Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 569-579, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.569

HEAT FLOW THROUGH A THIN COOLED PIPE FILLED WITH

A MICROPOLAR FLUID

Michal Beneš

Czech Technical University, Faculty of Civil Engineering, Czech Republic

Igor Pažanin

University of Zagreb, Faculty of Science, Zagreb, Croatia; e-mail address: pazanin@math.hr

Francisco Javier Suárez-Grau

Universidad de Sevilla, Facultad de Matemáticas, Sevilla, Spain

In this paper, a non-isothermal flow of a micropolar fluid in a thin pipe with circular cross-
-section is considered. The fluid in the pipe is cooled by the exterior medium and the heat
exchange on the lateral part of the boundary is described by Newton’s cooling condition.
Assuming that the hydrodynamic part of the system is provided,we seek for themicropolar
effects on the heat flow using the standard perturbation technique. Different asymptotic
models are deduced depending on the magnitude of the Reynolds number with respect to
the pipe thickness. The critical case is identified and the explicit approximation for the fluid
temperature is built improving the known result for the classical Newtonian flow as well.
The obtained results are illustrated by some numerical simulations.

Keywords: pipe flow, heat conduction, micropolar fluid, asymptotic analysis

1. Introduction

The Navier-Stokes model of classical hydrodynamics is based on the assumption that the fluid
particles do not posses any internal structure. However, in the case of fluids whose particles
have complex shapes (e.g. polymeric suspensions, liquid crystals, muddy fluids, animal blood,
even water in models with small scales), fluid particles can exhibit some microscopical effects
such as rotation and shrinking. For suchfluids, the local structure andmicro-motions of the fluid
elements cannotbe ignoredandoneof thebest-established theories covering thosephenomenae is
themicropolar fluid theory, introducedbyEringen (1966). Physically,micropolar fluids consist of
rigid, spherical particles suspended in a viscous mediumwhere the deformation of the particles
is ignored. The individual particles may rotate (independently of the movement of the fluid)
and, thus, a new vector field, the angular velocity field of rotation of particles (microrotation)
is introduced to classical pressure and velocity fields. Correspondingly, one new vector equation
is added expressing the conservation of the angular momentum. As a result, a non-Newtonian
model is obtained representing an important generalization of the Navier-Stokes equations. As
such, it describes the behavior of numerous real fluids better than the classical model, especially
when the characteristic dimensions of the flow (e.g. diameter of the pipe) become very small.
Here we investigate the non-isothermal 3D flow of a micropolar fluid in a thin (or long)

cylindrical pipe. The problem is described by a complex nonlinear system of PDEs in which
micropolar equations are coupled with the heat conduction equation (see Lukaszewicz, 1999).
The full coupled system is very difficult to be handled, especially if one wants to analytically
construct an asymptotic approximation of the flow. Therefore, in this paper, we are going to
consider only the heat flow in a thin pipe assuming that the velocity distribution is known.
Thatmeans that the governing problem is being described by the non-stationary heat equation



570 M. Beneš et al.

with given velocity in the convection term. In view of the applications we want to model (heat
exchangers, pipelines, etc.), we assume that the pipe is plunged in an ambient medium whose
temperature is different from the fluid temperature. The heat exchange between the fluid and
surroundingmedium is being described byRobin’s boundary condition resulting fromNewton’s
cooling law. Our goal is to derive a simplified mathematical model describing the asymptotic
behavior of the fluid temperature in such a situation.

Seeking primarily for the micropolar effects, one needs to focus on the convection term car-
rying the effects of the fluidmicrostructure in the velocity distribution. FollowingMarušić et al.
(2008), the idea is to assume that the Reynolds number may depend on the small parameter ε
(being the ratio between pipe thickness and its length) and to deduce variousmodels depending
on its magnitude with respect to ε. In the mentioned paper, classical Newtonian flow has been
treated and the critical case is identified in which the effects of the convection term and the
surrounding temperature are of the same order. Avoiding computation of the formal asymptotic
expansion, the method employed by Marušić et al. (2008) is very elegant but, unfortunately,
cannot be employed in the micropolar setting. The reason lies in the fact that the approxima-
tion for velocity feels themicrostructure effects in its corrector (see Appendix). In view of that,
we are forced to change the methodology and try to formally derive a higher-order asymptotic
approximation for the fluid temperature acknowledging the effects we seek for. Starting from the
non-dimensional setting and using the two-scale asymptotic technique, wemanage to construct
an explicit approximation in the critical case clearly showing in whichway the fluidmicrostruc-
ture and exterior temperature affect the heat flow inside the pipe. This is especially important
with regard to numerical computations.

The isothermal flow of amicropolar fluidwas successfully considered both in 2D, see Dupuy
et al. (2004, 2008) and in amore realistic 3Dcase, seePažanin (2011a,b). Taking into account the
thermal effects as well, non-isothermal flows have gainedmuch attention in the recent years, see
e.g. Prathap Kumar et al. (2010), Si et al. (2013), Ali and Ashraf (2014). To our knowledge, so
far only simplified 2D setting (in which themicrorotation is a scalar function) has been studied
and the influence of the surroundingmediumhas been neglected in the process. For that reason,
in the present paper we address the 3D problem in a cooled pipe describing a real-life situation.

2. Position of the problem

Let B =B(0,1)⊂ R2 be a unit circle. We study the flow in a straight pipe with length L and
cross-section diameter d given by Ω̂= {x̂=(x̂1, x̂2, x̂3)∈ R

3 : 0< x̂1 <L, (x̂2, x̂3)∈ dB}. We
take that the ratio ε= d/L is small meaning that we consider the problem in a cylindrical pipe
which is either very thin or very long. It is supposed that thepipe is filledwith an incompressible
micropolar fluid.Motivated by the framework of heat exchangers,we assume that thefluid inside
the pipe is cooled by the exteriormediumsoweprescribeNewton’s cooling condition on the pipe
lateral boundary.As explained in Introduction, our intention is to study the thermodynamicpart
of the system assuming that the velocity distribution is known and given by the approximation
provided in Appendix.

In such a situation, it is plausible to work with the problem written in a non-dimensional
form. In view of that, we introduce the non-dimensional variable x = (x1,x2,x3) = x̂/L and,
correspondingly, the domain Ωε = (0,1)× εB. We denote by Γε = (0,1)× ε∂B the lateral
boundary of the pipe. All physical properties of the fluid are assumed to be constant: ν – kine-
matic Newtonian viscosity, κ – thermal conductivity and cp – specific heat capacity at constant
pressure.We set density of the fluid to be equal to one, for the sake of notational simplicity. In
the non-dimensional framework, three characteristic numbers appear:Reε = ν−1U0L –Reynolds
number,Pr=κ−1νcp –PrandtlnumberandNu=βLκ

−1 –Nusselt number.HereU0 denotes the



Heat flow through a thin cooled pipe filled... 571

characteristic velocity of the fluid, while β stands for the heat transfer coefficient coming from
Newton’s cooling condition.We take the characteristic time of the process asT0 =L

2cpκ
−1 and,

correspondingly, introduce rescaled time as t= τ/T0. As a result, our problem for the unknown
fluid temperatureΦε(x,t) can be written as follows

∂Φε

∂t
−∆Φε+ReεPrvε ·∇Φε =0 in Ωε× (0,T)

Φε = θk for x1 = k (k=0,1)
(2.1)

∂Φε

∂n
=Nu(G−Φε) on Γε× (0,T) Φε(x,0)=Φ0(x) x∈Ωε (2.2)

The fluid velocity enters the above system as the known function vε(x1,x2,x3) =
uε(x1,x2/ε,x3/ε) with u

ε being provided in Appendix. Robin’s boundary condition (2.2)1 mo-
dels the heat exchange between the fluid inside the pipe and surrounding medium (n denotes
exterior unit normal on Γε). To simplify the calculations a little bit, exterior temperature G
and boundary temperatures θk are assumed to be independent of cross-section variables, i.e.
G=G(x1, t) and θk = θk(t) (k=0,1).
This is the formal setting of our problem. It consists of (linear) convection-diffusion equation

equipped with the appropriate mixed boundary conditions for temperature. The existence and
uniqueness issues for such a problem are resolved and well-known (see e.g. Ladyzhenskaya et
al., 1967) and, thus, are not going to be discussed here. Using asymptotic analysis with respect
to ε, we want to derive a macroscopic law describing the behavior of fluid temperature clearly
acknowledging the effects of the fluidmicrostructure and surrounding temperature.

3. Asymptotic analysis

Denotingx′ =(x2,x3), we introduce the fast variable y
′ =(y2,y3) as y

′ =x′/ε. Correspondingly,
we introduce the new unknown function ϕε(x1,y

′, t) = Φε(x1,εy
′, t). Taking into account that

uε(x1,y
′)=u0(y

′)+εu1(x1,y
′) (see Appendix),ϕε satisfies the following (rescaled) equation

∂ϕε

∂t
−
1

ε2
∆y′ϕ

ε−
∂2ϕε

∂x21
+PrReε

(
u10
∂ϕε

∂x1
+u21
∂ϕε

∂y2
+u31
∂ϕε

∂y3
+εu11

∂ϕε

∂x1

)
=0 (3.1)

The above equation is posed in Ω× (0,T), where Ω = (0,1)×B. Here and in the sequel, we
denote

∆y′φ=
∂2φ

∂y22
+
∂2φ

∂y23
∇y′φ=

∂φ

∂y2
e2+

∂φ

∂y3
e3

for a scalar function φ and Cartesian basis (e1,e2,e3). Newton’s cooling condition (2.2)1 after
rescaling has the following form

∇y′ϕ
ε ·y′ = εNu(G−ϕε) on Γ × (0,T) (3.2)

We express the unknown temperature ϕε as an asymptotic expansion in powers of the small
parameter ε

ϕε(x1,y
′, t)=ϕ0(x1,y

′, t)+εϕ1(x1,y
′, t)+ε2ϕ2(x1,y

′, t)+ . . . (3.3)

and substitute it in (3.1) and (3.2). It is essential to observe the following: ifwekept theReynolds
numberReε constant (independent of ε), thenwewould obtain no contribution of the convection
term in themacroscopic model. In fact, the exterior temperatureGwould completely dominate



572 M. Beneš et al.

the process and no effects of the fluidmicrostructure could be observed (see a) below). Thus, we
adopt the idea from Marušić et al. (2008) and use the fact that Reε can be compared with the
small parameter ε in a various way leading to possible different asymptotic models depending
on its order ofmagnitude. Indeed, plugging (3.3) in (3.1) and (3.2), after collecting equal powers
of ε, we detect two characteristic cases:

1. Reε ≪O
(1
ε

)

The lowest order approximation satisfies the following problem

1

ε2
: ∆y′ϕ0 =0 in B 1 : ∇y′ϕ0·y

′ =0 on ∂B, (x1, t)∈ (0,1)×(0,T)

(3.4)

providing ϕ0 =ϕ0(x1, t). The next term in the expansion is given by

1

ε
: ∆y′ϕ1 =0 in B 1 : ∇y′ϕ1 ·y

′ =Nu(G−ϕ0) on ∂B

For fixed (x1, t) ∈ (0,1)× (0,T), this is the standard Neumann problem for the Laplace
equation. Thus, it will be solvable if

∫
∂BNu(G−ϕ0) dy

′ = 0 implying ϕ0 =G. It means
that, for Reε ≪O(1/ε), the fluid inside the pipe assumes temperature of the surrounding
medium and that the effects of the boundary temperature θk are negligible. Moreover, the
micropolar effects do not appear even in higher-order terms.

2. Reε ≫O
(1
ε

)

In this case, the convection term becomes dominant, but∇y′ϕ0 ·y
′ =0 on ∂B (see (3.2)).

Therefore, we conclude that the effects of the surrounding temperatureGwould be negli-
gible on this assumption.

Taking into account the above discussion, we conclude that the most interesting case is when
Reε =O(1/ε) since it will lead to the asymptotic model in which all the effects we seek for are
balanced. Thus, in the sequel, we perform careful analysis in this critical case.

3.1. Critical case Reε =O
(1
ε

)

To simplify the notation let us take Reε = 1/ε. The zero order term in the asymptotic
expansion is described by problem (3.4) so, as above, we conclude ϕ0 =ϕ0(x1, t). However, the
next term is given by

1

ε
: ∆y′ϕ1 =Pru

1
0

∂ϕ0
∂x1

in B ε : ∇y′ϕ1 ·y
′ =Nu(G−ϕ0) on ∂B (3.5)

Taking into account that u10 =2Q(1−|y
′|2) (see Appendix), the compatibility condition for the

existence of the solution to problem (3.5) gives

PrQ
∂ϕ0
∂x1
−2Nu(G−ϕ0)=0 in (0,1)× (0,T) (3.6)

For fixed t ∈ (0,T), this is in fact, a linear ODE for ϕ0 (with respect to x1) which can be
easily solved. To assure uniqueness, we need one boundary condition and the natural choice is
ϕ0(0, t) = θ0. Such a choice of the boundary condition is obvious from the physical point of view
(the temperature of the fluid exiting the pipe should not be prescribed in advance) and has been



Heat flow through a thin cooled pipe filled... 573

rigorously justified in the case of Newtonian flow, see Marušić et al. (2008). Consequently, we
obtain

ϕ0(x1, t)= e
−2Nu
PrQ
x1

(
θ0(t)+

2Nu

PrQ

x1∫

0

e
2Nu

PrQ
ξ
G(ξ,t) dξ

)
(3.7)

In view of (3.6), the problem for ϕ1 becomes

∆y′ϕ1 =4Nu(G−ϕ0)(1−|y
′|2) in B

∇y′ϕ1 ·y
′ =Nu(G−ϕ0) on ∂B

(3.8)

It can be explicitly solved by passing to polar coordinates, leading to

ϕ1(x1,y
′, t)=Nu(G−ϕ0)

(
|y′|2−

1

4
|y′|4

)
+C(x1, t) (3.9)

HereC(x1, t) is a function to be determined after identifying the problem for ϕ2

1 :
∂ϕ0
∂t
−∆y′ϕ2−

∂2ϕ0
∂x21
+Pr

(
u10
∂ϕ1
∂x1
+u21
∂ϕ1
∂y2
+u31
∂ϕ1
∂y3
+u11
∂ϕ0
∂x1

)
=0 in B

ε2 : ∇y′ϕ2 ·y
′ =−Nuϕ1 on ∂B, (x1, t)∈ (0,1)× (0,T)

(3.10)

The above system will be solvable if and only if

π
∂ϕ0
∂t
−π
∂2ϕ0
∂x21
+Pr

∫

B

u10
∂ϕ1
∂x1
dy′+Pr

∫

B

u21
∂ϕ1
∂y2
dy′+Pr

∫

B

u31
∂ϕ1
∂y3
dy′

+Pr
∂ϕ0
∂x1

∫

B

u11 dy
′ =−Nu

∫

|y′|=1

ϕ1 dy
′

(3.11)

The components of the velocity corrector u1 appearing in (3.10)1 have the following form (see
Appendix)

u11(x1,y
′)=
1

8
(|y′|2−1)

[(df2
dx1
+2N2

g3
2R1+R2

)
y2+

(df3
dx1
−2N2

g2
2R1+R2

)
y3
]

u21(x1,y
′)=−

N2

8R1
g1(x1)(1−|y

′|2)y3

u31(x1,y
′)=

N2

8R1
g1(x1)(1−|y

′|2)y2

Taking into account expression (3.9), by direct integrationwe obtain that the last three integrals
on the left-hand side in (3.11) are equal to zero. Thus, from (3.11) we deduce an equation for
C(x1, t)

PrQ
∂C

∂x1
+2NuC =D

D(x1, t)=
∂ϕ0
∂t
−
∂2ϕ0
∂x21
+
7QNuPr

24

(∂ϕ0
∂x1
−
∂G

∂x1

)
+
3

2
Nu2(ϕ0−G)

Endowing it with the boundary condition C(0, t)= 0, we get

C(x1, t)=
1

PrQ
e
−2Nu
PrQ
x1

x1∫

0

e
2Nu

PrQ
ξ
D(ξ,t) dξ. (3.12)



574 M. Beneš et al.

Remark 1. Thecorrectorϕ1 is computed to satisfy equation (3.8)1 andcooling condition (3.8)2.
Note that the boundary condition atx1 =0could not be taken into account in the process.
As a consequence, ϕ1(0,y

′, t) 6=0 implying ϕ0+εϕ1 6= θ0, i.e. the boundary temperature
at the inlet is being modified by the higher-order term. We can fix that by introducing
the appropriate boundary layer corrector depending on the dilated variable (y1,y

′) =
(x1/ε,x

′/ε), see e.g.Marušić-Paloka andPažanin (2009). However, such a corrector would
have an exponential decay towards zero (as y1 → +∞) meaning that it would not affect
the approximation outside the boundary layer (it would only serve for the convergence
proof which is out of the scope of the present paper). Thus, there is no reason to formally
correct the approximation ϕ0 + εϕ1 in the vicinity of x1 = 0 since the effects of such
correction would not contribute to the macroscopic model except in the small boundary
layer near the pipe inlet.

The corrector ϕ1 is given in an explicit form by (3.9) and (3.12). We observe that there
are contributions of surrounding and boundary temperature, but we find no micropolar effects
at this order. Therefore, we have to continue computation and try to construct a higher-order
correctorϕ2. Taking into account (3.7) and (3.9) and the expression for the velocity distribution,
problem (3.10) can be written as

∆y′ϕ2 =
[
−
∂ϕ0
∂t
+
2Nu

PrQ

∂G

∂x1
+
(2Nu
PrQ

)2
(ϕ0−G)

]
(1−2|y′|2)−4NuC(1−|y′|2)

−Nu2(G−ϕ0)(1−|y
′|2)
(11
6
+4|y′|2−|y′|4

)

+
1

2
PrNuQ

∂G

∂x1
(1−|y′|2)

(
−
7

6
+4|y′|2−|y′|4

)

+
Nu

4Q
(ϕ0−G)(1−|y

′|2)H1y2+
Nu

4Q
(ϕ0−G)(1−|y

′|2)H2y3 in B

∇y′ϕ2 ·y
′ =−

3

4
Nu2(G−ϕ0)−NuC on ∂B, (x1, t)∈ (0,1)× (0,T)

(3.13)

Here we denote

H1(x1)=
df2
dx1
+2N2

g3
2R1+R2

H2(x1)=
df3
dx1
−2N2

g2
2R1+R2

coming fromu11. Problem (3.13) can be analytically solved by introducing six auxiliary problems
corresponding to each term on the right-hand side of (3.13)1 and passing to polar coordinates.
We leave the reader to confirm that

ϕ2(x1,y
′, t)=

[
−
∂ϕ0
∂t
+
2Nu

PrQ

∂G

∂x1
+
(2Nu
PrQ

)2
(ϕ0−G)

](1
4
|y′|2−

1

8
|y′|4

)

−NuC
(
|y′|2−

1

4
|y′|4

)
−Nu2(G−ϕ0)

(11
24
|y′|2+

13

96
|y′|4−

5

36
|y′|6+

1

64
|y′|8

)

+
1

2
PrNuQ

∂G

∂x1

(
−
7

24
|y′|2+

31

96
|y′|4−

5

36
|y′|6+

1

64
|y′|8

)

+
Nu

4Q
(G−ϕ0)

(df2
dx1
+
2N2g3
2R1+R2

)(1
6
−
1

8
|y′|2+

1

24
|y′|4

)
y2

+
Nu

4Q
(G−ϕ0)

(df3
dx1
−
2N2g2
2R1+R2

)(1
6
−
1

8
|y′|2+

1

24
|y′|4

)
y3

(3.14)

This is the end of the formal derivation. Our asymptotic solution has the following form

ϕεapprox(x1,y
′, t)=ϕ0(x1, t)+εϕ1(x1,y

′, t)+ε2ϕ2(x1,y
′, t)

(x1,y
′)∈Ω t∈ (0,T)

(3.15)



Heat flow through a thin cooled pipe filled... 575

Since the functionsϕi (i=0,1,2) are all given in the explicit form, see (3.7), (3.9), (3.14), we can
easily detect all the effectswewere originally interested for.Thefirst part of the solution, namely
ϕ0+εϕ1, is influencedby theeffects of coolingandupstreamboundarytemperature, and it canbe
viewed as an improvement of the result byMarušić et al. (2008) for the classical Newtonian flow.
Noeffects of thefluidmicrostructure canbe seen there.However, in the second-order correctorϕ2
we can clearly observe the correction coming due to the non-Newtonian (micropolar) nature of
the fluid.The correction, i.e. the difference betweenϕ0+εϕ1 andϕ

ε
approx can be clearly observed

in the numerical examples in the following Section as well.

4. Numerical examples

In this Sectionwe aim to visually compare the functionϕ0+εϕ1 (representing the approximation
for the Newtonian flow) with the asymptotic approximation ϕεapprox for the generalized micro-
polar flow. It must be emphasized that there is very little information in the existing literature
concerning the values ofmicropolar parameters.We use the reference byPapautsky et al. (1999)
where ν =2.9 ·10−3, νr =2.32 ·10

−4, ca+ cd =10
−6 are reported for the micropolar viscosity

constants. Furthermore, we neglect the body force f, while g is taken to be a point couple, see
e.g. Shu and Lee (2008) (sequence δn = n/[π(1+n

2x21)] is employed to approach the sifting
property of the Dirac delta function). Exterior and boundary temperatures are assumed to be
constant (G = 10, θ0 = 35), while for the remaining constants we take Nu = 3.66, Pr = 4.8
andQ=81. Polar coordinates (|y′|,ϑ) are used to describe the domain cross-sectionB, and 2D
comparisons for differentmagnitudes of the small parameter ε are presented (see Figs. 1 and 2).

Fig. 1. Temperature distribution for the variable x1 for a fixed |y
′| and polar angle ϑ=π/2

Fig. 2. Temperature distribution for the variable |y′| (with ϑ=π/2) and for a fixed x1



576 M. Beneš et al.

5. Concluding remarks

We derived the asymptotic model describing the heat flow through a cylindrical pipe (with
thickness O(ε)) in the critical case where the effects of micropolarity, surroundingmedium and
entering temperature are balanced. The discussion at the beginning of Sec. 3 suggested that the
critical case is reachedwhentheReynoldsnumber is of orderO(ε−1).Aspresented, the correctors
in the asymptotic expansion for thefluid temperature canbe explicitly computed clearly showing
the difference between the classical Newtonian and micropolar flow. It is important to observe
that such a more accurate approximation, is particularly interesting if ε is not too small (e.g.
ε = 0.2 or ε = 0.05 as chosen in the numerical examples). Indeed, the smaller ε becomes, the
difference between the Newtonian and micropolar flow becomes the less obvious (see Figs. 1
and 2).Moreover, by keeping ε in such a range, we ensure that the critical value of theReynolds
number is not exceeded, i.e. the flow stays in the laminar regime (not becoming turbulent, so
we can use the laminar velocity profile fromAppendix).

Let us discuss in more details the numerical results obtained in Sec. 4. First, in Fig. 1, we
fix the cross-section variable and consider the temperature variations for the variable x1. The
result clearly suggests that, except in the small boundary layer near the left end of the pipe, the
micropolar nature of the fluid enhances the cooling as we move along the pipe. Note that the
value of the temperature at x1 =0 is a bit lower than the imposed θ0 =35. That is essentially
due to the fact thatwehave computed the correctors to satisfy the heat equation and the cooling
condition, while the boundary condition at the pipe inlet has been neglected. Consequently, a
boundary layer appear (in the vicinity of x1 = 0) having no influence on the effective flow far
from the pipe inlet, as explained in Remark 1. On the other hand, by fixing x1, from Fig. 2
we deduce that the micropolar effects enhances the cooling far from the pipe lateral boundary.
Indeed, as we approach the boundary, those effects become negligible due to the influence of the
surrounding temperature.

To conclude, we believe that the above observations could improve the known engineering
practice by acknowledging the obtained micropolar effects on the cooling process.

Appendix A. Approximation for the velocity

For reader’s convenience, here we provide an asymptotic approximation for the fluid velocity
used in the main part of the paper. For detailed (formal) derivation, we refer the reader to
Pažanin (2013). Rigorous justification can be recovered fromPažanin (2011).

We start from the linearized micropolar equations (see Lukaszewicz, 1999)

− (ν+νr)∆û
ε+∇p̂ε =2νr rotŵ

ε+ f̂

divûε =0 in Ω̂ε

− (ca+ cd)∆ŵ
ε− (c0+ cd− ca)∇divŵ

ε+4νrŵ
ε =2νr rotû

ε+ ĝ

(A.1)

Theunknownsare: velocity ûε, pressure p̂ε andmicrorotation ŵε.Positive constants νr, c0, ca, cd
represent new viscosity coefficients coming from the asymmetry of the stress tensor. Due to the
pipe thickness, the body force f̂ = f̂(x̂1) and body couple ĝ = ĝ(x̂1) are assumed to depend
only on the variable going along the pipe. We use standard Dirichlet boundary conditions for
the velocity andmicrorotation and prescribe the pressure drop q̂0− q̂L between the pipe ends

û
ε =0 on Γ̂ε e1× û

ε =0

p̂ε = q̂i for x̂1 = i (i=0,L) ŵ
ε =0 on ∂Ω̂ε

(A.2)



Heat flow through a thin cooled pipe filled... 577

Since we need to work in non-dimensional framework, we introduce

x1 =
x̂1
L

y2 =
x̂2
εL

y3 =
x̂3
εL

u
ε =
ûε

U0
w
ε =
L

U0
ŵ
ε

pε =
ε2L

U0(ν+νr)
p̂ε f =

ε2L2

U0(ν+νr)
f̂ g=

ε2L

U0(ν+νr)
ĝ

When considering micropolar flows, three non-Newtonian characteristic numbers appear

R1 =
ca+cd
(ν+νr)L2

R2 =
c0+ cd− ca
(ν+νr)L2

N2 =
νr
ν+νr

(A.3)

The parameters Ri are related to characteristic length of the microrotation effects, while N
characterizes the coupling between the equations for velocity and microrotation. Note that for
N =0, the above system becomes decoupled and (A.1)1 and (A.1)2 reduce to classical Navier-
-Stokes equations. In view of that, equations (A.1) in the dimensionless form read

−∆y′u
ε−ε2

∂2uε

∂x21
+
1

ε
∇y′p

ε+
∂pε

∂x1
e1

=2N2
[
ε
(∂wε3
∂y2
−
∂wε2
∂y3

)
e1+εroty′w

ε
1−ε

2∂w
ε
3

∂x1
e2+ε

2∂w
ε
2

∂x1
e3

]
+ f

divy′u
ε+ε
∂uε1
∂x1
=0

−R1
(
∆y′w

ε+ε2
∂2wε

∂x21

)
−R2

[
∇y′(divy′w

ε)+ε
∂

∂x1
(divy′w

ε)e1

+ε∇y′
(∂wε1
∂x1

)
+ε2
∂2wε1
∂x21
e1

]
+4N2ε2wε

=2N2
[
ε
(∂uε3
∂y2
−
∂uε2
∂y3

)
e1+εroty′u

ε
1−ε

2∂u
ε
3

∂x1
e2+ε

2∂u
ε
2

∂x1
e3

]
+g

(A.4)

where

divy′v=
∂v2
∂y2
+
∂v3
∂y3

roty′vi =
∂vi
∂y3
e2−

∂vi
∂y2
e3 vi =v ·ei

Now we construct the approximation for the fluid velocity uε by plugging the asymptotic
expansions

u
ε(x1,y

′)=u0(x1,y
′)+εu1(x1,y

′)+ . . .

w
ε(x1,y

′)=w0(x1,y
′)+εw1(x1,y

′)+ . . .

pε(x1,y
′)= p0(x1)+εp1(x1,y

′)+ . . .

(A.5)

into system (A.4) and collecting the termswith equal powers of ε. The zero-order approximation
turns out to be the standard Poiseuille solution given by

u0(y
′)= 2Q(1−|y′|2)e1

Q=
1

π

∫

B

u10 dy
′ =
1

8

(
q0−q1+

1∫

0

f1(ξ) dξ

)
(A.6)

Since we observe no effects of the microstructure here, we have to derive a lower-order term u1
from the velocity expansion. Continuing computation, we arrive at



578 M. Beneš et al.

u11(x1,y
′)=
1

8
(|y′|2−1)

[(df2
dx1
+2N2

g3
2R1+R2

)
y2+

(df3
dx1
−2N2

g2
2R1+R2

)
y3
]

u21(x1,y
′)=−

N2

8R1
g1(x1)(1−|y

′|2)y3

u31(x1,y
′)=

N2

8R1
g1(x1)(1−|y

′|2)y2

(A.7)

Finally, our asymptotic approximation for the fluid velocity reads

u
ε(x1,y

′)=u0(y
′)+εu1(x1,y

′) (x1,y
′)∈Ω (A.8)

Acknowledgements

The first author of this work has been supported by the project GAČR 13-18652S. The second

author has been supported by the Croatian Science Foundation (project 3955: Mathematical modeling

and numerical simulations of processes in thin or porous domains). The third author has been supported

by the projects MTM2011-24457 of the Ministerio de Economía y Competitividad and FQM309 of the

Junta de Andalućia. The authors would like to thank the referee for his/her helpful comments and

suggestions.

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Manuscript received October 16, 2014; accepted for print December 9, 2014