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On the viscosity of a magnetic fluid
~ Iolumd magnetiofsm um.mc ELSEVIER
Journal of Magnetismand MagneticMaterials 153 (1996) 359-365
materials
On the viscosity of a magnetic fluid D.K.
Wagh *, Ashima Avashia
Department of Applied Mathematics, Shri G. S. Institute of Technology and Science, lndore, (M.P.), India
Received23 May 1995
Abstract
The formula for the viscosity of a magnetic fluid is derived considering viscosity as transport of momentum. The formula gives the viscosity in terms of microscopic parameters and the magnetic field gradient.
I. Introduction
The continuum approach has been very successful, fruitful and responsible for advances in the study of solid and fluid mechanics. However, the basic hypothesis of the continuous nature of matter has always pinched natural philosophers who wish to understand matter and related phenomena in terms of molecules, atoms and the forces that bind them. The well established fact that matter is made up of atoms, molecules which are bound together by intra-molecular forces has always urged philosophers to explain the various phenomena in terms of microscopic properties and led to the development of statistical physics, which now constitutes an advanced and important discipline. History tells us of the impact of these studies in the realm of solid and fluid mechanics. For instance, in 1828 Cauchy [1] tried to explain the notion of stress in a system of material point and derived the expression for stress in terms of molecular forces. However, no attention was paid to this idea until 1971, when Paria [2] and Wagh [3] revived the idea and used it to investigate the effect of initial stress. Similarly, in fluid mechanics the concepts of viscosity and thermal conductivity have been interpreted as transport of momentum and transport of thermal energy caused by the Brownian motion of the molecules. These few attempts appear to have been unused side tracks in the progress of continuum mechanics. In Ref. [4] Wagh attempted to extend these side tracks to derive a formula for the viscosity of a suspension because suspension flows are very important in the context of present day technology. The formula is obtained by including the transport of momentum caused by solid particles. It gives the viscosity of suspension in terms of microscopic parameters which characterise the suspension. Einstein's famous result for the viscosity of a suspension may be deduced as a particular case of specific types of suspensions from this formula. The result obtained motivated us to investigate the effect of a magnetic field on the viscosity of a magnetic suspension (magnetic fluid). In the present paper we propose a formula for the viscosity of a magnetic fluid in presence of a magnetic field.
* Correspondingauthor.
0304-8853/96/$15.00 © 1996ElsevierScienceB.V. All rights reserved SSDI 0304-8853(95)00541-2
360
D.K. Wagh, A. Avashia /Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
2. Derivation o f the formula for the viscosity of a magnetic fluid Consider a magnetic suspension. When no magnetic field is applied, let C~ and C 2 denote the speeds of a fluid molecule and a magnetic particle, respectively, which are in Brownian motion. Then C 1 ~-- ~/U~ + O~ "~ W~
(2.1)
C 2 = ~/u~ + v~ + w 2 ,
(2.2)
and
(ul, v l, % ) and (u 2 , v 2, w 2) being the components of the velocity in the x, y, z directions, respectively. These components may take the values from - o c to + % and C~ and C 2 may take values between 0 and m. When a magnetic field is applied to this suspension, a magnetic force acts on the magnetic particles, in addition to the random force due to collisions with the fluid molecules as well as magnetic particles. As a result there is a net progress in the direction of the field. In other words, we can say that there is a drift superposed on their random motion. The average drift velocity due to the magnetic force has the components [5]
I% M 3H r 2 Ox
tZo M OH r 2
3y
m2
I*o M OH r 2
m2
OZ
m2
where m 2 is the mass of each suspended magnetic particle and r 2 is the average time between two successive collisions of the magnetic particles. M is the magnetic moment of each particle and H is magnetic field. Thus the speed C 2 of the magnetic particle under the influence of the magnetic field becomes,
=
- m2
+
3x
v2 + -
me
-~y ]
+
w2 + -
m2
/ ~z ] ]
(2.3)
or
,2 =
L -.1-
2 t*o M % OH
~W2
m2
m2
[
0x ]
+
t77yJ
+
t~-z ]
I
+ _ _ m2
u2 + _ _ OX
m2
v2
3y (2.4)
,
3Z
Taking the mean of C~2, we get
-C-~2=u2+v2+w2+L m2
l/tTxj
t yj t-gzjj
or
C--~2= ~22v/i + 6 ,
(2.5)
3 = t*°Mr~--~ --~-2 { ( 3 H / 3 x ) 2 + ( 3 H / 3 y ) 2 + (OH/3z)2}.
(2.6)
where
C2m2
(Since (u 2, v 2, w 2) varies from particle to particle; for some it is positive and for others negative. Hence the average contribution due to the last three terms is zero.) u 2, v 2, and w~, are the a v e r a g e / m e a n of the square
D.K. Wagh, A. Avashia / Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
361
velocities of the magnetic particles in the x, y and z directions, respectively, in the absence of a magnetic field.) The mean speed of the fluid molecule is C---~= ~
+ v~ + w~.
(2.7)
Let n~ and n 2 be the numbers of fluid molecules and solid particles per unit volume of the suspension, respectively. Since all fluid molecules move in a random manner in all possible directions, n J 6 fluid molecules per unit volume have the velocity C l in the + z direction, and an equal number of molecules per unit volume have the velocity C--~in the - z direction. Similarly, n 2 / 6 solid particles per unit volume have the velocity ~22 in the + z direction and n 2 / 6 solid particles per unit volume have the velocity ~ in the - z direction. Now consider the layer (plane) AB. The number of fluid molecules which in unit time cross a unit area of the plane AB from below and above is the same and is equal to (1/6)nlC- T. Similarly, the number of solid particles which in unit time cross the unit area of the plane AB from below and above is equal and is (1/6)n2~22. The definition of the mean free path, l, implies that the molecule which crosses the plane AB must have experienced, on the average, a preceding collision at a distance l from AB. Let l~ and 12 be the mean free paths of the fluid molecules and solid particles, respectively. If u denotes the mean velocity of the suspension in the x-direction at AB and u / z is the velocity gradient, then the fluid molecules at a distance I I from AB have the momentum ml(u + I l Ou/Oz) or ml(u - I l Ou/az) depending on whether they are above or below the plane AB. The solid particles at a distance 12 have the momentum m2(u + 12 Ou/Oz) or m2(u -- 12 Ou//aZ), depending on whether they are above or below the plane AB. Hence the mean x-component of the momentum transported per unit time per unit area across the plane AB in the upward direction is
~n,C---~lm,( u -- l, Ou/Oz ) + l n2C-~2m2( u -- 12 a u / O z ) .
(2.8)
Similarly, by considering the molecule coming from above the plane AB, we find that the mean x-component of the momentum transported per unit time per unit area across the plane in downward is
6n,C--~trnl( u + l, ~u/Oz) + ~n2-~22mz( u + 12 ~ u / O z ) .
(2.9)
By subtracting (2.9) from (2.8) we obtain the net molecular transport of the mean x-component of the momentum per unit time per unit area from below to above the plane, and this gives the shear stress ~x- Thus
tTzx = - ~Tsa u / a z ,
(2.10)
where r/s is the coefficient of viscosity of magnetic suspension, which is given by
7Is = l / 3 ( nl'C-~lmlll + n2-~22m212 ).
(2.11)
3. Calculation of the mean free path
The mean free paths II and l 2 may be calculated as follows. Let us consider a fluid molecule moving with a mean speed C¿. During its motion it will collide with other similar fluid molecules and also with the suspended solid particles. The total collision frequency is
f, = [4va2ntV~l + Tr( a + b)2nz~22 ] ,
(3.1)
where Vj and V 2 denote the mean relative speeds between fluid-fluid and fluid-solid particles, respectively. The first term of (3.1) gives the frequency of collisions between fluid-fluid molecules and the second term gives the frequency of collisions between fluid and solid particles.
D.K. Wagh,A. Avashia/Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
362
V 2 = (u~i + v l j + wlk ) - {(u 2 + tzoMTz/m 2 • OH/Ox)i + (v 2 + tzoMT"2/m2 • 8 H / 3 y ) j + (w2 + ~o M~2/m2" ~n/~z) k},
11,'212=
( u, -
u2 - tzoM~'z/m2 " 3H/Ox) 2 + ( O 1
-I-- ( W I - - W 2 - -
--
0 2 --
IxoM~'2/m 2 • OH~Or) 2
t z o M T - 2 / m 2 . 0 H / 0 z ) 2.
Hence, taking the ensemble mean, we get
V-7=u~ + v~ + w~ + u~ + ~ +-~2 +(I~oMr2/m2)E{(aH/ax) 2 + (aH/ay) 2 + (aH/az) 2} = c ~,+ c~ + (~oM-~2/mz)Z{(aH/ax) ~ + (OH/Oy) 2 + (aH/az)2}. Hence,
V~2=f~l + C ~ + ( I z o M r 2 / m 2 ) 2 { ( a H / 3 x ) 2 + ( a n / O y ) 2 + ( i ~ H / b z ) Z }
= ~f-~l +-~2 z .
(3.2)
Now, putting ~22 = Cl and H = 0, we get Vl = Cl~r2 •
(3.3)
Putting for V--~and V--2from (3.3) and (3.2) in (3.1) we get,
fl='rr[4~a2nlv/2+(a+b)Zn2~]. Hence,
l[
l I = C-~l/f I = 41/~-,rra2n I 1 - - ~ - { ( a
]
+ b ) / a } 2 n 2 / n l V/1 +~2/~212 ,
(3.4)
neglecting squares and higher powers of n2/n v Now, consider a solid particle moving with the average speed ~22. During its motion, it will collide with solid particles as well as fluid molecules. The total collision frequency is [6].
f2 = 4'rrbZnzC22 + ~( a + b ) 2nl-~21,
(3.5)
where C22 is the mean relative speed between solid-solid particles, and C2~ is the mean relative speed between solid-fluid particles. Now Czl will be the same as V2; thus 6"21 = v 2 = ~ Putting
C---~= ~
+C22.
(3.6)
into the above we get the mean relative speed C2--" ~ between solid-solid particles; thus
C2--'~- = ~22v/-2- .
(3.7)
Substituting the values of C2~ and C22 in the expression (3.5) for f2, we get
f2=4wb2n2-~2v~ +'rr(a+b)2n,-~2~l
+ (C-"~j2/C'~2).
(3.8)
Hence,
6 = G/f~
=
1 -- 47~" n---Z-z n,
+
?l
b2 +
1
(3.9)
D.K. Wagh, A, Avashia /Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
363
We neglect squares and higher powers of n J n I. Substituting the values of I l and 12 in expression (2.11) for rls, and neglecting squares and higher powers of n2/n ~, we get mtC-~l [ n2 C2 ( h 2 m 2 / m t - 1 ) ( l +6)-(-C-~tt/-~2) ] r/s= 1 2 - - ~ a 2 1 + , nl ~ A1/(1+ 6) +C~/C22
(3.10)
where
6=
C2m2
{ ( o . / O x ) 2 + (OH/Oy) + (OH/az) 2}
and A = 4vl-2a2/(a + b) 2. Now for a particle of mass m which is in Brownian motion in a fluid at absolute temperature T, the mean square speed is given by [6] C 2 = 3KT/m. In the approximate calculations of this paper we make no distinction between the mean speed C and the root-mean-square speed (C2) 1/2. Thus we can write
-C-- 375-g-f/m. Using this result we find that the mean speed C I of a fluid molecule is
Cl= and the mean speed C 2 of a solid particle is
c-S-Hence,
c2/c,= Hence the expression (3.10) for rls can be written as
or
12V~-r/s'tr a 2 n 2 m~-'~m ~ mlC.--T = "OR= 1 + -nl-
m2/ra,(A z - 1) + 6 ( A 2 m 2 / m l - 1 ) - 1 h~/l + m2/m j + 6
(3.12)
4. Conclusions
Eq. (3.10) gives the viscosity of suspension of a magnetic fluid in terms of the microscopic parameters n2/n l, C 2 / C t, m2/m l, h and 6. Eq. (3.11) gives the viscosity of magnetic fluid in terms of four parameters n2/n ~, m2/m l, A and 6, since C~2/C~ can be replaced by m J m 2, as explained in Section 3. The term ¢1 + 8 gives the effect of the magnetic field on the viscosity since 6 depends upon the magnetic field gradient. If the
D.K. Wagh, A. Avashia/ Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
364
14-
Cl
12-
~1-~1= 10"3 2 = 1/4
10-
8-
l .-
42-
C2 0
-2 -4
Fig. 1. Variation of ~u = (~R -- 1)500 with ~3. field gradient is function of space coordinates x, y and z, then "os will vary from point to point, i.e. it will not be uniform. H o w e v e r , if OH/ax, aH/Oy, and OH/Oz are constants (i.e. the field gradient is constan0 then "os is the same throughout the fluid. In the absence of a magnetic field (5 is zero, and Eq. (3.12) reduces to
n2 'OR=l+
mm]-~l "m2/ml(A 2 - 1) - 1
-
n,
~1 + m2/m l
This is the same as the viscosity of suspension (non-magnetic suspension) obtained by Wagh [4]. For a clean fluid n J n I = O; therefore
mlC-T 'os
12V~- rr a 2 •
This is same as the viscosity of a clean fluid obtained in the literature [6] on the basis of the m i c r o s c o p i c approach. Fig. 1 gives the variation of "OM= ('OR -- 1)500 with t~. The curves C1, C2, C3 and C4 correspond to A = 4, 2, 1 and 1 / 2 , respectively. For A = 4, 2, "OM is positive, i.e. 'OR > 1, whereas for A = 1 and 1 / 2 , 7/, is negative, i.e. 'OR < 1 Fig. 2 s h o w s the variation of "oo = ('OR -- 1)1000 with 8 for different m2/rn I. The curves C1, C2 and C3 12 11"
//~
CI ~.--4
10"
1 1 2 =10-3 rll
9" 8"
~.
543-
J
"
-
~
C2
2" I0
Fig. 2. Variation of "rb,= (~?R-- 1)1000 with 8.
D.K. Wagh,A. Avashia/ Journal of Magnetism and Magnetic Materials 153 (1996) 359-365 0.3500 0.3000 0.2500 70.2000
~
C1
365
~.=4
m2=1/4 ml
tOl,OO! C2 C3
o.oo~ Fig. 3. Variation of ~,~ =
(9~R --
1)/2 with 3.
correspond to m 2 / m I = 1/2, 1/4 and 1/9. When m 2 / m I = 1/9, "or increases up to ~ = 2, but after that it decreases and then again increases. Fig. 3 shows the variation of 'ow = ('Or- 1)/2 with 8 for different n 2 / n r The curves C1, C2 and C3 correspond to n 2 / n I = 1/10, 1/100 and 1/1000. As n 2 / n I decreases, the variation of viscosity becomes insignificant. This is obvious because higher values of n 2 / n ~ indicate higher concentrations of magnetic particles and therefore the effect of the magnetic field should be predominant.
Acknowledgements Second author expresses her sincere thanks to Dr D.K. Wagh for guidance in the preparation of this paper.
References [1] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. (Cambrige University Press, 1927); Exercise de Mathematique 3 0828) 213. [2] G. Paria, Ind. J. Eng. Math. I (1968) 121; Ind. J. Math. 9 (1967) 197; Ind. J. Eng. Math. II (1969) 7. [3] D.K. Wagh, Ind. J. Mech. Math. 6 (1968) 107; J. Sci. Eng. Res. 13 (1969) 209; Ind. J. Theor. Phys. 17 (1969) 79; Gerlands Beitrage zur Geophys. 794 (1970) 289; Pure Appl. Geophys. Switz. 82 (1970) 62; Pure Appl. Geophys. Switz. 99 (1972) 49; J. Math. Phys. Sci. 3 (1969) 336; Ind. J. Eng. Math. 1 (1968) 28; J. Ind. Math. Soc. 33 (1969) 165; Torsional waves in an elastic cylinder having Cauchy's initial stress; PhD Thesis, University of Indore, June 1969. [4] D.K. Wagh, Viscosity of suspension: A molecular approach (paper presented at ICMF VII, January 1995). [5] R.P. Feynman, Lectures in Physics, vol. l (Addison-Wesley, Reading, MA), ch. 43. [6] F. Reif, Statistical Physics, Berkeley Physics Course Vol. 5 (McGraw-Hill, New York).
Journal of Magnetismand MagneticMaterials 153 (1996) 359-365
materials
On the viscosity of a magnetic fluid D.K.
Wagh *, Ashima Avashia
Department of Applied Mathematics, Shri G. S. Institute of Technology and Science, lndore, (M.P.), India
Received23 May 1995
Abstract
The formula for the viscosity of a magnetic fluid is derived considering viscosity as transport of momentum. The formula gives the viscosity in terms of microscopic parameters and the magnetic field gradient.
I. Introduction
The continuum approach has been very successful, fruitful and responsible for advances in the study of solid and fluid mechanics. However, the basic hypothesis of the continuous nature of matter has always pinched natural philosophers who wish to understand matter and related phenomena in terms of molecules, atoms and the forces that bind them. The well established fact that matter is made up of atoms, molecules which are bound together by intra-molecular forces has always urged philosophers to explain the various phenomena in terms of microscopic properties and led to the development of statistical physics, which now constitutes an advanced and important discipline. History tells us of the impact of these studies in the realm of solid and fluid mechanics. For instance, in 1828 Cauchy [1] tried to explain the notion of stress in a system of material point and derived the expression for stress in terms of molecular forces. However, no attention was paid to this idea until 1971, when Paria [2] and Wagh [3] revived the idea and used it to investigate the effect of initial stress. Similarly, in fluid mechanics the concepts of viscosity and thermal conductivity have been interpreted as transport of momentum and transport of thermal energy caused by the Brownian motion of the molecules. These few attempts appear to have been unused side tracks in the progress of continuum mechanics. In Ref. [4] Wagh attempted to extend these side tracks to derive a formula for the viscosity of a suspension because suspension flows are very important in the context of present day technology. The formula is obtained by including the transport of momentum caused by solid particles. It gives the viscosity of suspension in terms of microscopic parameters which characterise the suspension. Einstein's famous result for the viscosity of a suspension may be deduced as a particular case of specific types of suspensions from this formula. The result obtained motivated us to investigate the effect of a magnetic field on the viscosity of a magnetic suspension (magnetic fluid). In the present paper we propose a formula for the viscosity of a magnetic fluid in presence of a magnetic field.
* Correspondingauthor.
0304-8853/96/$15.00 © 1996ElsevierScienceB.V. All rights reserved SSDI 0304-8853(95)00541-2
360
D.K. Wagh, A. Avashia /Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
2. Derivation o f the formula for the viscosity of a magnetic fluid Consider a magnetic suspension. When no magnetic field is applied, let C~ and C 2 denote the speeds of a fluid molecule and a magnetic particle, respectively, which are in Brownian motion. Then C 1 ~-- ~/U~ + O~ "~ W~
(2.1)
C 2 = ~/u~ + v~ + w 2 ,
(2.2)
and
(ul, v l, % ) and (u 2 , v 2, w 2) being the components of the velocity in the x, y, z directions, respectively. These components may take the values from - o c to + % and C~ and C 2 may take values between 0 and m. When a magnetic field is applied to this suspension, a magnetic force acts on the magnetic particles, in addition to the random force due to collisions with the fluid molecules as well as magnetic particles. As a result there is a net progress in the direction of the field. In other words, we can say that there is a drift superposed on their random motion. The average drift velocity due to the magnetic force has the components [5]
I% M 3H r 2 Ox
tZo M OH r 2
3y
m2
I*o M OH r 2
m2
OZ
m2
where m 2 is the mass of each suspended magnetic particle and r 2 is the average time between two successive collisions of the magnetic particles. M is the magnetic moment of each particle and H is magnetic field. Thus the speed C 2 of the magnetic particle under the influence of the magnetic field becomes,
=
- m2
+
3x
v2 + -
me
-~y ]
+
w2 + -
m2
/ ~z ] ]
(2.3)
or
,2 =
L -.1-
2 t*o M % OH
~W2
m2
m2
[
0x ]
+
t77yJ
+
t~-z ]
I
+ _ _ m2
u2 + _ _ OX
m2
v2
3y (2.4)
,
3Z
Taking the mean of C~2, we get
-C-~2=u2+v2+w2+L m2
l/tTxj
t yj t-gzjj
or
C--~2= ~22v/i + 6 ,
(2.5)
3 = t*°Mr~--~ --~-2 { ( 3 H / 3 x ) 2 + ( 3 H / 3 y ) 2 + (OH/3z)2}.
(2.6)
where
C2m2
(Since (u 2, v 2, w 2) varies from particle to particle; for some it is positive and for others negative. Hence the average contribution due to the last three terms is zero.) u 2, v 2, and w~, are the a v e r a g e / m e a n of the square
D.K. Wagh, A. Avashia / Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
361
velocities of the magnetic particles in the x, y and z directions, respectively, in the absence of a magnetic field.) The mean speed of the fluid molecule is C---~= ~
+ v~ + w~.
(2.7)
Let n~ and n 2 be the numbers of fluid molecules and solid particles per unit volume of the suspension, respectively. Since all fluid molecules move in a random manner in all possible directions, n J 6 fluid molecules per unit volume have the velocity C l in the + z direction, and an equal number of molecules per unit volume have the velocity C--~in the - z direction. Similarly, n 2 / 6 solid particles per unit volume have the velocity ~22 in the + z direction and n 2 / 6 solid particles per unit volume have the velocity ~ in the - z direction. Now consider the layer (plane) AB. The number of fluid molecules which in unit time cross a unit area of the plane AB from below and above is the same and is equal to (1/6)nlC- T. Similarly, the number of solid particles which in unit time cross the unit area of the plane AB from below and above is equal and is (1/6)n2~22. The definition of the mean free path, l, implies that the molecule which crosses the plane AB must have experienced, on the average, a preceding collision at a distance l from AB. Let l~ and 12 be the mean free paths of the fluid molecules and solid particles, respectively. If u denotes the mean velocity of the suspension in the x-direction at AB and u / z is the velocity gradient, then the fluid molecules at a distance I I from AB have the momentum ml(u + I l Ou/Oz) or ml(u - I l Ou/az) depending on whether they are above or below the plane AB. The solid particles at a distance 12 have the momentum m2(u + 12 Ou/Oz) or m2(u -- 12 Ou//aZ), depending on whether they are above or below the plane AB. Hence the mean x-component of the momentum transported per unit time per unit area across the plane AB in the upward direction is
~n,C---~lm,( u -- l, Ou/Oz ) + l n2C-~2m2( u -- 12 a u / O z ) .
(2.8)
Similarly, by considering the molecule coming from above the plane AB, we find that the mean x-component of the momentum transported per unit time per unit area across the plane in downward is
6n,C--~trnl( u + l, ~u/Oz) + ~n2-~22mz( u + 12 ~ u / O z ) .
(2.9)
By subtracting (2.9) from (2.8) we obtain the net molecular transport of the mean x-component of the momentum per unit time per unit area from below to above the plane, and this gives the shear stress ~x- Thus
tTzx = - ~Tsa u / a z ,
(2.10)
where r/s is the coefficient of viscosity of magnetic suspension, which is given by
7Is = l / 3 ( nl'C-~lmlll + n2-~22m212 ).
(2.11)
3. Calculation of the mean free path
The mean free paths II and l 2 may be calculated as follows. Let us consider a fluid molecule moving with a mean speed C¿. During its motion it will collide with other similar fluid molecules and also with the suspended solid particles. The total collision frequency is
f, = [4va2ntV~l + Tr( a + b)2nz~22 ] ,
(3.1)
where Vj and V 2 denote the mean relative speeds between fluid-fluid and fluid-solid particles, respectively. The first term of (3.1) gives the frequency of collisions between fluid-fluid molecules and the second term gives the frequency of collisions between fluid and solid particles.
D.K. Wagh,A. Avashia/Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
362
V 2 = (u~i + v l j + wlk ) - {(u 2 + tzoMTz/m 2 • OH/Ox)i + (v 2 + tzoMT"2/m2 • 8 H / 3 y ) j + (w2 + ~o M~2/m2" ~n/~z) k},
11,'212=
( u, -
u2 - tzoM~'z/m2 " 3H/Ox) 2 + ( O 1
-I-- ( W I - - W 2 - -
--
0 2 --
IxoM~'2/m 2 • OH~Or) 2
t z o M T - 2 / m 2 . 0 H / 0 z ) 2.
Hence, taking the ensemble mean, we get
V-7=u~ + v~ + w~ + u~ + ~ +-~2 +(I~oMr2/m2)E{(aH/ax) 2 + (aH/ay) 2 + (aH/az) 2} = c ~,+ c~ + (~oM-~2/mz)Z{(aH/ax) ~ + (OH/Oy) 2 + (aH/az)2}. Hence,
V~2=f~l + C ~ + ( I z o M r 2 / m 2 ) 2 { ( a H / 3 x ) 2 + ( a n / O y ) 2 + ( i ~ H / b z ) Z }
= ~f-~l +-~2 z .
(3.2)
Now, putting ~22 = Cl and H = 0, we get Vl = Cl~r2 •
(3.3)
Putting for V--~and V--2from (3.3) and (3.2) in (3.1) we get,
fl='rr[4~a2nlv/2+(a+b)Zn2~]. Hence,
l[
l I = C-~l/f I = 41/~-,rra2n I 1 - - ~ - { ( a
]
+ b ) / a } 2 n 2 / n l V/1 +~2/~212 ,
(3.4)
neglecting squares and higher powers of n2/n v Now, consider a solid particle moving with the average speed ~22. During its motion, it will collide with solid particles as well as fluid molecules. The total collision frequency is [6].
f2 = 4'rrbZnzC22 + ~( a + b ) 2nl-~21,
(3.5)
where C22 is the mean relative speed between solid-solid particles, and C2~ is the mean relative speed between solid-fluid particles. Now Czl will be the same as V2; thus 6"21 = v 2 = ~ Putting
C---~= ~
+C22.
(3.6)
into the above we get the mean relative speed C2--" ~ between solid-solid particles; thus
C2--'~- = ~22v/-2- .
(3.7)
Substituting the values of C2~ and C22 in the expression (3.5) for f2, we get
f2=4wb2n2-~2v~ +'rr(a+b)2n,-~2~l
+ (C-"~j2/C'~2).
(3.8)
Hence,
6 = G/f~
=
1 -- 47~" n---Z-z n,
+
?l
b2 +
1
(3.9)
D.K. Wagh, A, Avashia /Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
363
We neglect squares and higher powers of n J n I. Substituting the values of I l and 12 in expression (2.11) for rls, and neglecting squares and higher powers of n2/n ~, we get mtC-~l [ n2 C2 ( h 2 m 2 / m t - 1 ) ( l +6)-(-C-~tt/-~2) ] r/s= 1 2 - - ~ a 2 1 + , nl ~ A1/(1+ 6) +C~/C22
(3.10)
where
6=
C2m2
{ ( o . / O x ) 2 + (OH/Oy) + (OH/az) 2}
and A = 4vl-2a2/(a + b) 2. Now for a particle of mass m which is in Brownian motion in a fluid at absolute temperature T, the mean square speed is given by [6] C 2 = 3KT/m. In the approximate calculations of this paper we make no distinction between the mean speed C and the root-mean-square speed (C2) 1/2. Thus we can write
-C-- 375-g-f/m. Using this result we find that the mean speed C I of a fluid molecule is
Cl= and the mean speed C 2 of a solid particle is
c-S-Hence,
c2/c,= Hence the expression (3.10) for rls can be written as
or
12V~-r/s'tr a 2 n 2 m~-'~m ~ mlC.--T = "OR= 1 + -nl-
m2/ra,(A z - 1) + 6 ( A 2 m 2 / m l - 1 ) - 1 h~/l + m2/m j + 6
(3.12)
4. Conclusions
Eq. (3.10) gives the viscosity of suspension of a magnetic fluid in terms of the microscopic parameters n2/n l, C 2 / C t, m2/m l, h and 6. Eq. (3.11) gives the viscosity of magnetic fluid in terms of four parameters n2/n ~, m2/m l, A and 6, since C~2/C~ can be replaced by m J m 2, as explained in Section 3. The term ¢1 + 8 gives the effect of the magnetic field on the viscosity since 6 depends upon the magnetic field gradient. If the
D.K. Wagh, A. Avashia/ Journal of Magnetism and Magnetic Materials 153 (1996) 359-365
364
14-
Cl
12-
~1-~1= 10"3 2 = 1/4
10-
8-
l .-
42-
C2 0
-2 -4
Fig. 1. Variation of ~u = (~R -- 1)500 with ~3. field gradient is function of space coordinates x, y and z, then "os will vary from point to point, i.e. it will not be uniform. H o w e v e r , if OH/ax, aH/Oy, and OH/Oz are constants (i.e. the field gradient is constan0 then "os is the same throughout the fluid. In the absence of a magnetic field (5 is zero, and Eq. (3.12) reduces to
n2 'OR=l+
mm]-~l "m2/ml(A 2 - 1) - 1
-
n,
~1 + m2/m l
This is the same as the viscosity of suspension (non-magnetic suspension) obtained by Wagh [4]. For a clean fluid n J n I = O; therefore
mlC-T 'os
12V~- rr a 2 •
This is same as the viscosity of a clean fluid obtained in the literature [6] on the basis of the m i c r o s c o p i c approach. Fig. 1 gives the variation of "OM= ('OR -- 1)500 with t~. The curves C1, C2, C3 and C4 correspond to A = 4, 2, 1 and 1 / 2 , respectively. For A = 4, 2, "OM is positive, i.e. 'OR > 1, whereas for A = 1 and 1 / 2 , 7/, is negative, i.e. 'OR < 1 Fig. 2 s h o w s the variation of "oo = ('OR -- 1)1000 with 8 for different m2/rn I. The curves C1, C2 and C3 12 11"
//~
CI ~.--4
10"
1 1 2 =10-3 rll
9" 8"
~.
543-
J
"
-
~
C2
2" I0
Fig. 2. Variation of "rb,= (~?R-- 1)1000 with 8.
D.K. Wagh,A. Avashia/ Journal of Magnetism and Magnetic Materials 153 (1996) 359-365 0.3500 0.3000 0.2500 70.2000
~
C1
365
~.=4
m2=1/4 ml
tOl,OO! C2 C3
o.oo~ Fig. 3. Variation of ~,~ =
(9~R --
1)/2 with 3.
correspond to m 2 / m I = 1/2, 1/4 and 1/9. When m 2 / m I = 1/9, "or increases up to ~ = 2, but after that it decreases and then again increases. Fig. 3 shows the variation of 'ow = ('Or- 1)/2 with 8 for different n 2 / n r The curves C1, C2 and C3 correspond to n 2 / n I = 1/10, 1/100 and 1/1000. As n 2 / n I decreases, the variation of viscosity becomes insignificant. This is obvious because higher values of n 2 / n ~ indicate higher concentrations of magnetic particles and therefore the effect of the magnetic field should be predominant.
Acknowledgements Second author expresses her sincere thanks to Dr D.K. Wagh for guidance in the preparation of this paper.
References [1] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. (Cambrige University Press, 1927); Exercise de Mathematique 3 0828) 213. [2] G. Paria, Ind. J. Eng. Math. I (1968) 121; Ind. J. Math. 9 (1967) 197; Ind. J. Eng. Math. II (1969) 7. [3] D.K. Wagh, Ind. J. Mech. Math. 6 (1968) 107; J. Sci. Eng. Res. 13 (1969) 209; Ind. J. Theor. Phys. 17 (1969) 79; Gerlands Beitrage zur Geophys. 794 (1970) 289; Pure Appl. Geophys. Switz. 82 (1970) 62; Pure Appl. Geophys. Switz. 99 (1972) 49; J. Math. Phys. Sci. 3 (1969) 336; Ind. J. Eng. Math. 1 (1968) 28; J. Ind. Math. Soc. 33 (1969) 165; Torsional waves in an elastic cylinder having Cauchy's initial stress; PhD Thesis, University of Indore, June 1969. [4] D.K. Wagh, Viscosity of suspension: A molecular approach (paper presented at ICMF VII, January 1995). [5] R.P. Feynman, Lectures in Physics, vol. l (Addison-Wesley, Reading, MA), ch. 43. [6] F. Reif, Statistical Physics, Berkeley Physics Course Vol. 5 (McGraw-Hill, New York).