#### Abstract

The aim of this paper is to present the result about asymptotic approximation of the micropolar fluid flow through a thin (or long) straight pipe with variable cross section. We assume that the flow is governed by the prescribed pressure drop between pipe's ends. Such model has relevance to some important industrial and engineering applications. The asymptotic behavior of the flow is investigated via rigorous asymptotic analysis with respect to the small parameter, being the ratio between pipe's thickness and its length. In the case of circular pipe, we obtain the explicit formulae for the approximation showing explicitly the effects of microstructure on the flow. We prove the corresponding error estimate justifying the obtained asymptotic model.

#### 1. Introduction

The Navier-Stokes model of classical hydrodynamics has a drastic limitation: it does not take into account the microstructure of the fluid. One of the best-established theories of fluids with microstructure is the theory of micropolar fluids, introduced by Eringen [1]. The mathematical model of micropolar fluid enables us to study many physical phenomenae arising from the local structure and micromotions of the fluid particles. It describes the behavior of numerous real fluids (such as polymeric suspensions, liquid crystals, muddy fluids, and animal blood) better than the classical Navier-Stokes model, especially when the characteristic dimensions of the flow (e.g., diameter of the pipe) become small. Due to its importance in industrial and engineering applications, there are large number of papers on micropolar fluid flow, mostly in the engineering literature (see, e.g., [2–7]). The monograph [8] provides a unified picture of the mathematical theory underlying the applications of this particular model. We would also like to point out two recent papers of Dupuy et al. [9, 10] in which the authors rigorously derive asymptotic models for two-dimensional micropolar flow through a periodically constricted tube and a thin curvilinear channel. It is important to emphasize that 2D setting (in which the microrotation is a scalar function) has often been employed, especially in blood motion modeling. However, in the present paper, our aim is to study 3D flow describing the real physical situation.

We consider one important application of micropolar fluids: laminar flow in a straight pipe with variable cross section. We suppose that the flow is stationary and governed by the prescribed pressure drop between pipe's ends. It is well-known that the stationary Navier-Stokes system describing the viscous flow in straight pipe with impermeable walls governed by the prescribed pressure drop has a solution in the form of the Poiseuille flow, which in case of pipe with constant circular cross section reads However, Poiseuille formula gives an exact solution only in case of laminar flow of Newtonian fluid through a pipe with constant cross section. If the pipe has a variable cross section or it is curved, one can only derive the approximation of the solution by a singular perturbation techniques (see, e.g., [11–14]). Here we deal with the micropolar fluid model (representing the generalization of the Navier-Stokes model) which introduces a new vector field, the angular velocity field of rotation of particles (microrotation). Correspondingly, one new vector equation is added to Navier-Stokes system, expressing the conservation of the angular momentum. Naturally, one cannot hope to obtain the exact solution of such (coupled) system of equations so our goal is to derive an asymptotic approximation of the solution and evaluate the difference between the exact solution of the governing problem (which we cannot find) and the asymptotic one. Generally, there are several methods that enables us to find the asymptotic behavior of the flow. By taking the average over the cross section of the pipe, we can obtain simple one-dimensional approximation, based on the assumption that, in case of very thin (or very long) pipe, the variations of the solution on the cross section are of no relevance for the global flow. However, obtained approximation would have low order of accuracy and gives no information about flow profile in the pipe. Another approach, which we use here, is based on the rigorous asymptotic analysis with respect to the small parameter , introduced as the ratio between pipe's thickness and its length. It relies on two-scale asymptotic expansions in powers of small parameter which, in our case, have the form The variable is directed along the pipe, while describes the cross section. The role of dilated (fast) variable is to capture the fast changes of the solution on the pipe's cross section. Plugging the above expansions in the governing system and collecting the terms with equal powers of , lead us to the recursive sequence of linear problems. Assuming that the pipe's cross section is circular (which is the most common case in real-life situations), we are in position to solve those problems explicitly and to clearly observe the influence of the microstructure on the effective flow. The main difficulty arises from the fact that the governing system is coupled so we have to simultaneously solve boundary-value problems for velocity and for microrotation. Furthermore, in some thin layer in the vicinity of pipe's ends we have some influence of the boundary condition for the microrotation which cannot be captured by the formal (interior) expansion, so we have to construct the appropriate boundary-layer correctors to fix our approximation.

The paper is organized as follows: in Section 2, we describe the geometry of our three dimensional domain and present the governing system of equations describing the fluid motion. After discussing its solvability, in Section 3, we write the problem in rescaled domain (independent of small parameter ) and construct an asymptotic expansion of the solution in terms of the pipe's thickness. The last section is devoted to rigorous justification of the derived asymptotic model. After deriving some *a priori* bounds for the original solution, we prove the error estimates in the appropriate norm. It turns out that our asymptotic solution approximates the flow with an error of order for the velocity and with an error of order for the microrotation.

#### 2. Position of the Problem

##### 2.1. The Geometry

In order to describe the thin pipe with a small parameter appearing explicitly, we first introduce where the family of bounded domains is chosen such that is locally Lipschitz. Now, we define our thin pipe with variable cross section and length by We are particularly interested in the case when the pipe has circular cross section, that is, when with being a strictly positive bounded function defined on . Finally, we denote the ends of the pipe by , , while its lateral boundary is given by

##### 2.2. The Governing Equations

The governing system of equations expresses the balance of momentum, mass, and angular momentum, which in stationary regime reads The unknown functions are , and standing for the velocity, the microrotation and the pressure of the fluid, respectively. The fields , represent given external forces and moments, respectievly and we assume . Viscosity coefficients read , , , , , where , , , , are the given positive constants ( is the usual Newtonian viscosity, is microrotation viscosity, , , are the coefficients of angular viscosities). Observe that if we put to be equal zero, then the system becomes decoupled and (2.5)-(2.6) reduce to classical Navier-Stokes equations. We refer the reader to [8] for a rigorous derivation of the above system from general conservation laws.

We complete the system (2.5)–(2.7) with the following boundary conditions where denotes the standard Cartesian basis.

*Remark 2.1. *By prescribing constant pressures , on , we assure that the fluid flow is governed by a pressure drop between pipe's ends. Condition (2.8) is the classical no-slip boundary condition for the velocity. Imposing that the tangential component of the velocity equals to zero is not a serious restriction since the only part that counts is the normal part, due to the Saint-Venant's principle for thin domains (see, e.g., [11]). The boundary conditions for the velocity and pressure as in (2.8), (2.9) are physically clear and justified. On the other hand, there exists no general agreement about the type of the boundary condition one should set for microrotation. The most commonly used throughout the literature is the one as in (2.10), although we can also find other types of boundary conditions (see, e.g., [15, 16]). Nevertheless, it must be emphasized that not much has been done in proving the well-posedeness of the corresponding boundary-value problems, except in the case of the classical Dirichlet condition (2.10).

In [8, Chapter 2, pages 60–69], the homogeneous Dirichlet boundary-value problem for an incompressible micropolar fluid is considered, with velocity prescribed on the whole boundary. Using fixed-point argument, the existence of its weak solution is proved (Theorem 1.1.1). Furthermore, such solution is shown to be unique if the viscosity is large enough (Theorem 1.1.2). In our setting (2.5)–(2.10), the only difference is that we prescribe the value of pressures at pipe's ends in order to consider the situation naturally arising in the applications. Pressure boundary condition (2.9)_{2} should be considered in view of the corresponding variational formulation: find in , on , on , , , such that
for any . As we can see, the nonlinear term in (2.11) does not vanish, causing the absence of the energy equality. Such technical difficulty can be elegantly overcome by prescribing dynamic (Bernoulli) pressure (which has no physical justification in the case of viscous fluid), or by restricting to the case of small boundary data. Indeed, from (2.11), it follows that we do not actually impose the value of the pressure at pipe's ends , but only the pressure drop . Following the approach first proposed in [17] (see also [18] for details) and supposing that the pressure drop is reasonably small, one can easily adapt the proof of Theorems and 1.1.2 from [8] to our situation and prove that the velocity is unique in some ball , with radius remaining bounded as . This fact is crucial for proving the a priori estimate for the velocity since it enable us to control the inertial term in (2.5) (see Section 4, Proposition 4.2).

#### 3. Asymptotic Analysis

##### 3.1. Rescaling of the Domain

Our main goal is to find the asymptotic behavior of the flow, as the thickness . To accomplish that, we first need to rescale the domain, that is, to write the governing problem on instead of . Introducing the new functions we can write the equations (2.5)–(2.7) in the following form: Here and in the sequel, we use the following notations for the formal partial differential operators:

##### 3.2. Asymptotic Expansions

In this section, we construct the formal asymptotic expansion of the solution in powers of small parameter . As mentioned in Introduction, we expand as follows:

###### 3.2.1. First-Order Approximation

Substituting the expansions (3.6) into the rescaled equations (3.2)–(3.4), after collecting the terms with equal powers of , we obtain the following problems for first-order approximation :
Here, we denote , . Notice that the problems for the velocity and the microrotation are, at this stage, decoupled. The system (3.7) can be solved by taking
where and denotes the solution of the auxiliary problem posed on the cross section :
If the pipe has circular cross section (2.3), we can compute explicitly from (3.10):
We still have to determine . The next term in (3.7)_{2} implies
Integration over with respect to yields
Introducing
from (3.9)_{1}, we deduce
It follows
with being an arbitrary constant. Taking into account the pressure boundary condition (2.9)_{2}, we get
Therefore, in the case of circular pipe, we have
Similarly, it can be verified that the problem (3.8) for microrotation will be satisfied for

###### 3.2.2. Correctors

Now, we compute the correctors. The term from momentum equation (3.2) gives The system is not decoupled anymore, so the effects of the microstructure on the fluid velocity occur. Inserting the expressions for and derived for circular pipe, we get the following problem for the first component: Let us introduce , as the solutions of the following two problems posed on : Taking into account (2.3) and using the polar coordinates yield We seek the solution of system (3.22) in the form implying For the other two velocity components from (3.21) and (3.3), we obtain where . Because is not divergence-free, it is not likely that the above system can be explicitly solved. However, it is important to emphasize that, since , such problem admits a unique solution (see Theorem IV.6.1. from [19]).

It remains to construct the corrector for the microrotation. From (3.4), we deduce If we write the above system by the components and take into account (3.19) and (3.20), we get Similarly as for , we obtain The problem satisfied by the other two components , must be solved carefully as a system implying

###### 3.2.3. Boundary Layers for Microrotation

It is important to notice that our approximation was computed to satisfy the governing equations and the boundary condition on , while the boundary conditions on pipe's ends were not taken into account. Consequently, the traces of on may be different from 0. Thus, before proving convergence, we need to correct our interior expansion in the boundary layer near and .

Near , we introduce the boundary layer correctors depending on the dilated variable , as the solutions of the following Dirichlet boundary-value problems posed in the semi-infinite strip : for and . Using the standard techniques (see [19, Chapter XI.4, pages 252–262] or [20, Appendix]), it can be proved that the system (3.33) admits a unique solution which is exponentially decaying to zero as (see, e.g., [19]). Analogously, the boundary layer correctors corresponding to the opposite side are constructed as the unique solutions of the following problems: for , , , , and its exponential decay to zero at infinity follows as well.

###### 3.2.4. Asymptotic Approximation

To conclude this section, let us write the obtained asymptotic approximation. For the microrotation, it has the following form: where and are given by the explicit formulae (3.20) and (3.30)–(3.32), respectively. On the other hand, the approximation for the velocity/pressure reads: The first term in the expansion , given by (3.19) is, in fact, the Poiseuille solution and we do not observe the effects of microstructure here. The Poiseuille flow is, therefore, corrected by a lower-order term which contains those effects (see (3.26) and (3.27)).

#### 4. Error Estimates

In this section, we rigorously justify the obtained asymptotic approximation. The first step is to derive the a priori estimates for the original solution. We start by a technical result.

Lemma 4.1. *There exists a constant , independent of , such that
**
for any , such that on .*

The above estimates can be verified by a simple change of variables (see, e.g., [11, Lemmas 7, 8]).

Proposition 4.2. *Let be the solution of the problem (2.5)–(2.10), then there exists , independent of , such that
*

* Proof. * Multiplying the equation (2.7) by and integrating over gives
First, we deduce
Using Poincaré's inequality (4.1), we get
Applying (4.5)–(4.7) into (4.4) implies
Now, we multiply (2.5) by and, after integrating over , we obtain
Employing the inequality (4.2), we have
Taking into account (4.1), we obtain
From the last assertion, in view of (4.8), we conclude that
Finally, as in (4.7) we get
Inserting the obtained estimates (4.10)–(4.13) in (4.9) yields
Due to pressure boundary condition, the velocity is unique in the ball with radius (see the discussion at the end of Section 2.2.). For that reason it is sufficient to choose such that in order to deduce (4.3)_{1} from (4.14). The estimate (4.3)_{2} follows immediately from (4.8).

The main result of this section can be stated as follows.

Theorem 4.3. *Let , , and be defined by (3.35), (3.36), and (3.37), respectively. Then the following estimates hold:
*

*Remark 4.4. *Since our domain is shrinking, the convergence in the norm would be worthless. Indeed, any -bounded function would converge to zero in such norm. Therefore, we express the error estimates in the rescaled norm , where stands for the Lebesgue measure of .

* Proof. * The function satisfies the following equation:
where , that is, . Now, we introduce
as the difference between the original solution and our asymptotic approximation. Subtracting the equations (2.7) and (4.18) gives
Multiplying the above equation by and integrating over lead to
As in (4.5), we have . Now, we carefully estimate each term on the right-hand side of (4.21)
Taking into account the obtained estimates (4.22), from (4.21), we obtain
The problem satisfied by is the following:
where and , . Before proceeding, it is important to notice that the norm of is not small enough to obtain satisfactory error estimate. Therefore, we need to construct the divergence correction. Since , we define as the solution of the problem
(here is treated only as a parameter). Taking into account (3.26), by a simple integration one can easily verify that , implying that such exists. Now we define our divergence correction as
and put
Such is divergence-free. Moreover, is chosen such that it keeps the estimate for , that is, . Denoting
we have
Now we introduce as the solution of the problem
If we suppose , such problem has at least one solution which satisfies
(see, e.g., Lemma 9 from [11]) Multiplying (4.29)_{1} by and integrating over , we obtain
We estimate the terms on the right-hand side in (4.32) using a priori estimates and Lemma 4.1,
Applying (4.33) into (4.32), we get
Now, we multiply the equation (4.29)_{1} by and integrate over to obtain
Analogously, we have
implying
The estimates for the velocity (4.16) and pressure (4.17) now follow directly from Poincaré's inequality (4.1) and (4.34). The estimate (4.15) for the microrotation then follows from (4.23) and the theorem is completely proved.

#### Acknowledgments

This research was supported by the Ministry of Science, Education and Sports, Republic of Croatia, Grants no. 037-0372787-2797. The author would like to thank the referees for their valuable remarks and comments.