Or 2004 Master Project

8/6/2019 Or 2004 Master Project

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Homogenization of the Acoustical Wave

Propagation in Magnetorheological Fluids

O. Reese

April 30, 2004

Abstract

We formulate a model for acoustic excitations in a magnetorhe-

ological fluid. Constitutive equations are derived for Navier-Stokes

flow coupled with Maxwells Equations. The viscosity of the fluid is

modified to reflect the dependence of waves propagating within the

fluid itself and in the case where they propagate along the network of

particles.

1 IntroductionOne of the defining characteristics of fluids is the inability to carry acous-tical shear wave modes. To enable a fluid to carry more than one longitu-dinal mode one of its properties must be changed. It has been experimen-tally proven that with the addition of small particles two longitudinal modesmay propagate through a system provided that the wavelength is compa-rable to the size of the particles or smaller [10, 20]. In the past decaderesearchers experimenting with magnetorheological fluids discovered in thelarge wavelength regime two acoustical longitudinal waves of differing speedand amplitude propagating simultaneously [13]. From their experimental

findings Nahmad-Molinari et al were able to show a strong behavioral de-pendence upon an external magnetic field in both waves. The exact natureof these waves have been debated [2, 14, 7] and several models have been pro-posed [13, 3, 4] of varying degrees of computational complexity and accuracy.The motivation of this study is to produce equations using homogenization

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theory which correctly model the behavior of acoustical waves in a magne-

torheological fluid.Magnetorheological fluids (MR fluids) are typically classified as Bingham

plastics. In the presence of an external magnetic field the effective viscosityof the fluid changes. The alteration in viscosity is sudden but not permanentand quickly returns to initial conditions once the magnetic field is removed.The variation of the viscosity is a product of the arrangement of the mag-netically active particles due to dipole forces [17]. These particles arrangethemselves to form web like structures [17, 21] and it is along these structuresthat the second longitudinal mode travels [13].

The study of the nucleation rates and responses to magnetic fields has a

much larger background than the study of the acoustical wave propagation.The interest there was how to best incorporate these fluids into moderntechnological uses. Many of these types of simulations were based either ona modified Bingham plastic [19], on a minimum energy approach [12, 1] orincorporated static Maxwells Equations to explain particle trajectories [11].This study departs from these earlier works formulating our equations basedupon the conservation of mass and momentum coupled with time dependentMaxwells equations.

Homogenization techniques have been used in the past to study magne-torheological materials [8, 16] and the comparable electrorheological mate-rials [15, 18]. Our model is based in part on prior work with waves in sus-pensions of particles [6] and structural deformations in MR fluids [8]. Thewave motion is taken to be a plain wave eliminating the need to include timestepping algorithms into the implementation of the equations. However, dueto the oscillatory nature of the acoustical excitations within the MR fluid weare not able to assume that the magnetic properties of the particles may berepresented by static Maxwells equations as was done by Levy. Furthermore,since the amplitudes of the oscillations are comparably small we are able toneglect non-linear terms in the fluid motion.

Several studies have modelled acoustical waves in MR fluids by partition-ing the system into two components: the fluid itself and the rigid skeleton

formed by the particles [3, 4]. In this way they obtain two waves whosecharacteristics approximate experimental results. However, their method-ology does not reflect the dependence upon the magnetic field observed inthe longitudinal mode propagating in the fluid. A more natural approachis to examine the entire system as a whole, weakening the viscosity of thefluid when modelling the waves traveling along the magnetic particles. The

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justification for this technique relies on the assumption that the two modes

will collapse into a single mode in the absence of a magnetic field. As thispaper will show not only does this happen but our equations reduce to amore familiar form of wave motion for an anisotropic viscous fluid.

2 Formalization of the Problem

We consider a suspension of magnetizable rigid spheres in a viscous incom-pressible fluid under the influence of a magnetic inductance, B. For our pur-poses we assume that the magnetic permeability of the particles is linear anduniform, S, and the electric permitivity is likewise linear and uniform, s.

Furthermore the particles carry no free charge and any accumulation of suchcharges is quickly dispersed by the surrounding fluid. We also assume thatthe dielectric polarization of the particles matches that of the fluid, renderingforce contributions by the induced electric field negligable.

The magnitization of each particle, M, is known and the magnetic forceson each particle are composed of the volumic density force (M) B andthe surface density force (nM) B. Generally the term M is denotedJB, reflecting the fact that it represents the density of the bound current ineach particle. Likewise, nM is commonly referred to as the bound surfacecurrent density, KB. This nomenclature shall be used for the remainder of

the study.The oscillatory nature of the fluids velocity, V, is taken to follow a plane

wave, V(x, t) = V(x)eit. Due to the fact that the currents within eachparticle are bound, the time dependence of the magnetic and electric fieldsmay be expressed as B(x, t) = B(x)eit and E(x, t) = E(x)eit respectively.The wavelength of the acoustical excitations is assumed to be larger than theparticle size. Thermal considerations are also neglected.

In the domain we have

E = iB (1)

E = 0 (2) B = 0 (3)

In the fluid we have

V = 0 (4)

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iFVi =ij

xj(5)

B = i00E (6)

where

ij = ijP + 2Dij(V)

Dij(V) =1

2

Vixj

+Vjxi

In each particle, S we have

Dij(V) = 0 (7) B = 0 M + i0SE (8)

On the boundary S the velocity, V, is continuous. While the magnetic andelectrical fields satisfy the relations

(BF BS) n = 0

n (1

0BF (

1

0BS M)) = n M

(0EF SES) n = 0

(EF ES) n = 0

The balance of forces is then expressed as

i

SsVdv =

S

(M) Bdv +

S(n M) Bd

Sijnj eid (9)

i

Ss(x xG) Vdv =

S

(x xG) (( M) B)dv

+

S

(x xG) ((n M) B)d

S(x xG) (ijnj ei)d (10)

The suspension of particles is assumed to be non-dilute and of infinitedimensions. The equilibrium state shall be used as the reference configurationwith the particles being distributed within a locally periodic unit cell under a

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uniform pressure P0. The length of this unit cell, l, compared to the length of

the macroscopic phenomena, L, denoted = l/L is small 1. We imposea double scale assymptotic expansion on the spacial coordinates within theunit cell. We define y = (x xG)/, where xG is the center of mass of thesolid, to be the scretched coordinates. The derivative operator takes on theform

x

x+

1

y

The functions are expanded in powers of epsilon in the limit 0,

V = V0(x) + Vr(x, y) + V1(x, y)...

E = E

0

(x) + E

1

(x, y) + ...B = B0(x) + B1(x, y) + ...

P = P0(x, y) + P1(x, y) + ...

where the functions Vr, V1, B1, and E1 are Y-periodic in y. Excluding pres-sure, the zeroth order functions are not arbitrairily chosen to be independentof y. It will be proved later that this initial assumption is valid.

The nature of the viscosity and the role that the viscosity has in thedetermination of the solution of the problem is of paramount concern in thisstudy. We take the viscosity of the fluid to be a function of ,

() =

where is a positive number. The physical interpritation of this mathemati-cal definition is to say that the contribution of forces resulting from the fluidsviscosity is relatively small compared to the magnetic forces when > 0.

3 Homogenized Maxwell Equations

3.1 Determination of E0 and B0

From eq(3) at order O(

1) we have

(y) B0 = 0 (11)

in F from eq(6) at order O(1) we have

(y) B0 = 0 (12)

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and in S from eq(8) at order O(1) we have

(y) B0 = 0 (13)

which combined with the boundary conditions allow us to conclude thatB0(x, y) B0(x).

The form of the electric field may be deduced from Maxwells equationsin relation to the magnetic induction. From eq(1) at order O(1) and eq(2)at order O(1) we have

(y) E0 = 0 (14)

(y) E0 = 0 (15)

which combined with the boundary conditions allows us to conclude thatE0(x, y) E0(x).

3.2 Determination of B1

From eq(3) at order O(1) we have

B0 = (y) B1 (16)

in F from eq(6) at order O(1) we have

B0 = i00E0 (y) B

1 (17)

and in S from eq(8) at order O(1) we have

B0 = 0JB + i00E0 (y) B

1 (18)

This allows us to write B1 as

B1 = y( B0) y (i00E0 B0) + 0[ei JB]

i (19)

with i = 13YS(y ei) where YS is the charateristic function of YS. Thestrength of the first order expansion of the magnetic induction is dependentupon the frequency of the propagating wave indicating some amount of non-linearity intrinsic within the electromagnetic forces.

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The integral of B1 over the surface of the particles is computed using

basic vector identities applied to eq(19) where we have grouped commonterms together:

B1dy =

ij(B0)eidy + (0[JB ei] + 3i00[E

0 ei])

idy

with

ij(B0) =

yj

B0jxi

yiB0jxj

yjB0ixj

4 Wave propagation in a viscous fluid

The propagation of waves within the fluid is modelled as a suspension ofmagnetic particles immersed in a non-magnetic fluid. The viscosity of thefluid is taken to be of the same order as the magnetic forces, = O(1).

4.1 Determination of V0

We get from eqn(4) at order O(1) and eqn(5) at order O(2) in F

(y) V0 = 0 (20)

yjDij(V

0) = 0 (21)

In S from (8) at the order O(1) we have

Dij(y)(V0) = 0 (22)

From eqn(9) and eqn(10) at order O() we have

2

Dij(y)(V0)nj eidy = 0 (23)

2

y Dij(y)(V0)nj eidy = 0 (24)

This implies that for some function Uad = { [H1(Y)], Y-periodic,

= 0 in F, Dij() = 0 i,j in YS}:YS

Dij(y)(V0)Dij(y)(u)dy = 0 (25)

u

with u = + y in S, and independent of y. We therefore concludethat V0(x, y) V0(x) and Vr(x, y) 0

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4.2 Determination of V1, and P0

We may choose to normalize P0 such that

YF

P0dy = 0

and with careful selection of normalization constants any periodic extensioninto the solid can produce [5]

Y

P0dy = 0 (26)

However, we choose not to explore this option for reasons that will becomeobvious later on.From eq(5) and eq(6) in F

(y) V1 = (x) V

0 (27)

yj0ij = 0 (28)

0ij = ijP0 + 2(Dij(x)(V

0) + Dij(y)(V1)) (29)

and from eq(8) in SDij(y)(V

1) = Dij(x)(V0) (30)

From eq(9) and eq(10) at order O(2) combined with the conditions ofeq(27) through eq(30)

0ij nj eidy = ||(KB B0)

y (0ijnj ei)dy = (KB B0)

ydy

We express the velocity of the particles as

V1 = V1

+ V1

where V1 reflects the macroscopic strains and V1 reflects the magneticforces on the particles.

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The equations governing the solution of V1, derived from eq(27) through

eq(30), are

V0

= (y) V1 (31)

yj0ij = 0 (32)

Dij(V0) = Dij(y)(V

1) (33)YS

0ijnj eidy = 0 (34)YS

y (0ijnj ei)dy = 0 (35)

As in Levy [8] we find the local variation of V1 as

V1(x, y) = Dij(V

0)Xij(y)

where Y

Xij(y)dy = 0

Xij Uad(Pij) = { [H1(Y)]3, Y-periodic, = 0 in YF, Dkl() =

Dkl(Pij) k,l in YS} with P

ij such that Pijk = yjik, andYF

2Dkl(Xij)Dkl( X

ij)dy = 0

For y Ys:

Xij =1

2(Pij + Pji) + ij + ij y

with ij and ij independent of the local variable y.The equations governing the solution ofV1, derived from eq(27) through

eq(30), are

(y) V1 = 0 (36)

yj0ij = 0 (37)

0ij = ijP + 2Dij(y)(V1) (38)

Dij(y)(V1) = 0 (39)

2

0ijnj eidy = ||(KB B0)

2

y (0ijnj ei)dy = (KB B0)

ydy

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This leads to a solution where we are looking for Uad = { [H1(Y)],

Y-periodic, = 0 in F, Dij() = 0 i,j in YS} excluding constant vectors,and

2

YF

Dij(V1)Dij(u)dy =

(KB B0) u

u

where u = + y for y YS.The answer is

V1(x, y, t) = [ei (KB B0)]wi (40)

with wi satisfying

Y

widy = 0

wi

such that

YF

2Dkj (wi)Dkj (u)dy =

ei udy

u

Note that for y YS, wi = i + i y where i and i are independent of

y.The asymptotic Y-periodic expansion of V written in terms of V0, B0

and KB up to an additive constant is

V(x, t) = V0(x, t) + Dij(V

0)Xij (y) + [ei (KB B0)]wi(y)

+A(x, t)

+ O(1)

4.3 Macroscopic Equations

In order to establish the correct homogenization results on the boundary wemust take an additional expansion of Vk in the form of a Taylor series [9, 18]:

Vk(x) = Vk(xG) +Vk(xG)

xj(xj xG ej ) + ...

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We have in the fluid, YF,

V1 = (y) V2 (41)

iFV0 =

xj0ij +

yj1ij (42)

where1ij = ijP

1 + 2(Dij(V1) + Dij(y)(V

2))

In the solid, YS, we have

Dij(V1) = Dij(y)(V

2) (43)

with the balance of forces given as

i

YS

sV0dy =

YS

JB B0dy +

KB B1dy

(0ijxk

yk + 1ij)nj eidy (44)

i

YS

sy V0dy =

YS

y (JB B0)dy +

y (KB B1)dy

y (0ijxk


We integrate eq(42) on YF and add it to eq(44) to get

i

YV0dy =

xj

YF

0ijdyei

xk

0ijyknj eidy

+

YS

JB B0dy +

KB B1dy (46)

where we take

=|YS|

|Y|S +

|YF|

|Y|F

The average of the equations is taken

i < V0 > i < E0 >= xl

< 0kl > + < fl >

requiring a computation of an extension of the macroscopic tensor, the aver-age velocity,

< V0 >=1

|Y|

Y

V0dy (47)

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the average of magnetic forces minus the frequency dependent terms,

< fl > =1

|Y|([KB ei] el)

ij(B0)dy +

|YS|

|Y|(KB B

0) el

+1

|Y|0[JB ei]

(KB i) eldy (48)

and the average of the frequency dependent terms of the magnetic forces

< E0 >=3

|Y|00[E

0 ei]

(KB i)dy (49)

which may be reguarded as a slight pertubation due to the induced electricfield by the particle oscillations.

4.4 Extension of < 0ij >

From the previous section we must determine < 0kl >, let us again split itinto two components reflecting the contributions of the magnetic fields andthose of the fluid.

< 0kl >=< 0kl >Y + <

0kl >Y + <

0kl > + <

0kl >

with0kl = klP

0(x) + 2[Dkl(V0) + Dkl(y)(V

1)]

and0kl = klP

0(x) + 2Dkl(y)(V1)

Therefore the sum of the fluid components is expressed up to an additiveconstant

< 0kl >Y + < 0kl >= kl

0(x) + aijklDij(V0) (50)

where

aijkl = 2|Y|

YFDpq(y)(Pij Xij )Dpq(y)(Pkl Xkl)dy

+2

||

(ikml + Dim(y)(Xkl))yjnmdy

and (x) an additive constant.

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To calculate < 0kl > we must extend 0kl over the entire period. This

extension, 0kl, satisfies the following conditions

0kl = 0kl in YF (51)

(y) 0kl = 0 in YS (52)

0klnlek = 0klnlek (KB B

0) on (53)

The mean volumic value is then given as

0kl =1

|Y|

YF

0kldy +

Y S0kldy

We have

YS

0kldy =

YS

yj(0kj yl)dy

YS

0kjyj

yldy

= kl0

0kj njyldy

Using eq(53) this becomes

YS

0kldy =

0kj njyldy +

(KB B0) ekyldy

=

YF0kldy + |Y| < 0kl >Y +

(KB B0) ekyldy

Hence

0kl =< 0kl >Y +

1

|Y|

(KB B0) ekyldy (54)

Then we follow with a direct calculation of < 0kl >Y:

YF

0kldy = kl

YF

P0dy + 2

YF

Dkl(y)(V1)dy

= 2[(KB B0) ei]

YF

Dkl(y)(wi)dy

YS

0kldy =

YS

0pqDpq(Xkl)dy

=

0klXkl

p nqdy

=

YF

0pqDpq(Xkl)dy + [KB B

0]

Xkldy

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which gives

0kl = kl0 + (KB B

0) 2ei|Y|

YF

Dkl(y)(wi)dy

+(KB B0)

ei

|Y|

(Xkl ei)dy (55)

We then give a direct calculation of < 0kl >

< 0kl >=2ei||

Dkm(y)(wi)ylnmdy (56)

Summing our results gives

< 0kl >= kl0 + aijklDij(V

0) + (KB B0) bkl (57)

with the vector bkl defined

bkl =2ei|Y|

YF

Dkl(y)(wi)dy +

ei

|Y|

(Xkl ei)dy

+2ei||

Dkm(y)(wi)ylnmdy (58)

5 Wave Propagation in a Slightly Viscous Fluid

The propagation of waves along the web like structure formed by the particlesunder the influence of a magnetic induction immersed in a non-magnetic fluidis modelled as a suspension of interacting particles in a slightly viscous fluid.The macroscopic stress is taken to be of a higher order, = O() resultingin the magnetic forces dominating the behavior of the acoustical waves.

5.1 Determination of V0 and Vr

from eqn(4) at order O(1

) and eqn(5) at order O(2

) in F

(y) (V0 + Vr) = 0

yjij = 0

0ij = ijP0 + 2Dij(y)(V

0 + Vr)

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In S we from (7) at the order O(1)

Dij(y)(V0 + Vr) = 0 (59)

From eqn(9) and eqn(10) at order O(2) we have

2

Dij(y)(V0 + Vr)dy =

KB B0dy (60)

2

y Dij(y)(V0 + Vr)dy = (KB B

0)

ydy (61)

Then for some function Uad = { [H1(Y)], Y-periodic, = 0 in F,

Dij() = 0 i,j in YS } we have

2

YF

Dij(y)(V0 + Vr)Dij(y)(u)dy =

(KB B0) udy (62)

u

with u = + y in S, and independent of y. We choose V0 to bethe terms which are independent of y. Hence eq(62) becomes by the homo-geneous nature of the suspension and the condition that V0 be independentof the local variable y:

2 YF

Dij(y)(Vr)Dij(y)()dy =

(KB B0) udy

This implies that Vr can be expressed as

Vr(x, y) = [ei (KB B0)]wi(y) (63)

with wi satisfying

Ywidy = 0

wi such that

YF

2Dkj (wi)Dkj (u)dy = ei

udy

u

Note that for y YS, wi = i + i y where i and i are independent of

y.

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5.2 Macroscopic Equations

We have in the fluid, YF,

V1 = (y) V2 (64)

iF(V0 + Vr) =

xj0ij +

yj1ij (65)

where1ij = ijP

1 + 2(Dij(V0 + Vr) + Dij(y)(V

1))

In the solid, YS, we have

Dij(V1) = D

ij(y)(V2) (66)

with the balance of forces given as

i

YS

s(V0 + Vr)dy =

YS

JB B0dy +

KB B1dy

(0ijxk


i

YS

sy (V0 + Vr)dy =

YS

y (JB B0)dy +

y (KB B1)dy

y (0ij

xk


We integrate eq(65) on YF and add it to eq(67) to get

i

Y(V0 + Vr)dy =

xj

YF

0ijdy +

xk

0ijyknj eidy

+

YS

JB B0dy +

KB B1dy (69)

where is previously defined.The average of the relative velocity is given by

< V

r

> =

1

|Y|

Y V

r

dy= 0 (70)

so that the macroscopic equation can be expressed as

i < V0 > i < E0 >=

xl< 0kl > + < fl > (71)

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where < fl > is the average frequency independent magnetic force contribu-

tions as defined in eq(48) and < E0 > is the average frequency dependentmagnetic force as define in eq(49).

5.3 Extention of < 0ij >

To calculate < 0kl > we must extend 0kl over the entire period. This exten-

sion, 0kl, satisfies the following conditions

0kl = 0kl in YF (72)

(y) 0kl = 0 in YS (73)

0klnlek =

0klnlek (KB B

0

) on (74)

The mean volumic value is then given as

0kl =1

|Y|

YF

0kldy +

Y S0kldy

We have

YS

0kldy =

YS

yj(0kj yl)dy

YS

0kjyj

yldy

=

0kj njyldy

Using eq(74) this becomes

YS

0kldy =

0kj njyldy +

(KB B0) ekyldy

=

YF

0kldy + |Y| < 0kl > +

(KB B0) ekyldy

Hence

0kl =< 0kl > +

1

|Y|

(KB B0) ekyldy (75)

Then we follow with a direct calculation of 0kl:

YF

0kldy = kl

YF

P0dy + 2

YF

Dkl(y)(Vr)dy

= klpi0 2[(KB B

0) ei]

YF

Dkl(y)(wi)dy

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YS

0kldy =

YS

0pqDpq(Xkl)dy

=

0klXkl

p nqdy

=

YF

0pqDpq(Xkl)dy + [KB B

0]

Xkldy

From this we say

0kl = kl0 + (KB B

0) 2ei|Y|

YF

Dkl(y)(wi)dy

+(KB B0)

ei

|Y|

(Xkl ei)dy

(76)

Using the result of < 0kl > in the previous section we may express themacroscopic stress tensor as

< 0kl >= kl0 + (KB B

0) bkl (77)

with the vector bkl defined

bkl =2ei|Y|

YF

Dkl(y)(wi)dy +

ei

|Y|

(Xkl ei)dy

+2ei||

Dkm(y)(w

i

)ylnmdy

6 Discussion of Results

The propagation of acoustical waves in a magnetorheological fluid is char-acterized by the macroscopic stress tensor, a symmetric tensor, plus someadditional force terms resulting from the electromagnetic fields. The volumefraction of the particles, |YS|/|Y|, and the geometry of the system plays animportant role in determining the constants of the constitutive equations.In the case of a viscous fluid the equations are eq(46), eq(57), eq(48), andeq(49)

i < V0 > i < E0 >=

xl< 0kl > + < fl >

where< 0kl >= kl

0 + aijklDij(V0) + (KB B

0) bkl

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We find that there is one dispersive, disipative wave within the fluid, magnetic

forces working against the waves propagation. In the abscence of a magneticfield the average electromagnetic force terms and the magnetic contributionof the macroscopic stress tensor disappear leaving an equation for acousticwaves propagating in a viscous anisotropic fluid.

In the case of a slightly viscous fluid the equations are eq(71) and eq(77),eq(48), and eq(49).

i < V0 > i < E0 >=

xl< 0kl > + < fl >

where

< 0kl >= kl0 + (KB B0) bkl

We find that there is one dispersive, disipative wave present which is predom-inately affected by the electromagnetic forces and lacks the standard viscoustensor. In the abscence of a magnetic field the right side of the equationreduces to a pressure term which may be selected so that it too disapears.Therefore, we conclude that this precludes a second longitudinal mode trav-elling along the network of particles unless an external magnetic induction isapplied to the system.

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