Effective Flow of Micropolar Fluid through a Thin or Long Pipe

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.


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 Δp: pressure drop, L, R: pipe s length and radius. 1.1 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 ε, introduced as the ratio between pipe's thickness and its length.It relies on twoscale asymptotic expansions in powers of small parameter which, in our case, have the form The variable x 1 is directed along the pipe, while x x 2 , x 3 describes the cross section.The role of dilated fast variable y x /ε 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 ε, 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 reallife 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 ε 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 ε 3 √ ε for the velocity and with an error of order ε 4 for the microrotation.

The Geometry
In order to describe the thin pipe with a small parameter ε appearing explicitly, we first introduce where the family of bounded domains {B x 1 } x 1 ∈ 0, ⊂ R 2 is chosen such that Ω is locally Lipschitz.Now, we define our thin pipe with variable cross section B x 1 and length by We are particularly interested in the case when the pipe Ω ε has circular cross section, that is, when with R being a strictly positive bounded function defined on R. Finally, we denote the ends of the pipe by Σ i ε εB i , i 0, , while its lateral boundary is given by

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 u ε , w ε and p ε standing for the velocity, the microrotation and the pressure of the fluid, respectively.The fields f f x 1 , g g x 1 represent given external forces and moments, respectievly and we assume f, g ∈ C 1 0, 3 .Viscosity coefficients read μ ν ν r , a 2ν r , α c a c d , β c 0 c d −c a , γ 4ν r , where ν, ν r , c 0 , c a , c d are the given positive constants ν is the usual Newtonian viscosity, ν r is microrotation viscosity, c 0 , c a , c d are the coefficients of angular viscosities .Observe that if we put ν r 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 e 1 , e 2 , e 3 denotes the standard Cartesian basis.
Remark 2.1.By prescribing constant pressures q 0 , q on Σ i ε , 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 e 1 × u ε 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 μ 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 for any v ∈ V ε .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 p 1/2 |u| 2 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 x 1 0, , but only the pressure drop q 0 − q .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 1.1.1 and 1.1.2from 8 to our situation and prove that the velocity u ε is unique in some ball B R 0 {v ∈ V ε ; ∇v L 2 Ω ε ≤ R 0 }, with radius R 0 remaining bounded as ε → 0. 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 .

Rescaling of the Domain
Our main goal is to find the asymptotic behavior of the flow, as the thickness ε → 0. To accomplish that, we first need to rescale the domain, that is, to write the governing problem on Ω instead of Ω ε .Introducing the new functions we can write the equations 2.5 -2.7 in the following form:

Mathematical Problems in Engineering
Here and in the sequel, we use the following notations for the formal partial differential operators: 3.5

Asymptotic Expansions
In this section, we construct the formal asymptotic expansion of the solution in powers of small parameter ε.As mentioned in Introduction, we expand as follows: 3.6

First-Order Approximation
Substituting the expansions 3.6 into the rescaled equations 3.2 -3.4 , after collecting the terms with equal powers of ε, we obtain the following problems for first-order approximation U 0 , P 0 , W 0 : Here, we denote Γ { x 1 , y ∈ R 3 : 0 < x 1 < , y ∈ ε∂B x 1 }.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 f i f • e i and χ x 1 , y denotes the solution of the auxiliary problem posed on the cross section B x 1 :

3.10
If the pipe has circular cross section 2.3 , we can compute χ explicitly from 3.10 :

3.11
We still have to determine P 0 x 1 .The next term in 3.7 2 implies

3.12
Integration over B x 1 with respect to y yields χ x 1 , y dy , 3.14 from 3.9 1 , we deduce

3.15
It follows with C 2 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

3.19
Similarly, it can be verified that the problem 3.8 for microrotation will be satisfied for 3.20

Correctors
Now, we compute the correctors.The O ε term from momentum equation 3.2 gives

3.21
The system is not decoupled anymore, so the effects of the microstructure on the fluid velocity occur.Inserting the expressions for P 1 and W 0 derived for circular pipe, we get the following problem for the first component:

3.22
Let us introduce χ i x 1 , y , i 2, 3 as the solutions of the following two problems posed on B x 1 :

3.23
Taking into account 2.3 and using the polar coordinates yield

3.24
We seek the solution of system 3.22 in the form

3.26
For the other two velocity components from 3.21 and 3.3 , we obtain

3.27
where Because U 1 is not divergence-free, it is not likely that the above system can be explicitly solved.However, it is important to emphasize that, since B x 1 ∂U 1 0 /∂x 1 0, such problem admits a unique solution It remains to construct the corrector for the microrotation.From 3.4 , we deduce

3.28
If we write the above system by the components and take into account 3.19 and 3.20 , we get

3.29
Similarly as for U 1 1 , we obtain

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The problem satisfied by the other two components W 2 1 , W 3 1 must be solved carefully as a system implying 3.31 3.32

Boundary Layers for Microrotation
It is important to notice that our approximation x /ε was computed to satisfy the governing equations and the boundary condition on Γ ε , while the boundary conditions on pipe's ends were not taken into account.Consequently, the traces of w 1 ε on Σ i ε i 0, may be different from 0. Thus, before proving convergence, we need to correct our interior expansion in the boundary layer near x 1 0 and x 1 .Near x 1 0, we introduce the boundary layer correctors B i i 0, 1 depending on the dilated variable y 1 , y x/ε, as the solutions of the following Dirichlet boundary-value problems posed in the semi-infinite strip G { y 1 , y ∈ R 3 : y 1 > 0, y y 2 , y 3 ∈ B y 1 }:

3.33
for i 0, 1 and ω { y 1 , y ∈ R 3 : y 1 > 0, y y 2 , y 3 ∈ ∂B y 1 }.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 B 0 ∈ H 1 G 3 which is exponentially decaying to zero as y 1 → ∞ see, e.g., 19 .Analogously, the boundary layer correctors H i i 0, 1 corresponding to the opposite side x 1 are constructed as the unique solutions of the following problems: and its exponential decay to zero at infinity follows as well.

Asymptotic Approximation
To conclude this section, let us write the obtained asymptotic approximation.For the microrotation, it has the following form:

3.35
where W 0 i W i 0 e i and W 1 i W i 1 e i 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:

3.37
The first term in the expansion u 0 ε x ε 2 U 1 0 x 1 , x /ε e 1 , 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 .

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
The above estimates can be verified by a simple change of variables see, e.g., 11, Lemmas 7, 8 .Proposition 4.2.Let u ε , p ε , w ε be the solution of the problem 2.5 -2.10 , then there exists C > 0, independent of ε, such that Proof.Multiplying the equation 2.7 by w ε and integrating over Ω ε gives

4.4
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 u ε and, after integrating over Ω ε , we obtain Employing the inequality 4.2 , we have Taking into account 4.1 , we obtain

4.11
Mathematical Problems in Engineering 13 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 u ε is unique in the ball B R 0 with radius R 0 see the discussion at the end of Section 2.2. .For that reason it is sufficient to choose ε such that CR 0 √ ε < 1/2 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.

4.17
Remark 4.4.Since our domain Ω ε is shrinking, the convergence in the norm • L 2 Ω ε would be worthless.Indeed, any L ∞ -bounded function would converge to zero in such norm.Therefore, we express the error estimates in the rescaled norm Proof.The function w 1 ε satisfies the following equation: 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 s ε and integrating over Ω ε lead to

4.21
As in 4.5 , we have Ω ε u ε ∇ s ε s ε 0. Now, we carefully estimate each term on the right-hand side of 4.21

4.22
Taking into account the obtained estimates 4.22 , from 4.21 , we obtain

4.23
The problem satisfied by u 1 ε , q ε is the following: Before proceeding, it is important to notice that the norm of π ε is not small enough to obtain satisfactory error estimate.Therefore, we need to construct the divergence correction.Since π ε ε 3 ∂U 1  1 /∂x 1 , we define Ψ x 1 , y as the solution of the problem

4.25
here x 1 is treated only as a parameter .Taking into account 3.26 , by a simple integration one can easily verify that B ∂U 1 1 /∂x 1 x 1 , y dy 0, 0 < x 1 < l implying that such Ψ exists.Now we define our divergence correction as Such v 0 ε is divergence-free.Moreover, Ψ ε is chosen such that it keeps the estimate for E ε , that is, we have

4.29
Now we introduce d ε as the solution of the problem div d ε r ε in Ω ε , d ε 0 on ∂Ω ε .

4.30
If we suppose Ω ε r ε 0, such problem has at least one solution which satisfies

4.32
We estimate the terms on the right-hand side in 4.32 using a priori estimates and Lemma 4.1, Analogously, we have

4.37
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.