Flow of an electrorheological °uids between eccentric ...ncmm.karlin.mff.cuni.cz/preprints/10206125456pr25.pdf · Flow of an electrorheological °uids between eccentric rotating

Nečas Center for Mathematical Modeling

Flow of an electrorheological fluidsbetween eccentric rotating cylinders

Vı́t Pr̊uša and K. R. Rajagopal

Preprint no. 2010-025

Research Team 1Mathematical Institute of the Charles University

Sokolovská 83, 186 75 Praha 8http://ncmm.karlin.mff.cuni.cz/

FLOW OF AN ELECTRORHEOLOGICAL FLUID BETWEEN ECCENTRIC ROTATING CYLINDERS

VÍT PRŮŠA AND K. R. RAJAGOPAL

Dedicated to professor R. R. Huilgol on the occasion of his seventieth birthday.

Abstract. Electrorheological fluids have numerous potential applications in vibration dampers, brakes, valves, clutches, exerciseequipment, etc. The flows in such applications are complex three dimensional flows. Most models that have been developedto describe the flows of electrorheological fluids are one dimensional models. Here, we discuss the behaviour of two fully threedimensional models for electrorheological fluids. The models are such that they reduce, in the case of simple shear flows with theintensity of the electric field perpendicular to the streamlines, to the same constitutive relation, but they would not be identical inmore complicated three dimensional settings. In order to show the difference between the two models we study the flow of thesefluids between eccentrically placed rotating cylinders kept at different potentials, in the setting that corresponds to technologicallyrelevant problem of flow of electrorheological fluid in journal bearing. Even though the two models have quite a different constitutivestructure, due to the assumed forms for the velocity and pressure fields, the models lead to the same velocity field but to differentpressure fields. This finding illustrates the need for considering the flows of fluids described by three dimensional constitutivemodels in complex geometries, and not restricting ourselves to flows of fluids described by one dimensional models or simple shearflows of fluids characterized by three dimensional models.

Contents

1. Introduction 12. Governing equations 33. Constitutive relations for the Cauchy stress tensor 34. Simple shear flows 44.1. Plane Poiseuille flow 44.2. Cylindrical Couette flow 45. Flow between eccentric cylinders 55.1. Dimensionless form of the governing equations 55.2. Bipolar coordinates 65.3. Explicit analytical solution for the electric field 85.4. Formulae for the divergence of the Cauchy stress tensor 85.5. On the influence of various terms on the velocity field 85.6. On the mean normal stress, the pressure and the Lagrange multiplier p 95.7. Numerical solution of the governing equations 96. Results 106.1. Convergence of the numerical method 116.2. Dependence on µ1 116.3. Dependence on ε̂0εr 116.4. Dependence on Re 117. Conclusion 14References 15

1. Introduction

Electrorheological fluids are fluids that change their material properties on the application of an external electric field. Thefact, that the material properties (particulary the viscosity) can be controlled by the external electric field leads, as it wasshown in the pioneering work by Winslow (1949), to many potential applications in industrial devices such as clutches, brakesand dampers, to name a few. The potential applications of the electrorheological fluids has led to intensive research in this fieldboth with regard to the manufacture of such fluids and the development of mathematical models to describe their response.

Date: July 23, 2010.

2000 Mathematics Subject Classification. 76W05, 65N35.Key words and phrases. electrorheological fluids, constitutive modelling, numerical simulation.Vı́t Pr̊uša thanks the Nečas Center for Mathematical Modeling (project LC06052 finaced by the MŠMT of the Czech republic) for its support.

1

2 VÍT PRŮŠA AND K. R. RAJAGOPAL

Since electrorheological fluids are usually suspensions of electrically active particles (typically zeolite or silica gel particles)in a non-conducting fluid (typically silicone or mineral based oils), it is possible to try to model the fluids using mixturetheory, see for example Rajagopal et al. (1994). This approach is however usually not preferred as there are problems withspecifying boundary conditions and the system of equations is also considerably complicated, and thus electrorheological fluidsare modelled as a homogenized single continuum. In such an approach, if one wants to fully describe a given fluid, it is sufficientto know a single tensorial constitutive relation of type (in the most simple cases)

f(T, D,E) = 0. (1.1)

that specifies the relation between the Cauchy stress T, the symmetric part of the velocity gradient D and the intensity of theelectric field E. (The boundary condition is also in general a constitutive specification that needs to be provided; it is howeverusually automatically assumed that the velocity field satisfies the no-slip boundary condition, and the “constitutive” nature ofthe boundary condition is ignored.)

Concerning constitutive equations for electrorheological fluids have been suggested by the experimentalists themselves, seefor example Shulman et al. (1989) or Choi et al. (2005) and references therein. In these works, however, the constitutive relationtakes the form of a scalar relation between the shear stress, shear rate and the intensity of the electric field—the reason is thatthe experiments are usually designed in such way that the fluid behaviour is investigated in a simple shear flow and the intensityof the electric field is perpendicular to the streamlines, and, in such a simple setting, only a scalar relation between the abovementioned quantities is necessary. Consequently, the constitutive theory is essentially one-dimensional and a generalization toa fully three-dimensional setting is not obvious in the sense that there exist many three-dimensional constitutive relations thatcan—in simple shear flows—lead to the same relation between the Cauchy stress, the shear rate and the intensity of the electricfield—a particular example is given in the present work.

On the other hand, it is possible to apply the classical tools that have been developed to generate constitutive relations(see for example Truesdell and Noll (1965) and Spencer (1971)) to derive fully three dimensional constitutive relations forelectrorheological fluids, see for example Wineman and Rajagopal (1995) and Rajagopal and Růžička (1996, 2001). Having aclass of three dimensional constitutive relations will allow one to assess the validity and usefulness of such models, and we dothis by considering special flows that correspond to flows in domains that are relevant to experiments. For instance, one canuse these constitutive relations to determine fluid behaviour in simple shear flows, viscometric flows and nearly viscometricflows1. We need to be in a position to differentiate between competing models and thus if different constitutive relationsare indistinguishable in simple shear flows, then it is necessary to find a different experimental setting that would allow usto differentiate between the constitutive relations that are indistinguishable in shear flows, and consequently would allow theexperimenter to decide which constitutive relation is the right one for description of the fluid in complicated flows that gobeyond simple shear flows.

In the present work we study the flow of electrorheological fluids between two eccentrically placed cylinders, the flow beingdriven by steady rotation of the inner cylinder. The aim of our study is threefold.

First, we want to investigate the flow of electrorheological fluids in a geometry that goes beyond a simple “shear flowgeometry” considered in the literature (see for example Atkin et al. (1991), Rajagopal and Wineman (1992), Gavin et al. (1996)and Choi et al. (2005)), but is still simple enough to allow us to determine the influence of the various parameters on theflow. Second, we want to illustrate the issues related to the constitutive modelling that we have pointed out in the previousparagraphs, namely we want to examine the possibility of an experimental design that would allow us to differentiate betweenthree dimensional models that are indistinguishable in simple shear flows. Third, the geometry corresponds to the journalbearing geometry and it is a technologically relevant geometry2. Note that although the electrorheological fluids are used inhydrodynamic lubrication, most results in this field are, however, based on crude approximations—see for example Bouzidaneand Thomas (2008)—and not on detailed modelling of the flow in relevant geometries.

The paper is organized as follows. In Section 2 we introduce the governing equations for electrorheological fluids and discussthe assumptions that lead to the system of governing equations. In Section 3 we introduce two particular fully three dimensionalmodels for the Cauchy stress tensor that are clearly different for general three dimensional flows and, in the next Section, wediscuss their behaviour in simple shear flows and we show that the two models coincide in simple shear flows. Finally, inSection 5, we propose the problem of flow of electrorheological fluids between eccentric rotating cylinders and we show thateven in this more complicated geometry the two models lead to the same velocity field though the pressure fields differ. Thispoints to the need for a fully three dimensional flow field to study the efficacy of the two models. The problem of flow in ajournal bearing is solved numerically and in Section 6 comment on the influence of various parameters on the solution.

1See for example Coleman et al. (1966), Truesdell (1974) and Pipkin and Owen (1967) for discussion on viscometric and nearly viscometric flows.2One of the biggest problems with large rotating machinery in power plants is the losses due to the decrease in the load carrying capacity of thebearing due to the decrease in the viscosity as a consequence of the increase in the temperature in the bearing, which is estimated in the billions ofdollars. One could compensate for this decrease in viscosity by applying an electric field, thereby increasing the viscosity.

FLOW OF AN ELECTRORHEOLOGICAL FLUID 3

2. Governing equations

If we assume that the electromagnetic field is weak and not changing rapidly and that the velocity of the fluid is small(compared to the speed of light) and the size of the domain is small3, then the flow of an isotropic, homogeneous, incompressible,non-conducting dielectric fluid is governed by the following set of equations (see for example Rajagopal and Růžička (1996, 2001)or monographs Penfield and Haus (1967), Hutter and van de Ven (1978) and Pao (1978) for details). For the electromagneticfield we can use a quasistatic approximation of Maxwell equations

div (ε0εrE) = 0, (2.1a)

rotE = 0, (2.1b)

where E denotes the intensity of the electric field, εr is the relative permittivity (a constant) and ǫ0 denotes the vacuumpermittivity (a constant). For the fluid (no external forces) we get the equations

ρdv

dt= div T + ǫ0εr [∇E] E, (2.2a)

T = T⊤, (2.2b)

div v = 0, (2.2c)

where ρ is the density of the fluid, v is the velocity field, T is the Cauchy stress tensor and D is the symmetric part of thevelocity gradient D = 1

2

(

∇v + ∇v⊤)

.In (2.1) and (2.2) we have already made a constitutive assumption that the polarization P is proportional to the intensity of

the electric field, thus P = ǫ0χeE, where χe denotes the electric susceptibility, εr =def χe + 1. Another constitutive assumptionthat is in fact tacitly introduced in (2.2) is the absence of internal body couples that leads to the symmetry (2.2b) of the Cauchystress tensor.

Note that the electric field E enters the balance of linear momentum (2.2a) in two ways, “implicitly” in the constitutiveequation for the Cauchy stress T, and “explicitly” as a volume force, [∇E] E. Obviously, the latter term vanishes if oneconsiders a homogeneous electric field, for example an electric field between two infinite plates, but this is not the case inmore complicated geometries. The term is nevertheless completely ignored in many theoretical and experimental works, see forexample Nikolakopoulos and Papadopoulos (1997) or Peng and Zhu (2006).

System (2.1) can be reduced to a single equation for the scalar potential φ,

∆φ = 0, (2.3)

where the intensity of the electric field E is related to the potential φ through the formula E = −∇φ. Obviously, the equationfor the potential can be solved without knowledge of the pressure and velocity field—the potential (intensity of the electricfield) therefore enters system (2.2) as a known “parameter”.

From the perspective of the full Maxwell equations, the study of flow of electrorheological fluids can be seen as a “counterpart”to magnetohydrodynamics. In the first case the magnetic field plays a secondary role and is in fact ignored, and the converseis true in the latter case. In magnetohydrodynamics, see for example Chandrasekhar (1961), the situation is however morecomplicated since the equations for the magnetic induction B can not be solved a priori without any reference to the velocityand pressure fields.

3. Constitutive relations for the Cauchy stress tensor

The experimental investigations show (see for example Shulman et al. (1989), Halsey et al. (1992), Kollias and Dimarogonas(1993), Martin et al. (1994), Abu-Jdayil and Brunn (1997), Krztoń-Maziopa et al. (2005), Belza et al. (2008) and reviews byGast and Zukoski (1989) and Zukoski (1993)) that electrorheological fluids can exhibit many non-newtonian characteristics:yield stress, shear thinning/thickening and normal stress differences.

In what follows, we focus on electrorheological fluids that do not exhibit yield stress. Although yield stress is observed inmany electrorheological fluids, there are electrorheological fluids which do not exhibit yield stress, see for example Abu-Jdayiland Brunn (1996). Note that a fluid by definition cannot support a shear stress and hence from a philosophical standpoint “yieldstress” is not a viable concept. However, it all depends on the time scale, length scale and force scale. A good example is the well

3Recall that the Lorentz transformation for the electromagnetic field reads

E′ = γ (E⊥ + w × B) + E‖, B′ = γ

„

B⊥ −1

c2w × E

«

+ B‖,

where w is the velocity of the reference frame S′ with respect to the reference frame S (it is assumed that coordinate axes in S′ and S remain parallel),

c is the speed of light, γ is the Lorentz factor 1γ

=def

q

1 −|w|2

c2, E‖ denotes the projection of the intensity of the electric field E to the direction

of the velocity w, E‖ =def1

|v|2(v • E)v, and E⊥ denotes the projection of the intensity of the electric field E to the direction perpendicular to

the direction of the velocity w, E⊥ =def E − E‖. (Similarly for B.) Requirement on the smallness of the velocity and the smallness of the domainensures that the Lorenz transformation can be approximated by a Galilean transformation. Note that the Lorentz transformation does not imply thatif B = 0 then B′ = 0, and similarly for E. If, however, E is not very strong and |w| is small compared to the speed of light c, then the implicationholds. In what follows, this allows us to consider the electric and magnetic field as different physical phenomena and not to consider them as relatedaspects of the electromagnetic field.


known experiment on asphalt—see for example Edgeworth et al. (1984)—that documents that the material that is apparentlysolid-like (if the observation time is in minutes or hours) in fact flows on a time scale of years. When yield stess is not considered,it can be shown (see for example Rajagopal and Wineman (1992), Wineman and Rajagopal (1995) and Rajagopal and Růžička(2001)) that a general constitutive relation for a homogeneous, isotropic, incompressible and non-conducting electrorheologicalfluid reads

T = −pI + S, (3.1a)

where T is the Cauchy stress, p is the Lagrange multiplier associated to the incompressibility constraint4, I denotes the identitytensor and S is the extra stress given by the formula

S = α2 (E, D) E⊗ E + α3 (E, D) D + α4 (E, D) D2 + α5 (E, D) (DE ⊗ E + E⊗ DE)

+ α6 (E, D)(

D2E⊗ E + E⊗ D2E)

, (3.1b)

where αi, i = 2, . . . , 6 are functions of combined invariants5 of the intensity of the electric field and symmetric part of the

velocity gradient D = 12

(

∇v + ∇v⊤)

. There are way too many material functions that appear in (3.1b) for us to outline asensible experimental program through which all of them can be characterized. In particular we will focus on two models thatbelong to class (3.1), namely the models

S1 = 2µ0D + µ1 (DE⊗ E + E⊗ DE) , (3.2a)

S2 = 2µ0

(

1 + β |E|2)

D, (3.2b)

where µ0, µ1 and β are constants. Note that in virtue of the constraint of incompressibility div v = Tr D = 0 we have Tr S2 = 0,but, in general, we can not expect Tr S1 = 0—we discuss the implications of this observation in Section 5.6.

The reason for the choice of these two models is the following. If we consider models of type (3.1b), and we are interestedin fluids where the generalized viscosity depends only on the intensity of the electric field, that do not exhibit normal stressdifferences effect, we see6 that S = α3 (E) D + α5 (E) (DE⊗ E + E⊗ DE) is—for these fluids—a natural choice for constitutiverelation for the extra stress. The model S = α3 (E) D + α5 (E) (DE⊗ E + E⊗ DE) however coincides—in simple shear flowswith the intensity of the electric field perpendicular to the streamlines—with a simpler model S = α̃3 (E) D, see Section 4.Clearly, the models can not be expected to be equivalent in more complicated settings, and one can ask what the effects of thetensorial term DE⊗ E + E⊗ DE are in such settings. Comparison of the models (3.2) gives us an opportunity to address thisquestion that has implications with respect to the constitutive modelling of electrorheological fluids, see Section 1. Let us nowbriefly comment on the behaviour of models (3.2a) and (3.2b) in simple shear flows.

4. Simple shear flows

4.1. Plane Poiseuille flow. Let us consider the flow of electrorheological fluids (3.2) between infinite parallel plates kept atdifferent potentials7 (see Figure 1c). Obviously, the intensity of the electric field is perpendicular to the plates—E = Eŷeŷ,where Eŷ is a constant. If we assume that the velocity field has the form v(x, y) = vx̂(y)ex̂, then the Cauchy stress is, accordingto models (3.2a) and (3.2b), given by the formula

T = −pI +(

µ0 +µ1

2

(

Eŷ)2)

[

0 dvx̂

dydvx̂

dy0

]

, and T = −pI + µ0

(

1 + β(

Eŷ)2)

[

0 dvx̂

dydvx̂

dy0

]

, (4.1)

respectively. Clearly, if we set β = µ12µ0

in (3.2b), then both models lead, in Poiseuille flow, to the same formula for the Cauchy

stress. In this sense, any experiment that uses measurements in Poiseuille flow can not be used to distinguish model (3.2a)from (3.2b).

4.2. Cylindrical Couette flow. Similarly, if we consider the flow between coaxial rotating cylinders kept at different potentials(see Figure 1a), we see that the intensity of the electric field is given by the formula E = E r̂(r)er̂ , where E

r̂(r) is a knownfunction of the radial variable r. If we assume that the velocity field has the form v(r, ϕ) = rω(r)eϕ̂, where ω(r) is the angularfrequency, then the Cauchy stress is, according to models (3.2a) and (3.2b), given by the formula

T = −pI +(

µ0 +µ1

2

∣

∣E r̂(r)∣

∣

2)

[

0 r dωdr

r dωdr

0

]

, and T = −pI + µ0

(

1 + β∣

∣E r̂(r)∣

∣

2)

[

0 r dωdr

r dωdr

0

]

, (4.2)

respectively. Again, if we set β = µ12µ0

in (3.2b), then both models lead to the same formula for the Cauchy stress, and the

models are indistinguishable in this type of flow.

4We deliberately use the term “the Lagrange multiplier associated with the constraint of incompressibility” instead of “the pressure”. Later we showthat these notions are not equivalent and that we must carefully differentiate between these two notions. For further discussion on this issue seealso Rajagopal and Srinivasa (2004), Rajagopal (2003) and Huilgol (2009).5The invariants are |E|2, Tr D, Tr

`

D2

´

, Tr`

D3

´

, Tr (DE ⊗ E) and Tr`

D2E ⊗ E

´

.6We refer the reader to Rajagopal and Wineman (1992) for details.7For a careful discussion of this type of problem see for example Rajagopal and Wineman (1992).


E

er̂

eϕ̂

x

R2

R1

y

Ω1

Ω2

(a) Cylindrical Couette flow (flow between concentriccylinders).

−eβ̂−eα̂

R2R1d1

a e

F1 F2

d2

x

y

β

(b) Flow between eccentric cylinders.

E

eŷ

ex̂h

y

x

(c) Plane Poiseuille/Couette flow.

Figure 1: Problem geometry.

5. Flow between eccentric cylinders

We have seen that models (3.2) lead to the same results in (idealized) Poiseuille and Couette type experiments. In bothexperiments, the intensity of the electric field is perpendicular to the streamlines. One needs to guess the flow field that couldbe used to differentiate between the two models. A problem that can be expected to help us to distinguish between the twomodels is for example the problem of flow between two eccentrically placed parallel cylinders, see Figure 1b. The cylinders arekept at different potentials, φ1 (inner cylinder) and φ2 (outer cylinder), and the cylinders are rotating with constant angularvelocities Ω1 and Ω2 about their axes. The radius of the inner cylinder is R1 and the radius of the outer cylinder is R2, thedistance between the axes of the cylinders (the eccentricity) is e. In the idealized case, we can assume that the cylinders areinfinite, and consequently we can essentially treat the problem as a two dimensional problem—there is no flow in the directionof axes of the cylinders and the velocity and pressure fields are the same on each cross section perpendicular to the axes of thecylinders. Furthermore, it is reasonable to assume that the flow is steady.

This type of problem has been already extensively studied within the context of electrically inactive fluids because it providesa simple model for the flow in a journal bearing (see for example Dai et al. (1992) and Gwynllyw et al. (1996)). We can in factuse much of the available “classical” apparatus to solve the problem for electrorheological fluids, this is especially true concerningthe numerical treatment of the problem. Since it is possible to find an analytical solution to the Laplace equation (2.3), we canexplicitly calculate all terms that include E, and we get almost the same problem as for the flow of the classical Navier–Stokesfluid in a journal bearing. The difference is that the viscosity is in our case a known function of E and hence a known functionof position.

5.1. Dimensionless form of the governing equations. Let us now convert (2.3) and (2.2) to a dimensionless form. LetL be a characteristic length scale, Eref the characteristic intensity of the electric field and V the characteristic velocity. Usingthese characteristic quantities a dimensionless version of the governing equations reads

dv⋆

dt⋆= div⋆ T⋆ + ε̂0εr [∇

⋆E⋆] E⋆, (5.1a)

div⋆ v⋆ = 0, (5.1b)

∆⋆φ⋆ = 0, (5.1c)


where the star denotes dimensionless quantities v⋆ = vV

, t⋆ = VL

t, x⋆ = xL

, E⋆ = EEref

, φ⋆ = φErefL

. Dimensionless form of

constitutive equations (3.2) is

T⋆ = −p⋆I + 2µ̂0D⋆ + µ̂1 (D

⋆E⋆ ⊗ E⋆ + E⋆ ⊗ D⋆E⋆) , (5.2a)

T⋆ = −p⋆I + 2µ̂0

(

1 + β̂ |E⋆|2)

D⋆, (5.2b)

where ε̂0 =ε0E

2

ref

ρV 2, µ̂0 =

µ0ρLV

, µ̂1 =µ1E

2

ref

ρLV, β̂ = βE2ref . Obviously some of these parameters can be used to determine the usual

dimensionless numbers, Re = 1µ̂0

is the Reynolds number (the ratio of inertial to the viscous forces), Ma = 1ε̂0

is the Mason

number (the ratio of inertial to the electrical forces). Let us further fix L = R1 and introduce the following notation

ηR =R1

R2, ηre =

e

R1, ηφ =

φ2

φ1, ηΩ =

Ω2Ω1

. (5.3)

Moreover, since we are interested in the situation where models (5.2a) and (5.2b) coincide in Poiseuille flow and cylindrical

Couette flow, we fix β̂ = µ̂12µ̂0

. Hereafter we only use dimensionless variables, and, for the sake of clarity, we omit the star

denoting the dimensionless quantities.

5.2. Bipolar coordinates. The bipolar coordinate system is the natural co-ordinate system within which to study the problem,see for example Jeffery (1921, 1922). Let us recall some basic formulae for the bipolar coordinate system8. (See Figure 1b forthe notation.) Bipolar coordinates (α, β) are related to the Cartesian coordinates (x, y) by the following formula

α + iβ = lny + i (x + a)

y + i (x − a). (5.4)

In what follows we call α the radial variable and β the azimuthal variable, similarly, components of vectors are referred to as theradial (component in the direction of eα̂) and the azimuthal (component in the direction of eβ̂) components. (See Section 5.2.1

for the formulae for normalized base vectors eα̂ and eβ̂ .) Using (5.4) it is easy to show that

x =a sinh α

cosh α − cos β, y =

a sin β

cosh α − cos β, (5.5)

and consequently (x − a coth α)2

+ y2 = a2

sinh2 α, x2 + (y − a cot β)

2= a

2

sin2 β. Obviously, the curves of constant α and β are

circles. Knowing the radii of the cylinders and the eccentricity, one can find parameter a, a > 0, and parameters αin, αout,0 < αout < αin such that the curves of constant α, α = αin and α = αout, are the corresponding circles (α = αin is the boundaryof the inner circle and α = αout is the boundary of the outer circle). The formulae for the parameters are the following

a =1

2ηre

√

((

1 +1

η2R

)

− (ηre)2

)2

−4

η2R, αin = arcsinh a, αout = arcsinh (ηRâ) . (5.6)

The gap between the circles is thus mapped to a rectangular domain [α, β] ∈ [αout, αin] × [0, 2π]. If we express the boundaryconditions for the velocity and the potential in the bipolar coordinates, we get

φ|α=αin = 1, φ|α=αout = ηφ, (5.7a)

and for components of the velocity we get

vα̂∣

∣

α=αin= 0, vβ̂

∣

∣

∣

α=αin= 1, vα̂

∣

∣

α=αout= 0, vβ̂

∣

∣

∣

α=αout=

ηΩ

ηR. (5.7b)

Let us recall that we study the problem with rotating inner cylinder and fixed outer cylinder, hence ηΩ = 0.

5.2.1. Tensor calculus. The tangent vectors are

gα =∂xi

∂αei =

a

(cosh α − cos β)2

[

1 − cosh α cos β− sinh α sin β

]

, gβ =∂xi

∂βei =

a


[

− sinh α sin βcosh α cos β − 1

]

,

and consequently the components of covariant and contravariant metric tensors are

[

gij]

=a2


[

1 00 1

]

,[

gij]

=(cosh α − cos β)2

a2

[

1 00 1

]

. (5.8)

The Christoffel symbols are

Γααα = −sinh α

cosh α − cos β, Γβββ = −

sin β

cosh α − cos β, (5.9)

8In some published papers, see for example Dai et al. (1992), there are misprints in the formulae for various differential operators in the bipolarcoordinates. Also, we need to use some of these formulae and therefore we feel the need to record the basic formulae.


and for the remaining nonzero Christoffel symbols we have Γβαβ = Γα

αα, Γα

ββ = −Γα

αα, Γβ

αα = −Γβ

ββ , Γα

αβ = Γβ

ββ. The

relations of the physical components9 v = vα̂eα̂ + vβ̂eβ̂ and components with respect to the non-normed basis v = v

αgα + vβgβ

are the following

vα̂ =a

cosh α − cos βvα, vα̂ =

cosh α − cos β

avα, v

α̂ = vα̂, (5.10)

and the same formulae hold also for vβ̂ since gαα = gββ . Moreover, since the metric tensor is a multiple of the identity tensor,

see (5.8), it is easy to see that for 1-1 tensors we have Aı̂k̂

= Aik.

5.2.2. Differential operators. Using formulae derived in Section 5.2.1, one can find expressions for the differential operators inthe bipolar coordinates. The gradient of a scalar function is10

∇φ =cosh α − cos β

a

[

∂φ∂α∂φ∂β

]

. (5.11)

The gradient of a vector field v and the symmetric part of the gradient of a vector field v are given by

∇v =

[

cosh α−cos βa

∂vα̂

∂α− sin β

avβ̂ cosh α−cos β

a∂vα̂

∂β+ sinhα

avβ̂

cosh α−cos βa

∂vβ̂

∂α+ sin β

avα̂ cosh α−cos β

a∂vβ̂

∂β− sinh α

avα̂

]

, (5.12)

1

2

(

∇v + ∇v⊤)

=

cosh α−cos βa

∂vα̂

∂α− sin β

avβ̂ 1

2

(

cosh α−cos βa

(

∂vα̂

∂β+ ∂v

β̂

∂α

)

+ sinhαa

vβ̂ + sin βa

vα̂)

1

2

(

cosh α−cos βa

(

∂vα̂

∂β+ ∂v

β̂

∂α

)

+ sinhαa

vβ̂ + sin βa

vα̂)

cosh α−cos βa

∂vβ̂

∂β− sinh α

avα̂

.

(5.13)

Consequently, the expression for the term [∇v] v reads

[∇v] v =

cosh α−cos βa

(

vα̂ ∂vα̂

∂α+ vβ̂ ∂v

α̂

∂β

)

− sin βa

vα̂vβ̂ + sinh αa

(

vβ̂)2

cosh α−cos βa

(

vα̂ ∂vβ̂

∂α+ vβ̂ ∂v

β̂

∂β

)

+ sin βa

(

vα̂)2

− sinh αa

vα̂vβ̂

. (5.14)

The divergence of a vector field v is

div v = Tr∇v =cosh α − cos β

a

(

∂vα̂

∂α+

∂vβ̂

∂β

)

−

(

sin β

avβ̂ +

sinh α

avα̂)

. (5.15)

Using (5.11) and (5.15) it can be shown that

∆φ =

(

cosh α − cos β

a

)2(

∂2φ

∂α2+

∂2φ

∂β2

)

. (5.16)

The divergence of a 1-1 tensor field A is

div A =cosh α − cos β

a

∂Aα̂α̂∂α

+∂Aα̂

β̂

∂β+ sinh α

cosh α−cos β

(

Aβ̂

β̂− Aα̂α̂

)

+ sin βcosh α−cos β

(

Aα̂β̂

+ Aβ̂α̂

)

∂Aβ̂

α̂

∂α+

∂Aβ̂

β̂

∂β+ sin β

cosh α−cos β

(

Aα̂α̂ − Aβ̂

β̂

)

− sinh αcosh α−cos β

(

Aα̂β̂

+ Aβ̂α̂

)

, (5.17)

in particular

∆v = div∇v =

(

cosh α−cos βa

)2 (∂2vα̂

∂α2+ ∂

2vα̂

∂β2

)

+ cosh α−cos βa

(

−2 sinβa

∂vβ̂

∂α+ 2 sinhα

a∂vβ̂

∂β− cosh α+cos β

avα̂)

(

cosh α−cos βa

)2 (∂2vβ̂

∂α2+ ∂

2vβ̂

∂β2

)

+ cosh α−cos βa

(

2 sin βa

∂vα̂

∂α− 2 sinhα

a∂vα̂

∂β− cosh α+cos β

avβ̂)

. (5.18)

Let us denote h =defcosh α−cos β

a, the scale factor that frequently occurs in the expressions for differential operators, and let us

note that, according to (5.11), we have

∇h2 = 2h2[

sinh αa

sin βa

]

. (5.19)

9By “physical” components we mean the components of the vector v with respect to the normed basis {eα̂, eβ̂} where eα̂ =defgα

|gα|, e

β̂=def

gβ

|gβ |.

Hereafter we will use the superscript hat to denote the physical components of the given vector or tensor. Obviously, vα̂ = vα |gα|, vα̂ = vα |gα| and

similarly for higher order tensors.10Components are, if not stated otherwise, always written with respect to the normed basis and we are using physical components of the tensors that

are involved. For example ∇v = (∇v)ı̂k̂eı̂ ⊗ e

k̂.


5.3. Explicit analytical solution for the electric field. Let us solve (5.1c). Using (5.16) it is easy to see that the solution

to (2.3) with boundary conditions (5.7a) reads φ(α, β) = Φα + ΦS , where Φ =1−ηφ

αin−αout, ΦS =

αinηφ−αoutαin−αout

. Consequently the

intensity of the electric field E, its gradient ∇E, and the force term [∇E] E in the balance of linear momentum (5.1a) are givenby

E = −∇φ = −hΦ

[

10

]

, ∇E = −hΦ

[

sinh αa

sin βa

sin βa

− sinhαa

]

, [∇E] E = (hΦ)2[

sinh αa

sin βa

]

. (5.20a)

Furthermore, using (5.19) it is easy to see that

∇|E|2

= 2 (hΦ)2

[

sinh αa

sin βa

]

. (5.20b)

5.4. Formulae for the divergence of the Cauchy stress tensor. Since we have a formula for the intensity of the electricfield, we can substitute for E in (3.2) and get a convenient formulae for div T in terms of the coordinates α, β and the fields pand v. Using (5.13) and (5.20) we see that

DE⊗ E + E⊗ DE = (hΦ)2 (D + F) , F =

[

h∂vα̂

∂α− sin β

avβ̂ 0

0 −(

h∂vβ̂

∂β− sinh α

avα̂)

]

, (5.21)

and finally, using (5.21) and (5.19), we obtain

div (DE⊗ E + E⊗ DE) = (hΦ)2(

2D

[

sinh αa

sin βa

]

+1

2∆v

)

+ (hΦ)2(

2F

[

sinh αa

sin βa

]

+ div F

)

. (5.22)

Another straightforward calculation yields

div

((

1 +µ̂1

2µ̂0|E|

2

)

D

)

=1

2

(

1 +µ̂1

2µ̂0|E|

2

)

∆v +µ̂1

µ̂0(hΦ)

2D

[

sinh αa

sin βa

]

. (5.23)

Divergence of the extra stress tensors (3.2) therefore reads

div S1 = µ̂0∆v +µ̂1

2(hΦ)

2∆v + 2µ̂1 (hΦ)

2D

[

sinh αa

sin βa

]

+ µ̂1 (hΦ)2

(

2F

[

sinh αa

sin βa

]

+ div F

)

, (5.24a)

div S2 = µ̂0∆v +µ̂1

2(hΦ)2 ∆v + 2µ̂1 (hΦ)

2D

[

sinh αa

sin βa

]

. (5.24b)

Obviously, the governing equations for models (3.2a) and (3.2b) are now, unlike in the cases discussed in Section 4, different,and one can expect that the velocity and pressure fields corresponding to the different models are different. This is, however,not completely true, it is possible to prove that the additional term in (5.24a)—compared to (5.24b)—does not have an influenceon the velocity field.

5.5. On the influence of various terms on the velocity field. Let us now formulate and prove a proposition concerningthe influence of various terms in the governing equations for the velocity field. We show that many terms can be, in ourparticular problem, rewritten as the gradient of a scalar function and consequently “absorbed in the pressure”—such terms donot have an influence on the velocity field.

Proposition 1. Let (p1,v1,E1) be a solution to (5.1) with T given by (5.2a), and subject to boundary conditions (5.7). Then(p2,v2,E2), where

p2 =def p1 − µ̂1 (hΦ)2

(

h∂vα̂1∂α

−sin β

av

β̂1

)

, (5.25)

and v2 =def v1, E2 =def E1, is a solution to (5.1) with T given by (5.2b) and subject to boundary conditions (5.7).Let εr = 0 and let (p0,v0,E0) be a solution to (5.1) with T given by (5.2a) or (5.2b), and subject to boundary conditions (5.7).

Then (pεr ,vεr ,Eεr) where

pεr =def p0 +1

2ε̂0εrΦ

2h2, (5.26)

and vεr =def v0, Eεr =def E0, is a solution to (5.1) with εr 6= 0 and with T given by (5.2a) or (5.2b) and subject to boundaryconditions (5.7).

Proof. Obviously, the governing equations for the fluid (5.2a) and (5.2b) differ in the term

µ̂1 (hΦ)2

(

2F

[

sinh αa

sin βa

]

+ div F

)

. (5.27)


This term can be however rewritten as a gradient of a scalar function. Using (5.19) we see that (5.27) can be rewritten asµ̂1Φ

2(

F(

∇h2)

+ h2 div F)

and consequently, (5.27) is equal to µ̂1Φ2 div

(

h2F)

. In virtue of the isochoricity condition div v = 0

and its expression in the coordinate system of interest (5.15), we see that F α̂α̂ − Fβ̂

β̂= 0, and hence

µ̂1Φ2 div

(

h2F)

= µ̂1Φ2 div

(

h2(

h∂vα̂

∂α−

sin β

avβ̂)[

1 00 1

])

= µ̂1Φ2∇

(

h2(

h∂vα̂

∂α−

sin β

avβ̂))

. (5.28)

Let us now consider the term [∇E] E, recalling formulae (5.19) and (5.20a) it immediately follows that [∇E] E = 12∇(

Φ2h2)

.The balance of linear momentum (5.1a) for the model (3.2a) therefore reads

dv

dt= −∇

(

p − µ̂1Φ2h2

(

h∂vα̂

∂α−

sin β

avβ̂)

−1

2ε̂0εrΦ

2h2)

+ µ̂0∆v +µ̂1

2(hΦ)2 ∆v + 2µ̂1 (hΦ)

2D

[

sinh αa

sin βa

]

, (5.29a)

and for the model (3.2b) we get

dv

dt= −∇

(

p −1

2ε̂0εrΦ

2h2)

+ µ̂0∆v +µ̂1

2(hΦ)

2∆v + 2µ̂1 (hΦ)

2D

[

sinh αa

sin βa

]

. (5.29b)

The results (5.25) and (5.26) now follow immediately. �

5.6. On the mean normal stress, the pressure and the Lagrange multiplier p. As we have already noted in theintroduction, if the fluid is specified by the constitutive relation (3.2a), we can not in general expect that Tr S = 0. Indeed,formula (5.21) gives11

T = −pI + 2µ̂0D + µ̂1 (hΦ)2

(D + F) = −pI +(

2µ̂0 + µ̂1 (hΦ)2)

D + µ̂1 (hΦ)2

(

h∂vα̂

∂α−

sin β

avβ̂)

1 0 00 1 00 0 0

, (5.30)

and consequently

Tr T = −3p + 2µ̂1 (hΦ)2

(

h∂vα̂

∂α−

sin β

avβ̂)

= −3p− 2µ̂1 (hΦ)2

(

h∂vβ̂

∂β−

sinh α

avα̂

)

, (5.31)

where the second equality follows from F α̂α̂ − Fβ̂

β̂= 0, see (5.27) and the following discussion. Note that as a consequence

of the boundary conditions (5.7) we have Tr T|α=αin,αout = −3p. On the other hand, it is obvious that if we use model (3.2b)we get Tr T = −3p. This shows that for electrorheological fluids we need to differentiate between the Lagrange multiplier passociated to the incompressibility constraint and the mean normal stress π =def −

1

3Tr T (the mechanical pressure). If we deal

with model (3.2a), then it is not, in general, true that π = p, although this equality can hold in special cases as Poiseuille andcylindrical Couette flow, see (4.1) and (4.2). In our particular case we have

π1 = p1 +2

3µ̂1 (hΦ)

2

(

h∂vα̂

∂α−

sin β

avβ̂)

, (5.32a)

for the mean normal stress in fluid (3.2a), and π2 = p2 for the mean normal stress in fluid (3.2b). Proposition 1 gives

π1 − π2 =5

3µ̂1 (hΦ)

2

(

h∂vα̂

∂α−

sin β

avβ̂)

(5.33)

or alternatively π1 =1

3(5p1 − 3p2).

5.7. Numerical solution of the governing equations. Although we are able to simplify the original system (5.1) by givingan explicit formula for E, the remaining equations (5.1a) and (5.1b) are still sufficiently complicated that we can not expect tofind an exact solution. (An exact solution is not known even for the classical Navier–Stokes fluid.) Therefore, we need to solvethe remaining part of the system numerically. In steady flow, the governing equations read

[∇v] v = −∇p + div S, (5.34a)

div v = 0. (5.34b)

We solve the problem in bipolar coordinates—this allows us to work with a rectangular domain (recall that [α, β] ∈ [αin, αout]×[0, 2π]) and consequently avoid difficulties with the curved boundary. Formulation in bipolar coordinates further enables astraightforward implementation of the boundary conditions (5.7b) and provides more transparent control of the different lengthscales in the “narrowest gap” and “widest gap” regions. Once the governing equations are converted to bipolar coordinates (seeformulae (5.11)–(5.18) and (5.24)) we can employ a simple linear transformation

α =1

αout − αin(−2α + (αin + αout)) , (5.35)

11In what follows the third spatial dimension (z coordinate) plays an important role, therefore we write all formulae in their fully three dimensionalform.


ui,j

pi,j vi,j i

1

2

3

M − 2

M − 1

M

0 2π

1 2 3 4 N

−1

1

α

β

j

(a) Computational tensor product grid.

β

α

β = 2πN

(N − 1)

β = 0

β = 2πN

eβ̂

eα̂

α = αout

α = αin

(b) Grid in the physical space.

Figure 2: Computational tensor product grid and its mapping onto the physical domain, N × M = 32 × 6 points, ηre = 0.5,ηR = 0.6.

Parameter ηφ ηre ηR ηΩ a αin αout

Value 2 0.5 0.6 0 1.15503 0.986866 0.64694

Table 1: Parameter values (geometry and boundary conditions). Note that a, αin and αout are not independent, they are givenby the formulae (5.6).

that maps [αout, αin] to the canonical interval [−1, 1]. The problem in the rectangular domain [α, β] ∈ [−1, 1] × [0, 2π] can benow solved by a straightforward application of a pseudospectral collocation method. Since the β coordinate is periodic, we useFourier interpolants in the β direction, and we use Chebyshev interpolants in the α direction. Consequently the tensor productgrid consist of Chebyshev Gauss–Lobatto/Fourier collocation points[αi, βj ] where

αi = cos

(

(k − 1)π

M − 1

)

, i = 1, . . . , M ; βj =2π

N(k − 1), k = j, . . . , N. (5.36)

See Figure 2a for the tensor product grid in the computational space and Figure 2b for its image in the physical space. In whatfollows we will use a shorthand notation wi,j = w(αi, βj) for values of the function w at the given grid point.

Since we are in principle dealing with a boundary value problem with variable coefficients, we work entirely in the physical(not Fourier) space, and for differentiation we use spectral differentiation matrices in the form given by Weideman and Reddy(2000). We use the same grid, both for the velocity and the pressure. The balance of linear momentum is enforced at all theinner collocation points, and the divergence equation is enforced at all the collocation points—in other words we use a QN–QNmethod, see Canuto et al. (2006, 2007). It is, however, well known—see for example Bernardi et al. (1988) and Schumack et al.(1991)—that although the QN–QN method gives (for the Stokes problem) the correct solution for the velocity field, the methodsuffers from spurious oscillations in the pressure p. This fact is especially inconvenient in our case, where the knowledge ofthe pressure field (the Lagrange multiplier p) constitutes the most important part of the solution to the problem. In order toeliminate the spurious oscillations, we have applied a filtration procedure that was developed by Phillips and Roberts (1993)that is based on detailed knowledge of the form of the possible spurious modes and on the singular value decomposition of themodified pressure Schur complement matrix (Uzawa operator).

The nonlinear term [∇v] v is treated by the standard Newton–Raphson iteration, see for example Thomasset (1981), and isalways considered in the calculations reported in Section 6, thus we do not treat the problem in the creeping flow approximation.The initial guess for the Newton–Raphson iteration is the solution to the Stokes problem (solution to the governing equationswithout the nonlinear term).

6. Results

The numerical results were obtained for a wide range of parameters. In what follows we mainly report results for a particularcase specified by the parameters summarized in Table 1. These parameters correspond to the situation where the diameter ofthe inner cylinder is “small” and the inner cylinder is placed eccentrically such that the width of the “widest” gap is comparableto the diameter of the inner cylinder. The inner cylinder is rotating and the outer cylinder is kept at rest. In this setting it ispossible to clearly observe (for the classical Navier–Stokes fluid) the well known recirculation phenomenon, see for example Szeri(1998).


The remaining parameters µ̂0 =1

Re, µ̂1 and ε̂0εr =

εrMa

are varied in order to determine the influence of the various terms inthe governing equations on the solution. In what follows, we assume that the assumptions that lead us to the model for the flowof electrorheological fluid in between the eccentrically placed cylinders are valid for all possible parameter values, namely wedo not take into account the possibility of dielectric breakdown due to a strong electric field, cavitation, and other phenomena.Note that when reporting results for the pressure p (more precisely the Lagrange multiplier), we in fact report results concerningthe difference of p from a reference level p0.

6.1. Convergence of the numerical method. The numerical method converges (as expected) in general very fast, whereasthe convergence is slower for electrorheological fluids and faster for the classical Navier–Stokes fluid. For small Reynolds numbers(up to approximately 100) we have found grid N × M = 30 × 25 to be sufficient (with an adequate safety margin) for all thecomputations with µ1 and ε̂0εr varying in the ranges specified in Section 6.2 and Section 6.3. Concerning the results reportedin Section 6.4 (dependence of the solution on the Reynolds number) we use a finer grid, namely N × M = 50 × 45.

For visualization purposes we reinterpolate all computed raw data to the fine grid N ×M = 120× 40, for reinterpolation weuse the method described and implemented by Weideman and Reddy (2000). Note that since we use a pseudospecral spectralmethod the raw data in principle correspond to coefficients in a Fourier series expansion, and consequently the reinterpolationis a minimal source of further numerical errors.

Concerning the Stokes problem (no nonlinearity due to the inertial term) we have found the residuum for the discrete problemto be of order 10−9 or lower, the residuum in the full nonlinear problem ranges from 10−6 to 10−8, the relative residuum isalways less than 10−9. Usually only two or three Newton–Raphson iterations are required to achieve the reported accuracy.

6.2. Dependence on µ1. Let us now discuss the behaviour of the solution with respect to parameter µ̂1. In this section wefix ε0εr = 0 in order to isolate effects due to the term ε0εr[∇E]E. (Note that in view of Proposition 1 the term ε0εr[∇E]Einfluences only the pressure field.) Since the governing equations—in this case—contain only the combination µ̂1Φ

2, increasingµ̂1 in fact either corresponds to increasing the value of material constant µ1 or potential ratio ηφ. Recall that µ1 = 0 correspondsto the classical Navier–Stokes fluid. All the results reported in this Section are for Re = 10, the described trends are neverthelessto a great extent valid for all Reynolds numbers that do not lead to dominant effects due to the nonlinearity [∇v] v.

As we have shown in Proposition 1, the velocity field is the same for both models (3.2), therefore we need to analyze thevelocity field only for one of the models. The plots of the velocity field, see Figure 3, show that increasing µ̂1 boosts the backflow,and it only slightly moves the point where the azimuthal velocity changes its orientation (see Figure 3b). The backflow boostis also clearly visible if we compare the “global” view of the velocity field, see Figure 3c (Navier–Stokes fluid) and Figure 3d(electrorheological fluid with µ̂1 = 10).

Concerning the pressure field p, model (3.2b) leads to a significant change in the pressure distribution in the domain. Highervalues of µ̂1 lead to formation of a steep pressure drop across the region close to the narrowest gap between the cylinders, seeFigure 4. With increasing µ̂1 the difference between the highest and lowest pressure increases and the pressure difference isconcentrated in a smaller region; the region of the highest pressure moves towards the narrowest gap, see especially Figure 4eand Figure 4f.

If we consider model (3.2a), the pressure (the mean normal stress) π does not coincide with the Lagrange multiplier p (seediscussion in Section 5.6). The Lagrange multiplier p exhibits nearly the same behaviour as p in model (3.2b), the magnitudeof the drop across the narrowest gap region remains almost the same, the difference between the models manifests itself mainlyin the internal structure of the p field, compare Figure 5a and Figure 5b. Concerning the mean normal stress π = − 1

2Tr T, the

magnitude is about 23

of p—this is a consequence of formula (5.33) and the fact that p2 (the Lagrange multiplier/pressure formodel (3.2b)) is nearly of the same magnitude as p1 (the Lagrange multiplier for model (3.2a)). Clearly, the distribution of themean normal stress is different for models (3.2a) and (3.2b), compare Figure 5a and Figure 5c.

6.3. Dependence on ε̂0εr. The dependence of the solution on ε̂0εr is illustrated for Re = 10 and µ̂1 = 1. Let us first considermodel (3.2b), the influence of ε̂0εr is best seen in the plot of the pressure in the computational space, see Figure 6c–e. Initially(for small values of ε̂0εr) the pressure field shows little variation in the radial direction, at least compared to variation in theazimuthal direction. Further, the minimal and maximal pressure are located on the surface of the inner cylinder. When ε̂0εrincreases the pressure field starts to vary even in the radial direction and finally the maximal pressure is located on the surfaceof the outer cylinder while the minimal pressure stays located on the surface of the inner cylinder. The plug flow region isshown in Figure 6f–h, pressure distribution along the cylinder walls is shown in Figure 6a and 6b. Obviously, if ε̂0εr is small,its influence on the pressure field is also small as expected.

Concerning model (3.2b), the influence of ε̂0εr on the Lagrange multiplier field p is similar (not shown).

6.4. Dependence on Re. All the results given in this section are, in contrast to the previous sections, computed on the gridN ×M = 50× 45 and reinterpolated to the grid N ×M = 120× 40. The reason is that for computations with higher Reynoldsnumbers (that are of interest in this section) we need a finer grid in order to achieve sufficient accuracy.

Let us first consider the classical Navier–Stokes fluid without any effects due to the electric field. As it is well known,the increase of the Reynolds number leads to increase of the role of nonlinearity [∇v] v. Plots of the velocity and pressurefield for various values of the Reynolds number are shown in Figure 7 and Figure 8. Recall that µ̂0 =

1

Reand consequently


µ̂1 = 0.1µ̂1 = 0.05µ̂1 = 0.01µ̂1 = 0.005µ̂1 = 0.001Navier–Stokes

α

vα̂(α

,β)

10.50-0.5-1

0

-0.0001

-0.0002

-0.0003

-0.0004

-0.0005

-0.0006

-0.0007

-0.0008

-0.0009

-0.001

(a) Radial velocity vα̂ in the section through the widest gap, β = 0(β = βj , j = 1).


α

vα̂(α

,β)

10.50-0.5-1

1

0.8

0.6

0.4

0.2

0

-0.2

-0.25-0.5-0.75

-0.11-0.12-0.13-0.14-0.15-0.16-0.17-0.18-0.19-0.2

(b) Azimuthal velocity vβ̂ in the section through the widest gap,β = 0 (β = βj , j = 1).

x

y

43.532.521.510.50

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

(c) Velocity in the physical space, µ̂1 = 0 (Navier–Stokes fluid).

x

y

43.532.521.510.50

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

(d) Velocity in the physical space, µ̂1 = 10.

Figure 3: Models (3.2a), (3.2b), dependence of the velocity field on µ̂1.

increasing the Reynolds number corresponds to decreasing µ̂0. The balance of linear momentum for the Navier–Stokes fluidreads [∇v] v = −∇p + µ̂0∆v, and the balance of linear momentum for electrorheological fluid (3.2b) is

[∇v] v = −∇p +

(

µ̂0 +µ̂1

2(hΦ)

2

)

∆v + 2µ̂1 (hΦ)2

D

[

sinh αa

sin βa

]

+ ε̂0εr [∇E] E, (6.1)

and therefore it is not very surprising that the change of the velocity field due to the decrease in the Reynolds number (seeFigure 7) is similar to the changes due to the increase of µ̂1 as discussed in Section 6.2, see especially Figure 3. In both caseswe change the magnitude of the coefficient in front of the Laplace operator and a similar influence on the velocity field can betherefore expected. (Remaining terms in (6.1) either do not have influence on the velocity field, see Proposition 1, or can beunderstood as a perturbation to the “dominant” Laplace operator.)

The pressure field is however completely different. As documented in Figure 8 increasing the Reynolds number leads tosubstantial changes in the distribution of the pressure in the domain. For small Reynolds numbers we get almost uniformdistribution of the pressure on the radial sections (at least compared to variations in the azimuthal sections), see Figure 8a.Further, the minimal and maximal pressure are located on the surface of the inner cylinder. When Re increases the pressurefield starts to vary even in the radial direction and finally the minimal pressure is located on the surface of the outer cylinder



β

p(α

,β)

2π32πππ

20

20

15

10

5

0

-5

-10

-15

-20

(a) Pressure on the inner cylinder wall, α = 1, (α = αk, k = 1).


β

p(α

,β)

2π32πππ

20

20

15

10

5

0

-5

-10

-15

-20

(b) Pressure on the outer cylinder wall, α = 1, (α = αk, k = M).

y

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

x

43.532.521.510.50

p(x

,y)

10

5

0

-5

-10

y

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

x

43.532.521.510.50

(c) Pressure in the physical space, µ̂1 = 0 (Navier–Stokes fluid).The region circumscribed with the dashed line corresponds to the“plug region” shown in Figure 4e.

y

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

x

43.532.521.510.50

p(x

,y)

10

5

0

-5

-10

y

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

x

43.532.521.510.50

(d) Pressure in the physical space, µ̂1 = 0.05. The region circum-scribed with the dashed line corresponds to the “plug region” shownin Figure 4f.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

10

8

6

4

2

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(e) Pressure in the physical space, µ̂1 = 0 (Navier–Stokes fluid), theplug region is magnified. Pressure isolines are shown in the rangefrom 2 to 2.8 in steps of 0.05.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

10

8

6

4

2

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(f) Pressure in the physical space, µ̂1 = 0.05, the plug region ismagnified. Pressure isolines are shown in the range from 8.5 to 10.5in steps of 0.25.

Figure 4: Model (3.2b), dependence of the pressure field p on µ̂1.


y0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

10

8

6

4

2

0

y0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(a) Model (3.2b), pressure (Lagrange multi-plier) p. Isolines are shown in the range from8.5 to 10.5 in steps of 0.25.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

10

8

6

4

2

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(b) Model (3.2a), Lagrange multiplier p. Iso-lines are shown in the range from 8.5 to 10.5in steps of 0.25.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

π(x

,y)

10

8

6

4

2

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(c) Model (3.2a), mean normal stress π =

− 13

Tr T. Isolines are shown in the range from5 to 7 in steps of 0.25.

Figure 5: Comparison of pressure/Lagrange multiplier fields for models (3.2a) and (3.2b). Plots in the physical space, the plugregion is magnified, µ̂1 = 0.05.

while the maximal pressure stays located on the surface of the inner cylinder, see Figure 8c. (Compare with the effect discussedin Section 6.3.) A magnification of the plug region is shown in Figure 8d–f.

Let us now focus on the electrorheological fluid (3.2b), fix µ̂1 = 1, ε̂rε0 = 10 and ε̂rε0 = 0.01 respectively, and let us considerthe behaviour of the solution with respect to the Reynolds number µ̂0 =

1

Re. In this case we can not expect the same behaviour

as in the classical case (Navier–Stokes fluid). Even if µ̂0 =1

Redecreases (Re increases) the character of the governing equation

(6.1) remains unchanged in the sense that the coefficient in front of the Laplace operator does not tend to zero for Re → +∞ asit happens in the Navier–Stokes equations. For electrorheological fluids, the Laplace operator does not cease to be the dominantoperator in the governing equation even for high Reynolds numbers12.

Plots of the velocity and pressure fields are shown in Figure 9, Figure 10 and Figure 11. (Note that as a consequenceof Proposition 1 the magnitude of ε̂rε0 is irrelevant with respect to the velocity field. It influences only the pressure field.)Clearly the behaviour with increasing Reynolds number is different from the behaviour of the classical Navier–Stokes fluid. Ifwe consider the case ε̂rε0 = 0.01, thus the case where the contribution of the volume force ε̂rε0 [∇E] E is small with respect toterm µ̂1(DE ⊗ E + E ⊗ DE), we see that with increasing Reynolds number the system reaches the state where its behaviouris dominated by µ̂1(DE ⊗ E + E ⊗ DE), and the change in magnitude of the Reynolds number has almost no effect, compareFigure 11b and Figure 11c.

On the other hand, if ε̂rε0 = 10 then the volume force dominates the “extra stress” term µ̂1(DE⊗E + E⊗DE), and we seethat on increasing the Reynolds number, the system reaches the state where its behaviour is determined mainly by the volumeforce, see Figure 10.

If we consider the Lagrange multiplier p, the behaviour of the model (3.2a) is similar to the behaviour of p in the model (3.2a).(Let us again recall that by virtue of Proposition 1 the velocity fields for (3.2a) and (3.2b) are identical.) The magnitude remainsalmost the same, the difference is in the internal structure of the p field, compare Figure 11a–c and Figure 12a–c.

7. Conclusion

We have studied the behaviour predicted by two models for the response of electrorheological fluids, which are expected tobe quite different within a fully three dimensional setting. Both models can be used to model the dependence of the viscosityon the intensity of the electric field. We have shown that although the models are not identical, they lead to the same velocityand pressure fields in simple shear flows (plane Poiseuille flow and cylindrical Couette flow). Next, we have considered flows ofthese fluids between eccentrically placed rotating cylinders, in a setting where the models can be expected to lead to differentpressure and velocity fields. We have proved that in the case of flow between eccentric cylinders the models lead to the samevelocity field but they lead to different pressure fields. Thus, even the flow due to cylinders rotating about non-coincidentaxes is insufficient to produce a problem wherein both the velocity and pressure fields are different and one would need a morecomplicated flow geometry to fully discriminate between the three dimensional constitutive models. In the present study, theproblem of flow between eccentric cylinders has been solved numerically and we have discussed the behaviour of the solutionwith respect to various parameters that have an effect on the flow.

12Note that if we were working with one material (fixed material constants) and the magnitude of the Reynolds number is adjusted by the choiceof the characteristic velocity (increasing/decreasing the angular velocity of the inner cylinder) then the experiment of the type “keep µ̂1 and ε̂0εrconstant and vary Re” would have been impossible even if we were allowed to change the characteristic intensity of the electric field, see the definitionof the dimensionless parameters in Section 5.1.


ε̂0εr = 20ε̂0εr = 10ε̂0εr = 5ε̂0εr = 1ε̂0εr = 0

β

p(α

,β)

2π32πππ

20

200

100

0

-100

-200

-300

-400

(a) Pressure on the inner cylinder wall, α = 1, (α = αk, k = 1).

ε̂0εr = 20ε̂0εr = 10ε̂0εr = 5ε̂0εr = 1ε̂0εr = 0

β

p(α

,β)

2π32πππ

20

200

150

100

50

0

-50

-100

-150

-200

-250

-300

(b) Pressure on the outer cylinder wall, α = −1, (α = αk, k = M).

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(c) Pressure field in the computational space,ε̂0εr = 0.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(d) Pressure field in the computational space,ε̂0εr = 1.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

150

100

50

0

-50

-100

-150

-200

-250

-300

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(e) Pressure field in the computational space,ε̂0εr = 10.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

180

160

140

120

100

80

60

40

20

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(f) Pressure field in the physical space, ε̂0εr =0, the plug region is magnified.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

160

140

120

100

80

60

40

20

0

-20

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(g) Pressure field in the physical space, ε̂0εr =1, the plug region is magnified.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

150

100

50

0

-50

-100

-150

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(h) Pressure field in the physical space, ε̂0εr =10, the plug region is magnified.

Figure 6: Model (3.2b), influence of term ε̂0εr [∇E] E.

The paper points out the necessity to consider fully three dimensional constitutive relations instead of one dimensionalconstitutive relations, and provides an explicit example and a thorough discussion of two different three dimensional constitutiverelations that coincide in simple shear flows but that are not identical in more complicated flows. Moreover, the flow geometrywe have used to document the difference between the considered models corresponds to a technologically relevant problem—theproblem of flow of an electrorheological fluid in a journal bearing.

References

Abu-Jdayil, B. and P. O. Brunn (1996). Effects of electrode morphology on the slit flow of an electrorheological fluid. J.Non-Newton. Fluid Mech. 63 (1), 45–61.

Abu-Jdayil, B. and P. O. Brunn (1997). Study of the flow behavior of electrorheological fluids at shear- and flow-mode. Chem.Eng. Process. 36 (4), 281–289.

Atkin, R. J., S. Xiao, and W. A. Bullough (1991, OCT). Solutions of the constitutive-equations for the flow of an electrorheo-logical fluid in radial configurations. J. Rheol. 35 (7), 1441–1461.


Re = 200Re = 150Re = 100Re = 50Re = 10Re = 0.01

α

vα̂(α

,β)

10.50-0.5-1

0

-0.002

-0.004

-0.006

-0.008

-0.01

-0.012

-0.014

-0.016


Re = 200Re = 150Re = 100Re = 50Re = 10Re = 0.01

α

vβ̂(α

,β)

10.50-0.5-1

1

0.8

0.6

0.4

0.2

0

-0.2

-0.25-0.5-0.75

-0.1-0.105-0.11

-0.115-0.12

-0.125-0.13

-0.135-0.14

-0.145-0.15

-0.155


Figure 7: Navier–Stokes fluid, dependence of the velocity field on Re.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

3000

2000

1000

0

-1000

-2000

-3000

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(a) Pressure field in the computational space,Re = 0.01.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(b) Pressure field in the computational space,Re = 50.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(c) Pressure field in the computational space,Re = 200.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

3000

2500

2000

1500

1000

500

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(d) Pressure field in the physical space, theplug region is magnified, Re = 0.01.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(e) Pressure field in the physical space, theplug region is magnified, Re = 50.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

0.35

0.3

0.25

0.2

0.15

0.1

0.05

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(f) Pressure field in the physical space, theplug region is magnified, Re = 200.

Figure 8: Navier–Stokes fluid, dependence of the pressure field on Re.

Belza, T., V. Pavĺınek, P. Sáha, and O. Quadrat (2008). Effect of field strength and temperature on viscoelastic properties ofelectrorheological suspensions of urea-modified silica particles. Colloid Surf. A-Physicochem. Eng. Asp. 316 (1-3), 89–94.

Bernardi, C., C. Canuto, and Y. Maday (1988). Generalized Inf-Sup conditions for Chebyshev spectral approximation of theStokes problem. SIAM J. Numer. Anal. 25 (6), 1237–1271.

Bouzidane, A. and M. Thomas (2008). An electrorheological hydrostatic journal bearing for controlling rotor vibration. Comput.Struct. 86 (3-5), 463–472.

Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang (2006). Spectral methods: Fundamentals in single domains. ScientificComputation. Berlin: Springer-Verlag.


Re = 100Re = 50Re = 10Re = 1Re = 0.1Re = 0.01

α

u(α

,β)

10.50-0.5-1

0

-5e-05

-0.0001

-0.00015

-0.0002

-0.00025

-0.0003

-0.00035

-0.0004

-0.00045


Re = 100Re = 50Re = 10Re = 1Re = 0.1Re = 0.01

α

v(α

,β)

10.50-0.5-1

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.25-0.5-0.75

-0.1-0.12-0.14-0.16-0.18-0.2

-0.22-0.24-0.26-0.28-0.3


Figure 9: Model (3.2b), dependence of the velocity field on Re, ε̂rε0 = 10.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

3000

2000

1000

0

-1000

-2000

-3000

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

-250

-300

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

150

100

50

0

-50

-100

-150

-200

-250

-300

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


Figure 10: Model (3.2b), dependence of the pressure field on Re, ε̂rε0 = 10.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

3000

2000

1000

0

-1000

-2000

-3000

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20


Figure 11: Model (3.2b), dependence of the pressure field on Re, ε̂rε0 = 0.01.

Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang (2007). Spectral methods: Evolution to complex geometries andapplications to fluid dynamics. Scientific Computation. Berlin: Springer.

Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. The International Series of Monographs on Physics.Clarendon Press, Oxford.

Choi, Y. T., J. U. Cho, S. B. Choi, and N. M. Wereley (2005, OCT). Constitutive models of electrorheological and magne-torheological fluids using viscometers. Smart Mater. Struct. 14 (5), 1025–1036.

Coleman, B. D., H. Markovitz, and W. Noll (1966). Viscometric flows of non-newtonian fluids. Theory and experiment. Berlin:Springer-Verlag.


α1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

3000

2000

1000

0

-1000

-2000

-3000

α1

0.5

0

-0.5

-1

β

2π32πππ

20

(a) Lagrange multiplier p field in the compu-tational space, Re = 0.01.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(b) Lagrange multiplier p field in the compu-tational space, Re = 10.

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

p(α

,β)

200

150

100

50

0

-50

-100

-150

-200

α

1

0.5

0

-0.5

-1

β

2π32πππ

20

(c) Lagrange multiplier p field in the computa-tional space, Re = 200.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

3000

2500

2000

1500

1000

500

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(d) Lagrange multiplier p field in the physicalspace, the plug region is magnified, Re = 0.01.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

180

160

140

120

100

80

60

40

20

0

-20

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(e) Lagrange multiplier p field in the physicalspace, the plug region is magnified, Re = 10.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

p(x

,y)

180

160

140

120

100

80

60

40

20

0

-20

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(f) Lagrange multiplier p field in the physicalspace, the plug region is magnified, Re = 200.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

π(x

,y)

2000

1800

1600

1400

1200

1000

800

600

400

200

0

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(g) Mean normal stress π field in the physicalspace, the plug region is magnified, Re = 0.01.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

π(x

,y)

120

100

80

60

40

20

0

-20

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

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0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(h) Mean normal stress π field in the physicalspace, the plug region is magnified, Re = 10.

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

π(x

,y)

120

100

80

60

40

20

0

-20

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

x

0.80.70.60.50.40.3

(i) Mean normal stress π field in the physicalspace, the plug region is magnified, Re = 200.

Figure 12: Model (3.2a), dependence of the pressure/Lagrange multiplier field on Re, ε̂rε0 = 0.01.

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Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Praha 8 – Karĺın, CZ 186 75, Czech Republic

E-mail address: [email protected]

Texas A&M University, Department of Mechanical Engineering, 3123 TAMU, College Station TX 77843-3123, United States of

America

E-mail address: [email protected]

1. Introduction2. Governing equations3. Constitutive relations for the Cauchy stress tensor4. Simple shear flows4.1. Plane Poiseuille flow4.2. Cylindrical Couette flow

5. Flow between eccentric cylinders5.1. Dimensionless form of the governing equations5.2. Bipolar coordinates5.3. Explicit analytical solu

Flow of an electrorheological °uids between eccentric ...ncmm.karlin.mff.cuni.cz/preprints/10206125456pr25.pdf · Flow of an electrorheological °uids between eccentric rotating

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