Solvability of the Non-Linearly Viscous Polymer Solutions Motion Model.

In this paper we consider the initial-boundary value problem describing the motion of weakly concentrated aqueous polymer solutions. The model involves the regularized Jaumann's derivative in the rheological relation. Also this model is considered with non-linear viscosity. On the basis of the topological approximation approach to the study of hydrodynamics problems the existence of weak solutions is proved. Also we consider an optimal feedback control problem for this initial-boundary value problem. The existence of an optimal solution minimizing a given performance functional is proved.

1. Introduction

In the fluid dynamics theory the motion of an incompressible fluid with a constant density can be described by the equations [1]: where is an unknown velocity field of the fluid; is an unknown pressure; is the external force; is an unknown deviator of the stress tensor. The divergence of the tensor is the vector with coordinates

System (1) and (2) describes the motion of all kinds of incompressible fluids. However, it is incomplete. As a rule, the additional relation between the deviator of the stress tensor and the strain rate tensor , . Such relations are known as constitutive or rheological laws. Choosing a rheological relation we specify a type of fluid (see [2]). Note that this relation should corresponds to the general requirements for a mathematical model. The main of which are the maximum proximity of the results obtained by this relation to the real fluid characteristics at the maximum simplicity of the relation itself.

This paper is devoted to the viscoelastic fluids. The rheological relation for such type of fluids is the following (see [3,4,5,6]): Here is the fluid viscosity and is the retardation time.

The rheological relation of viscoelastic fluids always contain the time parameter. From the mathematical point of view, it is possible to divide them into two groups: differential (coupling the instantaneous stress values with velocity gradient of the fluid) and integral (reflecting the dependence of the fluid stress from the prehistory of flow). In this paper the first group of rheological relations are considered.

Note that the relation (3) contains time derivative of the strain rate tensor. Mathematical studies of this model started with consideration in rheological relation (3) the partial derivative. This mathematical model (by analogy for the solid body model) received the name: Voigt model and have been studied in detail (see, for example [7,8]). Then one began consider the relation (3) with the total derivative. This model received the name of the Kelvin–Voigt model. The mathematical investigation of an initial–boundary value problem for this case is consider in many papers [9,10,11,12] and the solubility of the stationary case of the problem under consideration is proved in [13,14]. The next investigations of this model are connected with consideration in the relation (3) the objective derivative [15]. This leads to the fact that this rheological relation does not depend on the observer, i.e., this relation does not change under Galilean change of variables. The most general view of the objective derivative has the regularized Jaumann’s derivative (see [16]): where is a smooth function with compact support such that and for x and y with the same Euclidean norm; , is the vorticity tensor. Note that the rheological law (3) with the regularized Jaumann’s derivative is similar to a particular case of second grade fluids (e.g., see [17,18,19] and the bibliography therein). The weak solvability for this model is proved in [20]. Trajectory, global and pullback attractors for this model are considered in [21]. And finally an optimal feedback control problem for this model is investigated in [22].

At the same time according to Stokes’ hypothesis the stress tensor at a point at a given time is completely determined by the strain rate at the same point and at the same time. However, this relationship does not imply any restrictions associated with linearity, but it is believed that deformation occurring at some other point or at some other point in time prior to the considered one does not affect the value of stresses. The latter circumstance is taken into account in models of nonlinear viscoelastic media. The study of models with nonlinear viscosity, on the one hand, makes it possible to significantly expand the class of studied media, on the other hand, it significantly complicates the mathematical research of such initial–boundary value problems (due to the complexity of the problem). Note that many functions of nonlinear viscosity have been proposed in the literature. At this paper we will consider some of the natural viscosity constraints for real fluids proposed by V.G. Litvinov [23]: where the tensor is defined by the relation . Here we use the notation for arbitrary square matrices and of the same order.

Professor V.G. Litvinov gave examples of such fluids and natural restrictions on the viscosity of the fluid under consideration expressed via the properties of the function . This function must be continuously differentiable and satisfy the inequalities

Hereinafter, denotes various constants. Conditions have a clear physical meaning. Condition is connected with the existence of limit “Newtonian” viscosities for real fluids;  express the law that the shift stresses grow together with the deformation rates. Similar mathematical models with nonlinear viscosity have been considered in many papers (see, for example, Waele–Ostwald model, Norton–Hoff model, Sisko model et al.). In general, many types of function have been proposed in the literature, but most of them have been applied to the study of one–dimensional models. In the paper [23] it is shown that the given restrictions on the function are natural for real fluids and that the numerical results for this model is very close to the results of experimental studies.

Substituting the right–hand of (3) with nonlinear viscosity (5) and with the regularized Jaumann’s derivative (4) for in Equations (1) and (2), we obtain

For the system (6) and (7) we consider the initial–boundary value problem with the initial condition and the boundary condition

The obtained with such rheological relation mathematical model have to satisfy with the experimental data. The experimental data for this mathematical model have been also obtained. Obviously, if a small amount of polymer is added to the water, then the viscosity and the density of the resulting solution practically does not change and remain constant (in contrast to its rheological properties). It is fixed the reduction of friction resistance due to polymer additives. In such fluids the equilibrium state is not established immediately after a change in external conditions. It is established with some delay, which is characterized by the value of the retardation time. This delay explained by the processes of internal rearrangement. A group of scientists carried out experiments and proved that these mathematical model describes the flow of weakly concentrated water solutions of polymers, for example, solutions of polyethyleneoxide and polyacrylamid or solutions of polyacrylamide and guar gum [24,25]. Therefore, the model considered in this paper is also often called the model of aqueous polymers solutions motion.

Our aim is to investigate the weak solvability of this initial–boundary value problem (6)–(9) describing the motion of weakly concentrated aqueous polymer solutions with non-linear viscosity. Also we consider the existence of a feedback control problem for this model and prove the existence of an optimal solution of the problem under consideration minimizing a given bounded performance functional.

2. Preliminaries and Main Results

At the beginning we introduce some basic notations and auxiliary assertions.

Denote by the space of smooth functions with compact supports in and values in . Let be the set and let and be the closures of in and , respectively. We also use the space

We introduce a scale of space (see [26]). To do this, cosider the Leray projection and the operator defined on . This operator can be extended to a closed self–adjoint operator in (we denote the extension by the same letter). The extended operator A is positive and has a compact inverse. Let be the eigenvalues of By Hilbert’s theorem on the spectral decomposition of compact operators, the eigenfunctions of A form an orthonormal basis in . Put for the set of finite linear combinations of the vectors and define by the completion of with respect to the norm

Note that this norm for is equivalent to the norm of space (see [26]) and in the case of equals 0,1,3 is equivalent to following norms

By we denote the value of a functional on a function . We will need following two functional spaces and :

Weak solutions of the original initial–boundary value problem will be belong to the space and weak solutions of the approximation problem will be belong to the space . Now we ready to define weak solutions to the problem (6)–(9). Assume that and

A function v from the space and the initial condition

Our main result provides existence of weak solutions.

Let

Proof of this Theorem is based on the topological approximation approach used for studying mathematical problems of hydrodynamics (see [27,28]). First, we introduce a family of auxiliary problems which depend on a small parameter , obtain a priori estimates for solutions, and on the base of the theory of topological degree for maps of the Leray–Schauder type prove the existence of weak solutions to the auxiliary problem. Then, we pass to the limit using appropriate estimates.

3. Approximating Problem

Assume that external force and initial condition We consider the following auxiliary problem: find a function satisfies the initial condition , such that for any and a.a. the equality holds

Let us first give an operator statement of the problem under consideration. Consider the following operators:

Since is arbitrary in (3), this equality is equivalent to the following operator equation:

Thus a weak solution of the approximating problem is a solution of operator Equation (11) satisfying the initial condition .

We also define the following operators:

Thus our auxiliary problem can be rewritten in the following way: find a function satisfying the following operator equation:

Now we need following properties of the operators

The function

For any function

For any function Moreover, the inverse operator

For any function

For any function

For any function

Proofs of Lemmas 1–6 can be found, for example, in [10].

The operator

The operator

For any function

For any function

(1) We start by estimating and . Therefore, .

By definition, for any we have

This yields the estimate (17).

Now prove that the operator continuous. For any we have: Thus we get

Let the sequence converge to some limiting function Then the continuity of the mapping follows from the previous inequality.

(2) Let Then from (17) for almost all we get the estimate

Squaring this estimate and integrating with respect to t from 0 to T, we get This yields

Now prove that the continuity of the mapping

Let the sequence converge to some limit Square the inequality (19) and integrate with respect to t from 0 to Using the Hölder inequality, we obtain

We get that the left–hand side tends to zero. So we prove that is continuous.

(3) Finally, to prove (3) part we use the following Theorem

(Simon, [29]).  Let

the set F be bounded in

the set

Then the set F is relatively compact in

In our case let We have the compact embedding . So why F is embedded into compactly. Also we have that gives us that . Finally, we have Here the first embedding is continuous, the second embedding is compact and the mapping is continuous. Thus, for any function we see that the function and the mapping is compact.

Now prove the estimate (18). By (17), the estimate holds for all . Squaring it and integrating with respect to t from 0 to T, we get This yields the required estimate (18). □

If the function μ satisfies conditions with constants

The proof of this Lemma can be found in [30].

The operators L and K have the following properties

The operator

The operator

(1) To prove that the operator L is invertible it is sufficient to use Theorem 1.1 from [31] (Chapter 4). Since is continuous and monotone then all conditions of this Theorem 1.1 are hold. Applying this Theorem 1.1 shows that for each there exists solution and, hence, . Thus, the operator L is inverse.

(2) The complete continuity of the operator follows from the compactness of the operators Lemma 1; Lemma 5; Lemma 6; Lemma 6; Lemma 7. □

4. A Priori Estimate

Along with Equation (12) consider the following family of operator equations: which coincides with the Equation (12) for

If where

Let be a solution of (21). Then for any and almost all the following equation holds:

Note that

Then (24) can be rewritten in the form

Since the last equation holds for all it holds for as well:

We reduce the terms on the left-hand side of the Equation (25) in the following way: Here we take into account that the strain-rate tensor is symmetric and tensors and are skew-symmetric. Hence the Equation (25) can be rewritten in the following form:

Estimating of the right-hand side of the last equation from above as follows and the left-hand side from below as follows we see that

Integrating the last inequality with respect to t from 0 to where we obtain

The right-hand side of the last inequality can be estimated in the following way:

Here we used the Hölder inequality and the Cauchy inequality: for Thus, we get

Taking into account that we have

Hence, since and are positive we get the estimates

The right-hand side of the last two inequalities does not depend on Therefore, we can take the maximum with respect to on the left-hand side:

This proves (22) and (23). □

If where

Let be a solution of the problem (21). Then it satisfies the following operator equation

Therefore,

The right-hand side of the inequality can be estimated in the following way: using the continuous embedding and estimates (15), (16), (18) and (20) we have in view of a priori estimate (23)

Now using the estimate (13): on the left-hand side of (29), we get the required inequality (26).

The estimate (27) is obtained in the following way. As above, v satisfies the Equation (28), and, therefore,

Thus, we get the estimate

From this inequality and from estimate (14):

(27) follows. □

Theorems 3 and 4 imply the following Theorem:

If v is a solution of the operator Equation

Let

The prove of this Theorem is based on the Leray–Schauder topological degree theory for compact vector fields. If we consider the ball of radius then all solutions of the family of Equation (21) will be in this ball by virtue of a priori estimate (30).

We have the continuous operator and compact mapping (see Lemma 9). Thus, the mapping is compact in (jointly in and v). In other word we have the compact vector field

Therefore, the Leray–Schauder topological degree is defined. By the homotopy invariance and normalization condition of the degree we get

So why, the equation and, therefore, the Equation (12), and, therefore, the auxiliary problem, have a solution . □

Since the space is dense in there is a sequence converging to If then put But if then if m is sufficiently large. Then we set

Then the sequence converges to zero as The following estimate holds:

By Theorem 6 for any and there exists a weak solution of the approximation problem. Thus, each satisfies the equation and the initial condition

Since the sequence converges in it is bouned in the norm of Therefore, where is a constant which doesn’t depend on

Recall that the constant from the inequality (23) depends on

Hence, with the help of (34), it can be estimated in the following way:

Thus, in view of inequalities (31) and (35) from (23) we get that

Similarly from (31) and (35) with the help of inequalities (26) and (27) we get:

Since the embeddings are continuous, it follows from (36) and (38) that without loss of generality (passing to a subsequence if needed) we have

Then by definition of weak convergence as .

In view of the estimate (37) we have that converges weakly in . On the other hand, for any we have Since weakly converges to in (and, therefore, converges to in the sense of distributions) the latter expression equals

Thus, by uniqueness of the weak limit we get

Using Theorem 2 we have the compact embedding

Hence, taking into account estimates (36) and (38), we obtain

Thus, we get

For the remaining integrals we have as . Indeed, here the sequence converges to strongly in and converges to weakly in Thus, their product converges to the product of their limits.

In the last term we have

As above we get that strongly in and weakly in we obtain

Pass to limit in (32) as . We get a function satisfying

As we have a strong convergence (39) then we get that this obtained function satisfies the initial condition . So, we prove Theorem 1. □

7. Optimal Feedback Control Problem

In this section based on the topological approximation approach to mathematical hydrodynamics problems we prove the existence of an optimal feedback control for the (6)–(9) problem. First, we formulate the notion of a solution to the problem under consideration and the main result of this section.

Consider the multi-valued mapping as a control function. We will assume that satisfies the following conditions:

The mapping is defined on the space and has non-empty, compact, convex values;

The mapping is upper semicontinuous and compact;

The mapping is globally bounded, that is, there exists a constant such that

is weakly closed in the following sense: if and in then .

For completeness, we give an example of such a multi-valued mapping. Let continuous mappings satisfy the following conditions:

is globally bounded and makes a bounded set relatively compact;

—weakly closed, i.e., follows

We define a multimap with feedback as:

It is easy to see that U satisfies all the conditions of the multi-valued mapping

We will consider a weak formulation of the optimal feedback control problem for the initial–boundary value problem (6)–(9). By feedback, we mean the following condition:

We will assume that the initial condition belongs to the space

A pair of functions as well as the initial condition

The first result of this section is the following Theorem:

Let the mapping Ψ satisfy the conditions

Denote by the set of all weak solutions of the problem (6)–(9), (40). Consider an arbitrary functional satisfying the following conditions:

There exists a number such that for all

If in and in then

As an example of such a quality functional, consider: Here U and F are given speed and external force.

The main result of this section is the following Theorem.

If the mapping Ψ satisfies the conditions

To prove these Theorems we at first consider the auxiliary problem with some small parameter : We need to find a pair of functions satisfying for any and almost all the feedback condition (40), identity and the initial condition

Using the operator treatment (12) we can reformulate our auxiliary problem in the operator form. Thus, the problem of the existence of a feedback control for the approximation problem is equivalent to the problem of the existence of a solution satisfying the initial condition (44) of the following operator inclusion:

Let’s introduce the following operator: ; Then the problem of the existence of a solution of the approximation problem is equivalent to the problem of the existence of a solution for next inclusion

Since the operator is linear and continuous, and the operator K is compact, using the conditions – we obtain that the multi-valued mapping is compact and has non-empty, convex and compact values.

Consider also the following family of inclusions where .

Note that the left side of the operator inclusion

If v is a solution

The operator inclusion (

To prove this Theorem we use topological degree theory for multivalued vector fields (see, for example, [32]). By virtue of the a priori estimate (47), all solutions of the family of operator inclusions (46) lie in the ball of radius centered at zero. Hence for all . Using the degree homotopy invariance property and the degree normalization property, we obtain

Since this degree is nonzero, there exists at least one solution of the operator inclusion (45). □

Since there exists a solution of the inclusion (45), it follows from the above reasoning that the approximation problem has at least one solution

Using the results of Theorem 10, we completely repeat the proof of Theorem 1 with a small change related with the right-hand side. Taking into account the a priori estimates (36), (38) and the conditions , we can assume without loss of generality that there exists such that as . From this we obtain that there exists and satisfying (40), (2) and (42) which completes the proof of the Theorem 7.

From the Theorem 7 we get that the solution set is not empty. Therefore, there exists a minimizing sequence such that

As before, using the estimate (47), without loss of generality and passing to a subsequence if necessary, we can assume that *-weakly in ; is strong in ; is weak in ; is strong in for .

Whence, just as in the previous proof, we get weakly in ; weakly in ; strongly in ; weakly in ; weakly in ; weakly in for .

Passing to the limit in the relation we get that . Since the functional is lower semicontinuous with respect to the weak topology, we have which proves that is the required solution. This completes the proof of Theorem 8.

8. Conclusions

To summarize all calculations and proofs in this paper the mathematical model describing the motion of weakly concentrated water polymer solutions is investigated. This model contained the objective Jaumann derivative in the reological relation. Also this model is considered in the case on non-linear viscosity.

The main result of this paper is the solutions existence to initial–boundary value problem and to the feedback control problem for the mathematical model under consideration. Also the existence of an optimal solution to the problem under consideration that gives a minimum to a given bounded quality functional is proved. Results of this paper provide an opportunity for the future investigation of this model. Author proposes the following future research directions for the model under consideration—(1) the numerical analysis of the obtained solutions; (2) the consideration of a turbulence case of this problem; (3) the investigation of alpha-models for this problem and so forth.