 h
r mi
3ðm rÞ 5 3
4p
r
r
(1)
assuming that the origin of the coordinate system corresponds to the dipole location.
Eddy currents are induced in the bar when it crosses the
magnetic field lines. They create a secondary magnetic field b
which in this work is assumed to be just a very small perturbation 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 the
external 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
L ¼ h.18 The assumptions of the quasistatic approximations
are then satisfied when the magnetic Reynolds number
Rm ¼ l0 rvL is less than unity, where r denotes the electric
conductivity. 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 velocity equal to the bar velocity V0 ¼ v and the characteristic
magnetic field intensity B0 ¼ l0 md3 . This choice of the
characteristic parameters leads to the following expressions
for the current density j ¼ rvl0 md 3 j and Lorentz force
F ¼ rvl20 m2 d 3 F where “star” symbol represents nondimensional quantities. This presentation of the force F allows
us to express F as a function of the dipole orientations
k ¼ m ¼ ðkx ; ky ; kz Þ, distance between the dipole and the
bar h ¼ h=d and the dipole displacement in y direction
F ¼ F ðkx ; ky ; kz ; h ; y Þ. Similarly, the torque is defined as
T ¼ rvl20 m2 d 2 T and T ¼ T ðkx ; ky ; kz ; h ; y Þ. From now
only 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 /. The
induced 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 electrical potential can be obtained taking the divergence of Ohm’s
law (2). Since the magnetic field of the dipole (1) is irrotational (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 currents 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 and
currents 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
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113921-3
Kirpo et al.
Fbar ¼
Tbar ¼
J. Appl. Phys. 109, 113921 (2011)
ð
j B dV and
ðV
r ðj BÞdV:
(4)
V
By virtue of Newton’s third law, an opposite force and torque act on the dipole. This force and torque on the dipole are
of equal magnitude as Fbar and Tbar . They are given by the
formulas F ¼ ðk rÞb and T ¼ k b. The secondary magnetic field b in these expressions is produced by the induced
currents in the moving bar. It can be computed at the location 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.
TM
For this reason an automated Matlab script coupled with the
TM
Comsol FEM Laplace “pardiso” solver19 was used to solve
the problem numerically for the electrical potential using second-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
TM
implemented using built-in Comsol functions.
Preliminary computations showed that the accurate solution 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 refined
grid was used for simulations as shown in Fig. 2. We have
verified by a grid convergence study that computational accuracy is within 5% if the distance between the dipole and
the top surface of the moving bar h equals the doubled characteristic size of the element. The computational grid was
further refined for very small h 8 102 . The maximal
number 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 magnetohydrodynamic 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 equation is solved with the Poisson solver FishPack. Verifications of this code versus a spectral code and details on the
algorithms can be found in Krasnov et al.21 The code
computes 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 effects
in the case of large distances h the length of the solid bar
was chosen to be proportional to the distance to the dipole, i.
e., it was 7.5ph. The code used periodic boundary conditions
at x ¼6 7.5ph/2 for electrical potential. They did not influence the obtained solution due to sufficient length of the
computational domain. The numerical resolution of the uniform mesh was 8192 256 256 points in x, y, z in all simulations for h 1.
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) in
a plane x = const.
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 and
the y component of the torque are
"
#
1
h3
0
;
(5)
1
Fx ¼
128ph3
ð1 þ hÞ3
"
#
1
h2
0
:
(6)
1
Ty ¼ 128ph2
ð1 þ hÞ2
For this case, the components of the magnetic moment are
k ¼ (0, 0, 1) in our nondimensional representation, and we
can compare our simulation results directly by evaluating the
ratios Fx =F0x and Ty =Ty0 for different h, as shown in Fig. 3.
As expected, the force and torque ratios decrease monotonously as h increases. When h tends to zero, i. e., the magnetic dipole approaches the top surface of the bar, the curves
reach unity. This behavior agrees with our expectations
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113921-4
Kirpo et al.
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 values. This is because the “useful” volume, which the magnetic
dipole interacts with, decreases, too. This is, clearly, a distinct feature of the rectangular bar compared to the infinite
plate. The data for h > 2 indicate a transition to scaling
behavior for large distances, and the ratios Fx =F0x and Ty =Ty0
are related by a constant factor in this range. It can also be
noticed that the two curves start to decay at somewhat different values of h with the torque ratio Ty =Ty0 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 tabulated also for other selected ðkx ; ky ; kz Þ in Tables I and II. It
can be seen that the other orientations of the magnetic dipole
provide smaller values of the Lorentz forces compared with
FIG. 3. Ratios of nondimensional forces Fx =F0x and torques T=Ty0 acting on
a magnetic dipole located at x ¼ 0, y ¼ 0 for different h and oriented perpendicular to the closest surface, i. e., k ¼ (0, 0, 1).
J. Appl. Phys. 109, 113921 (2011)
FIG. 4. Nondimensional force Fx acting on the magnetic dipole located at
x ¼ 0, y ¼ 0 for different h and dipole orientations.
the vertically oriented dipole if it is placed in the lateral midplane of the bar y ¼ 0. For example, the Lorentz force on a
dipole 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 orientation. However, this argument seems misleading because
the same reduction in the field strength for the x orientation
reduces 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 horizontal extent and arbitrary depth produced by a dipole of
FIG. 5. Absolute values of nondimensional torque Ty acting on the magnetic
dipole with any k ¼ ðkx ; 0; kz Þ, kx2 þ kz2 ¼ 1 located at x ¼ 0, y ¼ 0 for different h.
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113921-5
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
TABLE I. Values of the nondimensional force Fx acting on the magnetic dipole located at x ¼ 0, y ¼ 0 for selected h and dipole orientations.
h
0.02
0.10
0.20
0.50
1.00
2.00
5.00
10.0
(0, 0, 1)
(0, 1, 0)
(1, 0, 0)
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð0; 0:5; 0:5Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0; 0:5Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0:5; 0Þ
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
ð 1=3; 1=3; 1=3Þ
3:11 102
2:46 100
2:80 101
1:01 102
4:12 104
9:53 106
3:51 108
3:74 1010
7:83 101
5:98 101
5:75 102
1:20 103
3:91 105
9:64 107
3:83 109
4:15 1011
2:34 102
1:85 100
2:02 101
6:56 103
2:59 104
5:99 106
2:22 108
2:37 1010
1:95 102
1:53 100
1:69 101
5:63 103
2:26 104
5:25 106
1:95 108
2:08 1010
2:73 102
2:16 100
2:41 101
8:31 103
3:35 104
7:76 106
2:87 108
3:05 1010
1:56 102
1:22 100
1:30 101
3:88 103
1:49 104
3:48 106
1:30 108
1:39 1010
2:08 102
1:64 100
1:80 101
5:94 103
2:37 104
5:50 106
2:04 108
2:17 1010
arbitrary orientation shows the same Lorentz force dependence on dipole orientation.24
Different orientations of the dipole can break the symmetry in current density distributions as shown in Figs. 6(d) and
6(e). In such cases not only the main Fx force
pffiffiffiffiffifficomponent
ffi pffiffiffiffiffiffiffi
appears. The simulations have shown, that for ð 0:5; 0:5; 0Þ
the force component Fy is present as well. Although it has a
is the
smaller absolute value than Fx, the order
pffiffiffiffiffiffiffiof magnitude
pffiffiffiffiffiffiffi
same. If the dipole is oriented as ð 0:5; 0; 0:5Þ, then the
force component Fy vanishes. Instead a smaller attractive force
Fz between the dipole and the bar appears. It is at least one
order of magnitude smaller than Fx.
It is also interesting to note that the maximal values of
the torque component Ty are obtained when the magnetic
dipole has ky ¼ 0, i. e., it is oriented in xz-plane only as it is
shown in Table II. Moreover, these Ty values are equal
within the accuracy of numerical results for different dipole
orientations with ky ¼ 0 and kx2 þ kz2 ¼ 1 in the xz-plane. The
translational motion of the conducting bar tries to rotate the
dipole located in xz-plane around the y axis with a constant
torque for given h. This result can be used for rotary flow
meters where flow rate is determined by measuring the rotation 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 velocity
of a long rotating cylindrical magnet above a translating conducting layer is obtained in two-dimensional approximation.
A partial explanation of these results follows simply
from inspection of the formula T ¼ k b for the torque on
the dipole. We can see that Ty ¼ kz bx kx bz vanishes if the
dipole is oriented in y direction, i. e., k ¼ (0, 1, 0). However,
the fact that the torque stays the same for any orientation
with ky ¼ 0 and kx2 þ kz2 ¼ 1 is not obvious. Equation (4)
gives
Ty ¼
ð
ðrz fx rx fz ÞdV;
(7)
V
where r is taken in the coordinate system whose origin corresponds to the dipole. The integrated torque Ty thus contains
contributions from spatial distributions of the Lorentz force
densities fx and fz in the bar. Figure 7 shows the force density
integrated in every cross section
ðð
fi dy dz; i ¼ x; y; z
(8)
vi ¼
depending on the coordinate x for the magnetic dipole with
k ¼ ð1; 0; 0Þ. It can be clearly seen that the integrals vx and
vz are approximately of same order of magnitude and the integral vy vanishes. The total force in vertical direction
ð
(9)
Fz ¼ vz dx
vanishes because vz is an antisymmetric function. But the
product rx fz always has a positive value which contributes to
the torque Ty. Both products rz fx and rx fz balance in a way
that the integrated torque Ty remains constant for any dipole
orientation with ðkx ; 0; kz Þ, kx2 þ kz2 ¼ 1. The same applies for
the infinite plate as shown by Priede et al.15 Their analytical
result for Ty depends on the sum ðkx2 þ kz2 Þ only.
After examination of the orientation, we finally comment on the asymptotic behavior of force and torque with the
distance h. The double logarithmic representations in Figs. 4
and 5 reveal different power law approximations for different distances h. The power law fitting provides Fx h3 ,
Ty h2 for small h and Fx h7 , Ty h6 for large h. It
can be noticed that these estimations exactly correspond to
Eqs. (5) and (6) for the infinite plate in small h region.
TABLE II. Values of the nondimensional torque Ty acting on the magnetic dipole located at x ¼ 0, y ¼ 0 for selected h and dipole orientations.
h
0.02
0.10
0.20
0.50
1.00
2.00
5.00
10.0
(0, 0, 1)
(0, 1, 0)
(1, 0, 0)
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð0; 0:5; 0:5Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0; 0:5Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0:5; 0Þ
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
ð 1=3; 1=3; 1=3Þ
6:22 100
2:38 101
4:52 102
2:99 103
2:01 104
8:10 106
6:70 108
1:37 109
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
6:20 100
2:30 101
4:62 102
3:01 103
2:01 104
8:11 106
6:71 108
1:37 109
3:11 100
1:19 101
2:26 102
1:50 103
1:00 104
4:05 106
3:35 108
6:84 1010
6:21 100
2:34 101
4:57 102
3:00 103
2:01 104
8:11 106
6:71 108
1:37 109
3:10 100
1:15 101
2:31 102
1:50 103
1:01 104
4:06 106
3:36 108
6:87 1010
4:14 100
1:56 101
3:05 102
2:00 103
1:34 104
5:41 106
4:47 108
9:13 1010
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113921-6
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
FIG. 6. Contours of the nondimensional current density magnitudes and current density vectors on the top surface (z ¼ 1/2) of the bar for h ¼ 0.2 and different
orientations of the magneticpdipole.
Only the central
of the surface is shown.
ffiffiffi
pffiffipart
ffi
pffiffiffi 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)/ 2 for case (d), (1,0,1)/ 2 for case (e), and (0,1,1)/ 2 for case (f).
However, the Lorentz force and torque decay faster for moving solid bar than for infinite plate for large h and the
Fx h7 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 should
change if the dipole is shifted from the symmetry plane
y ¼ 0. The presented numerical approach allows us to study
this dependence of Fx on the coordinates y and z of the
dipole. Pairs of y > 0 and z > 0 values are selected in the
x ¼ 0 plane, and the Lorentz force is computed for three
main 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 Fx ðy; zÞ
distributions. It can be seen that for the dipole with k ¼ (1, 0,
0) the distribution of Fx ðy; zÞ is completely symmetric about
the diagonal y ¼ z as shown in Fig. 8. For the dipole with
k ¼ (0, 0, 1) the force distribution Fx ðy; zÞ keeps the symmetry with respect to the y ¼ 0 plane only as it is shown on
Fig. 9. These distributions of Fx ðy; zÞ are compatible with the
previously described results: the Lorentz force has the smallest value if the dipole is oriented in y direction, i. e., the isolines of the same force magnitude are closer to the surface of
the bar. The force distribution Fx ðy; zÞ for the dipole with
k ¼ (0, 0, 1) also represents the results for the dipole with
k ¼ (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 component Fx. In the present mathematical model, the induced currents 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 quadratically on the applied field. The integrated Lorentz force components are therefore quadratic forms of the dipole
orientation vector ðkx ; ky ; kz Þ. In particular,
Fx ¼ kiT Aij kj ;
(10)
where summation on the repeated indices is understood.
If the eigenvalue problem Aa ¼ ka is solved,25 then the
eigenvector a which corresponds to the largest eigenvalue
max(k) of the matrix Aij provides orientation of the dipole
with the largest force component Fx. The matrix Aij is symmetric 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 ; kz Þmax , which provides the maximum force Fx, has kx ¼ 0
within the accuracy of numerical results. For different ðy; hÞ
pairs the components ky and kz, providing the maximum Lorentz force, are different, requiring tilting of the dipole by a
certain angle h between the z axis and direction of the dipole
which lies in x ¼ 0 plane. This definition of h is sketched in
Fig. 10.
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113921-7
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
ÐÐ
FIG. 7. Force density integrals vi ¼ fi dy dz, i ¼ x, y, z depending on the
coordinate x for the dipole with k ¼ (1, 0, 0) and h ¼ 0.2.
The computed eigenvectors with orientations ð0; ky ;
kz Þmax are shown in Fig. 11 where every arrow represents
direction of the optimal orientation for the dipole located at
certain y and h. The angle h between z axis and direction of
the eigenvector is shown in Fig. 12 for all studied ðy; hÞ values and in Fig. 13 for two selected distances h. The angle h
varies from almost zero for vertical dipole orientation k ¼ (0,
0, 1) at y ¼ 0 and any h to more than 50 for large y and
small h. If the dipole is placed very close to the surface of
the bar (h ! 0) then the angle h remains small with increasing y until the edge of the bar, i. e., while y < 0.5, and
changes rapidly to 50 at y > 0.5 as it is shown in Fig. 13. If
the dipole is located at larger h, then the angle h increases
monotonously with y. It can be seen that if y or h becomes
large, 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 obtained
dipole orientation provides the largest Lorentz force Fx. The
results of force comparison are combined into Table III and
clearly show that the Lorentz force for the optimal orientation 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 application
for very small velocity measurements or cases with small
electrical conductivity. However, in LFV applications it
could be difficult to realize different orientations of magnetization for the real magnets due to their finite size.
FIG. 8. Decimal logarithm of the force magnitude distribution for nondimensional force Fx ðy; zÞ, k ¼ (1, 0, 0). The gap in the isoline closest to the
bar is due to computational reasons.
explained with the help of asymptotic expansions in the
small parameter e: ¼ 1/h. Alternatively, the asymptotic
approach 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 assume
that the dipole is placed in the symmetry plane y ¼ 0 but
allow for arbitrary dipole orientation.
The asymptotic solution is based on the rescaled coordinates x ¼ h^
x, y ¼ y^, z ¼ ^z, whereby one can exploit the slow
IV. ASYMPTOTIC THEORY FOR LARGE h
The decay of the Lorentz force with the power h7 at
large distances is more rapid than one would expect from a
simple estimate. The straightforward estimation would
involve Fx B20 V based on a characteristic value B0 of the
magnetic field and an effective volume V of the bar affected
by the magnetic field. With the dipole field decaying according to B0 h3 and V ¼ hd2 one would obtain Fx h5 .
The decay with h7 is therefore not obvious. It can be
FIG. 9. Decimal logarithm of the force magnitude distribution for nondimensional force Fx ðy; zÞ, k ¼ (0, 0, 1). The gap in the isoline closest to the
bar is due to computational reasons.
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113921-8
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
FIG. 12. Angle h in degrees between the z axis and direction of the
eigenvector.
F2x
FIG. 10. Definition of the angle h between the z axis and direction of the
dipole.
variation in x when the parameter e tends to zero. The quantities of interest (i. e., B, /, j, F) are then represented as regular perturbation expansions in the small parameter e, e. g.,
for the magnetic field:
xÞ þ eB1 ð^
x; y^; ^zÞ þ e2 B2 ð^
x; y^; ^zÞ þ …: (11)
Bð^
x; y^; ^
zÞ ¼ B0 ð^
The superscripts denote the order of approximation for every
term. The expressions for B0 are given in Appendix A. The
velocity 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:
ð
0
(12)
Fx ¼ j0 B0 x dV;
F1x ¼
ð
j1 B0 þ j0 B1 x dV;
¼
ð
j2 B0 þ j1 B1 þ j0 B2 x dV:
(14)
Then the evaluation of Fx requires computation of these six
integrals. Each of them and details of the calculation are considered in the appendix. We only present the key steps of the
procedure at this point.
The Laplace equation for the leading term /0 of the
electrical potential is easily solved by /0 ¼ zB0y yB0z
þconst. Therefore j0 ¼ 0 and all integrals containing the current j0 vanish. The first-order term j1 of the current does not
vanish. We can determine its components in the yz plane by
a stream function representation. There is no contribution to
the Lorentz force from such a planar current distribution
interacting
a field B0 that is constant on each yz plane,
Ð with
1
i. e. ðj B0 Þx dV ¼ 0. However, there is a contribution
from the interaction with B1 :
ð
1
152 2:253
2
j B1 x dV ¼ ð5kx2 þ 7kz2 Þ:
(15)
e
220 ph7
(13)
FIG. 11. The eigenvectors show dipole orientations which provide the highest Lorentz force for different (y, h) pairs.
FIG. 13. Angle h in degrees between the z axis and direction of the eigenvector for two selected values of the distance h.
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113921-9
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
TABLE III. Nondimensional force Fx for different dipole orientations when
the dipole is located at selected points ðy; hÞ. Three last rows show the
Lorentz force magnitudes computed for the dipoles which are oriented to get
the highest force.
Fx at (y, h)
Dipole orientation
ðkx ; ky ; kz Þ
(0.56, 0.04)
(0.8, 0.8)
(0.4, 0.4)
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0:5; 0Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð 0:5; 0; 0:5Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
ð0; 0:5; 0:5Þ
(0.136, 0.725, 0.688)
(0, 0.563, 0.826)
(0, 0.400, 0.916)
0.678
0.671
0.633
0.678
0.659
1.007
1.010
2:10 104
1:38 104
2:39 104
1:74 104
2:24 104
3:18 104
0:88 102
0:38 102
1:17 102
0:63 102
1:03 102
1:20 102
3:28 104
1:36 102
FIG. 14. A comparison between numerically obtained values and large h asymptotic theory for two different dipole orientations.
For the last remaining integral we use the continuity equation
and the result for /0 to get
ð
15
e2 j2 B0 x dV ¼ 16 7 ð35kx2 þ 8ky2 þ 57kz2 Þ: (16)
2 ph
By combining all evaluated integrals we see that the leading
term of the Lorentz force is given by
Fx ¼ 15 2
2
2
45:561k
þ
8k
þ
71:785k
x
y
z :
216 ph7
(17)
We have thereby demonstrated that Fx h7 when h 1.
The dependence of the force on the orientation differs from
the limit h ! 0 as can be seen from the coefficients multiplying the components ki. In particular, the y orientation
becomes 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 distances exceeding h > 103 , while the commercial code used
in 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. The
observed 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 y ¼ 0 of
the bar. The force dependence on the distance h between the
dipole and the bar is governed by power laws when the distance 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 h7 , than
for 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 is
in a range of 105 N. The developed asymptotic theory can be
used 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, which
is proportional to h7 , is expected to be present in other similar problems involving laminar or turbulent flows in ducts or
pipes. Derivation of the asymptotic theory for the translational motion of a solid cylindrical body of round cross section 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 accuracy of numerical results for y ¼ 0, ky ¼ 0 and kx2 þ kz2 ¼ 1.
The discussed results allow us to evaluate the Lorentz
force magnitude and torque for specified h and m. If we take
an aluminum bar with v ¼ 1 m/s, d ¼ 5 102 m,
r ¼ 3:54 107 (X m)1 , h ¼ 102 m, and dipole moment
m ¼ (0, 0, 1) Am2, then the corresponding nondimensional
force value taken from Table I is Fx ¼ 0:28 and
Fx ¼ rvl20 m2 d 3 Fx ¼ 0:125 N. The analogous computation
for Ty with Ty ¼ 4:52 102 gives Ty ¼ 1:01 103 N
m. Such values are easily measured with typical laboratory
equipment. They show that the LFV approach is quite practical for the envisaged applications.
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113921-10
Kirpo et al.
J. Appl. Phys. 109, 113921 (2011)
Future numerical work will focus on cylindrical geometries 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 performed for different dipole orientations and (y, h) pairs by
members of our Research Training Group “Lorentz Force
Velocimetry and Lorentz Force Eddy Current Testing’’ project.
The solution for the potential is then given by
/0 ¼ ^zB0y y^B0z þ const;
which gives the currents j0 0. Hence, the integral (12) vanishes and F0x 0. Also, all the integrands involving j0 in
Eqs. (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 @j1y @j1z
þ
þ
¼ 0:
@^
x @^
y @^z
ACKNOWLEDGMENTS
The authors gratefully acknowledge financial support
from the Deutsche Forschungsgemeinschaft in the framework of the Research Training Group “Lorentz Force Velocimetry 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 ASYMPTOTIC
ANALYSIS
We assume that the dipole is placed in the symmetry
plane y ¼ 0 but allow arbitrary dipole orientation. In this situation, the leading order terms of the nondimensional magnetic field components are
1 kx ð2^
x2 1Þ þ 3kz x^
xÞ ¼
;
B0x ð^
4ph3
ð^
x2 þ 1Þ5=2
xÞ ¼
B0y ð^
ky
1
;
4ph3 ð^
x2 þ 1Þ3=2
2
B0z ð^
xÞ ¼
1 3kx x^ þ kz ð2 x^ Þ
:
4ph3
ð^
x2 þ 1Þ5=2
(A1)
They depend only on x^. The y component of the magnetic field
vanishes to leading order for dipoles with orientations k ¼ (0,
0, 1) and k ¼ (1, 0, 0), while B0z vanishes for k ¼ (0, 1, 0).
To obtain the currents j0 , j1 and j2 one has to solve the
Laplace 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 different 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
@ @ @
:
(A2)
r¼ e ; ;
@^
x @^
y @^z
There is no normal current at insulating walls and
@/0 ¼ B0z ;
@^
y y^¼60:5
@/0 ¼ B0y :
@^z ^z¼60:5
(A3)
ðr j1 Þx ¼
@j1z @j1y
@2w @2w
¼ 2 2:
@^
y @^z
@^
y
@^z
Ohm’s law (2) gives
@
@/1
@
@/1
þ B1y B1z
ðr j1 Þx ¼
@^z
@^
y
@^
y
@^z
0
@B
¼ x:
@^
x
(A7)
(A8)
The above result is obtained remembering that the magnetic
field of the dipole is solenoidal and therefore
@B0x @B1y @B1z
þ
þ
¼ 0:
@^
x
@^
y
@^z
(A9)
Hence, we have to solve the following Poisson equation for
the stream function
@ 2 w @ 2 w @B0x
;
þ 2 ¼
@^
x
@^
y2
@^z
(A10)
where wjwall ¼ 0. Equation (A10) also governs the laminar
flow profile in a rectangular duct.26 It can be solved using an
infinite series expansion, whereby one finds:
ð 0:5 ð 0:5
wd^
yd^z ¼
0:5 0:5
2:253 @B0x
:
x
64 @^
(A11)
This identity is used to calculate the x component of the first
term from integral (13) which also vanishes:
ð
(A4)
(A6)
This equation can be automatically satisfied for j0x ¼ 0 if a
stream function w ¼ wð^
x; y^; ^zÞ is introduced, i. e.,
y. We choose w to vanish at the
j1y ¼ @w=@^z and j1z ¼ @w=@^
walls and automatically satisfy boundary conditions for currents j n ¼ 0 because the boundary is a streamline of electric
current.
To obtain an equation for the stream function w we consider the x component of the current curl,
For the zero-order approximation the appropriate Laplace
equation becomes
@ 2 /0 @ 2 /0
þ
¼ 0:
@^
y2
@^z2
(A5)
ðj1 B0 Þx dV ¼ 2:253
64
ð1
1
B0x
@B0x
hd^
x 0:
@^
x
(A12)
The second-order terms Eq. (14) should be evaluated
Ð 2 to obtain
a nonvanishing
F
.
There
are
two
such
terms:
ðj B0 Þx dV
x
Ð 1
1
and ðj B Þx dV. The latter integral can be solved using the
stream function w by taking into account that
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113921-11
Kirpo et al.
ðj1 B1 Þx ¼
J. Appl. Phys. 109, 113921 (2011)
@
@
@B0
ðwB1y Þ þ ðwB1z Þ þ w x :
@^
x
@^
y
@^z
(A13)
By using Stokes’ theorem we see that the first two terms do
not contribute:
which can be computed analytically by integration by parts
and variable substitution. We thereby obtain the expression
ð
15 ðj2 B0 Þx dV ¼ 16 5 35kx2 þ 8ky2 þ 57kz2 : (A20)
2 ph
1
ð 0:5 ð 0:5 0:5 0:5
@
@
ðwB1y Þ þ ðwB1z Þ d^
yd^z
@^
y
@^z
0
1
þ
¼ @ w B1y d^z w B1z d^
yA ¼ 0:
|{z}
|{z}
¼0
(A14)
¼0
Then the integral can be transformed as
ð
ðj1 B1 Þx dV ¼
ððð
wð^
x; y^; ^zÞd^
yd^z
@B0x
hd^
x:
@^
x
(A15)
Using Eq. (A11), it integrates to
ð
ðj1 B1 Þx dV ¼ 152 2:253 2
5kx þ 7kz2 :
20
5
2 ph
(A16)
Ð
The integral ðj2 B0 Þx dV contains j2 . To compute the current at second order we again use the continuity equation in
the following form
@j2y @j2z
@j1
@ 2 /0
þ
¼ x¼
:
@^
y @^z
@^
x
@^
x2
(A17)
We can use this equation to obtain
@B0y
@B0
y^ z ;
@^
x
@^
x
2 0
1
1
@ Bz
j2y ¼
y^2 ;
x2
2
4 @^
1 2 1 @ 2 B0y
j2z ¼ ^z ;
x2
2
4 @^
j1x ¼ ^z
(A18)
which satisfy the boundary conditions for the current. The x
component of the first term of F2x in Eq. (14) is then given by
ð
2
0
ðj B Þx dV ¼
ð
j2y B0z j2z B0y dV;
(A19)
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18
When the dipole is far away from the bar the relevant diffusion time could
be based on the width of the bar d assuming that the dominant magnetic
field balance results from the cross stream diffusion. This situation would
be analogous to thin film flows. It would provide a different and weaker
constraint for the quasistatic approximation.
19
N. Petra and M. K. Gobbert, in Parallel Performance Studies for COMSOL Multiphysics Using Scripting and Batch Processing, edited by Yeswanth Rao (Proceedings of the COMSOL Conference, Boston, MA,
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As pointed out by the anonymous referee 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 have
computed 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 further since
we have performed our study with the electrical potential formulation.
25
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26
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Dynamics (Oxford University Press, New York, 1997)
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