Abstract

We investigate a problem for a model of a non-Newtonian micropolar fluid coupled system. The problem has been considered in a bounded, smooth domain of with Dirichlet boundary conditions. The operator stress tensor is given by . To prove the existence of weak solutions we use the method of Faedo-Galerkin and compactness arguments. Uniqueness and periodicity of solutions are also considered.

1. Introduction

Let be a bounded domain in with smooth boundary , and let . We denote by the time space cylinder , with lateral boundary , where is a time interval. The unsteady flows of incompressible fluids in a boundary domain , , are described by the system of equations where is the velocity, represents the pressure, is a positive constant determining the density of a material, stands for the given external body forces, denotes the extra stress tensor, denotes the symmetric part of the velocity gradient; that is, whose components are defined as in [1] by and represents the set of all symmetric matrices; that is, Note, for example, that when is of the form with , problem (1) turns into the Navier-Stokes system, which is a model for Newtonian fluids. In the expression (5), denotes the usual Euclidean matrix norm. We observe that (5) can be written in the form where , is the generalized viscosity function. Fluids constituted by (6) are sometimes named fluids with shear-dependent viscosity. Models belonging to this class of non-Newtonian fluid mechanics are frequently used in several fields of chemistry, glaciology, biology, and geology, as discussed by Malek et al. [2].

The first mathematical investigations of problem (1) was done by Ladyzhenskaya in , where she proposed to study system (1) with (5) and . Combining monotone operator theory and compactness arguments, she proved the existence of weak solution to model (1), if , and their uniqueness if . See also Lions [3] for another proof of the same results. More results are known about problem (1) obtained in a series of papers, including those of Malek et al. [2], Malek et al. [4], Frehse and Málek [5], Malek et al. [1], and other mathematicians.

The equations below describe the motion of Newtonian micropolar fluids: where and , denoting for , respectively, the unknown velocity, the microrotational velocity, and the hydrostatic pressure of the fluid and is a positive constant. The positive constants and are, respectively, the Newtonian and microrotational viscosity. The positive constants , , and are called coefficients of angular viscosities and satisfy .

The main difference with respect to modeled fluids by the Navier-Stokes is that the rotation of the particles is taken into account. The above approach was introduced by Eringen [6]. The nonlinear coupled system (7) can be used to model the behavior of liquid crystals, polymeric fluids, and blood under some circumstances (see, e.g., [7]). These systems have been mainly analyzed in the book of Lukaszewicz [8].

The problem that we study in this work consists in supposing that in system (7) the fluid is of the type (5). More precisely, we investigate the mixed problem: let be a bounded domain in with smooth boundary , and let . We denote by the time space cylinder , with lateral boundary , where is a time interval. We find that and solving the following system of equations: where the extra stress tensor is given by , as in (2) and (3), , , , and are positives constants, , and is given by the same holds for . Let us consider satisfying the hypothesis where , , and are positive constants. We observe that if is a constant function, then problem (8) reduces to problem (7).

2. Notation and Main Results

In order to solve problem (8) we need some notations about Sobolev spaces. We use standard notation of , , and for functions that are defined on and range in and the notation of , , and for functions that range in . We also work with the spaces or .

By we will represent the duality pairing between and , with   being the topological dual of the space . We Also define the followings spaces: is the closure of in the space , . In particular, . The norm of gradient in is given by The inner product and norm in is given, respectively, by is the closure of in the space , with inner product and norm defined, respectively, by

Remark 1. and are Hilbert’s spaces. We note that , where the first embeddings are compact.

We introduce the following bilinear and the trilinear forms, as well as the convention of summation of indices, that is, instead of : We note that (see Lions [3]) We also introduce the notations According to this, we have

Remark 2. We observe that implies for all that Therefore is a monotonous operator.

Definition 3. Let , , and . A weak solution to (8) is a pair of functions , such that satisfying the following identity:

Lemma 4 (Korn’s inequality). Let . Then, there exists a constant , such that the inequality is fulfilled for all satisfying either , where is open and bounded with .

Proof. The proof of this lemma can be found in [1], page 169.

Lemma 5. Let , , and and the assumptions below are satisfied for all and : Then, there exist positive constants , , such that

Proof. The proof of this lemma can be found in [4], page 263.

Lemma 6 (Vitali). Let be a bounded domain in and integrable for every . Assume that (1) exists and is finite for almost all ;(2)for every there exists such that then

Proof. The proof of this lemma can be found in [9], page 63.

Lemma 7. Consider and , with , , verifying . If , then for all , where is a constant independent of , , and .

Proof. The proof of this lemma can be found in [3], page 84.

Theorem 8. If , , , , and , then there exist a weak solution to problem (8).

Theorem 9. Under the assumptions of Theorem 8 with , problem (8) has a unique weak solution.

Theorem 10 (periodic solutions). Under the assumptions of Theorem 8 there exist a pair of functions such that

Theorem 11. Assuming that , , , , and there exist a unique weak solution to problem (8) such that

3. Proofs of the Results

Proof of Theorem 8. We will show the existence of a weak solution to system (8) employing the Galerkin approximations. For that purpose we consider , a basis of eigenvectors of the Stokes operator and a basis of eigenvectors of Lamé. We represent by the subspace generated by and the subspace generated by . Let us also consider the pair , such that are the solution of the approximate problem The system of ordinary differential equations (37) has a local solution on an interval , . The first estimate permits us to extend this solution to the whole interval .
First Estimate. We sometimes omit the parameter . Multiplying both sides of (37) by and (37) by , next adding from to , we obtain because , for all , for all (see Lions [3]), and (see Lukaszewicz [8]). Now using Young’s inequality we obtain from (38) and (39), respectively: From (27) (Korn’s inequality) and (11) we can get Adding inequalities (40) and (41) and integrating from to , with , we conclude By using Gronwall’s inequality, we can write Therefore, it follows from (43) that
Second Estimate. We consider as the orthogonal projections from to : We also consider the adjoint operator to which is . We note that . By the choice of the special basis , we obtain It follows from (37), (21), (22), and (23) that We have , for all . Therefore (47) implies Let . From (20), Hölder’s inequality, and (11) we take Therefore, from (47), we obtain From we derive . Using (17) and Hölder’s inequality we conclude for all . Therefore, from (47) On the other hand, let , and (see Lukaszewicz [8], pp. 116). It follows from (49) that It follows from (53)–(59), (51), and hypothesis about that
Analogously let be the orthogonal projections We also consider the adjoint operator to , which is . We have and by the choice of the special basis , we can get From (37), (21), (22), and (23) We note that , for all . Thus, (49) implies Analogously and by using the embedding we obtain On the other hand , for all . Now, by using (47), we have Finally assuming that , it follows from (18), (22), and Hölder’s inequality that for all . Therefore, .
Because if , then, (see Lions [3]). Now using Young’s inequality we get Therefore, (47)–(49) and (45) permit us to obtain Analogously and assuming that we obtain from Lemma 7, because . Thus, Therefore, (47)–(49) imply that It follows from (64)–(72) and hypothesis about that
We note that (45)–(49), (60), (73), and the Aubin-Lions lemma imply that there exist subsequences of and , still denoted by and , such that We note that (46) and (60) imply that . Similarly, (49) and (73) imply that . Thus, it does make sense to consider and .
In order to prove that we use (74) and (78) (see [1], pp. 210). Now, we note that or equivalently results from (80). The other terms of (37) are obtained in the usual manner. In order to prove that we use the fact a.e. in , (see [5] pp. 565-566). Therefore, that is, Since we obtain from (88) Thus, a.e. in and for all . Using (46) and (11) we obtain It follows that Moreover, if is a measurable set, we have from (11), (46), and Hölder’s inequality that Therefore, Assuming that is sufficiently small, we obtain for all . Now using (90), (92), (95), and Vitali’s lemma we can derive (86). Therefore, we can write in . The convergences (74)–(82) and (85) and (86) allow us to pass the limit on system (37), with and being fixed to obtain
This concludes the proof of Theorem 8.