CHAPTER-II

MODELLING OF

ELECTRORHEOLOGICAL FLUID

In this chapter a model is developed for electrorheological fluid

(ERF) suspension consisting of particles and a carrier oil which cap-

tures different features (cited in the first chapter). Here the ERF is

modelled in a homogenized sense within the framework of continuum

mechanics following the procedure from Rajagopal and Ruzicka (K.

R. Rajagopal, M. Ruizka 2001 [75]). Particularly the complex inter-

action of the electro-magnetic fields and the moving liquid is taken

into account, thus the electric field is treated as a variable that is

determined by Maxwell's equations. The final system describing the

motion of ERFs is derived from the general balance laws of ther-

modynamics and electrodynamics by a non-dimensionalization and

subsequent approximations which are realistic for electrorheological

fluids.

25

2.1 Modelling

Many electrorheological fluids (ERFs) are suspensions consist-

ing of particles and a carrier oil. These suspensions change their ma-

terial properties dramatically if they are exposed to an electric field.

The observed increase of the measured shear stresses (or the mea-

sured viscosity) is essentially due to the existence of particle struc-

tures forming in the presence of an electric field hindering the flow

and resulting in a higher, apparent viscosity. In this chapter attempt

is made to develop a model for electrorheological fluid which captures

the above described features. There are many ways to model ERFs

(K. R. Rajagopal, M. Ruizka, 2001(75]; M. Ruzika, 2000(791,[80]; W.

Eckart, 2000(35], [36]). Here we model the ERF in a homogenized

sense within the framework of continuum mechanics and follow the

procedure from Rajagopal and Ruzicka (K. R. Rajagopal, M. Ruizka

2001(75]). In particular we take into account the complex interaction

of the electro-magnetic fields and the moving liquid, thus treating the

electric field as a variable that is determined by Maixwell's equations.

The final system describing the motion of ERFs is derived from the

26

general balance laws of thermodynamics and electrodynamics by a

non-dimensionalization and subsequent approximations which are re-

alistic for electrorheological fluids.

We start by recording Maxwell's equations. Here we use the

so-called statistical formulation, which is based on a dipole-

current-loop model (A. C. Eringen, G. Maugin, 1989(37]; K. Hut-

ter, A. A. F. van de Ven,1978[52]; R. A. Grot, 1976[48l; Y. H. Pao,

1978[69l):

VxE=-f, (1)

V x H = - ^ + J, (2)

V.De = Qe (3)

V.B = 0 (4)

where E is the electric field, B the magnetic flux density, H is the

magnetic field given by H = /ig'^B — M with the magnetization

M, De is the dielectric displacement given by Dg = P H-eoE with

the electric polarization P, J the current density, Qe the density of

the free electric charges and SQ and ^o denote the dielectric constant

and the permeability in vacuum, respectively.

27

Now we state the thermo-mechanical balance laws. The balance of

mass and momentum respectively are

^ + ^V.v = 0 (5)

y9V-V.T = f + fe (6)

where the material time derivative is denoted by a superposed dot, p

is the mass density, T is the Cauchy stress tensor, f is the mechan-

ical force density and fg is the electro-magnetic force density. T is

introduced via t = T.n, where t is the Cauchy stress vector and n

the outer unit normal vector. Here we have also adopted the nota-

tion [Vv]w = (^j^)i=i,2,3? where the summation convention over

repeated indices is used. We will also use the convention through

out this thesis. The electro-magnetic force density is given by (Y. H.

Pao, 1978[69l)

fe = 9e^ + [ J + P - [VvlP + (V.v)P] X B + [VBi^M + [VE]V {7)

where £ is the effective electric field strength defined as

8 = E + vxB (8)

28

J the conductive current density given by

J = J - geV (9)

and M. the effective magnetization defined through

7W = M + v x P (10)

The balance of angular momentum takes the form

X X ^V - V.(X X T) = X X f + le (11)

in which Ig denotes the electro-magnetic torque density (Y. H. Pao,

1978[69]) given by

l e ^ x x f e + F x f ' + A ^ x B (12)

The balance of total energy takes the form

^Jt (^ "̂ h^-^ ^ ^- i^'^^ - q) + (f + fe).v + a; + ĉ e (13)

where e denotes the specific internal energy, q the heat flux, u the

mechanical energy production density and cjg the electro-magnetic

energy supply density which is given by (Y. H. Pao, 1978[69])

a;, = J.e + S.P - M.B + (£:.P)V.v (14)

29

Using equation (6) together with equation (14), we obtain from equa-

tion (13) the balance of internal energy as

ge + V.q = T.L + J.S + ^ .P - M . B + P.£:V.v + uj (15)

where L = Vv is the velocity gradient. Muller and his co-workers

have developed a general thermodynamic frame-work for studying

the thermodynamic response of materials that provides a system of

balance laws and the second law of thermodynamics(Mullar et al.,

1993 [68a]). The second law of thermodynamics in the form of the

Clausius-Duhem inequality (Mullar et al., 1993[68a]) is given by

^^ + V . | - ^ > 0 (16)

where 77 is the specific entropy and 9 the absolute temperature.

The set of equations(l)-(4), (5), (6), (15) and (16) which describes

the motion of the liquid has far more unknown quantities than equa-

tions. It is rendered determinate by providing appropriate constitu-

tive relations reflecting the material properties. The quantities

^ ,^ ,V0,v , D = i ( L 4 - L T ) , E , B (17)

can be assumed as the independent variables and thus it is tried to

30

find constitutive relations for

e,77,T,q,P,A^,J (18)

in the form

/ = / (^ ,^ ,V^,v,D,E,B) , (19)

where / stands for any of the quantities in (18).

Both the material and the balance equations are subject to invariance

requirements. It is well known that the mechanical balance laws (5),

(6) and (15) are form-invariant under Galilean transformations given

by

x* = Qx + Vot + bo, t * - t , (20)

where vo,bo are constant vectors and Q is a time independent or-

thogonal tensor, while Maxwell's equations (l)-(4) are form-invariant

under Lorentz transformations. We are interested in non-relativistic

effects and it is well-known that there are problems with consis-

tent invariance requirements for all thermo-mechanical and electro-

magnetic balance laws and constitutive equations in a non-relativistic

situation (R. A. Grot, 1976[48]; K. R. Rajagopal, M. Ruzicka, 2001[75l).

31

To avoid these difficulties we shall make the following invariance re-

quirements: We assume that the quantities (18), describing the ma-

terial properties, are invariant under Galilean transformations (A.

C. Eringen, G. Maugin, 1989[37]). Moreover we require that all

balance laws (5), (6), (15), (16) and (l)-(4) are form-invariant un-

der Galilean transformations (20). These two requirements imply

consistent transformation formulae for all necessary quantities (A.

C. Eringen, G. Maugin, 1989[37]). In particular, we obtain from

the invariance requirements that the constitutive relations (19) are

isotropic functions of their arguments and that (19) has to be re-

placed by (Grot, 1976[48])

/ = / (^ ,^ ,V^,D,E,B), (21)

where / stands for any of the quantities in (18). In addition to restric-

tions placed on the constitutive response functions by the invariance

requirements we have additional strictures due to the requirement

of the second law of thermodynamics. We shall now determine the

restrictions imposed by requiring that all admissible processes of the

body, i.e. processes compatible with the balance laws and the consti-

32

tutive response functions, meet the Clausius-Duhem inequality (16).

Introducing the specific Helmholtz (Mullar et al , 1993[68a]) free en-

ergy through

^ = e - 77̂ _ i f-.p, (22)

Q

and substituting it into (16) we obtain, with the help of the energy

balance (15) and the balance of mass (5), the dissipation inequality q V^

-^(1 + 7)9) + T.L - ^ ^ - £.F - M.B + J£>0 (23) u

From (21) and (22) we get that / = f{g, 6, V^, D, E, B). If we now

compute explicitly we can re-write (23), also using (5), as

Using the linearity of (24) with respect to the dotted quantities and

W and their independence on the arguments appearing in the consti-

tutive relations (21) one easily deduces (B. D. Coleman and W. Noll,

1963[15J; C. Truesdell and W. Noll, 1965(93]; R. A. Grot, 1976(481)

"^ 89' dV9 '

33

di) dip

^ = -M^ T̂ = T (25) dip

and the reduced dissipation inequality

(T + , ^ g l ) . D - 3 | ? + ^ . f > 0 (26)

where, 77, P and M. are functions of p, 9, S, and B only.

2.2 Electrorheological approximation

The equations derived in the last section may be simphfied

in view of electrorheological applications. Towards this end it is

preferable to carry out an appropriate non-dimensionalization with

a subsequent approximation. All assumptions made in this section

are based upon the understanding of the behaviour of ERFs, both

from the theoretical and experimental point of view (R. Blood worth,

1994[13]; R. Bloodworth, E. Wendt, 1996[14l).

Firstly, we shall assume that the Cauchy stress tensor T does not

depend on the magnetic flux density B, i.e.

T = t (^ ,^ ,V^,D,£) . (27)

34

This assumption reflects the observation that the material properties

of an ERF do not change if a magnetic field is applied, because surely

the particles in an ERF bear no magnetic properties. Secondly, we

shall assume that we are dealing with a dielectricum, i.e.

A^ = 0 where M^M-\-v xF (28)

It is worth noting here that this assumption ensures that an apparent

magnetization can only be generated by a moving polarized fluid

(Grot, 1976[48]). This common assumption is a crucial point for

deriving the so-called quasi-electro static equations. In view of (25)

the assumption (28) also implies that the Helmholtz free energy , and

thus also the polarization P and the entropy 77, are only functions of

Q, 9 and E.

Thirdly, we shall assume that the fluid is electrically non-conducting,

i.e.

J = 0 (29)

This assumption may not be fully justified in general, because some

ERFs exhibit a certain electrical conductivity which is often due to

the content of water. However, many of them are free of water and

35

have very low electrical conductivity (for example the polyurethane

dispersions), and thus we may restrict ourselves to such a class.

In order to reach the final electrorheological approximation and

to determine and retain terms that are dominant and discard oth-

ers that are insignificant, we will carry out a dimensional analy-

sis which follows closely the one in (K. R. Rajagopal and Ruzicka,

1996(74]; Ruzicka, 2000(80]). Let us introduce the following dimen-

sionless quantities which will be used in our calculation :

e ^ E - B q,

Eo Bo 0̂ .^ T _ V _ X

lo Vo IJO

to eo-ĉ o lo ^0 ^0

where the quantities with the subscript 0 are appropriate character-

istic quantities of the problem in question and dimensionless quanti-

ties and operators are denoted by a superposed bar. This form of the

non-dimensionalization was chosen in order to evaluate the relative

importance of the various terms tha t occur in the equation retaining

the stress tensor in generality. In typical problems and for many

ERFs (Bloodworth, 1994(13]; Bloodworth and Wendt, 1996(14]), we

36

envisage that

EQ ~ 3.(10^ - 10 )̂ V m-\ Vo ~ (10-3 - 1) m s-\

Lo - 5.(10-4 - 10-3), m 7?o - (10-2 - 10-^) kg (m s)-\

to ~ (10-3 - 1) s, 0̂ ~ 103 kg m-3. (31)

The time to may be either a characteristic electric or hydrodynamic

time, depending on the specific problem. Moreover, ô -̂nd T/Q are

the density and the dynamic viscosity of the fluid in the absence

of an electric field, respectively. Using (31), the Reynolds number

Re = (goLoVo)/r]o and the Strouhal number Str = Lo/{Voto) lie in

the range

5 . 10-3

that

IS;-^< '̂ (̂ )̂ resulting in

Bo ~ (10-^^ - 10-'") V s/m' (34)

Recalling that c^ 3 . 10^ ms~^ denotes the speed of electromagnetic

waves in vacuum we see that (33) is consistent with the assumption

that the magnetic flux density is only induced by oscillations of the

electric field and/or the motion of a polarized body. Let us introduce

a small non-dimensional number e through

€ = 10-^ (35)

which measures the importance of the terms. The situation de-

scribed above together with an assumption that there are only few

free charges in the fluid can thus be summarized as

ClQ C

^ = 0(e-^) - 0(.), * ^ = 0(. ' ) ,

U=0(.VO(.^), g^Od). (36) 38

The non-dimensionalized system of balance laws may then be approx-

imated by retaining terms up to order e^ , while neglecting terms of

higher order.

Firstly, let us discuss the role of S in the constitutive relations. It

follows from the definition of £ that

e = ^ = E + ^ v x B = E + 0(e^), (37)

where we used that

0{e') - 0{e') (38) VoBo

Thus, we can replace

where the quantity underbraced on the right hand side is of the order

ofe^

V.B = 0

V x B + ^ V x ( v x P ) = f A | ( E + P ) -

0(1) ^~o(xr

where the underbraced quantity on the left side and first underbraced

quantity on the right hand side are of the order of 1 and second

underbraced quantity on the right hand side is of the order of e^.

Where we also used the relation EQIIQ = c~ .̂

Neglecting terms of O(e^), we obtain the electrorheological ap-

proximation of Maxwell's equations according to

V.(eoE + P) = 0 (39)

V X E = 0 (40)

V.B = 0 (41)

l v x B + V x ( v x P ) . = ? i ^ ^ 5 ± ^ (42) /io ^ ' dt

where F = F{g,e,E).

40

Now we turn to the approximation of the thermo-mechanical bal-

ance laws. The conservation of mass (5) remains unaffected. In the

momentum equation (6) we re-write the electro-magnetic force fe

using (8), (28), (29) and then use (36) and (37), which leads to

eoEito^'di eoEi / \

QeE + - ^ geV X B

V J

^QEQ

+ l?^f xB ^Eoto^ dt

K)5, 0 + -ir^{[VP]v + (V.v)P X B -h V X ([VBJP)} + [VElP + O(e^)

0(e5)

(43)

We see that all underbraced terms on the right-hand side of (43)

have to be neglected. We shall retain the mechanical force term and

the term with the Cauchy stress. Furthermore, one easily computes

that

QoVpLc eoEoto

0(1) - 0(62) if E^^9x 10̂ 2 y2 ^-2^

0(e-^) - O(e^) if E^^9x 10̂ ° V^ m-^

0(e-2) - 0(1) if El^9x 10̂ V^ m'^ (44)

41

and

^ = 0(1) - 0{^) if El^Qx 10^21/2 ^-2^

= 0(e-^) - 0{e^) if El^9x 10̂ ° F^ ^-2^

= 0(e-2) - 0(1) if El^9x 10« V̂2 ^ - 2 4̂5̂

Therefore also the first and the second term on the left-hand side of

(43) have to be kept. With regard to the approximation of the other

thermo-mechanical non- dimensionalized equations, we only replace

^ by E since we have no indication of the behaviour of the other

quantities.

Therefore, the electrorheological approximation of the thermo-

mechanical balance laws is given by

g -H ^V.v = 0, (46)

^v - V.T = f + [V.EJP, (47)

c^ge - kAe - ( ^ . E + ^trl)^ 9 = {T - 7rI).D + a;, (48)

( T - 7 r I ) . D - - ^ ^ ^ > 0 , (49) u

where we used the definition of the specific heat Cy and of the ther-

42

modynamic pressure TT according to

5 ^ 2di{j

Moreover Ct,,P,7r and •0 are functions of Q^9 and E; while we have

for the Cauchy stress T = T(^, 0, VO, D, E).

2.3 Constitutive relations

Now we will develop a constitutive theory for ERFs. In order to keep

the already very long and complicated formulae as simple as possible

we keep the dependence on V^ only in the constitutive relation for

the heat flux q and assume that

q = -kV9, (50)

where the thermal conductivity /c is a positive constant. In all other

constitutive relations we drop the dependence on V^. We also restrict

ourselves to the case of an incompressible ERF, i.e.

trD = 0, (51)

and consequently we also drop the dependence on g in all consti-

tutive relations. Moreover we assume a linear dependence of the

43

polarization P on the electric field E, i.e.

P - XE(^)E, (52)

where XE is the dielectric susceptibility. The Cauchy stress can be

sphted according to T = -TTI + S. From the above assumptions and

(27) we get that the extra stress tensor S is of the form

S = S(^,D,E). (53)

From representation theorems (A. C. Eringen, G. Maugin, 1989[37))

it follows that the most general form for S is given by

S = a2E 0 E + QsD + a4D^ + a5(D.E O E + E 0 D.E) +

a6(D^E (8) E + E O D^E) (54)

where ai,i = 2,..., 6 may be functions of the invariants

e, |E|2, t r D ^ t r D ^ t r (DE®E) , tr(D2E(8)E). (55)

In view of certain peculiarities in the behaviour of the normal stress

differences in the case 0:4 7̂ 0 even in the absence of an electric

field (J. Malek, K. R. Rajagopal, and M. Ruzicka, 1995[65]) and due

to previous mathematical investigations for shear dependent viscous

44

fluids, which suggests that terms involving D^ can be treated as a

perturbation, we assume that

a4 = 0, as = 0 (56)

Based on experimental data (T. C. Halsey, J. E. Martin, and D.

Adolf, 1992[49]; T. Wunderlich, P. O. Brunn, 2000[97];) we assume

that in the presence and the absence of an electric field the ERF

behaves like a generalized Newtonian fluid with power p, where the

power p can depend on the magnitude of the electric field |E|^ .

Moreover, we restrict ourselves to the case that the material functions

Q!2,0:3 and as depend only on the invariants ^, |D|^ and |Ep and that

all terms have the same growth behaviour. Thus we deal with the

following model for the extra stress tensor S

S = a2i{(l + |D|2)(P-^)/^ - 1}E O E + (^31 + a33|E|2)

(1 + jD|2)(P-^)/^D + a5i(l + lD|2)(P-^)/^(DE 0 E + E O DE),

(57)

where aij are constants and p = p{\E\^) is a function such that

1 < Poo < P(|E|2) < p,. (58)

45

To ensure the validity of the Clausius-Duhem inequahty we further

require that the constant coefficients Q;̂ and the function p are such

that (M. Ruzicka, 2000[79l)

4 0̂ 31 > 0, Q33 > 0, ^33 + oQfsi > 0, (59)

4 k(PQ)\ 1 is a computable constant for

PQ > 3. It is to be noted here that these requirements ensure that

the operator induced by —V.S(D,E) is coercive.

In this chapter, we have developed the governing equations for

an electrorheological fluid within the context of the full electro-

dynamical theory, thus setting the stage for a detailed enquiry into

mathematical issues such as existence, uniqueness and stability of the

flow of such materials. General representations for the stress have

been derived and approximations have been provided based on cer-

tain growth conditions. A detailed non-dimensionalization has been

carried out, on the basis of which the various governing equations

for the mechanical and electrical fields have been simplified.

46