Problems on Electrorheological Fluid Flows

COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 1, Number 4, October 2002 pp. 1–??

PROBLEMS ON ELECTRORHEOLOGICAL FLUID FLOWS

R.H.W. HOPPE AND W.G. LITVINOV

RONALD H.W. HOPPE Department of Mathematics University of Houston Houston, TX 77204-3008, USA and Institute of Mathematics University of Augsburg D-86159 Augsburg, Germany

WILLIAM G.LITVINOV Institute of Mathematics University of Augsburg D-86159 Augsburg, Germany (Communicated by ???)

ABSTRACT. In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity field. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.

1. Introduction. Electrorheological fluids are smart materials which are concentrated suspensions of polarizable particles in a nonconducting dielectric liquid. In moderately large electric fields, the particles form chains along the field lines, and these chains then ag- gregate to form columns (see Fig. 1 , taken from [20]). These chainlike and columnar structures cause dramatic changes in the rheological properties of the suspensions. The fluids become anisotropic, the apparent viscosity (the resistance to flow) in the direction orthogonal to the direction of electric field abruptly increases, while the apparent viscosity in the direction of the electric field does not change drastically.

1991 Mathematics Subject Classiﬁcation. 37C45. Key words and phrases. electrorheological ﬂuids, boundary value problems, existence and uniqueness, numerical solution.

1 2 R.H.W. HOPPE AND W.G. LITVINOV

The chainlike and columnar structures are destroyed under the action of large stresses. Then, the apparent viscosity of the fluid decreases and the fluid becomes less anisotropic. Constitutive relations for electrorheological fluids in which the stress tensor σ is an isotropic function of the vector of electric field strength E and the rate of strain tensor ε were derived in [21]. For an incompressible fluid, the following equation has been obtained: 2 2 2 σ = −pI1 +α2E⊗E+α3ε+α4ε +α5(εE⊗E+E⊗εE)+α6(ε E⊗E+E⊗ε E). (1)

Here, p is the pressure, I1 the unit tensor, αi are scalar functions of six invariants of the tensors ε, E ⊗ E, and mixed tensors. Note that the αi , 1 ≤ i ≤ 6, have to be determined experimentally. In case of simple shear flow, when the vectors of velocity v and the electric field E are orthogonal and E is in the plane of flow, the terms with coefficients α2, α4, α5, α6 give rise to two normal stresses differences (see [21]). But these terms lead to incorrectness of the boundary value problems for the constitutive equation (1). Indeed, very restrictive conditions have to be imposed on the coefficients α2, α4, α5, α6 in order to get an operator satisfying the conditions of coerciveness and monotonicity (the condition of coerciveness is almost similar to the Clausius-Duhem inequality following from the second law of ther- modynamics, and the condition of monotonicity means that stresses increase as the rate of strains increases). The constitutive equation (1) does not describe anisotropy of the fluid. For simple shear flow, (1) gives the same values of the shear stresses in the cases, when the vectors of velocity and electric field are orthogonal and parallel (σ is an isotropic function of E and ε in (1)). Stationary and nonstationary mathematical problems for the special case of (1) have been studied in [22]. It is assumed in [22] that the velocities are equal to zero everywhere on the boundary and the stress tensor is given by

2 k−1 σ = −pI1 + γ1((1 + |ε| ) 2 − 1)E ⊗ E 2 2 k−2 2 k−2 +(γ2 + γ3|E| )(1 + |ε| ) 2 ε + γ4(1 + |ε| ) 2 (εE ⊗ E + E ⊗ εE), (2)

2 Pn 2 where |ε| = i,j=1 εij , n being the dimension of the flow domain, γ1 −γ4, are constants, and k is a function of |E|2. The constants γ1 − γ4 and the function k have to be determined by the approximation of flow curves which have been obtained experimentally for different values of the vector of the electric field E (see Subsection 2.2). But the conditions of coerciveness and monotonicity of the operator − div(σ + pI1) impose severe constraints on the constants γ1 − γ4 and on the function k (cf. [22]) such that under these restrictions one cannot obtain a good approximation of a flow curve, to say nothing about the approximation of a set of flow curves corresponding to different values of E. In Section 2 below, we develop a constitutive equation of electrorheological fluids in such a way that the fluid is considered as a viscous one with the viscosity depending on the second invariant of the rate of strain tensor, on the module of the vector of the electric field strength, and on the angle between the vectors of velocity and electric field strength. This constitutive equation describes the main peculiarities of electrorheological fluids. It can be identified so that a set of flow curves corresponding to different values of E is approximated with a high degree of accuracy. Furthermore, it leads to well-posed mathematical problems. In Section 3, we present auxiliary results, and in Sections 4–8 we study problems on stationary flow of such fluids under nonhomogeneous mixed boundary conditions. Here, ELECTRORHEOLOGICAL FLUID FLOWS 3

(a)(b) FIGURE 1. Fibrous structure formed by the electric field for alumina particles we prescribe values of velocities and surface forces on different parts of the boundary and ignore inertial forces. Two cases are studied: namely the case where the viscosity function is continuous and the case where it is singular (equal to infinity, when the second invariant of the rate of strain tensor is equal to zero). In the latter case the problem reduces to a variational inequality. By using fixed point methods, monotonicity and compactness arguments, we prove existence results for regular and singular viscosity functions. In the latter case, existence results are obtained under more restrictive assumptions. Here, the singular viscosity is approximated by a continuous bounded one with a regularization parameter, and a solution of the variational inequality is obtained as a limit of the solutions of regularized problems. Section 9 is concerned with the numerical solution of the problems on stationary flows of electrorheological fluids with regular viscosity function. We consider methods of augmented Lagrangians, Birger-Kachanov, contraction and gradient. Numerical results are given for the compression and rebound stage of electrorheological shock absorbers.

2. Constitutive equation. 2.1. The form of the constitutive equation. It has been found experimentally that the shear stress and the viscosity of electrorheological fluids depend on the shear rate, the module of the vector of the electric field strength, and the angle between the vectors of fluid velocity and the electric field strength (see [20, 24]). On the basis of these experimental results we introduce the following constitutive equation

σij (p, u, E) = −pδij + 2ϕ(I(u), |E|, µ(u, E))εij(u), i, j = 1, . . . , n, n = 2 or 3. (3) Here, σij (p, u, E) are the components of the stress tensor which depend on the pressure p, the velocity vector u = (u1, . . . , un) and the electric ﬁeld strength E = (E1,...,En), δij is the Kronecker delta, and εij (u) are the components of the rate of strain tensor

1¡ ∂ui ∂uj ¢ εij (u) = + . (4) 2 ∂xj ∂xi Moreover, I(u) is the second invariant of the rate of strain tensor Xn 2 I(u) = (εij(u)) , (5) i,j=1 4 R.H.W. HOPPE AND W.G. LITVINOV and ϕ denotes a viscosity function depending on I(u), |E| and µ(u, E). The function µ is introduced in the constitutive equation (3) in order to take into account the anisotropy of the electrorheological fluid under which the viscosity of the fluid depends on the angle between the vector of electric field and the vector of velocity with respect to the charged electrode (the counter electrode is usually not charged). The electrode can move relative to the body of an electrorheological device, and hence, we consider that the electrode can move relative to the reference frame under consideration. Let uˇ(x, t) = (ˇu1(x, t), uˇ2(x, t), uˇ3(x, t)) be a vector of transfer velocity and uˇ(x, t) a velocity of a point of the electrode which coincides with the point x of the frame at a moment t. It is assumed that uˇ is a known function. We define the function µ as the square of the cosine of the angle between the vector of the electric field and the vector of the velocity relative to the electrode, i. e.,

³ ´2 2 u − uˇ E ((ui − uˇi)Ei) µ(u, E) = , = ³ P ´³ P ´. (6) |u − uˇ| |E| R3 3 2 3 2 i=1(ui − uˇi) i=1 Ei Here and below, Einstein’s summation convention over repeated indices is applied, and we 3 refer to (., .)R3 as the scalar product in R . If the electrode does not move relative to the reference frame, then uˇ = 0 and the function µ takes the form ³ u E ´2 µ(u, E) = , . (7) |u| |E| R3 In the general case, the function uˇ is deﬁned as follows: ◦ uˇ(x, t) = u(t) + w(x, t), (8) ◦ ◦ ◦ ◦ where u(t) = (u1(t), u2(t), u3(t)) is a vector of translation velocity, and w(x, t) = (w1(x, t), w2(x, t), w3(x, t)) is a vector rotational velocity,

w1(x, t) = ω2(t)x3 − ω3(t)x2,

w2(x, t) = ω3(t)x1 − ω1(t)x3,

w3(x, t) = ω1(t)x2 − ω2(t)x1. (9)

Here, ω(t) = (ω1(t), ω2(t), ω3(t)) stands for a vector of angular velocity at time instant t. It follows from (3), (6), (8), and (9) that for given functions of the electric ﬁeld E and the velocity u − uˇ relative to the electrode, the value of the function µ(u, E) and the deviator of the stress tensor

3 {2ϕ(I(u), |E|, µ(u, E))εij (u)}i,j=1, do not depend on the velocity uˇ of the electrode (see [8] for details). The function µ defined by (6), (7) is not specified at zero values of the functions E and v = u − uˇ. Moreover, there does not exist an extension of µ by continuity to the zero values of E and v. However, for zero value of E there is no influence of the electric field, and the function µ does not have to be specified for E = 0. Let Qt be the set of points (x, t) such that v(x, t)=u(x, t) − uˇ(x, t)=0 for (x, t) ∈ Qt. If the measure of Qt is zero, we do not have to specify µ for v = 0. ◦ Assume that the measure of Qt is positive, and let Qt be the interior of Qt. Then, by (8), (9), we have ◦ (εij (u))(x, t) = 0 in Qt, i, j = 1, 2, 3. (10) ELECTRORHEOLOGICAL FLUID FLOWS 5

◦ Therefore, the deviator of the stress tensor is equal to zero in Qt, i.e., ◦ (ϕ(I(u), |E|, µ(u, E))εij (u))(x, t) = 0 in Qt, i, j = 1, 2, 3. (11) ◦ Since the measure of the set Q \ Qt is zero, there is no need to specify the function µ in Qt and hence, for v = 0.

2.2. Assumptions on the viscosity function. Flow curves of electrorheological ﬂuids obtained experimentally for µ(u, E) = 0 have the form as displayed in Fig. 2 (cf.,e.g.,[24]). τ 4

τ0 1

γ 0 γ0 γ1 FIGURE 2

These curves define the relationship between the shear stress τ = σ12 and the shear rate 1 du1 γ = ε12(u) = for a flow that is close to simple shear flow. Line 1 is the flow curve 2 dx2 for |E| = 0, and lines 2–4 represent the flow curve for increasing |E|. Flow curves are obtained in some region, say γ0 ≤ γ ≤ γ1, γ0 > 0. Experimental results for small γ are not precise, and one has to extend the flow curves to R+. It is common to extend flow curves by straight lines over the region γ1 < γ < ∞. One can extend flow curves in [0, γ0) in such a way that either τ = τ0 for γ = 0 or τ = 0 for γ = 0 (see the dash and dot-dash lines in Figure 2). The viscosity η(γ, E) of the fluid is determined as 1 τ η(γ, E) = , (12) 2 γ and it is defined by the approximation of the lines 1–4 extended to R+. Generalizing (12) to an arbitrary flow, we take

³ ´ 1 ³³ ´ 1 ´ 1 2 1 2 γ = I(u) , ϕ(I(u), |E|, µ(u, E)) = η I(u) ,E . (13) 2 2

If the ﬂow curve is extended by the straight line τ = c1 + c2γ, γ ∈ (γ1, ∞) , we obtain ( 2 ϕ1(I(u), |E|, µ(u, E)),I(u) ∈ [0, 2γ1 ], ϕ(I(u), |E|, µ(u, E)) = 1 (14) 1 1 − 2 2 2 (c2 + c1( 2 I(u)) ),I(u) ∈ (2γ1 , ∞). 6 R.H.W. HOPPE AND W.G. LITVINOV

Here, the coefficients c1 and c2 depend on |E| and µ(u, E). The viscosity function ϕ is 2 continuous in R+ × [0, 1], if the flow curve is extended in [0, γ0] by the dot-dash line, and it has the form b(|E|, µ(u, E)) ϕ(I(u), |E|, µ(u, E)) = 1 + ψ(I(u), |E|, µ(u, E)) (15) I(u) 2 − 1 with b(|E|, µ(u, E)) = 2 2 τ0, if the flow curve is extended by the dash line in [0, γ0], ψ 2 being a function continuous in R+ × [0, 1]. We note that if ψ(I(u), |E|, µ(u, E)) = b1(|E|, µ(u, E)), then (15) is the viscosity function of an extended Bingham electrorheological fluid. In the case that the flow curve is extended in [0, γ0] by the dot-dash line, the viscosity function can be written as follows: − 1 ϕ(I(u), |E|, µ(u, E)) = b(|E|, µ(u, E))(λ + I(u)) 2 + ψ(I(u), |E|, µ(u, E)), (16) where λ is a small positive parameter. Obviously, for λ = 0 the function ϕ defined by (16) is the same as the one defined by (15). Moreover, for I(u) = 0 we have ϕ(0, |E|, µ(u, E)) = ∞ for (15), − 1 ϕ(0, |E|, µ(u, E)) = b(|E|, µ(u, E))λ 2 + ψ(0, |E|, µ(u, E)) for (16). Flow problems for fluids with a constitutive equation (15) reduce to the solution of variational inequalities. Such problems are considerably more complicated than problems for fluids with finite viscosity. In particular, this applies in case of fluids with a constitutive equation as given by (16). From a physical point of view, (16) with a finite, but possibly large viscosity for I(u) = 0 seems to be more reasonable than (15). We will study problems in the case where ϕ is a continuous bounded function of its arguments and in the case where ϕ is singular and the singular part of the function ϕ is − 1 equal to b(|E|, µ(u, E))I(u) 2 . In the first case, we assume that ϕ satisfies one of the following conditions (C1), (C2), (C3): 2 (C1): ϕ :(y1, y2, y3) → ϕ(y1, y2, y3) is a function continuous in R+ × [0, 1]. For an arbitrarily fixed (y2, y3) ∈ R+ ×[0, 1], the function ϕ(., y2, y3): y1 → ϕ(y1, y2, y3) is continuously differentiable in R+. Moreover, the following inequalities hold true:

a2 ≥ ϕ(y1, y2, y3) ≥ a1 (17) ∂ϕ ϕ(y1, y2, y3) + 2 (y1, y2, y3)y1 ≥ a3 (18) ∂y1 ¯ ¯ ¯ ∂ϕ ¯ ¯ (y1, y2, y3)¯y1 ≤ a4, (19) ∂y1 where ai, 1 ≤ i ≤ 4, are positive numbers. 2 (C2): ϕ :(y1, y2, y3) → ϕ(y1, y2, y3) is a function continuous in R+ × [0, 1]. For an arbitrarily ﬁxed (y2, y3) ∈ R+ × [0, 1], (17) and the following inequality hold true:

2 2 2 2 [ϕ(z1 , y2, y3)z1 − ϕ(z2 , y2, y3)z2](z1 − z2) ≥ a3(z1 − z2) ∀(z1, z2) ∈ R+. (20) 2 (C3): ϕ :(y1, y2, y3) → ϕ(y1, y2, y3) is a function continuous in R+ × [0, 1]. For an arbitrarily ﬁxed (y2, y3) ∈ R+ × [0, 1], (17) holds true and the function z → 2 ϕ(z , y2, y3)z is strictly increasing in R+, i.e., the conditions z1, z2 ∈ R+, z1 > z2 2 2 imply ϕ(z1 , y2, y3)z1 > ϕ(z2 , y2, y3)z2. ELECTRORHEOLOGICAL FLUID FLOWS 7

Let us dwell on the physical sense of these inequalities. (17) indicates that the viscosity is bounded from below and from above by positive constants. The inequality (18) implies that for fixed values of |E| and µ(u, E) the derivative of the function I(v) → G(v) is positive, where G(v) is the second invariant of the stress deviator G(v) = 4[ϕ(I(v), |E|, µ(u, E))]2I(v). This means that in case of simple shear flow the shear stress increases with increasing shear ∂ϕ rate. (19) is a restriction on for large values of y1. These inequalities are natural from ∂y1 a physical point of view. The assumptions (C2) and (C3) indicate that in case of simple shear flow the shear stress must increase with increasing shear rate. 2 Moreover, let λ(z) = ϕ(z , y2, y3)z. Assume that the function λ is continuously dif- dλ dλ ferentiable in R+. It follows from (C2) that dz ≥ a3 for all z ∈ R+. Calculating dz (z), we obtain (18) from (20). On the other hand, (20) follows from (18). The assumption (C3) dλ indicates that dz (z) > 0 for all z ∈ R+ or, equivalently ∂ϕ ϕ(y1, y2, y3) + 2 (y1, y2, y3)y1 > 0, y1 ∈ R+. ∂y1 The following assumption is concerned with the function (coefficient) b in (15), (16):

(C4): b : y1, y2 → b(y1, y2) is a function continuous in R+ × [0, 1] and satisﬁes

0 ≤ b(y1, y2) ≤ a5, (y1, y2) ∈ R+ × [0, 1], (21)

where a5 is a positive number. Generally, the continuous function ϕ is expressible in the form Xm ϕ(I(u), |E|, µ(u, E)) = ei(|E|, µ(u, E))βi(I(u)). (22) i=1

Polynomials or splines can be used to represent the functions βi. The flow curves obtained for various values of E can be approximated within an arbitrary accuracy. We note that (22) can also be used for an identification of the function ψ. In the general case, the coefficients ei as well as the viscosity function ϕ depend on the temperature and these coefficients can be determined by an approximation of the corresponding flow curves as well.

2.3. General problems. Fig. 3 below gives an example of an electrorheological ﬂuid ﬂow.

Ω2 Γ0 S2

F Ω1 Ω3 uˆ ∆U(t)

Γ1

S1 = S \ S2

FIGURE 3 8 R.H.W. HOPPE AND W.G. LITVINOV

Here, the domain Ω of fluid flow consists of three parts Ω1, Ω2, Ω3. A fluid flows from part Ω1 across Ω2 into Ω3. Electrodes are placed on Γ0 and Γ1 of the boundary of Ω2, and an electric field E is generated by applying voltages 4U(t) to electrodes at time t. Generally, there may be k pairs of electrodes, and voltages 4Ui(t) are applied to the i-th pair of electrodes, i = 1, . . . , k. The boundary S of the domain Ω consists of two parts S1 and S2. Surface forces F = (F1,...,Fn) act on S2, whereas the velocity uˆ = (û1,..., uˆn) is given on S1. The equations of motion and the incompressibility condition read as follows: ³ ´ h i ∂ui ∂ui ∂p ∂ ρ + uj + − 2 ϕ(I(u), |E|, µ(u, E))εij(u) ∂t ∂xj ∂xi ∂xj

= Ki in Q = Ω × (0,T ), i = 1, . . . , n, (23) Xn ∂u div u = i = 0 in Q. (24) ∂x i=1 i

Here, Ki are the components of the volume force K, ρ is the density, and T denotes a positive constant. In (23) and below, Einstein’s summation convention over repeated indices is applied. n We assume that Ω is a bounded domain in R , n = 2 or 3. Suppose that S1 and S2 are open subsets of S such that S = S1 ∪ S2 and S1 ∩ S2 = ∅. The boundary and initial conditions are the following: ¯ u¯ =u, ˆ (25) S1×(0,T ) ¯ [−pδij + 2ϕ(I(u), |E|, µ(u, E))εij (u)]νj¯ = Fi i = 1, . . . , n, (26) S2×(0,T ) u(., 0) = u0 in Ω. (27)

Here, Fi and νj are the components of the vector of surface force F = (F1,...,Fn) and the unit outward normal ν = (ν1, . . . , νn) to S, respectively. We consider Maxwell’s equations in the following form (see e.g. [10]): 1 ∂B curl E + = 0, div B = 0, c ∂t 1 ∂D curl H − = 0, div D = 0. (28) c ∂t Here E is the electric field, B the magnetic induction, D the electric displacement, H the magnetic field, c the speed of light. We can assume that D = ²E, B = µH, (29) where ² is the dielectric permittivity, µ the magnetic permeability. Since electrorheological fluids are dielectrics, the magnetic field H can be neglected. Then (28), (29) lead to the following relations curl E = 0, (30) div(²E) = 0. (31) It follows from (30) that there exists a potential θ such that E = − grad θ, (32) and (31) implies div(² grad θ) = 0. (33) ELECTRORHEOLOGICAL FLUID FLOWS 9

The boundary conditions are the following:

θ = Ui(t) on Γi, i = 1, . . . , k, (34)

θ = 0 on Γi0, (35) [k ν · ² grad θ = 0 on S\( (Γi ∪ Γi0)). (36) i=1

Here, Γi and Γi0 are the surfaces of the i-th control and null electrodes respectively, and it is assumed that Γi, Γi0 are open subsets of S. We suppose

² ∈ L∞(Ω), e1 ≤ ² ≤ e2 a.e. in Ω, (37) e1, e2 being positive constants. We further assume

1 2 Ui(t) ∈ H00(Γi), t ∈ [0,T ], i = 1, . . . , k, (38) where 1 ¯ ¯ 2 1 H (Γi), {ψ | ψ = v¯ , v ∈ H (Ω), v¯ = 0}. (39) 00 Γi S\Γi 1 2 The space H00(Γi) is equipped with the norm ¯ ¯ 1 ¯ ¯ kψk 1 = inf{kvkH1(Ω), v ∈ H (Ω), v = ψ, v = 0}. (40) 2 Γi S\Γi H00(Γi) Let θˇ be a function such that ¯ ¯ ˇ 1 ˇ ˇ θ ∈ H (Ω), θ¯ = Ui(t), θ¯ = 0, i = 1, . . . , k. (41) Γi Γi0 We deﬁne a space V˜ and a bilinear form a : H1(Ω) × V˜ → R as follows: ¯ ¯ V˜ = {v ∈ H1(Ω), v¯ = 0}, (42) ∪k (Γ ∪Γ ) Z i=1 i i0 ∂v ∂h a(v, h) = ² i dx, v ∈ H1(Ω), h ∈ V,˜ (43) Ω ∂xi ∂xi and we consider the problem: Find u satisfying u ∈ V˜ a(u, h) = −a(θ,ˇ h) h ∈ V.˜ (44) By use of Green’s formula, it can be seen that, if u is a solution of problem (44), then the function θ = u + θˇ is a solution of (33)–(36) in the sense of distributions. On the other hand, if θ is a solution of (33)–(36), then u = θ − θˇ is a solution of the (44). Therefore, the function θ = u + θˇ is a generalized solution of (33)–(36).

THEOREM 2.1. Assume that conditions (37), (38) are satisﬁed. Then, there exists a unique generalized solution θ of problem (33)–(36). The function θ admits the representation θ = θˇ + u, where θˇ satisﬁes (41) and u is the solution of (44). Proof. In view of (37), the bilinear form a is continuous and coercive in V˜ . Conse- quently, there exists a unique solution u of (44), and the function θ = θˇ+u is a generalized solution of (33)–(36). ˇ ˇ ˇ Let θ1 = θ1 + u1 and θ2 = θ2 + u2 be two generalized solutions of (33)–(36), where θ1 ˇ and θ2 satisfy (41), and u1, u2 ∈ V˜ . Then θ1−θ2 = w ∈ V˜ , and (44) implies a(w, w) = 0. Hence, w = 0. • 10 R.H.W. HOPPE AND W.G. LITVINOV

The functions of volume force K and surface force F in (23) and (26) are represented in the form K = K˜ + Ke,F = F˜ + Fe, (45) where K˜ and F˜ are the main volume and surface forces, Ke and Fe are volume and surface forces generated by the vector of electric field E. Considering the electrorheological fluid n as a liquid dielectric, we represent the stress tensor σe = {σeik}i,k=1 induced by the electric field as follows (see [10]): |E|2 ¡ ∂² ¢ ² σ = − ² − ρ δ + E E . (46) eik 8π ∂ρ ik 4π i k Taking (30) into account, we obtain the following formula for the vector of volume force

Ke = (Ke1,...,Ken), 2 ³ ´ ∂σeij |E| ∂² 1 ∂ 2 ∂² Kei = = − + |E| ρ , i = 1, . . . , n. (47) ∂xj 8π ∂xi 8π ∂xi ∂ρ The vector of surface forces is given by

Fe = (Fe1,...,Fen),Fei = σeikνk. (48) Consequently, the systems (23)–(27) and (33)–(36) are separated. Hence, one can ﬁrst solve the quasi-static system (33)–(36) and then solve the problem (23)–(27), (45).

3. Auxiliary results. Let Ω be a bounded domain in Rn with a Lipschitz continuous boundary S, n = 2 or 3, and let S1 be an open non-empty subset of S. We consider the following spaces: 1 n X = {u|u ∈ H (Ω) , u|S1 = 0}, (49) V = {u|u ∈ X, div u = 0}. (50) By means of Korn’s inequality, the expression µZ ¶ 1 2 kukX = I(u)dx (51) Ω deﬁnes a norm on X and V being equivalent to the H1(Ω)n-norm. In the sequel we will use the following notations: If Y is a normed space, we denote by Y ∗ the dual of Y , and by (f, h) the duality between Y ∗ and Y , where f ∈ Y ∗, h ∈ Y . In n particular, if f ∈ L2(Ω) or f ∈ L2(Ω) , then (f, h) is the scalar product in L2(Ω) or in n L2(Ω) , respectively. The sign * denotes weak convergence in a Banach space. We further consider three functions v,˜ v1, v2 such that 1 n v˜ ∈ H (Ω) , v1 ∈ L2(Ω), v1(x) ≥ 0 a. e. in Ω,

v2 ∈ L∞(Ω), v2(x) ∈ [0, 1] a.e. in Ω. (52) ∗ We set v = (˜v, v1, v2) and deﬁne an operator Lv : X → X as follows: Z

(Lv(u), h) = 2 ϕ(I(u +v ˜), v1, v2)εij (u +v ˜)εij(h)dx u, h ∈ X. (53) Ω LEMMA 3.1. Assume that (C1) and (52) are satisﬁed. Then, the following inequalities hold true 2 (Lv(u) − Lv(w), u − w) ≥ µ1ku − wkX , u, w ∈ X, (54)

kLv(u) − Lv(w)kX∗ ≤ µ2ku − wkX , u, w ∈ X, (55) ELECTRORHEOLOGICAL FLUID FLOWS 11 where

µ1 = min(2a1, 2a3), µ2 = 2a2 + 4a4. (56) Proof. Let u, w be arbitrarily ﬁxed functions in X and h = u − w. (57) We introduce the function γ as follows: Z

γ(t) = ϕ(I(˜v + w + th), v1, v2)εij(˜v + w + th)εij(e)dx, t ∈ [0, 1], e ∈ X. Ω (58) It is obvious that 1 γ(1) − γ(0) = (L (u) − L (w), e). (59) 2 v v By classical analysis, it follows that γ is differentiable at any point t ∈ (0, 1). Therefore, dγ γ(1) = γ(0) + (ξ), ξ ∈ (0, 1), (60) dt where Z dγ (ξ) = [ϕ(I(˜v + w + ξh), v1, v2)εij(h)εij (e) dt Ω ∂ϕ +2 (I(˜v + w + ξh), v1, v2)εkm(˜v + w + ξh)εkm(h)εij (˜v + w + ξh)εij (e)]dx. ∂y1 (61) Observing the inequality

1 1 |εij(˜v + w + ξh)εij(h)| ≤ I(˜v + w + ξh) 2 I(h) 2 and (17), (19), we see that (55) is a direct consequence of (57)–(61). We consider the function g deﬁned according to ( ∂ϕ ∂ϕ (α, v1(x), v2(x)), if (α, v1(x), v2(x)) < 0, ∂y1 ∂y1 g(α, x) = ∂ϕ 0, if (α, v1(x), v2(x)) ≥ 0, ∂y1 where α ∈ R+, x ∈ Ω. Then, taking e = h in (61) and applying (18) we get Z dγ (ξ) = [ϕ(I(˜v + w + ξh), v1, v2)I(h) dt Ω ∂ϕ 2 2 +2 (I(˜v + w + ξh), v1, v2)(εij(˜v + w + ξh)εij(h)) ]dx ≥ min(a1, a3)khkX , ∂y1 (62) and (54) follows from (62). •

LEMMA 3.2. Let the function ϕ satisfy condition (C2) and assume that (52) holds true. ∗ Then, the operator Lv is a continuous mapping from X into X and Z 1 1 2 (Lv(u) − Lv(w), u − w) ≥ 2a3 [I(u +v ˜) 2 − I(w +v ˜) 2 ] dx, u, w ∈ X. (63) Ω Moreover,

(Lv(u) − Lv(w), u − w) = 0 ⇐⇒ u = w. (64) 12 R.H.W. HOPPE AND W.G. LITVINOV

1 1 1 1 Proof. We set u = u +v ˜, w = w +v ˜. Taking into account that [εij(u )εij (w )] ≤ 1 1 1 1 I(u ) 2 I(w ) 2 , we obtain 1 1 (Lv(u) − Lv(w), u − w) = (Lv(u) − Lv(w), u − w ) Z 1 1 1 = 2 [ϕ(I(u ), v1, v2)I(u1) + ϕ(I(w ), v1, v2)I(w ) Ω 1 1 1 1 1 1 −ϕ(I(u ), v1, v2)εij(u )εij (w ) − ϕ(I(w ), v1, v2)εij(w )εij (u )]dx Z 1 1 1 1 1 1 1 1 1 1 ≥ 2 [ϕ(I(u ), v1, v2)I(u ) 2 − ϕ(I(w ), v1, v2)I(w ) 2 ][I(u ) 2 − I(w ) 2 ]dx (65) Ω Observing (20), (65), we deduce (63). Now, assume

(Lv(u) − Lv(w), u − w) = 0. (66) Then, by (63) we have 1 1 I(u +v ˜) = I(w +v ˜) a.e. in Ω, ϕ(I(u ), v1, v2) = ϕ(I(w ), v1, v2) a.e. in Ω. (67)

Taking (17), (53), (66), and (67) into account, we get ku − wkX = 0. The continuity of the mapping Lv follows from the continuity of the Nemytskii operator (see [26], and [15], Lemma 8.2 in Chapter 2). •

LEMMA 3.3. Assume that (C3) is satisﬁed and (52) holds true. Then

(Lv(u) − Lv(w), u − w) ≥ 0, u, w ∈ X. (68) ∗ Moreover, (64) is valid, and the operator Lv is a continuous mapping from X into X . Proof. Indeed, (68) follows from (C3) and (65). Assume that (66) is valid. Then (C3) and (65) imply (67), and by (17) we obtain u = w. • We introduce the set U as follows

U = {h ∈ L∞(Ω), 0 ≤ h(x) ≤ a5 a.e. in Ω}. (69) 1 n ∗ For v˜ ∈ H (Ω) and a given constant λ > 0, we deﬁne an operator Lλ : U × X → X as follows Z − 1 (Lλ(h, u), w) = h(λ + I(˜v + u)) 2 εij (˜v + u)εij(w)dx. (70) Ω LEMMA 3.4. For an arbitrary λ > 0 and an arbitrarily ﬁxed h ∈ U the following inequalities hold true

(Lλ(h, u1) − Lλ(h, u2), u1 − u2) ≥ 0, u1, u2 ∈ X, (71) − 1 kLλ(h, u1) − Lλ(h, u2)kX∗ ≤ αku1 − u2kX , α = 2a5λ 2 , (72) µZ ¶ 1 2 2 kLλ(h, u)kX∗ ≤ α1, α1 = h dx . (73) Ω Moreover, the conditions

{hm} ⊂ U, hm → h a.e. in Ω,

∂umi ∂ui um → u in X, um → u a.e. in Ω, → a.e. in Ω, i, j = 1, . . . , n, (74) ∂xj ∂xj imply ∗ Lλ(hm, um) → Lλ(h, u) in X . (75) ELECTRORHEOLOGICAL FLUID FLOWS 13

Proof. The viscosity function associated with the operator Lλ(h, .): u → Lλ(h, u) has the form 1 − 1 ϕ(y) = h(λ + y) 2 , y ∈ R , (76) 2 + where y plays the role of the second invariant of the rate of strain tensor. We have

dϕ 1 − 1 −1 ϕ(y) + 2 (y)y = h(λ + y) 2 [1 − (λ + y) y] > 0, y ∈ R , λ > 0. (77) dy 2 + The left-hand side of (77) represents the derivative of the function g : z → g(z) = ϕ(z2)z, z2 = y. Therefore, the function g is increasing, and (71) follows from the proof of Lemma 3.3. By (69), (70) we obtain Z µZ ¶ 1 2 − 1 1 1 2 |(Lλ(h, u), w)| ≤ h(λ + I(˜v + u)) 2 I(˜v + u) 2 I(w) 2 dx ≤ h dx kwkX , Ω Ω which readily gives (73). Moreover, (69) and (76) yield ¯ ¯ 1 − 1 ¯dϕ ¯ 1 − 1 ϕ(y) ≤ a λ 2 , ¯ (y)¯y ≤ a λ 2 , y ∈ R , (78) 2 5 dy 4 5 + and hence, observing Lemma 3.1 we get (72). Assuming (74), it follows that

kLλ(hm, um) − Lλ(h, u)kX∗ ≤ kLλ(hm, um) − Lλ(hm, u)kX∗

+kLλ(hm, u) − Lλ(h, u)|X∗ . (79) Further, Z − 1 (Lλ(hm, um) − Lλ(hm, u), w) = hm[(λ + I(˜v + um)) 2 εij (um − u)εij (w) Ω − 1 − 1 +((λ + I(˜v + um)) 2 − (λ + I(˜v + u)) 2 )εij (˜v + u)εij(w)]dx, whence ·Z ¸ 1 2 2 −1 kLλ(hm, um) − Lλ(hm, u)kX∗ ≤ hm(λ + I(˜v + um)) I(um − u)dx Ω ½Z ¾ 1 2 1 1 2 − 2 − 2 2 + hm[(λ + I(˜v + um)) − (λ + I(˜v + u)) ] I(˜v + u)dx . (80) Ω Obviously, the ﬁrst term of the right-hand side in (80) tends to zero. By (74) we have I(˜v + um) → I(˜v + u) a.e. in Ω, and by the Lebesgue theorem we obtain that the second term of the right-hand side in (80) tends to zero. The second term of the right-hand side in (79) also tends to zero. Thus, (75) is satisﬁed which concludes the proof. •

LEMMA 3.5. Let Ω be a bounded domain in Rn, n = 2 or 3, with a Lipschitz continuous boundary S. Moreover, let the operator B ∈ L(X,L2(Ω)) be deﬁned as follows Bu = div u. (81) Then, the inf-sup condition (Bv, µ) inf sup ≥ β1 > 0 (82) µ∈L2(Ω) v∈X kvkX kµkL2(Ω) 14 R.H.W. HOPPE AND W.G. LITVINOV

⊥ ⊥ holds true. The operator B is an isomorphism from V onto L2(Ω), where V is the orthogonal complement of V in X, and the operator B∗ that is adjoint to B, is an isomorphism from L2(Ω) onto the polar set V 0 = {f ∈ X∗, (f, u) = 0, u ∈ V }. (83) Moreover,

−1 1 ⊥ kB kL(L2(Ω),V ) ≤ , (84) β1 ∗ −1 1 0 k(B ) kL(V ,L2(Ω)) ≤ . (85) β1 For a proof see [2]. Lemma 3.5 is a generalization of the inf-sup condition in case that 1 the operator div acts in the subspace H0 (Ω) (see [7]). This result was ﬁrst established in an equivalent form by Ladyzhenskaya and Solonnikov in [9]. ∞ ∞ Let {Xm}m=1, {Nm}m=1 be sequences of ﬁnite-dimensional subspaces in X and L2(Ω), respectively, such that

lim inf ku − zkX = 0, u ∈ X, (86) m→∞ z∈Xm

lim inf kw − ykL2(Ω) = 0, w ∈ L2(Ω). (87) m→∞ y∈Nm ∗ We introduce the operators Bm ∈ L(Xm,Nm) according to Z

(Bmu, µ) = µ div u dx, u ∈ Xm, µ ∈ Nm. (88) Ω ∗ ∗ ∗ We refer to Bm ∈ L(Nm,Xm) as the adjoint operator of Bm with (Bmu, µ) = (u, Bmµ) for all u ∈ Xm and all µ ∈ Nm. 0 We further introduce the spaces Vm and Vm by

Vm = {u ∈ Xm, (Bmu, µ) = 0, µ ∈ Nm}, (89) 0 ∗ Vm = {q ∈ Xm, (q, u) = 0, u ∈ Vm}. (90) The following Lemma is valid (see [2]).

∞ ∞ LEMMA 3.6. Let {Xm}m=1, {Nm}m=1 be sequences of ﬁnite-dimensional subspaces in X and L2(Ω) and assume that the discrete inf − sup condition (LBB condition) (B u, µ) inf sup m ≥ β > 0, m ∈ N (91) µ∈N m u∈Xm kukX kµkL2(Ω) ∗ 0 holds true. Then, the operator Bm is an isomorphism from Nm onto Vm, and the operator ⊥ ∗ ⊥ Bm is an isomorphism from Vm onto Nm, where Vm is the orthogonal complement of Vm in Xm. Moreover,

∗ −1 1 −1 1 k(B ) kL(V 0 ,N ) ≤ , kB kL(N ∗ ,V ⊥) ≤ , m ∈ N. (92) m m m β m m m β

Consider a functional Ψ: U × X → R+ of the form Z 1 Ψ(h, u) = hI(u) 2 dx h ∈ U, u ∈ X, (93) Ω where U is as in (69). ELECTRORHEOLOGICAL FLUID FLOWS 15

LEMMA 3.7. For an arbitrarily ﬁxed h ∈ U, the functional Ψ(h, .): u → Ψ(h, u) is continuous in X, and the conditions

{hm} ⊂ U, hm → h a.e. in Ω, um * u in X (94) imply

lim inf Ψ(hm, um) ≥ Ψ(h, u). (95)

Proof. Let um → u in X. We have

Z ³ Z ´ 1 ³ Z ´ 1 1 2 2 2 hI(um − u) 2 dx ≤ h dx I(um − u)dx . Ω Ω Ω Therefore, Z 1 lim hI(um − u) 2 dx = 0, (96) Ω and Z ¯ Z Z ¯ 1 ¯ 1 1 ¯ hI(um − u) 2 dx ≥ ¯ hI(um) 2 dx − hI(u) 2 dx¯. Ω Ω Ω

Consequently, lim Ψ(h, um) = Ψ(h, u). Let α ∈ [0, 1], u, v ∈ X. Then

Xn I(αu + (1 − α)v) = I(αu) + 2α(1 − α) εij(u)εij (v) i,j=1 1 1 2 +I((1 − α)v) ≤ [αI(u) 2 + (1 − α)I(v) 2 ] . (97)

Hence, Z 1 Ψ(h, αu + (1 − α)v) = hI(αu + (1 − α)v) 2 dx ≤ αΨ(h, u) + (1 − α)Ψ(h, v), Ω (98) which shows that

Ψ(h, .): u → Ψ(h, u) is a convex functional in X. (99)

Let now (94) be fulﬁlled. We have Z 1 1 Ψ(hm, um) = [hI(um) 2 + (hm − h)I(um) 2 ] dx, (100) Ω ¯ Z ¯ ¯ 1 ¯ 2 ¯ (hm − h)I(um) dx¯ ≤ khm − hkL2(Ω)kumkX . (101) Ω In view of (94), the right-hand side of (101) tends to zero as m → ∞. Hence, (99), (100), and the continuity of the functional Ψ(h, .) imply

lim inf Ψ(hm, um) = lim inf Ψ(h, um) ≥ Ψ(h, u), (102) so that the lemma is proved. • 16 R.H.W. HOPPE AND W.G. LITVINOV

4. The stationary problem. We consider stationary flow problems of electrorheological fluids under the Stokes approximation, i.e., we ignore inertial forces. Such an approach is reasonable, because the viscosities of electrorheological fluids are large, and the inertial terms have a small impact. We deal with the following problem: Find a pair of functions u, p satisfying ∂p ∂ − 2 [ϕ(I(u), |E|, µ(u, E))εij (u)] = Ki in Ω, i = 1, . . . , n, (103) ∂xi ∂xj div u = 0 in Ω, (104) ¯ ¯ u¯ =u, ˆ (105) S1 ¯ ¯ [−pδij + 2ϕ(I(u), |E|, µ(u, E))εij (u)]νj¯ = Fi, i = 1, . . . , n. (106) S2 We assume that 1 n uˆ ∈ H 2 (S1) . (107) Then, there exists a function u˜ such that ¯ ¯ u˜ ∈ H1(Ω)n, u˜¯ =u, ˆ divu ˜ = 0. (108) S1 We further assume n n K = (K1,...,Kn) ∈ L2(Ω) ,F = (F1,...,Fn) ∈ L2(S2) . (109) In view of (15), we choose the viscosity function ϕ of the following form b(|E|, µ(u, E)) ϕ(I(u), |E|, µ(u, E)) = 1 + ψ(I(u), |E|, µ(u, E)), (110) I(u) 2 where ψ is a function satisfying one out of the conditions (C1), (C2), (C3) with ϕ replaced by ψ, and b satisfies (C4). We refer to the fluid with the viscosity function ϕ defined by (110) as a generalized Bingham electrorheological fluid. We define a functional J on the set X × X and an operator L : X → X∗ as follows Z 1 J(v, h) = 2 b(|E|, µ(˜u + v, E))I(˜u + h) 2 dx, v, h ∈ X. (111) Z Ω

(L(v), h) = 2 ψ(I(˜u + v), |E|, µ(˜u + v, E))εij(˜u + v)εij (h)dx, v, h ∈ X. (112) Ω We use the notations Z Z

(K, h) = Kihidx, (F, h) = Fihids, h ∈ X. (113) Ω S2 THEOREM 4.1. Let Ω be a bounded domain in Rn, n = 2 or 3, with a Lipschitz continuous boundary S. Assume that (108), (109) are satisﬁed and (u, p) with u =u ˜ + v is a regular solution of (103) –(106), where the viscosity function ϕ is deﬁned by (110) with ψ meeting one out of the conditions (C1), (C2), (C3) (ϕ replaced by ψ) and b satisfying (C4). Then v ∈ V, (114) J(v, h) − J(v, v) + (L(v), h − v) ≥ (K + F, h − v), h ∈ V. (115) Proof. Multiplying (103) by h − v and integrating over Ω, the proof is an easy consequence of Green’s formula. • Let v be a solution of the problem (114), (115) such that I(˜u + v) 6= 0 a.e. in Ω. (116) ELECTRORHEOLOGICAL FLUID FLOWS 17

We replace h in (115) by v + λh, λ > 0, from which λ−1[J(v, v + λh) − J(v, v)] + (L(v), h) ≥ (K + F, h). (117) For λ → 0, we get µ ¶ ∂J (v, v), h + (L(v), h) ≥ (K + F, h), h ∈ V, (118) ∂h ∂J where ∂h (v, v) is the partial Gateauxˆ derivative of the functional J with respect to the second argument µ ¶ Z ∂J − 1 (v, v), h = 2 b(|E|, µ(˜u + v, E))I(˜u + v) 2 εij(˜u + v)εij (h)dx. (119) ∂h Ω Replacing h by −h in (118), we obtain µ ¶ ∂J (v, v), h + (L(v), h) = (K + F, h), h ∈ V, (120) ∂h and Lemma 3.5 gives ∂J (v, v) + L(v) − K − F = B∗p, p ∈ L (Ω), (121) ∂h 2 i.e., µ ¶ ∂J (v, v), h + (L(v), h) − (B∗p, h) = (K + F, h) ∀h ∈ X. (122) ∂h It follows from (122) that the pair (u, p) with u =u ˜ + v is a solution of (103)–(106) in the sense of distributions.

5. Problem for the ﬂuid with constitutive equation (16). 5.1. Existence theorem. We deﬁne the following functional on X × X Z 1 Jλ(v, h) = 2 b(|E|, µ(˜u + v, E))(λ + I(˜u + h)) 2 dx, λ > 0. (123) Ω

Obviously, Jλ(v, h) = J(v, h) for λ = 0. Note that the functional Jλ is Gateauxˆ differentiable in X with respect to the second argument for λ > 0, but not for λ = 0. ∂Jλ The partial Gateauxˆ derivative ∂h is given by ³ ´ Z ∂Jλ − 1 (v, h), w = 2 b(|E|, µ(˜u + v, E))(λ + I(˜u + h)) 2 εij (˜u + h)εij (w)dx, ∂h Ω v, h, w ∈ X, λ > 0. (124)

We consider the following problem: Find vλ such that

vλ ∈ V, (125) ³∂J ´ λ (v , v ), w + (L(v ), w) = (K + F, w), w ∈ V. (126) ∂h λ λ λ If vλ is a solution of (125), (126), then Lemma 3.5 implies the existence of a function pλ such that the pair (vλ, pλ) solves the following problem

(vλ, pλ) ∈ X × L2(Ω), (127) ³∂J ´ λ (v , v ), w + (L(v ), w) − (B∗p , w) = (K + F, w), w ∈ X, (128) ∂h λ λ λ λ (Bvλ, q) = 0, q ∈ L2(Ω). (129) 18 R.H.W. HOPPE AND W.G. LITVINOV

We remark that equations (127)–(129) represent the ﬂow of an electrorheological ﬂuid with the constitutive equation (16). We are looking for an approximate solution of (127)–(129) of the form

(vm, pm) ∈ Xm × Nm, (130) ³∂J ´ λ (v , v ), w + (L(v ), w) − (B∗ p , w) = (K + F, w), w ∈ X , (131) ∂h m m m m m m (Bmvm, q) = 0, q ∈ Nm, (132) where Xm and Nm are ﬁnite dimensional subspaces in X and L2, (Ω), respectively, and Bm is deﬁned as in (88).

THEOREM 5.1. Assume that the conditions (C4), (108), (109) are satisﬁed and the function ψ meets one of the conditions (C1), (C2), (C3) (ϕ replaced by ψ). Let {Xm}, {Nm} be sequences of ﬁnite-dimensional subspaces in X and L2(Ω), respectively, such that (86), (87), (91) hold true and

Xm ⊂ Xm+1,Nm ⊂ Nm+1, m ∈ N. (133)

Then, for an arbitrary λ > 0 there exists a solution (vλ, pλ) of (127)–(129). Moreover, for m ∈ N and λ > 0 there exists a solution (vm, pm) of (130)–(132), and a subsequence {(vk, pk)} can be extracted from the sequence {(vm, pm)} such that vk * vλ in X, pk * pλ in L2(Ω).

Proof. It follows from (89), (130)–(132) that vm is a solution of the problem ³∂J ´ v ∈ V , λ (v , v ), w + (L(v ), w) = (K + F, w), w ∈ V . (134) m m ∂h m m m m Taking (21) into account, we obtain ¯³ ´¯ ¯ Z ¯ ¯ ∂Jλ ¯ ¯ εij (˜u + e)εij(e) ¯ ¯ (e, e), e ¯ = 2¯ b(|E|, µ(˜u + e, E)) 1 dx¯ ∂h (λ + I(˜u + e)) 2 Z Ω 1 ≤ 2 b(|E|, µ(˜u + e, E))I(e) 2 dx ≤ c1kekX , λ > 0, e ∈ X, (135) Ω where 1 c1 = 2a5(mes Ω) 2 . (136) By (17), (108), (109), (112), and (135), for an arbitrary e ∈ X we get ³∂J ´ z(e) = λ (e, e), e +(L(e), e)−(K +F, e) ≥ 2a kek2 −ckek , e ∈ X, λ > 0, ∂h 1 X X (137) c giving z(e) ≥ 0 for kekX ≥ r = . 2a1 The corollary of Brouwer’s ﬁxed point theorem (cf.[6]) implies the existence of a solution of (134) with

kvmkX ≤ r, kL(vm)kX∗ ≤ c2, m ∈ N, (138) where the second inequality follows from (17) and (108). For an arbitrary f ∈ X∗ we ∗ denote by Gf the restriction of f to Xm. Then Gf ∈ Xm, and by (90), (134) we obtain ³∂J ´ G λ (v , v ) + L(v ) − K − F ∈ V 0 . (139) ∂h m m m m Therefore, there exists a unique pm ∈ Nm (see Lemma 3.6) such that ³∂J ´ B∗ p = G λ (v , v ) + L(v ) − K − F . (140) m m ∂h m m m ELECTRORHEOLOGICAL FLUID FLOWS 19

Consequently, the pair (vm, pm) is a solution of (130)–(132). Due to (21), (109), (138) and Lemmas 3.4, 3.6 we get

kpmkL2(Ω) ≤ c, m ∈ N. (141)

By (138), (141) we can extract a subsequence {vη, pη} such that

vη * v0 in X, (142) n vη → v0 in L2(Ω) and a.e. in Ω, (143)

pη * p0 in L2(Ω), (144) ∗ L(vη) * χ in X , (145) ∂J λ (v , v ) * χ in X∗. (146) ∂h η η 1

Let η0 be a ﬁxed positive number and w ∈ Xη0 , q ∈ Nη0 . Observing (142), (144)– (146), we pass to the limit in (131), (132) with m replaced by η, and obtain ∗ (χ1 + χ − B p0, w) = (K + F, w), w ∈ Xη , Z 0

q div v0 dx = 0, q ∈ Nη0 . (147) Ω

Since η0 is an arbitrary positive integer, by (86), (87) ∗ χ1 + χ − B p0 = K + F, (148)

div v0 = 0. (149) We represent the operator L(v) in the form L(v) = L(v, v), (150) where the operator (v, w) → L(v, w) is considered as a mapping of X × X into X∗ according to Z

(L(v, w), h) = 2 ψ(I(˜u + w), |E|, µ(˜u + v, E))εij (˜u + w)εij (h)dx. (151) Ω We get µ ¶ ∂J ∂J X (w) = λ (v , v ) + L(v , v ) − λ (v , w) − L(v , w), v − w , w ∈ X. η ∂h η η η η ∂h η η η (152) Lemmas 3.3, 3.4 imply

Xη(w) ≥ 0, η ∈ N, w ∈ X. (153) We have ° ° °∂Jλ ∂Jλ ° ° (vη, w) − (v0, w)° ∂h ∂h X∗ h Z i 1 2 2 ≤ 2 [b(|E|, µ(˜u + vη,E)) − b(|E|, µ(˜u + v0,E))] dx . (154) Ω (21), (143), (154) and the Lebesgue theorem give ∂J ∂J λ (v , w) → λ (v , w) in X∗. (155) ∂h η ∂h 0 Likewise, we obtain ∗ L(vη, w) → L(v0, w) in X . (156) 20 R.H.W. HOPPE AND W.G. LITVINOV

Observing (Bηvη, pη) = 0, by (131), (142), (144) we obtain ³∂J ´ λ (v , v ) + L(v ), v = (K + F, v ) → (K + F, v ), (157) ∂h η η η η η 0 and µ ¶ ∂J lim λ (v , v ) + L(v ), w − (B∗p , w) = (K + F, w), w ∈ X. (158) ∂h η η η 0 In view of (155)–(158) and passing to the limit in (152), by (149), (153) we get µ ¶ ∂J K + F − λ (v , w) − L(v , w) + B∗p , v − w ≥ 0, w ∈ X. (159) ∂h 0 0 0 0

We choose w = v0 − γh, γ > 0, h ∈ X, and consider γ → 0. Then, Lemmas 3.3, 3.4 give µ ¶ ∂J K + F − λ (v , v ) − L(v , v ) + B∗p , h ≥ 0. (160) ∂h 0 0 0 0 0 This inequality holds for any h ∈ X. Hence, replacing h by −h shows that we do have equality. Consequently, the pair (vλ, pλ) with vλ = v0 and pλ = p0 solves (127)–(129). •

5.2. On the uniqueness of the solution. For the sake of simplicity we assume that the transfer velocity uˇ is equal to zero (see (6)), and the velocity of the ﬂuid on S1 is also equal to zero, that is u˜ = 0. Let vλ, wλ be two solutions of (127)–(129) and

e = wλ − vλ. (161) Suppose that the functions b and ψ are continuously differentiable and that

¯ ¯ 1 ¯ ∂ψ ¯ 2 ¯ (y1, y2, y3)¯y1 ≤ c,ˇ (y1, y2, y3) ∈ R+ × R+ × [0, 1], (162) ∂y3 ¯ ¯ ¯ ∂b ¯ ¯ (y1, y2)¯ ≤ c,ˆ (y1, y2) ∈ R+ × [0, 1]. (163) ∂y2 We observe that (162) is a restriction on the behavior of the function ∂ψ for large values ∂y3 of y1. This inequality is natural from a physical point of view, since the structure of the electrorheological fluid is destroyed at large shear rates and the fluid gets close to isotropic. Let Ω1 be a subdomain of Ω in which the electric field is equal to zero. Let Ω0 be a small vicinity of the solid boundary S1. Ω0 is a domain with a boundary layer. We set

Ω2 = Ω0\{Ω0 ∩ Ω1}.

We assume that the direction of the velocity vector is given in Ω2, i.e.,

vλ(x) wλ(x) = a.e. in Ω2, (164) |vλ(x)| |wλ(x)| −1 and that the function x → vλ(x)|vλ(x)| is given in Ω2. The direction of the velocity vector in Ω2 is parallel to a tangent to the boundary S1. Let Ω3 = Ω\{Ω1 ∪ Ω2}. We suppose that

|vλ(x) + te(x)| ≥ l a.e. in Ω3, t ∈ [0, 1], (165) where l is a positive constant. The condition (165) indicates that the vectors vλ(x) and wλ(x) are not equal to zero in Ω3 and not opposite in direction, i.e., (vλ(x), wλ(x))Rn > 0 almost everywhere in Ω3. ELECTRORHEOLOGICAL FLUID FLOWS 21

This is natural from a physical point of view. We deﬁne an operator M : X × X → X∗ by means of Z − 1 (M(v, w), g) = 2 [b(|E|, µ(w, E))(λ + I(v)) 2 + ψ(I(v), |E|, µ(w, E))]εij(v) εij(g) dx, Ω v, w, g ∈ X. (166) It follows from (112) and (124) that ∂J M(v, v) = λ (v, v) + L(v) (167) ∂h at uˇ = 0 and u˜ = 0. Since vλ and wλ are the solutions of the problem (127)–(129), we have

(M(wλ, wλ)−M(vλ, vλ), e) = (M(wλ, wλ)−M(vλ, wλ), e)+(M(vλ, wλ)−M(vλ, vλ, e) = 0. (168) Lemmas 3.1 and 3.4 (see (54) and (71)) imply 2 (M(wλ, wλ) − M(vλ, wλ), e) ≥ µ1||e||X . (169) We introduce a function t → η(t), t ∈ [0, 1]: Z − 1 η(t) = 2 [b(|E|, µ(vλ + te, E))(λ + I(vλ)) 2 Ω3 +ψ(I(vλ), |E|, µ(vλ + te, E))]εij (vλ)εij(e) dx. (170) In view of (162), (163), and (165), the function η is differentiable at any point t ∈ (0, 1). Consequently, dη η(1) = η(0) + (ξ), ξ ∈ (0, 1), (171) dt where Z h dη ∂b − 1 (ξ) = 2 (|E|, µ(vλ + ξe, E)) fξ(λ + I(vλ)) 2 dt Ω3 ∂y2 ∂ψ i + (I(vλ), |E|, µ(vλ + ξe, E))fξ εij(vλ)εij(e) dx, (172) ∂y3 and ³ ´ h³ ´ vλ + ξe E e E fξ = 2 , , |vλ + ξe| |E| Rn |vλ + ξe| |E| Rn i −3 E −|v + ξe| (v + ξe )e (v + ξe, ) n . (173) λ λi i i λ |E| R It follows that |e| |fξ| ≤ 4 . (174) |vλ + ξe| Taking into account (164), (166), (170) and (171), we obtain dη (M(v , w ) − M(v , v ), e) = η(1) − η(0) = (ξ). (175) λ λ λ λ dt (172) along with the inequalities (162), (163), (174) yield ¯ ¯ Z 1 ¯dη ¯ |e|(I(e)) 2 −1 ¯ (ξ)¯ ≤ 8(ˆc +c ˇ) dx ≤ 8(ˆc +c ˇ)|||vλ + ξe| ||L4(Ω3)|||e|||L4(Ω)||e||X dt Ω3 |vλ + ξe| −1 2 2 ≤ 8(ˆc +c ˇ)c0|||vλ + ξe| ||L4(Ω3)||e||X = k||e||X , (176) 22 R.H.W. HOPPE AND W.G. LITVINOV where c0 is the constant in the inequality

|||e|||L4(Ω) ≤ c0||e||X . (177) By (168), (169), (175) and (176), we obtain 2 0 = (M(wλ, wλ) − M(vλ, vλ), e) ≥ (µ1 − k)||e||X . (178)

Consequently, e = wλ − vλ = 0 provided that µ1 > k. We have thus proved the following result:

THEOREM 5.2. Assume that the conditions (C1) (ϕ replaced by ψ), (C4), (109), (162), (163) are satisﬁed. Let vλ, wλ be two solutions of the problem (127)–(129) such that (164), (165) hold true and µ1 > k. Then vλ = wλ. If

|vλ(x) + te(x)| ≥ l a.e. in Ω\Ω1, t ∈ [0, 1], (179) −1 then the function x → vλ(x)|vλ(x)| has not to be assigned in Ω2 and vλ = wλ at µ1 > k.

REMARK 1. In realistic settings, the electric ﬁeld is large between the electrodes where the electric vector is perpendicular to the velocity vector. The electric ﬁeld rapidly decays as −1 the distance to the electrodes increases. Because of this meas(Ω3)(meas(Ω)) is typically small, so that k is small (see(176)) and µ1 > k.

6. Variational inequality for the extended Bingham electrorheological fluid. We now consider a problem on stationary flow of the extended Bingham electrorheological fluid. The constitutive equation of this fluid is the following: b(|E|, µ(u, E)) ϕ(I(u), |E|, µ(u, E)) = 1 + b1(|E|, µ(u, E)). (180) I(u) 2 We deal with the problem (103)–(106) and assume that (107) and (109) are satisfied. Then, according to Remark 4.1, the generalized solution of our problem is u =u ˜ + v, where u˜ is a function satisfying (108) and v is a solution of the problem v ∈ V, (181)

J(v, h) − J(v, v) + (L1(v), h − v) ≥ (K + F, h − v), h ∈ V. (182) ∗ Here, J is the functional given by (111), and the operator L1 : X → X is deﬁned by Z

(L1(v), h) = 2 b1(|E|, µ(˜u + v, E))εij (˜u + v)εij(h)dx, v, h ∈ X. (183) Ω

The function b1 is subject to the following condition:

(C5): b1 :(y1, y2) → b1(y1, y2) is a continuous function on R+ × [0, 1] and satisﬁes

a6 ≤ b1(y1, y2) ≤ a7, (y1, y2) ∈ R+ × [0, 1], (184)

with positive constants a6 and a7.

We approximate the functional J by Jλ as given by (123). Replacing J by Jλ, arguing in the same way as in the proof of Theorem 4.1, we obtain the following problem: Find vλ such that

vλ ∈ V, (185) ³∂J ´ λ (v , v ), w + (L (v ), w) = (K + F, w), w ∈ V, (186) ∂h λ λ 1 λ

∂Jλ with ∂h given by (124). ELECTRORHEOLOGICAL FLUID FLOWS 23

THEOREM 6.1. Assume that conditions (108), (109), (C4), (C5) are satisﬁed. Then there exists a solution v of (181), (182). Moreover, for an arbitrary λ > 0 there exists a solution of (185), (186). Moreover, there exists a function pλ such that the pair (vλ, pλ) is a solution of the problem

(vλ, pλ) ∈ X × L2(Ω), (187) ³∂J ´ λ (v , v ), w + (L (v ), w) − (B∗p , w) = (K + F, w), w ∈ X, (188) ∂h λ λ 1 λ λ (Bvλ, q) = 0, q ∈ L2(Ω). (189)

A subsequence can be extracted from the sequence {vλ}, again denoted by {vλ}, such that n vλ * v in X and vλ → in L2(Ω) as λ → 0. (190) If I(˜u + v) 6= 0 almost everywhere in Ω, the functional h → J(v, h) is Gateauxˆ differentiable in v, and there exists a function p ∈ L2(Ω) such that the pair (v, p) is a solution of the problem

v ∈ V, p ∈ L2(Ω), (191) ³∂J ´ (v, v), h + (L (v), h) − (B∗p, h) = (K + F, h), h ∈ X, (192) ∂h 1 ∂J with ∂h given by (119).

Proof. The existence of a solution vλ of (185), (186) follows from Theorem 5.1 as well as the existence of a function pλ such that (187)–(189) hold true. It follows from the proof of Theorem 5.1 (see (137)) that

vλ remains in a bounded set of V independent of λ. (193)

Therefore, from {vλ} we can select a subsequence, again denoted by {vλ}, such that

vλ * v in X as λ → 0, (194) n vλ → v in L2(Ω) and a.e. in Ω. (195) For h ∈ V , we introduce

Zλ = (L1(vλ), h − vλ) + Jλ(vλ, h) − Jλ(vλ, vλ) − (K + F, h − vλ). (196) Using (188), we see that ³∂J ´ Z = − λ (v , v ), h − v + J (v , h) − J (v , v ). (197) λ ∂h λ λ λ λ λ λ λ λ It follows from (123), (124) and Lemma 3.4 (cf. (71)) that for an arbitrarily ﬁxed w ∈ X the functional u → Jλ(w, u) is convex, whence

Zλ ≥ 0. (198) (C5), (195) and the Lebesgue theorem give

b1(|E|, µ(˜u + vλ,E))εij (h) → b1(|E|, µ(˜u + v),E))εij(h) in L2(Ω). (199) (194), (199) imply

lim(L1(vλ), h) = (L1(v), h). (200) We have

(L1(vλ), vλ) = A1λ + A2λ, (201) 24 R.H.W. HOPPE AND W.G. LITVINOV where Z

A1λ = 2 b1(|E|, µ(˜u + vλ,E))εij(˜u)εij (vλ)dx, (202) ZΩ

A2λ = 2 b1(|E|, µ(˜u + vλ,E))I(vλ)dx. (203) Ω (199) still holds true, if the function h is replaced by u˜. Consequently, (194) implies Z

lim A1λ = 2 b1(|E|, µ(˜u + v, E))εij (˜u)εij(v). (204) Ω It follows from (184) and (195) that 1 1 [b1(|E|, µ(˜u + vλ,E))] 2 w → [b1(|E|, µ(˜u + v))] 2 w in L2(Ω), w ∈ L2(Ω). Hence, (194) yields Z Z 1 1 lim [b1(|E|, µ(˜u + vλ,E))] 2 εij(vλ)wdx = [b1(|E|, µ(˜u + v, E))] 2 εij (v)wdx, Ω Ω w ∈ L2(Ω). It follows that 1 1 [b1(|E|, µ(˜u + vλ,E))] 2 εij(vλ) * [b1(|E|, µ(˜u + v, E))] 2 εij (v) in L2(Ω). (205) (203) and (205) give Z

lim inf A2λ ≥ 2 b1(|E|, µ(˜u + v, E))I(v)dx. (206) Ω By (201), (204), and (206) we obtain

lim inf(L1(vλ), vλ) ≥ (L1(v), v). (207) (123), (195) and the Lebesgue theorem give

lim Jλ(vλ, h) = J(v, h). (208) Setting

bλ = b(|E|, µ(˜u + vλ,E)), b0 = b(|E|, µ(˜u + v, E)), (209)

Iλ = I(˜u + vλ),I0 = I(˜u + v), (210) we have Jλ(vλ, vλ) = Jλ(v,λ ) + B1λ, (211) where Z 1 B1λ = 2 (bλ − b0)(λ + Iλ) 2 dx, (212) Ω and ³ Z ´ 1 ³ Z ´ 1 2 2 2 |B1λ| ≤ 2 (λ + Iλ) dx |bλ − b0| dx . (213) Ω Ω (194), (195) and (21) imply lim B1λ = 0. (214)

(111) and (123) yield Jλ(v, vλ) ≥ J(v, vλ), whence

lim inf Jλ(v, vλ) ≥ lim inf J(v, vλ). (215) (194) and Lemma 3.7 give

lim inf J(v, vλ) ≥ J(v, v). (216) ELECTRORHEOLOGICAL FLUID FLOWS 25

(211), (214), (215), and (216) result in

lim inf Jλ(vλ, vλ) ≥ J(v, v). (217) (189), (194) imply (181). Using (196), (198), (200), (207), (208), and (217), we obtain (182). It follows from Remark 4.1 that if I(˜u + v) 6= 0 almost everywhere in Ω, then there exists a function p such that (191) and (192) hold true. •

7. General variational inequality. We assume that the function µ in the operator L de- ﬁned by (112) is replaced by a function µ1 such that

um * u in X ⇒ µ1(um,E) → µ1(u, E) in L∞(Ω). (218)

∞ ω ∈ C (R+), supp ω = [0, a], ω(z) ≥ 0 z ∈ R+, Z ω(|x|)dx = 1, a is a small positive constant. (221) Rn In (220) we assume that the function u is extended to Rn. For the function µ1 condition (218) is satisfied. From the physical point of view (219) means that the value of the function µ1 and therefore the viscosity of the fluid at a point x depends on the angle between the vectors of velocity and the electric field strength at points belonging to some small vicinity of the point x, implying that the model is not local. This seems to be natural, since electrorheological properties of the fluid are linked with the presence of small solid particles in the fluid. The mean dimension of these particles can be taken as the regularization parameter a. In the case under consideration the operator L is defined as follows Z

(L(v), h) = 2 ψ(I(˜u + v), |E|, µ1(˜u + v, E))εij(˜u + v)εij (h)dx, v, h ∈ X. (222) Ω We also assume that the following condition of uniform continuity of the function ψ holds true (C6): For an arbitrary γ > 0, there exists ε > 0 such that the conditions

0 00 0 00 y3, y3 ∈ [0, 1], |y3 − y3 | ≤ ε, y1, y2 ∈ R+ imply 0 00 |ψ(y1, y2, y3) − ψ(y1, y2, y3 | ≤ γ.

In the functionals J and Jλ the function µ can as well be replaced by the function µ1. The following theorem does not assume this, although it remains valid in this case. 26 R.H.W. HOPPE AND W.G. LITVINOV

THEOREM 7.1. Assume that the conditions (C4), (108), (109) are satisﬁed. Moreover, suppose that the function ψ meets one of the conditions (C1), (C2), (C3) (ϕ replaced by ψ) and that (C6) holds true. Further, assume that the function µ1 meets condition (218), and the operator L is given by (222). Then, for an arbitrary λ > 0 there exists a solution (vλ, pλ) of (127)–(129), and there exists a solution v of (114), (115). A subsequence can be selected from the sequence {vλ}, again denoted by {vλ}, such that n vλ * v in X and vλ → v in L2(Ω) as λ → 0.

Proof. 1) The existence of a solution (vλ, pλ) of (127)–(129) follows from Theorem 5.1. The proof of this theorem (see (137)) implies that vλ remains in a bounded set of V independent of λ. Therefore, from the sequence {vλ} we can select a subsequence, again denoted by {vλ}, such that

vλ * v in V as λ → 0, (223) n vλ → v in L2(Ω) and a.e. in Ω. (224)

For every vλ, we deﬁne a functional Ψλ as follows

Ψλ(v) = Jλ(vλ, v) + Φ(vλ, v) − (K + F, v), v ∈ V, (225) where Z ³ Z I(˜u+v) ´ Φ(vλ, v) = ψ(ξ, |E|, µ1(˜u + vλ,E))dξ dx. (226) Ω 0 We consider the problem: Find a function v˜ satisfying

v˜ ∈ V, Ψλ(˜v) = min Ψλ(v). (227) v∈V If v˜ is a solution of (227), then we have ¯ ³ ´ d ¯ ∂Jλ v˜ ∈ V, Ψλ(˜v+th)¯ = (vλ, v˜), h)+(L(vλ, v˜), h −(K+F, h) = 0, h ∈ V. dt t=0 ∂v (228) Here, L(vλ, v˜) is the Gateauxˆ derivative of the functional Φ(vλ,.): u → Φ(vλ, u) in v˜, i.e., ∂Φ (v , v˜) = L(v , v˜). ∂v λ λ ∗ The operator L(vλ,.): X 3 u → L(vλ, u) ∈ X has the form Z

(L(vλ, u), h) = 2 ψ(I(˜u + u), |E|, µ1(˜u + vλ,E))εij(˜u + u)εij(h)dx, u, h ∈ X, Ω (229) and (222) yields L(v, v) = L(v). (230)

It follows from (125), (126) that the function v˜ = vλ is a solution of (228). By means of Lemmas 3.1–3.4 the functional Ψλ is strictly convex. Therefore, there exists a unique solution v˜ = vλ of (228), and problems (227) and (228) are equivalent. (227) implies

Jλ(vλ, h) + Φ(vλ, h) − Jλ(vλ, vλ) − Φ(vλ, vλ) ≥ (K + F, h − vλ), h ∈ V. (231) 2) It follows from (223), (224) and the proof of Theorem 6.1 (see (208), (217)) that

lim Jλ(vλ, h) = J(v, h), lim inf Jλ(vλ, vλ) ≥ J(v, v). (232) We have Z Z

Φ(vλ, h) = fλdx, Φ(v, h) = f dx, (233) Ω Ω ELECTRORHEOLOGICAL FLUID FLOWS 27 where Z I(˜u+h)(x) fλ(x) = ψ(ξ, |E(x)|, µ1(˜u + vλ,E)(x))dξ, 0 Z I(˜u+h)(x) f(x) = ψ(ξ, |E(x)|, µ1(˜u + v, E)(x))dξ. 0

(224) and (C6) imply fλ → f almost everywhere in Ω, and (17) yields

|fλ| ≤ a2I(˜u + h). Thus, (233) and the Lebesgue theorem give

lim Φ(vλ, h) = Φ(v, h). (234) It is obvious that

Φ(vλ, vλ) = Φ(v,λ ) + αλ, (235) Z Z n I(˜u+vλ) αλ = Φ(vλ, vλ) − Φ(v,λ ) = [ψ(ξ, |E|, (µ1(˜u + vλ,E)) Ω 0 o −ψ(ξ, |E|, µ1(˜u + v, E))]dξ dx. (218), (223) and (C6) imply Z

αλ ≤ βλ I(˜u + vλ)dx, Ω and lim βλ = 0. Therefore lim αλ = 0. The functional Φ(v, .): u → Φ(v, u) is continuous in X. Indeed, let um → u in X. We have

¯ ¯ ¯ Z ³ Z I(˜u+um) ´ ¯ ¯ ¯ ¯ ¯ ¯Φ(v, um) − Φ(v, u)¯ = ¯ ψ(ξ, |E|, µ1(˜u + v, E))dξ dx¯ Ω I(˜u+u) ¯ Z ¯ ¯ ¯ ≤ a2¯ (I(˜u + um) − I(˜u + u))dx¯, (236) Ω and the right hand side of this inequality tends to zero as m → ∞. By Lemmas 3.1, 3.2, and 3.3 the functional Φ(v, .) is convex in X. Therefore, Φ(v, .) is lower semi-continuous with respect to the weak topology on X, and (223), (235) imply

lim inf Φ(vλ, vλ) ≥ Φ(v, v). (237) By (232), (234), (237) we pass to the limit as λ → 0 in (231). This gives J(v, h) + Φ(v, h) − J(v, v) − Φ(v, v) ≥ (K + F, h − v), h ∈ V. (238) Taking into account (238) and the convexity of the functional J(v, .): u → J(v, u), we get J(v, v) + Φ(v, v) − (K + F, v) ≤ J(v, (1 − θ)v + θh) + Φ(v, (1 − θ)v + θh) − (K + F, (1 − θ)v + θh) ≤ (1 − θ)J(v, v) + θJ(v, h) + Φ(v, (1 − θ)v + θh) − (K + F, (1 − θ)v + θh), θ ∈ (0, 1), whence Φ(v, (1 − θ)v + θh) − Φ(v, v) + J(v, h) − J(v, v) − (K + F, h − v) ≥ 0. (239) θ For θ → 0 we obtain (115) with the operator L deﬁned by (229), (230). • 28 R.H.W. HOPPE AND W.G. LITVINOV

REMARK 2. We have shown that (238) implies (115). Let us show that (115) yields (238), that is problems (114), (115) and (114), (238) are equivalent. Let (114), (115) be valid. Obviously, J(v, h) + Φ(v, h) − J(v, v) − Φ(v, v) − (K + F, h − v) = J(v, h) − J(v, v) + (240) (L(v, v), h − v) + Φ(v, h) − Φ(v, v) − (L(v, v), h − v) − (K + F, h − v) The functional u → Φ(v, u) is convex, whence Φ(v, h) − Φ(v, v) − (L(v, v), h − v) ≥ 0. Together with (115), (230), and (240) this gives (238).

REMARK 3. By comparing Theorems 5.1 and 7.1 we observe that a solution of the operator equations for a fluid with bounded viscosity function (16) exists under less restrictive conditions than the conditions for the existence of a solution of the variational inequality (115) for a fluid with unbounded viscosity function (15). In addition, for (15) the pressure is only well defined in case that (116) holds true, i.e., when the viscosity function does not take infinite values. But in this case, the variational inequality reduces to operator equations as outlined in Section 4.

8. Problems with given function µ. In the case where the distance between the electrodes is small compared with the lengths of the electrodes, we can assume that in between the electrodes the velocity vector is orthogonal to the vector of electric field strength, and the electric field strength is equal to zero in the remaining part of the domain under consideration. In this case, µ(u, E) is a known function of x so that the viscosity functions (15) and (16) take the form e(|E|, x) ϕ(I(u), |E|, x) = 1 + ψ1(I(u), |E|, x), (241) I(u) 2 − 1 ϕ(I(u), |E|, x) = e(|E|, x)(λ + I(u)) 2 + ψ1(I(u), |E|, x), (242) and the constitutive equation is defined by (3). The dependence of the viscosity function on x in (241), (242) is related to the anisotropy of the fluid. If the direction of the velocity vector in x with E(x) 6= 0 is known, then the viscosity functions (15) and (16) reduce to (241) and (242). We assume the function ψ1 to satisfy

(C0): For almost all x ∈ Ω the function ψ1(., ., x):(y1, y2) → ψ1(y1, y2, x) is contin- 2 2 uous in R+. For an arbitrarily ﬁxed (y1, y2) ∈ R+, the function ψ1(y1, y2,.): x → ψ1(y1, y2, x) is measurable in Ω.

We also assume that for almost all x ∈ Ω and all y2 ∈ R+, the function ψ1(., y2, x): y1 → ψ1(y1, y2, x) satisﬁes one of the following conditions (C1a), (C2a), (C3a):

(C1a): ψ1(., y2, x) is continuously differentiable in R+, and the following inequalities hold true

a2 ≥ ψ1(y1, y2, x) ≥ a1, (243)

∂ψ1 ψ1(y1, y2, x) + 2 (y1, y2, x) ≥ a3 (244) ∂y1 ¯ ¯ ¯∂ψ1 ¯ ¯ (y1, y2, x)¯y1 ≤ a4. (245) ∂y1 ELECTRORHEOLOGICAL FLUID FLOWS 29

2 (C2a): (243) is satisﬁed, and for an arbitrary (z1, z2) ∈ R+ the following inequality is valid

2 2 2 [ψ1(z1 , y2, x)z1 − ψ1(z2 , y2, x)z2](z1 − z2) ≥ a3(z1 − z2) . (246) 2 (C3a): (243) is fulﬁlled, and the function z → ψ1(z , y2, x)z is strictly increasing in 2 2 R+, i.e., the conditions z1, z2 ∈ R+, z1 > z2 imply ψ1(z1 , y2, x)z1 > ψ1(z2 , y2, x)z2. (C1a), (C2a), (C3a) are analogues of conditions (C1), (C2), (C3). An analogue of (C4) is the following condition.

(C4a): For almost all x ∈ Ω, the function e(., x): y → e(y, x) is continuous in R+. For an arbitrarily ﬁxed y ∈ R+, the function e(y, .): x → e(y, x) is measurable in Ω and

0 ≤ e(y, x) ≤ a5. (247)

We consider functionals Y and Yλ, λ > 0 deﬁned according to Z 1 Y (u) = 2 e(|E|, x)I(˜u + u) 2 dx, u ∈ X, (248) Z Ω 1 Yλ(u) = 2 e(|E|, x)(λ + I(˜u + u)) 2 dx, u ∈ X. (249) Ω ∗ We also introduce an operator L2 : X → X by means of Z

(L2(u), h) = 2 ψ1(I(˜u + u), |E|, x)εij (˜u + u)εij (h)dx, u, h ∈ X. (250) Ω We consider the following two problems: Problem 1. Find a pair of functions (vλ, pλ) such that

vλ ∈ X, pλ ∈ L2(Ω), (251) ³∂Y ´ λ (v ), h + (L (v ), h) − (B∗p , h) = (K + F, h), h ∈ X, (252) ∂u λ 2 λ λ (Bvλ, q) = 0, q ∈ L2(Ω). (253) Problem 2. Find a function v such that v ∈ V, (254)

Y (h) − Y (v) + (L2(v), h − v) ≥ (K + F, h − v), h ∈ V. (255)

∂Yλ ∗ Here, the operator ∂u : X → X is given by ³ ´ Z ∂Yλ − 1 (u), h = 2 e(|E|, x)(λ + I(˜u + u)) 2 εij (˜u + u)εij (h)dx, u, h ∈ X. ∂u Ω (256) If (vλ, pλ) is a solution of Problem 1, then the pair (u˜ + vλ, pλ) is a generalized solution of (103)–(106) with the viscosity function deﬁned by (242). If v is a solution of Problem 2, then u˜ + v is a generalized solution of (103)–(106) with the viscosity function deﬁned by (241).

THEOREM 8.1. Assume that the conditions (108), (109), (C4a) are satisﬁed, and let ψ1 satisfy both (C0) and one of the conditions (C1a), (C2a), (C3a). Then, for an arbitrary λ > 0, there exists a unique solution (vλ, pλ) of (251)–(253). Moreover, there exists a unique solution v of (254)–(255). In addition vλ * v in V as λ → 0. 30 R.H.W. HOPPE AND W.G. LITVINOV

Proof. The existence of a solution (vλ, pλ) of (251)–(253) for an arbitrary λ > 0 follows 1 1 2 2 from Theorem 5.1. Let (vλ, pλ) and (vλ, pλ) be two solutions of (251)–(253). By (252) we obtain ³∂Y ∂Y ´ λ (v1) + L (v1) − λ (v2) − L (v2), v1 − v2 = 0. (257) ∂u λ 2 λ ∂u λ 2 λ λ λ

∂Yλ By Lemmas 3.1–3.4, the operator ∂u + L2 is strictly monotone . Consequently, (257) 1 2 1 2 implies vλ = vλ, whence pλ = pλ. Theorem 7.1 implies that, considering the sequence {vλ}, a subsequence, again denoted by {vλ}, can be selected such that vλ * v in X, where v is a solution of (254), (255). Also, Remark 7.1 infers

v ∈ V,Y (h) + Φ1(h) − Y (v) − Φ1(v) ≥ (K + F, h − v), h ∈ V, (258) where Z ³ Z I(˜u+u) ´ Φ1(u) = ψ1(ξ, |E|, x) dξ dx, u ∈ V, (259) Ω 0 and ³∂Φ ´ 1 (u), h = (L (u), h). (260) ∂u 2 In addition, problems (254), (255), and (258) are equivalent. The functional Y is convex, and the functional Φ1 is strictly convex. Therefore, the functional

Ψ1(u) = Y (u) + Φ1(u) − (K + F, u), u ∈ V, is strictly convex. Hence, if v1, v2 are two solutions of the problem (258), then we have 1 1 1 Ψ1( (v1 + v2)) < Ψ1(v1) + Ψ1(v2) = inf Ψ1(h), 2 2 2 h∈V whence v1 = v2. •

9. Numerical solution of stationary problems.

9.1. Convergence Theorem. Let {Xm}, {Nm} be sequences of ﬁnite-dimensional subspaces of X and L2(Ω), respectively, such that (86), (87), (91), and (133) hold true. We set ∂Y L = λ + L , (261) 3 ∂u 2

∂Yλ where L2 and ∂u are the operators deﬁned by (250) and (256), respectively. We want to compute an approximate solution (vm, pm) of (251)–(253) of the form

(vm, pm) ∈ Xm × Nm, (262) ∗ (L3(vm), h) − (Bm pm, h) = (K + F, h), h ∈ Xm, (263)

(Bm vm, q) = 0, q ∈ Nm. (264) ∗ We remark that the operator Bm is deﬁned by (88), and Bm denotes the adjoint of Bm.

THEOREM 9.1. Assume that conditions (108), (109), and (C4a) are satisﬁed, and let the function ψ1 satisfy both (C0) and (C3a). We also assume that (86), (87), (91), and (133) are satisﬁed. Then, for an arbitrary m ∈ N, there exists a unique solution of (262) –(264) and

vm * vλ in X, pm * pλ in L2(Ω), (265) ELECTRORHEOLOGICAL FLUID FLOWS 31 where vλ, pλ is the solution of (251)–(253). If in addition ψ1 satisﬁes (C1a) or (C2a), then

vm → vλ in X, (266)

pm → pλ in L2(Ω). (267)

Proof. The existence of a unique solution (vm, pm) of (262)–(264) as well as the relations (265) follow from Theorems 5.1, 8.1. The following equalities are also consequences of the proof of Theorem 5.1 (see (157), (158))

(L3(vm), vm) = (K + F, vm) → (K + F, vλ), (268) ∗ lim(L3(vm), h) − (B pλ, h) = (K + F, h), h ∈ X. (269) (252), (268), and (269) give

lim(L3(vm) − L(vλ), vm − vλ) = 0. (270)

Assume that (C2a) is satisfied: If ψ1 fulfills (C1a), it also satisfies (C2a) (see Subsection 2.2). Then, observing Lemmas 3.2, 3.4, we obtain Z 1 1 2 (L3(vm) − L3(vλ), vm − vλ) ≥ 2a3 [I(˜u + vm) 2 − I(˜u + vλ) 2 ] dx. Ω Moreover, (270) implies Z 1 1 2 [I(˜u + vm) 2 − I(˜u + vλ) 2 ] dx → 0. (271) Ω R 2 Taking into account that the function u → Ω u dx is a continuous mapping from L2(Ω into R, we obtain Z Z

I(˜u + vm)dx → I(˜u + vλ)dx. (272) Ω Ω It is obvious that Z Xn Z Xn 2 2 2 kvm − vλkX = (εij (vm − vλ)) dx = [εij (˜u + vm) − εij(˜u + vλ)] dx Ω i,j=1 Ω i,j=1 Z Z Z

= I(˜u + vm)dx + I(˜u + vλ)dx − 2 εij (˜u + vm) εij(˜u + vλ)dx. (273) Ω Ω Ω In view of (265) and (272), the right-hand side in (273) converges to zero. Consequently, (266) holds true. It follows from (252) and (261) that ∗ (L3(vλ), h) − (B pλ, h) = (K + F, h), h ∈ Xm, (274) Observing (263) and (274), we get ∗ ∗ (B (pm − µ), h) = (L3(vm) − L3(vλ), h) + (B (pλ − µ), h), h ∈ Xm, µ ∈ Nm. This equality, together with (91), yields ∗ (B (pm − µ), h) kpm − µkL2(Ω) ≤ sup h∈Xm βkhkX −1 ∗ ≤ β (kL3(vm) − L3(vλ)kX + ckpλ − µkL2(Ω)), µ ∈ Nm, (275) where ∗ ∗ c = kB kL(L2(Ω),X ) = kBkL(X,L2(Ω)). 32 R.H.W. HOPPE AND W.G. LITVINOV

Hence,

kpλ − pmkL2(Ω) ≤ kpλ − µkL2(Ω) + kpm − µkL2(Ω) −1 −1 ∗ ≤ β kL3(vm) − L3(vλ)kX + (cβ + 1) inf kpλ − µkL2(Ω). (276) µ∈Nm ∗ Lemmas 3.2, 3.4 and (266) imply L3(vm) → L3(vλ) in X , and (267) follows from (276) and (87). •

9.2. A saddle-point approach. We introduce two functionals J : X → R, Ψ: X × L2(Ω) → R according to Z ³ Z I(˜u+u) ´ − 1 J(u) = [e(|E|, x)(λ + ξ) 2 + ψ1(ξ, |E|, x)]dξ dx − (K + F, u), Ω 0 (277) Ψ(u, ν) = J(u) − (B∗ν, u). (278) We consider the problem: Find a saddle-point of the Lagrangian Ψ, i.e.,

vλ, pλ ∈ X × L2(Ω), (279)

Ψ(vλ, ν) ≤ Ψ(vλ, pλ) ≤ Ψ(u, pλ), u ∈ X, µ ∈ L2(Ω). (280)

THEOREM 9.2. Assume that conditions (108), (109), and (C4a) are satisfied. Moreover, suppose that the function ψ1 satisfies (C0) and one of the conditions (C1a), (C2a), (C3a). Then, for an arbitrary λ > 0, there exists a unique solution vλ, pλ of problem (279), (280) which also solves (251)–(253). Hence, (251)– (253) and (279), (280) are equivalent. Proof. Let h be an arbitrary fixed element of X. Consider the function g : t → g(t) = Ψ(vλ + th, pλ), t ∈ R. It follows from the second inequality in (280) that g(0) ≤ g(t), dg t ∈ R. The function g is differentiable in R, whence dt (0) = 0. This equality is equivalent to (252). By Lemmas 3.1–3.4, the operator L3 is monotone. Hence, the functional u → Ψ(u, q) is convex for an arbitrarily fixed q ∈ L2(Ω) (see e.g.[6]). Therefore, the minimum of the functional u → Ψ(u, pλ) is characterized by (252). The first inequality in (280) gives (Bvλ, pλ − ν) ≤ 0 for all ν ∈ L2(Ω). Consequently, we get (253). However, in this case Ψ(vλ, pλ) = Ψ(vλ, ν) for all ν ∈ L2(Ω). Therefore, (251)–(253) and (279), (280) are equivalent. The existence and uniqueness of the solution of (279), (280) follows from Theorem 8.1. • Now, we introduce the augmented Lagrangian as follows: r Ψ (u, ν) = Ψ(u, ν) + (Bu, Bu), (u, ν) ∈ X × L (Ω). (281) 1 2 2

It is obvious that the pair (vλ, pλ) is also the saddle point of the functional Ψ1, where r is an arbitrary positive constant. We study the following method of augmented Lagrangians: Find a sequence {vm, pm} satisfying

(vm+1, pm+1) ∈ X × L2(Ω), (282) ∗ (L3(vm+1), h) − (B pm, h) + r(Bvm+1, Bh) = (K + F, h), h ∈ X, (283)

(pm+1 − pm, ν) + ρm(Bvm+1, ν) = 0, ν = L2(Ω). (284)

Here, ρm is a positive constant.

THEOREM 9.3. Assume that the conditions (108), (109), and (C4a) are satisﬁed, and let the function ψ1 satisfy (C0) and one of the conditions (C1a), (C2a). Assume that (v0, p0) ELECTRORHEOLOGICAL FLUID FLOWS 33 is an arbitrary pair in X × L2(Ω). Then, for an arbitrary m, there exists a unique pair (vm+1, pm+1) satisfying (282)–(284). Moreover, if

0 < inf ρm ≤ sup ρm < 2r, (285) m∈N m∈N then vm → vλ in X, pm → pλ in L2(Ω), (286) where (vλ, pλ) is the solution of (251)–(253). Proof. We set um = vm − vλ, qm = pm − pλ. (287) By subtracting (252) from (283) and (253) from (284), we get ∗ (L3(vm+1) − L3(vλ), h) + r(Bum+1, Bh) = (B qm, h), h ∈ X, (288)

(qm+1 − qm, ν) = −ρm(Bum+1, ν), ν ∈ L2(Ω). (289)

Choosing ν = 2qm+1 in (289), we obtain kq k2 − kq k2 + kq − q k2 = −2ρ (Bu , q ). (290) m+1 L2(Ω) m L2(Ω) m+1 m L2(Ω) m m+1 m+1

Now, we choose h = um+1 in (288). Then, (288)–(290) imply kq k2 − kq k2 + kq − q k2 m+1 L2(Ω) m L2(Ω) m+1 m L2(Ω) +2ρ (L (v ) − (L (v ), u ) + 2ρ rkBu k2 m 3 m+1 3 λ m+1 m m+1 L2(Ω)

= −2ρm(Bum+1, qm+1 − qm). (291) (291) and Lemma 3.2 (see (63)) give 2 2 2 kqm+1k − kqmk + kqm+1 − qmk L2(Ω)Z L2(Ω) L2(Ω) 1 1 2 +4ρma3 [I(˜u + vm+1) 2 − I(˜u + vλ) 2 ] dx Ω +2ρ rkBu k2 ≤ 2ρ kBu k kq − q k . (292) m m+1 L2(Ω) m m+1 L2(Ω) m+1 m L2(Ω) By applying the inequality 2ab ≤ a2 + b2 to the right-hand side of (292), we obtain Z 2 2 1 1 2 kq k − kq k + 4ρ a [I(˜u + v ) 2 − I(˜u + v ) 2 ] dx m+1 L2(Ω) m L2(Ω) m 3 m+1 λ Ω +ρ (2r − ρ )kBu k2 ≤ 0. (293) m m m+1 L2(Ω)

In view of (285), there exists δ > 0 such that ρm(2r − ρm) ≥ δ for any m. Moreover (293) gives kq k ≥ kq k . Hence, the sequence {kq k2 } converges, m L2(Ω) m+1 L2(Ω) m L2(Ω) i.e., lim kq k2 = α ≥ 0, and (293) yields m L2(Ω) Z 1 1 2 [I(˜u + vm+1) 2 − I(˜u + vλ) 2 ] dx → 0, (294) Ω Bum → 0 in L2(Ω). (295) R 2 Since the function u → Ω u dx is a continuous mapping from L2(Ω) into R, from (294) we obtain Z Z

I(˜u + vm)dx → I(˜u + vλ)dx. (296) Ω Ω Therefore, kvmkX ≤ c, m ∈ N. (297) 34 R.H.W. HOPPE AND W.G. LITVINOV

Moreover, (283) and (85) yield kpmkL2(Ω) ≤ c for all m. Therefore, a subsequence {vη, pη} can be extracted such that vη * v0 in X and pη * p0 in L2(Ω). We pass to the limit as before. Then, we get v0 = vλ, p0 = pλ. Due to (296) and by the uniqueness of the solution of (251)–(253), we obtain (see (273))

vm → vλ in X. (298) It follows from (288) and (85) that ∗ ∗ ∗ ∗ kL3(vm+1) − L3(vλ) + rB Bum+1kX = kB qmkX ≥ β1kqmkL2(Ω).

This inequality and (295),(298), yield qm → 0 in L2(Ω). Therefore, (286) holds true. • 9.3. Solving a nonlinear problem. We consider two methods for the solution of the nonlinear problem (283), namely the Birger-Kachanov method and the contraction method. Both methods transform a nonlinear problem into a sequence of linear problems. We consider the problem: Find a function u satisfying

u ∈ X, (L3(u), h) + r(Bu, Bh) = (f, h), ∀h ∈ X, (299) where f ∈ X∗. For an arbitrary v ∈ X, we deﬁne the operator M(v) ∈ L(X,X∗) according to Z − 1 (M(v)w, h) = 2 [e(|E|, x)(λ + I(˜u + v)) 2 εij(˜u + w) εij (h) Ω +ψ1(I(˜u + v), |E|, x) εij (˜u + w) εij (h)]dx + r(Bw, Bh). (300)

The Birger-Kachanov method consists in constructing a sequence {um} such that

um+1 ∈ X, (M(um)um+1, h) = (f, h), ∀h ∈ X. (301) Convergence criteria for the Birger-Kachanov method have been established in [5]. The results [5], [15], and [18] imply the following theorem.

THEOREM 9.4. Assume that the conditions (108), (109), and (C4a) are satisﬁed. Further, suppose that ψ1 is a nonincreasing function satisfying conditions (C0) and (C1a). More- over, let λ > 0 and choose u0 as an arbitrary element of X. Then, for any m there exists a unique solution vm+1 of (301) and um → u in X, where u solves (299). We now consider the contraction method: Let A be a linear continuous selfadjoint and coercive mapping from X into X∗, i.e.

(Au, h) = (u, Ah), |(Au, h)| ≤ b1kukX khkX , u, h ∈ X, 2 (Au, u) ≥ b2kukX , u ∈ X, (302) where b1, b2 are positive constants. In view of (302) the expression 1 kuk1 = (Au, u) 2 (303) 1 n deﬁnes a norm in X which is equivalent to k.kX and to the H (Ω) -norm. We denote by X1 the space X equipped with the inner product

(u, h)X1 = (Au, h) (304) and with the norm (303). We study the following iterative method: Find um+1 ∈ X such that

(Aum+1, h) = (Aum, h) − t[L3(um, h) + r(Bum, Bh) − (f, h)], ∀h ∈ X, (305) where t is a positive constant. ELECTRORHEOLOGICAL FLUID FLOWS 35

We deﬁne the operator A by Z ∂u ∂h (Au, h) = g i i dx, u, h ∈ X, (306) Ω ∂xj ∂xj where g ∈ C(Ω), g(x) ≥ c0 > 0 for all x ∈ Ω. Then, taking h = (h1, 0) and h = (0, h2) 2 3 in the case that Ω ⊂ R , and h = (h1, 0, 0), h = (0, h2, 0)¯ , h = (0, 0, h3) for Ω ⊂ R , 1 where hi are arbitrary functions from U = {η ∈ H (Ω), η¯ = 0}, we split the problem S1 (305) and obtain independent problems for the computation of umi, i = 1, . . . , n.

LEMMA 9.1. Assume that the conditions (108), (C0), (C1a), (C4a), and (302) are satis- ﬁed. Then, 2 (L3(v) − L3(h), v − h) ≥ q1kv − hk1, (307) kL (v) − L (h)k ∗ ≤ q kv − hk , (308) 3 3 X1 2 1 where −1 q1 = µ1b1 , 1 − 1 − 2 2 q2 = (µ2 + 4a5λ )b2 . (309)

Note that µ1 and µ2 are deﬁned by (56). Proof. Taking into account the inequalities of (302), and applying Lemmas 3.1 and 3.4, we obtain (307), (308) with q1, q2 deﬁned by (309). •

THEOREM 9.5. Assume that the conditions (108), (C0), (C1a), and (C4a) are satisﬁed. Let f ∈ X∗, and suppose that the operator A ∈ L(X,X∗) satisﬁes (302). Let also λ > 0 −2 and u0 be an arbitrary element of X. Then, for t ∈ (0, 2 q1 q3 ), where ∗ q = q + rkB Bk ∗ , (310) 3 2 L(X1,X1 ) and for any m, there exists a unique solution um+1 of (305). The following estimate holds true m k(t) ∗ kum − uk1 ≤ kL3(u0) + rB B u0 − fkX∗ , (311) 1 − k(t) 1 where 1 2 2 2 k(t) = (1 − 2q1t + q3 t ) < 1, (312) and u is the solution of (299). 2 −2 1 −2 The function k takes its minimal value k(t0) = (1 − q1 q3 ) 2 in t0 = q1 q3 . ∗ Proof. Let N = L3 + rB B. By Lemma 9.1 we have 2 (N(v) − N(h), h − v) ≥ q1kv − hk1, kN(v) − N(h)k ∗ ≤ q kv − hk , v, h ∈ X, (313) X1 3 1 ∗ where q = q + rkB Bk ∗ . 3 2 L(X1,X1 ) ∗ We refer to J as the Riesz operator J ∈ L(X1 ,X1), i.e., ∗ (Jg, h) = (g, h), g ∈ X , h ∈ X , kJgk = kgk ∗ . (314) X1 1 1 1 X1

It is obvious that (299) is equivalent to the computation of a ﬁxed point u = Ut(u), where Ut : X1 → X1 is given by

Ut(h) = h − tJ(N(h) − f). (315) 36 R.H.W. HOPPE AND W.G. LITVINOV

In view of (313)–(315), we have 2 2 kUt(v) − Ut(h)k1 = kv − h − tJ(N(v) − N(h))k1 2 2 2 = kv − hk − 2t(N(v) − N(h), v − h) + t kN(v) − N(h)k ∗ 1 X1 2 2 2 2 2 2 ≤ kv − hk1 − 2tq1kv − hk1 + t q3kv − hk1 = (k(t)kv − hk1) , −2 where k(t) is deﬁned by (312) and k(t) < 1, if t ∈ (0, 2q1q3 ). Therefore, the mapping Ut is a contraction, and the existence of a unique solution u of (299) and the estimate (311) follow from the ﬁxed point theorem (see e.q. [23]). •.

9.4. Numerical simulation of electrorheological shock absorbers. Based on a discretiza- tion of the flow model by Q2 − P1 Taylor-Hood elements with respect to a simplicial triangulation and the solution of the discretized problem by the technique of augmented Lagrangians, we have performed simulations of the compression and rebound stage of an electrorheological shock absorber. As shown in Figure 4, the absorber consists of two fluid chambers filled with an electrorheological fluid and a piston with two transfer ducts whose inner walls serve as electrodes and counter-electrodes, respectively. The viscosity function in the constitutive relation has been approximated by splines fitted to experimentally obtained data from flow curves for the electrorheological fluid (Rheobay TP AI 3565, polyurethane) under consideration [1].

FIGURE 4. Schematic representation of an electrorheological shock absorber)

We note that the electric field is perpendicular to the flow in the ducts, but it is no longer perpendicular to the flow field neither in the vicinity of the inflow and outflow boundaries nor in the other parts of the fluid chambers. We emphasize that the operational behavior ELECTRORHEOLOGICAL FLUID FLOWS 37

FIGURE 5. Vertical component of the velocity during compression

FIGURE 6. Vertical component of the velocity during rebound 38 R.H.W. HOPPE AND W.G. LITVINOV

FIGURE 7. Contour lines of the electric potential

(characteristics) of the ERF shock absorber strongly depends on the flow conditions at the inflow and outflow boundaries. Figures 5 and 6 display the computed vertical component of the velocity for three different positions of the piston during the compression stage (Figure 5) and the rebound stage (Figure 6). The applied voltage was chosen as U = 3kV in all cases. Figure 7 shows the computed contour lines of the electric potential indicating clearly that the generated electric field is only perpendicular to the velocity field within the transfer ducts. We note that the obtained numerical results allow to compute the characteristics (dependence of the damper force on the velocity of the piston) of the absorber which are in good agreement with experimentally determined characteristics.

Acknowledgments. The work of the authors has been supported by the German National Science Foundation (DFG) within the Collaborative Research Center SFB 438.

REFERENCES

[1] Bayer AG, Provisional Product Information. Rheobay TP AI 3565 and Rheobay TP AI 3566. Bayer AG, Silicones Business Unit, No. AI 12601e, Leverkusen, 1997 [2] Belonosov M.S., Litvinov W.G., Finite element methods for nonlinearly viscous fluids, Z.angew. Math.Mech. 76, 307–320, 1996. [3] Ceccio S., Wineman A., Influence of orientation of electric field on shear flow of electrorheological fluids, J.Rheol. 38, 453–463, 1994. [4] Duvaut G., lions J.L., Inequalities in Mechanics and Physics, Springer-Verlag, 1976. [5] Fucikˇ S., Kratochvil A., Necasˇ J., Kachanov-Galerkin method, Comment. Math. Univ. Carolinae 14, 651– 659, 1973. [6] Gajewski H., Groger¨ K., Zacharias K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichun- gen, Akademie-Verlag, Berlin, 1974. ELECTRORHEOLOGICAL FLUID FLOWS 39

[7] Girault V., Raviart P., Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin 1986. [8] Hoppe R.H.W., Litvinov W.G., Flow of electrorheological ﬂuids under the conditions of slip on the boundary, Institute of Mathematics, University of Augsburg, 2002. [9] Ladyzhenskaya O., Solonnikov V., Some problems of vector analysis and generalized formulation of boundary value problems for the Navier-Stokes equations, Zap.Nauchn.Sem.Leningrad. Otdel.Mat.Inst. Steklov (LOMI), 59,81–116,1976 (in Russian). [10] Landau L.D., Lifshitz E.M., Electrodynamics of Continuous Media, Pergamon, Oxford, 1984. [11] Langenbach A.,Monotone Potentialoperatoren in Theorie und Anwendung, Berlin,1975. [12] Lions J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [13] Lions J.L., Quelques Methodes´ de Resolution´ des Problemes` aux limites non lineaires,´ Dunod, Paris, 1969. [14] Litvinov W.G., Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics, Birkhauser, 2000. [15] Litvinov W.G., Motion of Nonlinearly Viscous Fluid, Moscow, Nauka, 1982 (in Russian). [16] Maugin G.A., Continuum Mechanics of Electro-Magnetic Solids, North- Holland, Amsterdam, 1988. [17] Mosolov P.P., Miasnikov V.P., Proof of the Korn inequality, Dokl.Akad.Nauk SSSR, Matem., 201, 36– 39,1971 (in Russian). [18] Necasˇ J., Hlavacekˇ I., Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction, Elsevier Scientiﬁc Publishing Company, Amsterdam, 1981. [19] Panagiotopoulos D., Inequality Problems in Mechanics and Applications, Birkhauser,¨ 1985. [20] Parthasarathy M., Klingenberg D.J., Electrorheology: mechanisms and models. Material Science and Engi- neering, R17, 57–103, 1996. [21] Rajagopal K., Wineman A., Flow of electrorheological materials, Acta Mechanica 91, 57–75, 1992. [22] Ru˙ziˇ ckaˇ M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer, 2000. [23] Schwartz L., Analyse Mathematique´ 1, Hermann, 1967. [24] Shulman Z.P., Nosov B.M., Rotation of Nonconducting Bodies in Electrorheological suspensions, Nauka i Technika, Minsk, 1985 (in Russian). [25] Temam R., Navier-Stokes Equations, North-Holland Publishing Company, Amsterdam, 1979. [26] Vainberg M.M., Variational Methods for the Study of Nonlinear Operators, Holden Day, San Francisco, 1964. Received December 0000; revised April 0000. E-mail address: [email protected] E-mail address: [email protected]