Deformation of the free surface of horizontal magnetic fluid layer by current-induced magnetic fields

Fluid DynamicsResearch 10 (1992) 1-9 North-Holland

FLU I D D Y N A M I C S RESEARCH

Deformation of the free surface of horizontal magnetic fluid layer by current-induced magnetic fields Kanefusa Gotoh, Youichi Murakami and Toshiyuki Ohkita Department of Mathematical Sciences, College of Engineering, University of Osaka Prefecture, Sakai, Osaka 591, Japan

Received 20 December 1991 Abstract. Static deformation of free surface is investigated for horizontal magnetic fluid layer in the presence

of applied magnetic field induced by circular electric current situated horizontally below the layer. Shape of the deformed free surface is found by numerical calculation for various current intensity and size of circuit. The problems for a straight line current and a magnetic dipole are solved separately.

1. Introduction

It is more than a quarter of a century since fluids with large response to magnetic field, which are nowadays called magnetic fluids, began to be developed in the early 1960s. While uses of magnetic fluids have been extended to a wider range of engineering, their basic dynamic theory has been steadily established (Rosensweig, 1985), and recently the behaviour of a concentrated dense fluid has been attractive. But yet the theory of dilute fluid needs quantitative data on various problems to justify itself in comparison to experiments. One of these problems is the variation of the interface between the magnetic and the non-magnetic fluids in the presence of a magnetic field. One exact approach to this problem constitutes the so-called Stefan problem, that is, the field quantities should be determined together with the unknown interface shape. When the susceptibility of the magnetic fluid is small, we can use the so-called (magnetic) induction free approximation, and the problem can be reduced to a fixed boundary value problem for the shape of interface. In this paper we consider a horizontal layer of magnetic fluid bounded from below by a rigid wall and examine the shape of the free surface in the presence of various magnetic field induced by an electric current situated horizontally outside the layer. This phenomenon is closely related to that of an isolated spike of magnetic fluid which is substantial in the discharge of a magnetic fluid drop. It is applicable to flow control and a plug by magnetic fluid in a pipe and so on. It may technologically contribute to some improvement in equipment designed to collect magnetic fluids. The shape of the free boundary is obtained numerically for various intensities of linear current and for various diameters in the case of a circular current. The result for a magnetic dipole (in the limiting case of a small circular circuit) is compared to that by Ageev et al. Correspondence to: K. Gotoh, Department of Mathematical Sciences, College of Engineering, University of Osaka Prefecture, Sakai, Osaka 591, Japan.

0169-5983/92/$03.25 © 1992 - The Japan Societyof Fluid Mechanics. All rights reserved

2

K. Gotoh et al. / Deformation of magnetic fluid layer

(1989). The excess volume is found to increase almost linearly with the intensity of electric current in both cases, and in the case of circular current the excess volume per unit length of circuit increases monotonically with the diameter of the circuit. The discussions on the results will be given in the last section.

2. Formulation of the problem We take the x- and y-axes of a Cartesian coordinate system in the rigid bottom wall of the horizontal layer of magnetic fluid and the z-axis vertically upward (fig. 1). In quiescent magnetic fluid the distribution of the pressure p and the magnetization M are given by

p

-pgz

+

ofM d H -

1

~izoMH,

(2.1)

M = chMsL ( t z o m H / k T ) ,

(2.2)

where p is the density of the fluid, g the acceleration due to gravity, /x 0 the magnetic permeability of the vacuum, H the magnitude of magnetic field, ~b the volumetric concentration of magnetic particles dispersed in the fluid, M S the saturation value of the magnetization of magnetic particles in unit volume, L ( X ) is the Langevin function defined by coth(X) - X - 1, m is the magnetic moment of a particle, k the Boltzmann constant and T the temperature. Since the difference in pressure across the free boundary is balanced by the surface tension, the dynamical condition at interface is given by

2crK + pgz = ( c b M s k T / m ) log[(sinh X ) / X ] + pgh,

(2.3)

with the boundary condition

z=h

as

x 2 + y 2--'>oo,

where o" is the coefficient of surface tension, K the curvature of free surface, X = ~ o m H / k T , h the depth of the fluid layer far from the current and use has been made of eq. (2.2). In eq. (2.3) the term izoM2~/2 has been neglected in comparison with the term izofM dH where M n is the magnitude of magnetization normal to the interface #t In a strict approach to this problem, the magnetic field H is unknown a priori and it must be determined by Maxwell's equations (rot H = 0, div B = 0, and B = ~ 0 ( H + M)) and the conditions at the interface (continuities of the tangential component of H and of the normal component of B across the interface). In a fluid with small susceptibility M / H , however, the induced magnetic field is negligibly small everywhere, and we can neglect it in the evaluation Z

O

o i

Ih" r,×

IO Fig. 1. Configuration of the problem. #1 The term ~zoM2/2 is exclusively essential in the problem of uniform applied magnetic field in which the term tzofM dH has no contribution to eq. (2.3).

K Gotohet al. / Deformationof magneticfluidlayer

3

of M in (2.2) and (2.3). This treatment is called the induction free approximation, and by making use of this approximation the right-hand side of eq. (2.3) is reduced to a known function of position. In this paper we examine the magnetic field induced by (i) a straight line current along y-axis through the point (x, z ) = (0, - b ) , (ii) a circular current in the plane z = - b with diameter 2a and its center on z-axis, and (iii) a magnetic dipole situated at the point (x, y, z) = (0, 0, - b ) along the z-axis. In case (i) the magnetic field is given by

H = I / 2 w [ x 2 + ( z + b)2] 1/2,

(2.4)

where I is the intensity of electric current. In case (ii), by introducing cylindrical polar coordinates (r, O, z) we have (see Landau and Lifshitz, 1960)

Mr no=O

(

l(z+b) 2wrD

-K+

a2+r2+(z+b) 2 (a-r)2+(z+b)

2

E

) '

,

a2-r2-(z Hz - 2wD

+b) 2

K + (a-r)2+(z+b)2E)

'

(2.5)

where K = f0~/2(1 - k 2 sin 2 0) -1/2 dO,

(2.6)

E = f 0 / 2 ( 1 - k 2 sin 2 0) 1/2 dO,

(2.7)

k 2 = 4ar/D 2,

(2.8)

D=[(r+a)2+(z+b)

21 1 / 2

]

,

(2.9)

and in case (iii), which is the limiting case of a ~ 0 and a2I = 4mg ( = const.) in (ii), we have the dipole magnetic field,

H r = 3mg(Z + b ) r / [ ( z + b) 2 + r2] 5/2, 14o=0,

By assuming spatial symmetry of ~" and substituting the expression of the curvature into K in eq. (2.3), we have for the deviation ((x) or st(r) of the free surface from original level z = h the following equations: In case (i), t

(;'/¢1 + (,2) _ Bo ~'= - B o m log[(sinh X ) / X ] ,

(2.11)

with the boundary condition ~"(0) = ( ( ~ ) = 0,

(2.12)

4

K. Gotoh et al. / Deformation of magneticfluid layer

where

X = C / [ x 2 + (~ + b + 1) 2] 1/2,

(2.13)

C = IzornI/2whkT,

(2.14)

the pr,me denotes the differentiation with respect to x and ~', h and b have been made nondimensional by making use of h. The nondimensional parameters Bo and Bom are defined as the Bond number, Bo = pgh2/o ",

(2.15)

and the magnetic Bond number, Bom = c~MshkT/m~r.

(2.16)

In case (ii), the equation for ~'(r) is t

(r~,/i 1 + if,2)/r-

Bo ~ = - B o m log[(sinh X ) / X ] ,

(2.17)

with the same boundary conditions as (2.12), where the prime denotes differentiation with respect to r and the nondimensionalization has been made as in the case (i). The variable X is given by X = C[ Hr .2 + H z*2]'/2,

(2.18)

where ~'+b + 1 - K +

H~*=

rD*

1(

K+

D*

=

a2+r2(~+b+1)2 (a-r)2+(~+b+l)

E] 2 ]

a2-r2-(~+b+ l) 2 t 2 E ( a - r ) + ( ~ ' + b + l ) 2 )'

21 1/2

[(r + a ) 2 + ( ~ ' + b + 1) ]

'

(2.19)

with K from (2.6), E from (2.7) and k 2 = 4ar/D .2 instead of (2.8). In case (iii), the equation for ~" is the same as (2.17) but X is replaced by

X=Cd[r2+4(~ +b+ 1)2]'/2/[r2+(~+b+

1)2] 2,

(2.20)

where

C d = mmg/kTh 3 = wa2C/2h 2.

(2.21)

The nonlinear ordinary differential equations (2.11) and (2.17) can be solved by the conventional method of numerical integration.

3. Numerical calculation and results

Equations (2.11) and (2.17) are solved by a shooting method of integration as follows. For given values of Bo, BOm, b and C (or Cd), we guess a trial value of ~'0 of ~'(0), then integrate the equation by the fourth-order Runge-Kutta method with quadruple precision or the Adams-Bashforth method with double precision from x (or r ) = 0 where ~'(0)= if0 and

K. Gotoh et aL / Deformation of magnetic fluid layer Table 1 Precision of the initial values of integration

c

¢(0)

0.5 1.0 2.0 3.0

0.061437477197 0.204071488820 0.550150807980 0.889051273975

'(0) = 0. For the first guessed value of (0 the integration ~'(x) does not eventually converge to 0 even if the integration is p e r f o r m e d , forever. T h e n we adjust a n o t h e r value of ~r0 appropriately and carry out the integration again. This p r o c e d u r e is r e p e a t e d until ~'(x) converges to zero as x increases. T h e convergence d e p e n d s sensitively on the value of ~(0). T h e finally fixed values of it are t a b u l a t e d in table 1. In the axisymmetric cases (ii) and (iii), eq. (2.17) has a singularity at r = 0, and the p r o g r a m does not run as directly as in the case (i). This trouble is, however, easily o v e r c o m e by starting integration from r = Ar, instead of 0, with the starting values ~'(Ar) and ~r,(Ar) which can be evaluated analytically be expanding the solution into a Taylor series. Because of the slower decrease in the solution in case (i) utilization of greater precision than in the case (ii) or (iii) is needed. T h e step-size used in the execution is Ax = 10 -8 in most runs. In fig. 2 the shapes of the free surface are shown for various values of C or C d (and a in the case (ii)) and fixed values of other p a r a m e t e r s as Bo = 1.0, Bo m = 3.0 and b = 0.1. In the cases of small circuit (figs. 2b, 2c), the p e a k of the free surface ~'max a p p e a r s at r = 0 as in the cases of line current (fig. 2a) and magnetic dipole (fig. 2e). A shift of the p e a k a p p e a r s for rather large circuit, a > 3 w h e n C = 1.0 (fig. 2d). In fig. 2d the distribution of H along the free surface is depicted which shows that the p e a k of free surface shifts in accordance with that of the intensity of magnetic field there. T h e dashed curves in fig. 2e are r e p r o d u c e d from A g e e v et al. (1989). They solved eq. (2.17) with (2.20) and the b o u n d a r y conditions ~'(0)' = g'(3.0) = 0 by a finite-difference method. It is not clear in their p a p e r how the singularity was dealt with. Figure 2e shows that their assumption on the d i a m e t e r of the edge of elevation ( r = 3.0) is insufficient. T h e variations in ~'max and the excess volume V beyond the level z = h are depicted in fig. 3. T h e s e quantities increase as C 2 (or C 2) for fairly small values of C (or C d) as confirmed analytically by considering the linearized p r o b l e m of eq. (2.11) or (2.17). F o r m o d e r a t e values of C (or Cd), ~rmax increases almost linearly and for larger values m o r e slowly. Despite this fact, an almost exactly linear increase in V is evidenced by the result that rising part b e c o m e s fat as shown in fig. 4 where ~'(r)/~'ma x is plotted for various values of C. For the same value of,C, the d e f o r m a t i o n of the free surface covers a wider region in case (i) than in the other cases. This is a consequence from the asymptotic decay of the magnetic field; 1/x in case (i) and 1/r 3 in (ii) and (iii), and for this reason we had to use the greater precision in (i) m e n t i o n e d above. T h e variations in various quantities with the radius of circuit a are plotted in fig. 5, in which we present the centre level, ~(0), the peak, ~'max r e p r o d u c e d from fig. 2d, the excess volume per unit length of circuit, V / 2 w a , and the excess volume V itself. Figure 5 shows that ~'(0) and ~'max decrease as a increases over 2.6, while V and V / 2 w a increase monotonically, and the latter a p p r o a c h e s its asymptotic value. Therefore, the most effective contribution of the electric current to the excess volume is attained by the straight current. Practically 90% contribution is, however, obtainable by a circular current, for example a = 3.4 when C = 1.0.

6

K. Gotoh et al. / Deformation of magnetic fluid layer

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F \ 0 3 ~-------~-£~

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(e)

0.5

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l

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shapes

due to ( a ) s t r a i g h t c u r r e n t , (b, c, d ) c i r c u l a r c u r r e n t s , ( e ) m a g n e t i c dipole. ) ~'(r), ( - - - ) g ( r ) for C = 1.0 and the n u m b e r s a t t a c h e d to curves d e n o t e the values of a.

of the free surface

C = I~oml/2"rrhkT, C d = m m g / k T h 3 and X = I ~ o m H / k T . In (d), (

K. Gotoh et al. / Deformation of magnetic fluid layer

O.q 1.

(a)

q

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~

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a=o.5

1.2

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t

0.3

1.0 0.8

/ ' 1 0 / - / V / 7 0.2

O.q

0 ]

0.2 0.0

1

0

1.8

(~)~ I

'

i

,,I .... q

3

2

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'

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O- _~r'l

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a=l.0

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C Cd Fig. 3. Height of peak ~,,ax and excess volume V. C = # o m l / 2 w h k T and CO= m m g / k T h 3. (a) straight current, (b, c) circular currents, (d) magnetic dipole.

1.0

,~,,

. . . .

i

....

I

. . . .

I

]

l 0

r ( ")a

0.9

0

0.8

0

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O.B

0

O.S

~C=3

~ X

2

i

0 5

,

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,

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(b)

C=6

0 LI

O.q 0.3

0 3

0.5

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0.1 0.0

,

0 2 0 l I

0

I

0.0 5

10

lS

20

0

l

2

3

q

5

X

Fig. 4. Shapes of free surface normalized by ~'max"

C =

t~oml/2~rhkT. (a) straight current, (b) circular current,

8

K. Gotoh et aL / Deformation ofmagneticfiuid layer

4. Conclusion and discussion

In this paper we have investigated the shape of the free surface of a horizontal magnetic fluid layer in the presence of a magnetic field induced by electric current situated outside the layer (fig. 1). Numerical computation has been carried out for a straight line current and a circular current in the horizontal plane below the layer and the following results have been obtained. (1) A peak of free surface appears at center of symmetry when the radius of the circuit is small. In the case of a large circuit, the peak shapes a somma with the same radius as the circuit (fig. 2). (2) The height of the peak increases almost linearly with increasing electric current (fig. 3). (3) For strong magnetic field the major part of the free surface is similar in shape to the intensity distribution of the magnetic field on the free surface (fig. 2d). (4) The excess volume of fluid increases in proportion to the current intensity (fig. 3). (5) The area supporting the excess volume is practically kept constant (for fixed values of radius a in the case of a circular current) (fig. 4). (6) The excess volume of fluid increases monotonically with increasing a (fig. 5). (7) The excess volume per unit length of current also increases with increasing a (fig. 5), and reaches its maximum value in the case of a line current, but in practice it is obtained by the circuits of finite radii (in the case of C = 1.0 the maximum value is 0.88 and V/2~a = 0.79, 90% of the maximum, is realized for a = 3.4). It has been pointed out that the deformation of the free surface is closely related to the intensity distribution of the magnetic field along the deformed surface (fig. 2d). So it is interesting to know how much and where the effect of the surface tension appears. In order to make this point clear, we evaluate the magnitude of (BOm/BO) log[(sinh X)/X] along the deformed surface. The result is shown in fig. 6 in comparison with the full solutions. It shows no peculiar contribution of the surface tension; it lowers down the peak and elevates up the valley to minimize the area of the surfaces as an inherent effect of surface tension. In the low Bond number problem of extremely shallow layer, the effects of gravity, the surface tension and the magnetic force appear somewhat separately and the deformed free

0clo/

0.3

....

I ....

~

,

~

'

' I ....

0.q

t

'

I ~ I '

0.3

]

[ '

/

\

/

\

0.2

o. 1

.~

\

z

O. 1

0 . 0 ~

0

ih",'~'~

I l'l

1

I '

I l [ ....

2

[ ....

J I I I

3

q

El Fig, 5. V a r i a t i o n s in the c e n t e r level ~'(0), t h e p e a k ~'max, t h e excess v o l u m e V w i t h t h e results o f circuit a. ( 0 ) ~'(0), ( + ) r . . . . (zx) V/IO 2 a n d ( O ) V/lO'rca.

0.0

\\

I

1

,

I

2

~

I

3

,

I

q

,

1

S

,

I

6

,

I

7

r Fig. 6. C o n t r i b u t i o n o f t h e s u r f a c e t e n s i o n . D a s h e d curves are the values of (Bom/BO)log[(sinh X ) / X ] e v a l u a t e d o n t h e d e f o r m e d f r e e s u r f a c e w h e r e X is t h e f u n c t i o n o f p o s i t i o n g i v e n by (2.18) a n d (2.19) with C = 1.0 a n d a = 5.0.

K. Gotoh et aL / Deformation of magnetic fluid layer

9

surface is f o r m e d by t h r e e parts: t h e p e a k , the l o g a r i t h m i c tail and t h e g r a v i t y - d o m i n a t e d part. T h e analysis o f this p r o b l e m will b e r e p o r t e d in detail in o u r s e p a r a t e p a p e r .

Acknowledgement This w o r k has b e e n p a r t i a l l y s u p p o r t e d by G r a n t - i n - A i d for Scientific R e s e a r c h (A) 02302042 f r o m the M i n i s t r y of E d u c a t i o n , Science a n d C u l t u r e .

References Ageev, V.A., V.V. Balyberdin, I.I. levlev and V.I. Legeida (1989) Magnitnaya Gidrodinamika 25, 118-121 (Magnetohydrodynamics (1990) 25, 385) Landau, L.D. and E.M. Lifshitz (1960) Electrodynamics of Continuous Media (Pergamon Press, Oxford) p. 125. Rosensweig, R.E. (1985) Ferrohydrodynamics (Cambridge Univ. Press, Cambridge). Murakami, Y., T. Ohkita and K. Gotoh (1991) submitted to J. Phys. Soc. Japan.