Electromagnetic drag on a magnetic dipole near a ... · Electromagnetic drag on a magnetic dipole near a translating conducting bar Maksims Kirpo, Saskia Tympel, ... and in other

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

JOURNAL OF APPLIED PHYSICS 109, 113921 (2011)

Downloaded 13 Jun 2011 to 141.24.172.15. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

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).

113921-2 Kirpo et al. J. Appl. Phys. 109, 113921 (2011)


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.



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.



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


; 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


Þ ðffiffiffiffiffiffiffi0:5p




; 0Þ ðffiffiffiffiffiffiffiffi1=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






; 0Þ ðffiffiffiffiffiffiffiffi1=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



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).



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.



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.



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


; 0Þ 0.678 1:74� 10�4 0:63� 10�2ðffiffiffiffiffiffiffi0:5p


Þ 0.659 2:24� 10�4 1:03� 10�2ð0;



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.



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



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)

1A. Thess, E. Votyakov, and Y. Kolesnikov, Phys. Rev. Lett. 96, 164501(2006).

2Y. Kolesnikov, C. Karcher, and A. Thess, “Lorentz force flowmeter for

liquid aluminum: Laboratory experiments and plant tests,” Metall. Mater.

Trans. B. 42B, 441-450 (2011).3S. Argyropoulos, Scan-d. J. Metall. 30, 273 (2000).4R. Ricou and C. Vives, Int. J. Heat Mass Transfer 25, 1579 (1982).5J. Szekely, C. Chang, and R. Ryan, Metall. Trans. B 8, 333 (1977).6Y. Takeda, Nucl. Technol. 79, 120 (1987).7S. Eckert, G. Gerbeth, and V. Melnikov, Exp. Fluids 35, 381 (2003).8J. Shercliff, ed., The Theory of Electromagnetic Flow Measurement (Cam-bridge University Press, Cambridge, UK, 1962).

9M. Bevir, J. Fluid Mech. 43, 577 (1970).10F. Stefani, T. Gundrum, and G. Gerbeth, Phys. Rev. E70, 056306 (2004).11J. Reitz, J. Appl. Phys. 41, 2067 (1970).12B. Palmer, Eur. J. Phys. 25, 655 (2004).13A. Thess, E. Votyakov, B. Knaepen, and O. Zikanov, New J. Phys. 9, 299

(2007).14J. Priede, D. Buchenau, and G. Gerbeth, Magnetohydrodynamics 45, 451

(2009).15J. Priede, D. Buchenau, and G. Gerbeth, “Single-magnet rotary fowmeter

for liquid metals,” J. Appl. Phys. (submitted).16J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).17P. Roberts, ed., An Introduction to Magnetohydrodynamics (American

Elsevier, New York, 1967).18When 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 magneticfield 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.19N. Petra and M. K. Gobbert, in Parallel Performance Studies for COM-

SOL Multiphysics Using Scripting and Batch Processing, edited by Yes-wanth Rao (Proceedings of the COMSOL Conference, Boston, MA,

2009), pp. 1-6.20COMSOL Multiphysics Reference Guide, ver. 3.5a (Comsol AB, 2008).21D. Krasnov, O. Zikanov, and T. Boeck, “Comparative study of finite dif-

ference formulations to simulation of magnetohydrodynamic turbulence at

low magnetic Reynolds number,” Comput. Fluids (submitted).22Y. Morinishi, T. Lund, O. Vasilyev, and P. Moin, J. Comput. Phys. 143,

90 (1998).23M.-J. Ni, R. Munipalli, N. Morley, P. Huang, and M. Abdou, J. Comput.

Phys. 227, 174 (2007).24As 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 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 further since

we have performed our study with the electrical potential formulation.25G. Strang, ed., Linear Algebra and Its Applications (Thomson Brooks/

Cole, Belmont, 2006).26C. Pozrikidis, ed., Introduction to Theoretical and Computational Fluid

Dynamics (Oxford University Press, New York, 1997)



http://dx.doi.org/10.1103/PhysRevLett.96.164501http://dx.doi.org/10.1007/s11663-011-9477-6http://dx.doi.org/10.1007/s11663-011-9477-6http://dx.doi.org/10.1034/j.1600-0692.2001.300501.xhttp://dx.doi.org/10.1016/0017-9310(82)90036-9http://dx.doi.org/10.1007/BF02657664http://dx.doi.org/10.1007/s00348-003-0606-0http://dx.doi.org/10.1017/S0022112070002586http://dx.doi.org/10.1103/PhysRevE.70.056306http://dx.doi.org/10.1063/1.1659166http://dx.doi.org/10.1088/0143-0807/25/5/008http://dx.doi.org/10.1088/1367-2630/9/8/299http://dx.doi.org/10.1006/jcph.1998.5962http://dx.doi.org/10.1016/j.jcp.2007.07.025http://dx.doi.org/10.1016/j.jcp.2007.07.025

s1cor1s2E1E2E3F1E4s3s3AE5E6F2F3F4F5E7E8E9T1T2s3BE10F6s4F7F8F9E11E12E13E14E15F10F11F12F13E16E17s5T3F14EA1EA2EA3EA4EA5EA6EA7EA8EA9EA10EA11EA12EA13EA14EA15EA16EA17EA18EA19EA20B1B2B3B4B5B6B7B8B9B10B11B12B13B14B15B16B17B18B19B20B21B22B23B24B25B26

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