Problems on electrorheological fluid flows

We develop a model of an electrorheological fluid such that the fluid is considered as an anisotropic one with the viscosity depending on the second invariant of the rate of strain tensor, on the module of the vector of electric field strength, and on the angle between the vectors of velocity and electric field. We study general problems on the flow of such fluids at nonhomogeneous mixed boundary conditions, wherein values of velocities and surface forces are given on different parts of the boundary. We consider the cases where the viscosity function is continuous and singular, equal to infinity, when the second invariant of the rate of strain tensor is equal to zero. In the second case the problem is reduced to a variational inequality. By using the methods of a fixed point, monotonicity, and compactness, we prove existence results for the problems under consideration. Some efficient methods for numerical solution of the problems are examined.


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 aggregate to form columns (see Fig. 1 , taken from [18]). 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 changes not so drastically.
The chainlike and columnar structures are destroyed under the action of large stresses, and 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 [19], and for an incompressible fluid there was obtained the following equation: (1.1) Here p is the pressure, I 1 the unit tensor, α i are scalar functions of six invariants of the tensors ε, E ⊗ E, and mixed tensors; α i are to be determined by experiments.
In the condition of simple shear flow, when the vectors of velocity v and electric field E are orthogonal and E is in the plane of flow, the terms with coefficients α 2 , α 4 , α 5 , α 6 give rise to two normal stresses differences (see [19]). But these terms lead to incorrectness of the boundary value problems for the constitutive equation (1.1), and very restrictive conditions should be imposed on the coefficients α 2 , α 4 , α 5 , α 6 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 thermodynamics, and the condition of monotonicity denotes that stresses increase as the rate of strains increase).
The constitutive equation (1.1) does not describe anisotropy of the fluid; in the case of simple shear flow (1.1) 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 (1.1)).
Stationary and nonstationary mathematical problems for the special case of (1.1) are studied in [20]. It is supposed in [20] that velocities are equal to zero everywhere on the boundary and the stress tensor is given by where |ε| 2 = n i,j=1 ε 2 ij , n being the dimension of a domain of flow, γ 1 − γ 4 , are constants, and k is a function of |E| 2 .
The constants γ 1 − γ 4 and the function k are determined by the approximation of flow curves which are obtained experimentally for different values of the vector of electric field E (see Subsection 2.2). But the conditions of coerciveness and monotonicity of the operator − div(σ + pI 1 ) impose severe constraints on the constants γ 1 − γ 4 and on the function k, see [20], such that with these restrictions one cannot obtain a good approximation of a flow curve, to say nothing of approximation of a set of flow curves corresponding to different values of E. Below in Section 2, we develop a constitutive equation of electrorheological fluids such that a 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 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, and 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, and it leads to correct mathematical problems.
In Section 3, we present auxiliary results, and in Sections 4-8 we study problems on stationary flow of such fluids at nonhomogeneous mixed boundary conditions. Here we prescribe values of velocities and surface forces on different parts of the boundary and ignore the inertial forces. The cases where the viscosity function is continuous and singular, equal to infinity, when the second invariant of the rate of strain tensor is equal to zero, are studied. In the second case the problem is reduced to a variational inequality.
By using the methods of a fixed point, monotonicity, and compactness we prove existence results for the regular and singular viscosity functions. In the second case existence results are obtained at more restrictive assumptions. Here the singular viscosity is approximated by a continuous bounded one with a parameter of regularization, and a solution of the variational inequality is obtained as a limit of the solutions of regularized problems.
Section 9 is concerned with numerical solution of the problems on stationary flows of electrorheological fluids with regular viscosity function. We consider here methods of the augmented Lagrangian, Birger-Kachanov, contraction and gradient.
In Sections 10 and 11 we study problems on flow of electrorheological fluids in which inertial forces are taken into account. Here we consider nonhomogeneous boundary conditions in the case that velocities are given on the whole of the boundary and in the case that velocities and surface forces are prescribed on different parts of the boundary. With some suppositions existence results are proved.

Constitutive equation.
2.1. The form of the constitutive equation. It has been found experimentally that the shear stress and accordingly the viscosity of electrorheological fluids depend on the shear rate, the module of the vector of electric field strength, and the angle between the vectors of fluid velocity and electric fields strength (see [18,22]). Thus, on the basis of experimental results we introduce the following constitutive equation σ ij (p, u, E) = −pδ ij + 2ϕ(I(u), |E|, µ(u, E))ε ij (u), i, j = 1, . . . , n, n = 2 or 3. (2.1) Here, σ ij (p, u, E) are the components of the stress tensor which depend on the pressure p, the velocity vector u = (u 1 , . . . , u n ) and the electric field strength E = (E 1 , . . . , E n ), δ ij is the Kronecker delta, and ε ij (u) are the components of the rate of strain tensor ε ij (u) = 1 2 Moreover, I(u) is the second invariant of the rate of strain tensor and ϕ the viscosity function depending on I(u), |E| and µ(u, E), where .

(2.4)
So µ(u, E) is the square of the scalar product of the unit vectors u |u| and E |E| . The function µ is defined by (2.4) in the case of an immovable frame of reference. If the frame of reference moves uniformly with a constant velocityǔ = (ǔ 1 , . . . ,ǔ n ), then we set: As the scalar product of two vectors is independent of the frame of reference, the constitutive equation (2.1) is invariant with respect to the group of Galilei transformations of the frame of reference that are represented as a product of time-independent translations, rotations and uniform motions. It is obvious that µ(u, E)(x) ∈ [0, 1], and for fixed y 1 , y 2 ∈ R + , where R + = {z ∈ R, z ≥ 0}, the function y 3 → ϕ(y 1 , y 2 , y 3 ) reaches its maximum at y 3 = 0 and its minimum at y 3 = 1 when the vectors u(x) +ǔ and E are correspondingly orthogonal and parallel.
The function µ defined by (2.4), (2.5) is not specified at E = 0 and at u = 0, and there does not exist an extension by continuity to the values of u = 0 and E = 0. However, at E = 0 there is no influence of the electric field. Therefore, (2.6) and the function µ(u, E) need not be specified at E = 0. Likewise, in case that the measure of the set of points x at which u(x) = 0 is zero, the function µ need not also be specified at u = 0. But in the general case we should specify µ for all values of u. Because of this we assume that the function µ is defined as follows: whereĨ denotes a vector with components equal to one, and α is a small positive constant. If u(x) = 0 almost everywhere in Ω, we may choose α = 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., [22]). These curves define the relationship between the shear stress τ = σ 12 and the shear rate γ = ε 12 (u) = 1 2 du 1 dx 2 for a flow that is close to simple shear flow. Line 1 is the flow curve for |E| = 0, and lines 2-4 represent the flow curve for increasing |E|.
Flow curves are obtained in some region, say γ 0 ≤ γ ≤ γ 1 , γ 0 > 0. Experimental results for small γ are not precise, and one has to extend the flow curves to R + . It is customary to extend flow curves by straight lines over the region γ 1 < γ < ∞. One can prolong flow curves in [0, γ 0 ) such that either τ = τ 0 for γ = 0 or τ = 0 for γ = 0 (see the dash and dot-dash lines in Fig.2).
Flow problems for fluids with a constitutive equation (2.11) reduce to the solution of variational inequalities. Such problems are considerably more complicated than problems for fluids with finite viscosity, in particular, for fluids with a constitutive equation as given by (2.12). From a physical point of view, (2.12) with a finite, but possibly large viscosity for I(u) = 0 seems to be more reasonable than (2.11).
The assumptions (C2) and (C3) indicate that in case of simple shear flow, the shear stress must increase with increasing shear rate.
The following assumption is concerned with the function (coefficient) b in (2.11), (2.12): a 5 being a positive number. Generally, the continuous function ϕ is expressible in the form Polynomials or splines can be used to represent the functions β i . The flow curves obtained for various values of E can be approximated with an arbitrary accuracy. We note that (2.18) may also be used for an identification of the function ψ.
In the general case, the coefficients e i 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.
2.3. General problems. Fig. 3 below gives an example of an electrorheological fluid flow. Here, the domain Ω of fluid flow consists of three parts Ω 1 , Ω 2 , Ω 3 . A fluid flows from the part Ω 1 across Ω 2 in the part Ω 3 . Electrodes are placed on parts Γ 0 and Γ 1 of the boundary of Ω 2 , and an electric field E is generated by applying voltages △U (t) to electrodes at time t. Generally it may be a k pairs of electrodes and voltages △U i (t) are applied to i-th pair of electrodes, i = 1, . . . , k. The boundary S of the domain Ω consists of two parts S 1 and S 2 . Surface forces F = (F 1 , . . . , F n ) act on S 2 , and the distribution of velocitiesû = (û 1 , . . . ,û n ) is given on S 1 .
The equations of motion and the incompressibility condition read as follows: Here, K i are the components of the volume force vector K, ρ is the density, T a positive constant. In (2.19) and below Einstein's convention on summation over repeated index is applied. We assume that Ω is a bounded domain in R n , n = 2 or 3. Suppose that S 1 and S 2 are open subsets of S such that S = S 1 ∪ S 2 and S 1 ∩ S 2 = ∅. The boundary and initial conditions are the following: [−pδ ij + 2ϕ(I(u), |E|, µ(u, E))ε ij (u)]ν j S 2 ×(0,T ) = F i i = 1, . . . , n, Here, F i and ν j are the components of the vector of surface force F = (F 1 , . . . , F n ) and the unit outward normal ν = (ν 1 , . . . , ν n ) to S, respectively.
We consider Maxwells equations in the following form (see e.g. [8]): Here E is the electric field, B the magnetic induction, D the electric displacement, H the magnetic field, c the speed of light. One can assume that The boundary conditions are the following: Here Γ i and Γ i0 are the surfaces of the i-th control and null electrodes respectively, and it is supposed that Γ i , Γ i0 are open subset of S. We assume ǫ ∈ L ∞ (Ω), e 1 ≤ ǫ ≤ e 2 a.e. in Ω, (2.33) e 1 , e 2 being positive constants. Suppose also that where Letθ be a function such thať Define a spaceṼ and a bilinear form a : H 1 (Ω) ×Ṽ → R as follows: Proof. By virtue of (2.33) the bilinear form a is continuous and coercive inṼ . Therefore there exists a unique solution u of problem (2.40), and the function θ =θ +u is a generalized solution of problem (2.29)-(2.32).
The functions of volume force K and surface force F in (2.19) and (2.22) are represented in the form whereK andF are the main volume and surface forces, K e and F e are volume and surface forces generated by the vector of electric field E. Considering electrorheological fluid as a liquid dielectric we present the stress tensor σ e = {σ eik } n i,k=1 induced by electric field as follows (see [8]): Taking into account (2.26), we obtain the following formula for the vector of volume force K e = (K e1 , . . . , K en ), The vector of surface forces is given by

Auxiliary results.
Let Ω be a bounded domain in R n with a Lipschitz continuous boundary S, n = 2 or 3. Let S 1 be an open non-empty subset of S. We consider the following spaces: By means of Korn's inequality, the expression defines a norm on X and V being equivalent to the norm of H 1 (Ω) n . Everywhere below we 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 particular, if f ∈ L 2 (Ω) or f ∈ L 2 (Ω) n , then (f, h) is the scalar product in L 2 (Ω) or in L 2 (Ω) n , respectively. The sign ⇀ denotes weak convergence in a Banach space.
We further consider three functionsṽ We set v = (ṽ, v 1 , v 2 ) and define an operator L v : X → X * as follows: where Proof. Let u, w be arbitrarily fixed functions in X and We introduce the function γ as follows: It is obvious that By classical analysis it follows that γ is differentiable at any point t ∈ (0, 1). Therefore Taking note of the inequality and (2.13), (2.15) we get (3.7) as a direct consequence of (3.9)-(3.13).
Define the function g as follows: Then, taking e = h in (3.13) and applying (2.14) we get dγ dt and (3.6) follows from (3.14).

Lemma 3.3. Assume that (C3) is satisfied and (3.4) holds true. Then
(3.20) Moreover, (3.16) is valid, and the operator L v is a continuous mapping from X into X * .
Define the set U as follows a.e. in Ω}, (3.21) and letṽ ∈ H 1 (Ω) n . For a given constant λ > 0 define an operator L λ : U × X → X * as follows: For an arbitrary λ > 0 and an arbitrarily fixed h ∈ U the following inequalities hold: Moreover, the conditions Proof. The viscosity function associated with the operator L λ (h, .) : u → L λ (h, u) has the form where y plays the role of the second invariant of the rate of strain tensor. We have The left-hand side of (3.29) represents the derivative of the function g : z → g(z) = ϕ(z 2 )z, z 2 = y. Therefore, the function g is increasing, and (3.23) follows from the proof of Lemma 3.3. By (3.21), (3.22) we obtain which readily gives (3.25). Moreover, (3.21) and (3.28) yield and hence, observing Lemma 3.1 we get (3.24). Assuming (3.26), it follows that Further, Obviously, the first term of the right-hand side in (3.32) tends to zero. By (3.26) we have in Ω, and by the Lebesgue theorem we obtain that the second term of the right-hand side in (3.32) tends to zero. The second term of the right-hand side in (3.31) also tends to zero. Thus (3.27) is satisfied, and the lemma is proven.
Let Ω be a bounded domain in R n , n = 2 or 3 with a Lipschitz continuous boundary S, and let the operator B ∈ L(X, L 2 (Ω)) be defined as follows: holds true. The operator B is an isomorphism from V ⊥ onto L 2 (Ω), where V ⊥ is the orthogonal complement of V in X, and the operator B * that is adjoint to B, is an isomorphism from L 2 (Ω) onto the polar set Moreover, For a proof see in [1]. Lemma 3.5 is a generalization of the inf-sup condition in case that the operator div acts in the subspace H 1 0 (Ω) (see [6]). This result was first established in an equivalent form by Ladyzhenskaya and Solonnikov in [7].
be sequences of finite-dimensional subspaces in X and L 2 (Ω), respectively, such that Define the operators B m ∈ L(X m , N * m ) as follows: We introduce the spaces V m and V 0 m by The following Lemma is valid (see [1]).
be sequences of finite-dimensional subspaces in X and L 2 (Ω) and assume that the discrete inf − sup condition (LBB condition) where U is as in (3.21).
Here and below the sign ⇀ designates the weak convergence. Therefore, and Therefore, In view of (3.46), the right-hand side of (3.53) tends to zero as m → ∞. Hence (3.51), (3.52), and the continuity of the functional Ψ(h, .) imply and the lemma is proved.
where h 1 , h 2 are positive constants. Then the expression Ω hI(u) defines a norm on X. Note that this norm is not equivalent to the norm of the space W 1 1 (Ω) n . However, is a norm on X, which is equivalent to the norm of W 1 p (Ω) n for p > 1 (cf., e.g., [15]).

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 We assume thatû Then there exists a functionũ such that Suppose also In line with (2.11), we choose the viscosity function ϕ of the following form: 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 (4.8) as a generalized Bingham electrorheological fluid. Define a functional J on the set X × X and an operator L : X → X * as follows: We use the notations The following assertion holds.
We multiply (4.1) with h i − v i , sum over i and integrate over Ω. By Green's formula and (4.4), (4.8), we obtain (4.14) We use the relations so that the first addendum of the left-hand side of (4.14) is majorized by Then, (4.14) implies (4.13), and the theorem is proved.
Let v be a solution of the problem (4.12), (4.13) such that We replace h in (4.13) by v + λh, λ > 0, from which Replacing h by −h in (4.18) we obtain and Lemma 3.5 gives It follows from (4.22) that the pair (u, p) with u =ũ + v is a solution of (4.1)-(4.4) in the sense of distributions. Thus, we have proved the following: If v is a solution of (4.12), (4.13) that satisfies (4.16), then there exists a function p ∈ L 2 (Ω) such that the pair (u, p) with u =ũ + v is a solution of (4.1)-(4.4) in the distributional sense. In view of this and Theorem 4.1 it is reasonable to refer to the function u =ũ + v as a generalized solution of (4.1)-(4.4). 5. Problem for the fluid with constitutive equation (2.12).

Existence theorem.
We define the following functional on X × X: Note that the functional J λ is Gâteaux differentiable in X with respect to the second argument for λ > 0, but not for λ = 0. The partial Gâteaux derivative ∂J λ ∂h is given by Consider the following problem: , then there exists a function p λ such that the pair (v λ , p λ ) is a solution of the following problem: We remark that (5.5)-(5.7) represent the flow of an electrorheological fluid with the constitutive equation (2.12). We seek an approximate solution of the problem (5.5)-(5.7) of the form where X m and N m are finite dimensional subspaces in X and L 2 , (Ω), respectively, and B m is defined as in (3.40). giving z(e) ≥ 0 for e X ≥ r = c 2a 1 . From the corollary of Brouwer's fixed point theorem (cf. [5]) it follows that there exists a solution of (5.12) with where the second inequality follows from (2.13) and (4.6).
For an arbitrary f ∈ X * we denote by Gf the restriction of f to X m . Then Gf ∈ X * m , and by (3.42), (5.12) we obtain Therefore, there exists a unique p m ∈ N m (see Lemma 3.6) such that Let η 0 be a fixed positive number and w ∈ X η 0 , q ∈ N η 0 . Observing (5.20), (5.22)-(5.24) we pass to the limit in (5.9), (5.10) with m replaced by η, and obtain Since η 0 is an arbitrary positive integer, by (3.38), (3.39) We present the operator L(v) in the form where the operator (v, w) → L(v, w) is considered as a mapping of X × X into X * according to We get Likewise we obtain Taking into account that (B η v η , p η ) = 0, by (5.9), (5.20), (5.22) we obtain Observing (5.33)-(5.36) and passing to the limit in (5.30), by (5.27), (5.31) we get We choose w = v 0 − γh, γ > 0, h ∈ X, and consider γ → 0. Then, Lemmas 3.3, 3.4 give This inequality holds for any h ∈ X. Therefore, replacing h by −h shows that equality holds true. Consequently, the pair (v λ , p λ ) with v λ = v 0 and p λ = p 0 solves (5.5)-(5.7). The theorem is proved.
We now consider a problem on stationary flow of the extended Bingham electrorheological fluid. The constitutive equation of this fluid is the following: We deal with the problem (4.1)-(4.4) and assume that (4.5) and (4.7) are satisfied. Then, according to Remark 4.1 the generalized solution of our problem is u =ũ + v, whereũ is a function satisfying (4.6) and v is a solution of the problem v ∈ V, (6.2) Here, J is the functional given by (4.9) and the operator L 1 : X → X * is defined as follows The function b 1 is subject to the following condition: is a continuous function on R + × [0, 1] and satisfies with positive constants a 6 and a 7 .
However, (6.15), (6.16) do not imply lim inf(L(v λ ), v λ ) ≥ (L(v), v) (compare with ((6.28)), and we cannot assert that v is a solution of (4.12), (4.13). In the next section we prove the existence of a solution of (4.12), (4.13) under conditions which are more restrictive than those of Theorem 5.1.

General variational inequality.
We assume that the function µ in the operator L defined by (4.10) is replaced by a function µ 1 such that According to (2.7) we may define µ 1 as follows: where α, β are small positive constants,Ĩ is a vector with components equal to one, and P an operator of regularization given by where 3) we assume that the function u is extended to R n . In case that P u(x) = 0 a.e. in Ω we may choose α = 0, if P E(x) = 0 a.e. in Ω we may choose β = 0 . For the function µ 1 condition (7.1) is satisfied. From the physical point of view (7.2) means that the value of the function µ 1 and therefore the viscosity of the fluid at a point x depends on the angle between the vectors of velocity and 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.
Proof. 1) The existence of a solution (v λ , p λ ) of (5.5)-(5.7) follows from Theorem 5.1, and it is inferred from the proof of this theorem (see (5.15)), that v λ remains in a bounded set of V independent of λ. Therefore, from the sequence {v λ } we can select a subsequence, again denoted by {v λ }, such that v λ ⇀ v in V as λ → 0, (7.6) v λ → v in L 2 (Ω) n and a.e. in Ω.
Let (4.12), (4.13) be valid. Obviously, Moreover, from a physical point of view a fluid with finite viscosity (2.12) seems to be more reasonable than a fluid with unbounded viscosity (2.11).

Problems with given function µ.
In the case that the distance between the electrodes is small compared with the lengths of the electrodes, one can assume that in between the electrodes the velocity vector is orthogonal to the vector of electric field strength, and the electric fields strength is equal to zero in the remaining part of the domain under consideration.
9. Numerical solution of stationary problems.
We study Algorithm of the augmented Lagrangian: find a sequence {v m , p m } satisfying Here ρ m is a positive constant. then v m → v λ in X, p m → p λ in L 2 (Ω), (9.26) where (v λ , p λ ) is the solution of (8.11)-(8.13).
Proof. We set By subtracting (8.12) from (9.23) and (8.13) from (9.24), we get Take h = u m+1 in (9.28). Then (9.28)-(9.30) give (9.31) (9.31) and Lemma 3.2 (see (3.15)) imply By applying the inequality 2ab ≤ a 2 + b 2 to the right-hand side of (9.32), we obtain By virtue of (9.25) there exists δ > 0 such that ρ m (2r − ρ m ) ≥ δ for any m, and by (9.33), q m L 2 (Ω) ≥ q m+1 L 2 (Ω) . Thus, the sequence { q m 2 L 2 (Ω) } converges, i.e. lim q m 2 L 2 (Ω) = α ≥ 0, and (9.33) yields Bu m → 0 in L 2 (Ω). (9.35) Since the function u → Ω u 2 dx is a continuous mapping from L 2 (Ω) into R we obtain from (9.34) that and by (9.23), (3.37) we get p m L 2 (Ω) ≤ c for all m. Therefore, a subsequence {v η , p η } can be extracted such that v η ⇀ v 0 in X, p η ⇀ p 0 in L 2 (Ω). We pass to the limit by analogy with the above. Then we get v 0 = v λ , p 0 = p λ . Due to (9.36) and by the uniqueness of the solution of (8.11)-(8.13), we obtain by analogy with the above (see (9.13)) that v m → v λ in X. (9.38) It follows from (9.28) and (3.37) that This inequality, together with (9.35) and (9.38), yields q m → 0 in L 2 (Ω). Therefore (9.26) holds true. 9.3. Solving a nonlinear problem. We consider two methods for solving the nonlinear problem (9.23), 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 For an arbitrary v ∈ X we define the operator M (v) ∈ L(X, X * ) as follows: The Birger-Kachanov method consists in constructing a sequence {u m } such that The conditions for the convergence of the Birger-Kachanov method in the general situation were established in [4].
Theorem 9.4. Suppose the conditions (4.6), (4.7), (C4a) are satisfied. Assume that ψ 1 is a nonincreasing function meeting the conditions (C0) and (C1a). Let also λ > 0 an u 0 be an arbitrary element of X. Then for any m there exists a unique solution v m+1 of the problem (9.41) and u m → u in X, where u is the solution of (9.39).
Consider now the contraction method. Let A be a linear continuous selfadjoint and coercive mapping from X into X * , i.e. defines that norm in X that is equivalent to the norm . X and to the norm of H 1 (Ω) n . By X 1 we denote the space X equipped with the scalar product (u, h) X 1 = (Au, h) (9.44) and with the norm (9.43).
We study the following iterative method: where t is a positive constant. We may define the operator A by Proof. Taking into account the inequalities of (9.42), and applying Lemmas 3.1 and 3.4, we obtain (9.47), (9.48) with q 1 , q 2 defined by (9.49).

50)
and for any m there exists a unique solution u m+1 of problem (9.45) and the following estimate holds and u is the solution of (9.39).
The function k takes its minimal value k(t 0 ) = (1 − q 2 1 q −2 3 ) 1 2 at the point t 0 = q 1 q −2 3 . Proof. Let N = L 3 + rB * B. By Lemma 9.1 we have where q 3 = q 2 + r B * B L(X 1 ,X * 1 ) . Denote by J the Riesz operator J ∈ L(X * 1 , X 1 ) that is defined as follows (Jg, h) It is obvious that the problem (9.39) is equivalent to finding a fixed point u = U t (u), where U t : By (9.53)-(9.55) we have where k(t) is defined by (9.52) and k(t) < 1, if t ∈ (0, 2q 1 q −2 3 ). Therefore the mapping U t is a contraction, and the existence of a unique solution u of problem (9.39) and estimate (9.51) follow from the fixed point theorem (see e.q. [21]). . Remark 9.1. Equalities (9.49) and (9.50) imply that q 3 → ∞ as λ → 0. Therefore 2 the minimal value of k(t) tends to unit as λ → 0, and the iterative method (9.45) provides slow convergence at small value of λ. The reason of this is that the differentiable functional Y λ tends to nondifferentiable functional Y (see (8.8), (8.9)) as λ → 0. 9.4. Solving the problem (5.8)-(5.10). For the case that the conditions of Theorem 5.2 are satisfied, the operators ∂J λ ∂h and L are strictly monotone and Lipschitz continuous. Therefore the algorithm of the augmented Lagrangian (see (9.22)-(9.24)) can be used for the solution of the problem (5.8)-(5.10), and the corresponding nonlinear systems can be solved by the Birger-Kachanov method and by the contraction method.
Let us consider the general case that the conditions of Theorem 5.2 are not satisfied.
be bases in the spaces X m and N m , respectively. Let also k = k 1 (m) + k 2 (m). Define a mapping M : R k → R k as follows: The problem (9.59) is equivalent to the following one: In the case that the functions ψ and b are continuously differentiable the functional Φ 2 is continuously differentiable in R k , and gradient method can be applied for calculation of a solution of the problem (9.61). Derivative of the mapping v → ∂J λ ∂h (v, v) + L(v) is defined in 5.2 (see (5.43)-(5.48)). 10. Stationary problem with consideration for the inertia forces.
10.1. Basic equations and auxiliary results. The equations of motion with regard for the inertia forces read as follows: The condition of incompressibility is div u = 0. (10.2) We assume that velocities are specified on the boundary S of Ω, i.e. We assume also (C7): Ω is a bounded domain in R n , n = 2 or 3. The boundary S of Ω belongs to the class C 2 and consists of l connected components Γ 1 , . . . , Γ l (l ≥ 1), and suppose thatû 4) and the function ϕ is defined by (2.12).
The following lemma follows from the known results (see e.g. [11,23]). We set Obviously Therefore, the trilinear form q is continuous in H 1 (Ω) n × H 1 (Ω) n × H 1 (Ω) n . We consider the following spaces X = H 1 0 (Ω) n with the norm · X = · X , (10.9) V = {w ∈ X , div w = 0} with the norm · V = · X , (10.10) , Ω w dx = 0} with the norm · N = · L 2 (Ω). (10.11) It is easy to verify that q(z, w, h) = −q(z, h, w), z ∈ V, w, h ∈ H 1 (Ω) n , n = 2 or 3, q(z, h, h) = 0. (10.12) Define a trilinear form q 1 as follows: It is evident that 14) Then, for an arbitrary fixed h ∈ H 1 0 (Ω) n the following relations hold In this case if v ∈ V, then Application Green's formula gives and by analogy with the stated above we obtain Proof. By Green's formula we obtain It follows from here that It follows from (4.7) and (10.5) that y ∈ X * . The left-hand side of (10.28) belongs to the polar set V • = {f ∈ X * , (f, w) = 0, w ∈ V}.
Therefore, there exists a function p ∈ N such that the pair (v, p) is a solution of the following problem: (v, p) ∈ X × N , Proof. Define an operator M : X → X * by (M (g), w) = ∂J λ ∂h (g, g), w + (L(g), w) + q 1 (g, q, w) + q 1 (ũ, g, w) + q 1 (g,ũ, w), g, w ∈ X . For an arbitrary f ∈ X * we denote by Gf the restriction of f to X m . Then Gf ∈ X * m , and by (10.43) we obtain 11. Stationary problem with consideration for the inertia forces. Mixed problem.
11.1. Formulation of the problem and an existence result. As before we consider that S 1 and S 2 are open subsets of the boundary S of Ω such that S 1 is non-empty, S 1 ∩ S 2 = ∅ and S 1 ∪ S 2 = S. We study the problem on searching for a pair of functions (u, p) which satisfy the motion equations (10.1), the condition of incompressibility (10.2) and the mixed boundary conditions, wherein velocities are specified on S 1 and surface forces are given on S 2 , i.e. It is obvious that in the special case that S 2 is an empty set this problem transforms into the problem considered in Section 10. We assume that ϕ is defined by (2.12) andû ∈ H 1 2 (S 1 ). Then there exists a functionũ satisfying (4.6).
Remark 11.1. In the general case we can consider that the density of an electrorheological fluid depends on the module of the vector of electric field strength, i.e. ρ = ρ(|E|), and ρ 2 ≥ ρ(y) ≥ ρ 1 , y ∈ R + , (11.31) where ρ 1 , ρ 2 are positive constants. It is easy to see that all results of Sections 10 and 11 still stand valid in the case that ρ is a function satisfying the condition (11.31).