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Some remarks on the boundary conditions for magnetic fluids
iti. J. &gag Sci. Vol. M, No. 10, pp. 1451-1457, Printed in Great Britain. At1rightsreserved
1992
oO2wI225~92 $5.00 f 0.00 1992 PergamonPressLtd
Copyright@
SOME REMARKS ON THE BOUNDARY FOR MAGNETIC FLUIDS
CONDITIONS
P. N. KALONI Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada NQB 3P4 Abstract-Within the continuum description for the flow dynamics of magnetic fluids, the equations of motion are found to involve, apart from the magnetic field, a fluid velocity v and a particle rotation rate (spin) ar. By emplo~ng the standard “no slip” and “no spin” boundary conditions, theoretical behavior predicted from these equations, in a certain boundary value problem, is found to be inconsistent with the experimental results. In the present note, by applying variant boundary conditions, both for velocity and spin, to the same boundary value problem, we show that an agreement, between the theoretical predictions and experimental observations, is possible.
1. INTRODUCTION
A magnetic fluid is a colloidal solution, a stable suspension of fine ferromagnetic particles, such as magnetite in nonconducting liquids (Rosensweig [l]). These liquids are composed of small magnetic particles (about as many as lo’* in 1 cm”) coated with a molecular layer of a dispersant suspended in a liquid carrier. This coating prevents the particles from sticking to each other and the thermal agitation keeps the particles suspended because of the Brownian motion. These fluids are attracted strongly by an external magnetic field with forces that are large enough to overcome gravity. The magnetic force originates in the particles and is transmitted to the surrounding carrier liquid by viscous interaction. In the absence of an applied field, the particles in a colloidal magnetic fluid are randomly oriented and thus the fluid has no net magnetization, that is the magnetic moment of the unit volume is equal to zero. When such a fluid is placed in a homogeneous magnetic field, the latter causes a partial orientation of the magnetic moments. In variable fields, magnetization is relaxational. By assuming the magnetic fluid to be a homogenous isotropic continuum, the balance equations for mass, momentum and angular momentum have been proposed which are patterned after polar, micropolar, dipolar theories (Rosensweig [l,Z]). These equations are then supplemented by the constitutive equations for the stress, couple stress, magnetization, etc., and the Maxwell’s equation in the absence of electric currents. Rosensweig [2] list a combination of 41 equations for 41 unknowns. There is, however, little discussion concerning all the hydrodynamic and magnetic boundary conditions. Tbe complete study in this subject is then divided into two groups. The first one called equilib~um magnetization in which case the magnetic field intensity H is assumed parallel to the magnetization vector M. In this case the stress tensor becomes symmetric and the couple stress disappears altogether. In the second group, which is more realistic, one assumes that M X H does not vanish and this leads to the existence of a couple stress and asymmetric stress tensor. We are re-examining [3] the complete set of the above equations and the boundary ~nditions in the light of modern continuum mechanics arguments but our study to be reported here is not yet complete. In a recent article, Rosensweig et al. [4] have considered both theoretically and experimentally the magnetic fluid motion in a rotating field. The earlier experiments concerning this problem were carried out by Moskowitz and Rosensweig [5] and by Brown and Horsnell [6]. The interesting major finding of the latter authors was that when a beaker containing a ferrofluid was rotated in the direction of the field, the fluid inside the beaker rotated in the opposite direction. Rosensweig et al. [4] carried out various tests to confirm the above prediction and also did tests when the magnetic fluid is confined in the annular region between concentric cylinders. These authors also report theoretical calculations based upon the spin 1451
P. N. KALONI
1452
diffusion theory equations (which for completeness are presented in the next section), and the “no slip” and “no spin” boundary conditions. They concluded that “In spite of all the evidence, the spin diffusion theory suffers from a fatal flaw-we find that the direction of flow is the reverse in each instance from that which is predicted.” Rosensweig et al. [4] attributed tangential free-surface-stress as the dominant mechanism in the coupling of uniform rotary magnetic fields to the spin up motion of colloidal magnetic fluid. Our purpose, in this note, is to show that by employing the same set of equations of spin diffusion theory, but by appropriately applying new boundary conditions both for the velocity and the spin vector, the correct direction of flow, as observed in experiments, can be demonstrated.
2. BASIC
EQUATIONS
AND BOUNDARY
CONDITIONS
The continuum equations describing the flow dynamics of a magnetic fluid may be written as (Rosensweig [ 11):
$+v.(pv)=o
(1) (2)
$+V.(pw)=V*T+pF ~+v.(pv~)=V-c+A+pG.
(3)
Here p is the mass density, v the velocity, s the angular momentum density, T the stress tensor, F the external force per unit mass, C the couple stress tensor, G the body couple per unit mass and A is the skew symmetric part of the stress tensor. The field s is given by s = lo where UI is the angular velocity of the micro scale rotation of particles and I is the average moment of inertia per unit volume. We remark that the above set of equations is quite the same as that for micropolar materials developed by Eringen [7,8]. The constitutive equations for the field variables T, C and A are given as (Rosensweig [l]): T=-pI+A(trD)I+2qD-IXA-
[
iH’I+BH
1
C = n’(tr D’)I + 2q’D’ A = c(sL - 20),
where S2-VXv.
(4) (5) (6)
Here p is the pressure, 3Lthe bulk viscosity, tl the ordinary viscosity, 5 the rotational viscosity, il’ the bulk spin dilfusion coefficient, r,r’ the spin viscosity, H the magnetic field, B = &H + M) the magnetic induction, and M the magnetization. Moreover, G is given by G = PO(M X H),
(7)
and 2D = Vv + (Vv)‘,
2D’ = Vo + (VU)=.
(8)
When (4)-(8) are substituted in (2) and (3), the equations of motion take the form p~=-V~+pF+2~(VXo)+(l+q-C)VV*v+(<+~)Vt+/@IVH,
(9)
,oZz=2t(VXv-2w)+(I’+rj’)VVo+~‘V20,+@IXH.
(10)
Boundary conditions for magnetic fluids
1453
We again remark that the above equations, apart from the contribution due to the magnetic stress and magnetic couple terms, have the same structure as those in the theory of micropolar fluids (also referred to as polar fluids, fluids with antisymmetric stress and asymmetric hydromechanics) . The above equations are to be supplemented by Maxwell’s equations in the magneto-static limit and an equation accounting for the relaxation of M. However, since for the problem at hand the effect of H and M does not contribute to (9), and M X H in (10) takes a constant value for a given H, these equations will not be reported here. We, however, refer the reader to Refs [l, 21. The question of prescribing a physically well-founded boundary condition for the microrotation vector o is still being discussed in the polar fluid theory literature. Since the motivation and development of such theories are similar to those of magnetic fluid theory and since the structure of the equations of motion is similar in the two cases, we believe that the standard “no spin” boundary condition, so far used in magnetic fluid theories, should be applicable in only very limited cases. With regard to the boundary condition on the velocity v, most of the authors in all the above theories, however, have used the adhesion condition, implying the equality of v on a solid surface with the velocity of the boundary itself.? However, in view of the size of the magnetic particles we believe that the boundary condition on v also needs some modification. This is particularly so because the number of magnetic particles in a certain volume are comparable with the number of molecules of air in the same volume, and hence the molecular constitution cannot be ignored in seeking a boundary condition for v. We recall that Stokes, using a continuum approach and Navier, using a molecular approach, arrived at the same form of Navier-Stokes equations. In solving boundary-value problems in gas dynamics and in fluid dynamics we, however, use different boundary conditions. For the spin vector o the most general and useful boundary condition is that proposed by Aero et al. [ll]. It, however, involves a second rank tensor with nine parameters. However, because of the unknown nature of these parameters it becomes difficult to draw a definite conclusion by using this boundary condition. It is our belief that a simple boundary condition of the type
(11) would suffice for our purposes. A slight modification of (ll), of the form [CD]= ((u/2)(V x v) + S2(1- a), where B is the angular velocity of the boundary surface, has been suggested by Migun [ 121. For the velocity v we follow Lamb [13] ( as was done by Brunn [14]) and assume that slipping is resisted by a tangential force proportional to the relative velocity. Thus if V, be the velocity of the boundary surface and n be the outward drawn normal to the surface we require the following to hold on the boundary V*n=V,.n V-V,=$[nX(n*TXn)],
or in the components
tFor an interesting
form
iconoclastic
account,
however, see Brunn (91 and Brenner [lo].
(124
P. N. KALONI
1454
where v and T denote the far away velocity and stress field respectively and /3 is the coefficient of sliding friction.
3. MAGNETIC
FLUID
BETWEEN
TWO
COAXIAL
CYLINDERS
When the applied rotating fluid is spatially uniform and horizontally oriented, the solution of this problem by using “no slip, no spin” conditions has been given in [4]. However, for the reason stated earlier we believe that these boundary conditions are very much restricted. We assume cylindrically symmetric flow and let RI be the radius of the inner cylinder and R2 be that of the outer one. The vectors v and o in this case are 0 = o,(r)e,.
v = rM)e0,
. The differential equations (9) and (10) for an incompressible and negligible moment of inertia take the form
(13) magnetic fluid with no body force
04) (15) where we have written ve = IJand w, = w, and where pG = pJ4H sin (Ywith (Ythe lag angle of M relative to H. The solution of the simultaneous differential equations (14) and (15) can be written as 0
(rl+C)@ -
= -
E; u=pGr+C 4t)
Cl
+
+
C,&(kr)
+
(16)
C,K,(kr),
4tl r+C,+*=2 T 1 r
Wr) - ACJ k K,(b),
(17)
where C1, C2, C3, and C, are integration constants, Z,, Zr, K,, K1 are modified Bessel functions of the zero and first order and of the first and second kind respectively, and where A=sy5
,
k2=
4t7g
(18)
rl’(?l+ c3’
On applying the boundary conditions (11) and (12) we obtain following four equations for the four unknowns C,-C,:
PC(+r1+5
G
+ cl + C,&,(kR,) + C,K,(kR,)
= a[ $+
CI + + G&(kRd
+ ; G&(kRd], (19)
PG (+t1+5 + C, + C2&(kR,) + CJG(kR2) = a[ 5
G
@R, + C,Rl + 7
Z*(kR,) - 7
K,(kR,)
4rl
KGR2+C,Rz+ 411
*+
Z,(kR,) - 4 GKdkR2)
+ Cl + ; C&(kR,)
+ 8 W,,(kRz)],
Boundary conditions for magnetic fluids
The solution of (19)-(22)
1455
is
’
[A;'-A,']+P+Q
c, = -[A* + C’A’]
2,
c, = -[A, + CJ,]
2,
C
- A+
K,(kR,)]{
B + 2,
+ GRIP],
(23)
where A = [Zo(kR,)Ko(kR,) A, = A2 A’
=
- ~WWG(~RI)~~
K&R,) - K,(kR,),
= Z&R,)- k,(kR,),
(rl + Ml - 4 3’ + 50 - 4 ’
+$(y),
N =--
I
-- &‘K,(kR,) R: kRz
kR,
’
I I ’
(24)
*
Equation (23) and (24), when used in (16) and (17), give the complete microscale rotation component o and the velocity component u. These however, lengthy and complicated and we, therefore, consider the case cylindrical enclosure 0 I r I R. We remark that this latter case was studied
solutions for the expressions are, of the complete experimentally in
[5,61.
Thus by setting RI + 0, R2 = R, C4 = 0, and noting that K,,, KI are unbounded origin and hence that C3 = 0, we find PG[rl + 0 - 451
c1=-4rl
1
X(W
_
c
kRZ,(kR)
~G(tl+f)[As-(1-41 -
c *-
4?l
Zo(kR)
f
I[
2Z,(kR) -’ ” - (’ - cu)kRZ,(kR) I ’
near the
(25)
(26)
where 1
II+(l-df
=
(27)
5 1 +$rl)f [
(
1.
P. N. KALONI
1456
On substituting the above values in (16) and (17) we get
(29) where 2MkR) ’ = kRZ,(kR) ’
(30)
From (28) and (29) we note that the quantities pG, w, and u will be of the same sign provided each of the square brackets on the right side of these equations have positive values. A close examination of the expressions involved shows that this is not the case. If we define u=
2rl 1+-/3-l, R
(
>
a=-1
(31)
we note that, since C#J < 1, 0 5 (Y5 1, the quantity [A, - (1 - (Y)@]> 0 provided ;>
(1 - &)(9X7 - 1).
(32)
Since the magnitude of the viscosity n is always much larger than the vortex viscosity C;, we assume that (32) is always satisfied for reasonable value of u. In the no slip case we have @-’ = 0 and o = 1 and, therefore, (32) is always satisfied. Now, on rewriting (29) in a slightly different form we get
_
Irb?+(l--05‘m-l) 05;
.
(33)
II
The following observations are in order. First of all we note from (28) that w is not always in the same direction as pG, the applied field. In fact very close to the boundary when r = R, we have o negative, that is o is in the opposite direction of pG. However, when r decreases, (1 - Z,,(kr)/Zo(kR)) gradually increases and, eventually, w then becomes positive and attains the same direction as that of pG. We remark that this behavior is only possible if we discard the “no spin” boundary condition, i.e. assume LYf 0. A similar observation is true for the velocity u in (33). Here again near the boundary we again find v to be in the opposite direction to pG. As a matter of fact at the boundary r = R we find u = _
[rl + (1- a)tl(u 4rl[h - (I-
l)pGR L1_ ql +P,l
(34)
We note again that this opposite direction is possible only because of not employing the “no slip” condition. If we had assumed to slip, i.e. @-’ = 0, u = 1 we would have obtained, as expected, v = 0 as it should be.? The above is a sketchy explanation of the “Wrong Way Round” problem. We are in the process of numerically calculating v and w, for different possible values of the material tWhiIe presenting the paper at the Eringen Symposium, Dr T. Chang of M.I.T. pointed out that they have also observed similar opposite flow phenomenon in the kinetic theory of evaporation. Their paper [15] discusses the interesting inverted temperature profiles in the saturation vapor problem. Dr Chang also suggested for me to look at the non-local micropolar theory constitutive equations.
Boundary conditions for magnetic fluids
1457
parameters, for both cases of the flow in a circular vessel as well as for the flow in a circular annulus. We are very hopeful of explaining theoretically the complete experimental findings of Rosensweig et al. [4]. Acknowledgement-The work reported in this note was supported by the grant #A7728 of N.S.E.R.C. author gratefully acknowledges this support.
of Canada. The
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
R. E. ROSENSWEIG, Ann. Rev. Fluid Mech. 19, 437 (1987). R. E. ROSENSWEIG, Ferrohydrodynamics, p. 344. Cambridge University Press (1985). P. N. KALONI, Magnetic and electric effects on micropolar fluids. In preparation. R. E. ROSENSWEIG, J. POPPLEWELL and R. J. JOHNSTON, .r. Mugn. Magn. Mat. 85, 171 (1990). R. MOSKOWITZ and R. E. ROSENSWEIG, Appl. Phys. Lett. 11,301(1967). R. BROWN and T. S. HORSNELL, Electr. Rev. 183,235(1969). A. C. ERINGEN, ht. J. Engng Sci. 2,205 (1964). A. C. ERINGEN, I. Math. Mech. 16, 1 (1966). P. BRUNN, Rheol. Acta 14, 1039 (1975). H. BRENNER, Int. J. Engng Sci. 22,645 (1984). E. L. AERO, A. N. BULYGIN and E. V. KUVSHINSKII, Prikl. Mat. Mekh. 29,297 (1%5). N. P. MIGUN, Rheol. Acta 23,575 (1984). H. LAMB, Hydrodynamics, 6th edn, p. 576. Cambridge University Press (1932). P. BRUNN, Int. J. Multiphase Flow 7, 221 (1981). J. R. THOMAS JR., T. S. CHANG and C. E. SIEWERT, Phys. Reu. Len. 33, 680 (1974).
1992
oO2wI225~92 $5.00 f 0.00 1992 PergamonPressLtd
Copyright@
SOME REMARKS ON THE BOUNDARY FOR MAGNETIC FLUIDS
CONDITIONS
P. N. KALONI Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada NQB 3P4 Abstract-Within the continuum description for the flow dynamics of magnetic fluids, the equations of motion are found to involve, apart from the magnetic field, a fluid velocity v and a particle rotation rate (spin) ar. By emplo~ng the standard “no slip” and “no spin” boundary conditions, theoretical behavior predicted from these equations, in a certain boundary value problem, is found to be inconsistent with the experimental results. In the present note, by applying variant boundary conditions, both for velocity and spin, to the same boundary value problem, we show that an agreement, between the theoretical predictions and experimental observations, is possible.
1. INTRODUCTION
A magnetic fluid is a colloidal solution, a stable suspension of fine ferromagnetic particles, such as magnetite in nonconducting liquids (Rosensweig [l]). These liquids are composed of small magnetic particles (about as many as lo’* in 1 cm”) coated with a molecular layer of a dispersant suspended in a liquid carrier. This coating prevents the particles from sticking to each other and the thermal agitation keeps the particles suspended because of the Brownian motion. These fluids are attracted strongly by an external magnetic field with forces that are large enough to overcome gravity. The magnetic force originates in the particles and is transmitted to the surrounding carrier liquid by viscous interaction. In the absence of an applied field, the particles in a colloidal magnetic fluid are randomly oriented and thus the fluid has no net magnetization, that is the magnetic moment of the unit volume is equal to zero. When such a fluid is placed in a homogeneous magnetic field, the latter causes a partial orientation of the magnetic moments. In variable fields, magnetization is relaxational. By assuming the magnetic fluid to be a homogenous isotropic continuum, the balance equations for mass, momentum and angular momentum have been proposed which are patterned after polar, micropolar, dipolar theories (Rosensweig [l,Z]). These equations are then supplemented by the constitutive equations for the stress, couple stress, magnetization, etc., and the Maxwell’s equation in the absence of electric currents. Rosensweig [2] list a combination of 41 equations for 41 unknowns. There is, however, little discussion concerning all the hydrodynamic and magnetic boundary conditions. Tbe complete study in this subject is then divided into two groups. The first one called equilib~um magnetization in which case the magnetic field intensity H is assumed parallel to the magnetization vector M. In this case the stress tensor becomes symmetric and the couple stress disappears altogether. In the second group, which is more realistic, one assumes that M X H does not vanish and this leads to the existence of a couple stress and asymmetric stress tensor. We are re-examining [3] the complete set of the above equations and the boundary ~nditions in the light of modern continuum mechanics arguments but our study to be reported here is not yet complete. In a recent article, Rosensweig et al. [4] have considered both theoretically and experimentally the magnetic fluid motion in a rotating field. The earlier experiments concerning this problem were carried out by Moskowitz and Rosensweig [5] and by Brown and Horsnell [6]. The interesting major finding of the latter authors was that when a beaker containing a ferrofluid was rotated in the direction of the field, the fluid inside the beaker rotated in the opposite direction. Rosensweig et al. [4] carried out various tests to confirm the above prediction and also did tests when the magnetic fluid is confined in the annular region between concentric cylinders. These authors also report theoretical calculations based upon the spin 1451
P. N. KALONI
1452
diffusion theory equations (which for completeness are presented in the next section), and the “no slip” and “no spin” boundary conditions. They concluded that “In spite of all the evidence, the spin diffusion theory suffers from a fatal flaw-we find that the direction of flow is the reverse in each instance from that which is predicted.” Rosensweig et al. [4] attributed tangential free-surface-stress as the dominant mechanism in the coupling of uniform rotary magnetic fields to the spin up motion of colloidal magnetic fluid. Our purpose, in this note, is to show that by employing the same set of equations of spin diffusion theory, but by appropriately applying new boundary conditions both for the velocity and the spin vector, the correct direction of flow, as observed in experiments, can be demonstrated.
2. BASIC
EQUATIONS
AND BOUNDARY
CONDITIONS
The continuum equations describing the flow dynamics of a magnetic fluid may be written as (Rosensweig [ 11):
$+v.(pv)=o
(1) (2)
$+V.(pw)=V*T+pF ~+v.(pv~)=V-c+A+pG.
(3)
Here p is the mass density, v the velocity, s the angular momentum density, T the stress tensor, F the external force per unit mass, C the couple stress tensor, G the body couple per unit mass and A is the skew symmetric part of the stress tensor. The field s is given by s = lo where UI is the angular velocity of the micro scale rotation of particles and I is the average moment of inertia per unit volume. We remark that the above set of equations is quite the same as that for micropolar materials developed by Eringen [7,8]. The constitutive equations for the field variables T, C and A are given as (Rosensweig [l]): T=-pI+A(trD)I+2qD-IXA-
[
iH’I+BH
1
C = n’(tr D’)I + 2q’D’ A = c(sL - 20),
where S2-VXv.
(4) (5) (6)
Here p is the pressure, 3Lthe bulk viscosity, tl the ordinary viscosity, 5 the rotational viscosity, il’ the bulk spin dilfusion coefficient, r,r’ the spin viscosity, H the magnetic field, B = &H + M) the magnetic induction, and M the magnetization. Moreover, G is given by G = PO(M X H),
(7)
and 2D = Vv + (Vv)‘,
2D’ = Vo + (VU)=.
(8)
When (4)-(8) are substituted in (2) and (3), the equations of motion take the form p~=-V~+pF+2~(VXo)+(l+q-C)VV*v+(<+~)Vt+/@IVH,
(9)
,oZz=2t(VXv-2w)+(I’+rj’)VVo+~‘V20,+@IXH.
(10)
Boundary conditions for magnetic fluids
1453
We again remark that the above equations, apart from the contribution due to the magnetic stress and magnetic couple terms, have the same structure as those in the theory of micropolar fluids (also referred to as polar fluids, fluids with antisymmetric stress and asymmetric hydromechanics) . The above equations are to be supplemented by Maxwell’s equations in the magneto-static limit and an equation accounting for the relaxation of M. However, since for the problem at hand the effect of H and M does not contribute to (9), and M X H in (10) takes a constant value for a given H, these equations will not be reported here. We, however, refer the reader to Refs [l, 21. The question of prescribing a physically well-founded boundary condition for the microrotation vector o is still being discussed in the polar fluid theory literature. Since the motivation and development of such theories are similar to those of magnetic fluid theory and since the structure of the equations of motion is similar in the two cases, we believe that the standard “no spin” boundary condition, so far used in magnetic fluid theories, should be applicable in only very limited cases. With regard to the boundary condition on the velocity v, most of the authors in all the above theories, however, have used the adhesion condition, implying the equality of v on a solid surface with the velocity of the boundary itself.? However, in view of the size of the magnetic particles we believe that the boundary condition on v also needs some modification. This is particularly so because the number of magnetic particles in a certain volume are comparable with the number of molecules of air in the same volume, and hence the molecular constitution cannot be ignored in seeking a boundary condition for v. We recall that Stokes, using a continuum approach and Navier, using a molecular approach, arrived at the same form of Navier-Stokes equations. In solving boundary-value problems in gas dynamics and in fluid dynamics we, however, use different boundary conditions. For the spin vector o the most general and useful boundary condition is that proposed by Aero et al. [ll]. It, however, involves a second rank tensor with nine parameters. However, because of the unknown nature of these parameters it becomes difficult to draw a definite conclusion by using this boundary condition. It is our belief that a simple boundary condition of the type
(11) would suffice for our purposes. A slight modification of (ll), of the form [CD]= ((u/2)(V x v) + S2(1- a), where B is the angular velocity of the boundary surface, has been suggested by Migun [ 121. For the velocity v we follow Lamb [13] ( as was done by Brunn [14]) and assume that slipping is resisted by a tangential force proportional to the relative velocity. Thus if V, be the velocity of the boundary surface and n be the outward drawn normal to the surface we require the following to hold on the boundary V*n=V,.n V-V,=$[nX(n*TXn)],
or in the components
tFor an interesting
form
iconoclastic
account,
however, see Brunn (91 and Brenner [lo].
(124
P. N. KALONI
1454
where v and T denote the far away velocity and stress field respectively and /3 is the coefficient of sliding friction.
3. MAGNETIC
FLUID
BETWEEN
TWO
COAXIAL
CYLINDERS
When the applied rotating fluid is spatially uniform and horizontally oriented, the solution of this problem by using “no slip, no spin” conditions has been given in [4]. However, for the reason stated earlier we believe that these boundary conditions are very much restricted. We assume cylindrically symmetric flow and let RI be the radius of the inner cylinder and R2 be that of the outer one. The vectors v and o in this case are 0 = o,(r)e,.
v = rM)e0,
. The differential equations (9) and (10) for an incompressible and negligible moment of inertia take the form
(13) magnetic fluid with no body force
04) (15) where we have written ve = IJand w, = w, and where pG = pJ4H sin (Ywith (Ythe lag angle of M relative to H. The solution of the simultaneous differential equations (14) and (15) can be written as 0
(rl+C)@ -
= -
E; u=pGr+C 4t)
Cl
+
+
C,&(kr)
+
(16)
C,K,(kr),
4tl r+C,+*=2 T 1 r
Wr) - ACJ k K,(b),
(17)
where C1, C2, C3, and C, are integration constants, Z,, Zr, K,, K1 are modified Bessel functions of the zero and first order and of the first and second kind respectively, and where A=sy5
,
k2=
4t7g
(18)
rl’(?l+ c3’
On applying the boundary conditions (11) and (12) we obtain following four equations for the four unknowns C,-C,:
PC(+r1+5
G
+ cl + C,&,(kR,) + C,K,(kR,)
= a[ $+
CI + + G&(kRd
+ ; G&(kRd], (19)
PG (+t1+5 + C, + C2&(kR,) + CJG(kR2) = a[ 5
G
@R, + C,Rl + 7
Z*(kR,) - 7
K,(kR,)
4rl
KGR2+C,Rz+ 411
*+
Z,(kR,) - 4 GKdkR2)
+ Cl + ; C&(kR,)
+ 8 W,,(kRz)],
Boundary conditions for magnetic fluids
The solution of (19)-(22)
1455
is
’
[A;'-A,']+P+Q
c, = -[A* + C’A’]
2,
c, = -[A, + CJ,]
2,
C
- A+
K,(kR,)]{
B + 2,
+ GRIP],
(23)
where A = [Zo(kR,)Ko(kR,) A, = A2 A’
=
- ~WWG(~RI)~~
K&R,) - K,(kR,),
= Z&R,)- k,(kR,),
(rl + Ml - 4 3’ + 50 - 4 ’
+$(y),
N =--
I
-- &‘K,(kR,) R: kRz
kR,
’
I I ’
(24)
*
Equation (23) and (24), when used in (16) and (17), give the complete microscale rotation component o and the velocity component u. These however, lengthy and complicated and we, therefore, consider the case cylindrical enclosure 0 I r I R. We remark that this latter case was studied
solutions for the expressions are, of the complete experimentally in
[5,61.
Thus by setting RI + 0, R2 = R, C4 = 0, and noting that K,,, KI are unbounded origin and hence that C3 = 0, we find PG[rl + 0 - 451
c1=-4rl
1
X(W
_
c
kRZ,(kR)
~G(tl+f)[As-(1-41 -
c *-
4?l
Zo(kR)
f
I[
2Z,(kR) -’ ” - (’ - cu)kRZ,(kR) I ’
near the
(25)
(26)
where 1
II+(l-df
=
(27)
5 1 +$rl)f [
(
1.
P. N. KALONI
1456
On substituting the above values in (16) and (17) we get
(29) where 2MkR) ’ = kRZ,(kR) ’
(30)
From (28) and (29) we note that the quantities pG, w, and u will be of the same sign provided each of the square brackets on the right side of these equations have positive values. A close examination of the expressions involved shows that this is not the case. If we define u=
2rl 1+-/3-l, R
(
>
a=-1
(31)
we note that, since C#J < 1, 0 5 (Y5 1, the quantity [A, - (1 - (Y)@]> 0 provided ;>
(1 - &)(9X7 - 1).
(32)
Since the magnitude of the viscosity n is always much larger than the vortex viscosity C;, we assume that (32) is always satisfied for reasonable value of u. In the no slip case we have @-’ = 0 and o = 1 and, therefore, (32) is always satisfied. Now, on rewriting (29) in a slightly different form we get
_
Irb?+(l--05‘m-l) 05;
.
(33)
II
The following observations are in order. First of all we note from (28) that w is not always in the same direction as pG, the applied field. In fact very close to the boundary when r = R, we have o negative, that is o is in the opposite direction of pG. However, when r decreases, (1 - Z,,(kr)/Zo(kR)) gradually increases and, eventually, w then becomes positive and attains the same direction as that of pG. We remark that this behavior is only possible if we discard the “no spin” boundary condition, i.e. assume LYf 0. A similar observation is true for the velocity u in (33). Here again near the boundary we again find v to be in the opposite direction to pG. As a matter of fact at the boundary r = R we find u = _
[rl + (1- a)tl(u 4rl[h - (I-
l)pGR L1_ ql +P,l
(34)
We note again that this opposite direction is possible only because of not employing the “no slip” condition. If we had assumed to slip, i.e. @-’ = 0, u = 1 we would have obtained, as expected, v = 0 as it should be.? The above is a sketchy explanation of the “Wrong Way Round” problem. We are in the process of numerically calculating v and w, for different possible values of the material tWhiIe presenting the paper at the Eringen Symposium, Dr T. Chang of M.I.T. pointed out that they have also observed similar opposite flow phenomenon in the kinetic theory of evaporation. Their paper [15] discusses the interesting inverted temperature profiles in the saturation vapor problem. Dr Chang also suggested for me to look at the non-local micropolar theory constitutive equations.
Boundary conditions for magnetic fluids
1457
parameters, for both cases of the flow in a circular vessel as well as for the flow in a circular annulus. We are very hopeful of explaining theoretically the complete experimental findings of Rosensweig et al. [4]. Acknowledgement-The work reported in this note was supported by the grant #A7728 of N.S.E.R.C. author gratefully acknowledges this support.
of Canada. The
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