International Journal of Engineering Science 72 (2013) 107–116

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International Journal of Engineering Science

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A one-dimensional model for unsteady axisymmetric swirlingmotion of a viscous fluid in a variable radius straight circulartube

0020-7225/$ - see front matter � 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.ijengsci.2013.06.010

⇑ Corresponding author. Tel.: +351 266745370; fax: +351 266745393.E-mail addresses: flc@uevora.pt (F. Carapau), jjanela@iseg.utl.pt (J. Janela).

Fernando Carapau a,⇑, João Janela b

a Universidade de Évora, Dept. Matemática e CIMAUE, Rua Romão Ramalho 59, 7001-671 Évora, Portugalb Universidade Técnica de Lisboa, ISEG, Dept. Matemática e CEMAPRE, Rua do Quelhas 6, 1200-781 Lisboa, Portugal

a r t i c l e i n f o

Article history:Available online 17 August 2013

Keywords:One-dimensional modelSwirling motionUnsteady flowHierarchical theory

a b s t r a c t

A one-dimensional model for the flow of a viscous fluid with axisymmetric swirling motionis derived in the particular case of a straight tube of variable circular cross-section. Themodel is obtained by integrating the Navier–Stokes equations over cross section the tube,taking a velocity field approximation provided by the Cosserat theory. This procedureyields a one-dimensional system, depending only on time and a single spatial variable.The velocity field approximation satisfies exactly both the incompressibility conditionand the kinematic boundary condition. From this reduced system, we derive unsteadyequations for the wall shear stress and mean pressure gradient depending on the volumeflow rate, the Womersley number, the Rossby number and the swirling scalar function overa finite section of the tube geometry. Moreover, we obtain the corresponding partial differ-ential equation for the scalar swirling function.

� 2013 Published by Elsevier Ltd.

1. Introduction

In this paper we present a one-dimensional model for the swirling motion of a viscous fluid, based on the Cosserat theory– also called director theory. The swirling features in flow fields are commonly called vortices. For most purposes (see e.g.,Lugt, 1972; Kitoh, 2006; Nissan and Bressan, 1961; Moene, 2003), a vortex is characterized by a swirling motion of fluidaround a central region. The swirling flow through a straight tube of variable circular cross-section is a complex turbulentflow and it is still challenging to predict and it is computationally demanding to simulate the full three-dimensional equa-tions for swirling flows, which makes the direct 3D numerical simulation infeasible in many relevant situations. In recentyears, the computational dynamics of two-dimensional swirling flows has been studied extensively with the purpose of bet-ter understanding the underlying physical phenomena and getting insight on important applications like the study of hur-ricanes and tornadoes (see e.g., Guinn and Shubert, 1993; Lewellen, 1993). Here we apply the Cosserat theory (see Caulk andNaghdi (1987)) to reduce the full three-dimensional system of fluid equations to a one-dimensional system of partial differ-ential equations, which depend only on time and on a single spatial variable. The basis of this theory (see Duhem (1893) andCosserat and Cosserat (1908)) is to consider an additional structure of deformable vectors (called directors) assigned to eachpoint on a spatial curve (the Cosserat curve). The use of directors in continuum mechanics goes back to Duhem (1893), whoregarded a body as a collection of points, together with associated directions. Theories based on such models of an orientedmedium were further developed by Cosserat and Cosserat (1908). This theory has also been used by several authors in

108 F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116

studies of rods, plates and shells (see e.g., Ericksen and Truesdell (1958), Truesdell and Toupin(1960), Green et al. (1968);Green et al., 1974 and Naghdi (1972)). An analogous hierarchical theory for unsteady/steady flows has been developed byCaulk and Naghdi (1987) in straight tubes of variable circular cross-section and by Green and Naghdi (1984) in channels.Applications to unsteady viscous flows in curved tubes of elliptic cross-section were presented by Green et al. (1993). Re-cently, this theory has been applied to several models arising in haemodynamics. We refer to a survey by Robertson andSequeira (2005), and an application by Carapau and Sequeira (2006a). Also, Carapau and co-authors (see Carapau and Seque-ira, 2006b, 2006c; Carapau et al., 2007; Carapau and Sequeira, 2008; Carapau, 2008a, 2008b; Carapau, 2009; Carapau, 2010a,2010b) have analysed extensions of the theory to deal with several non-Newtonian fluid models in different geometries.Regarding the swirling motion, this hierarchical theory was used to study a Rivlin–Ericksen fluid (complexity n ¼ 2) withaxisymmetric swirling steady motion flowing in a straight tube of variable circular cross-section (see Carapau, 2009). Thistheory was validated in straight tubes of constant circular cross-section for Newtonian fluids (see Caulk and Naghdi,1987) and for some non-Newtonian fluids (see Carapau and Sequeira, 2006a, 2006b). Another validation was provided inthe case of a particular non-Newtonian fluid flow in a linearly tapered tube for (see Carapau, 2010a). The advantage of usinga theory of directed curves is not so much getting an approximation of the three-dimensional system, but rather in using it asan independent framework to predict some properties of the full three-dimensional problem. The main features of the direc-tor theory are: (i) it incorporates all components of the linear momentum equations; (ii) it is a hierarchical theory, making itpossible to increase the accuracy of the model; (iii) there is no need for closure approximations, i.e. additional relations be-tween variables in the 1D model; (iv) invariance under superposed rigid body motions is satisfied at each order; (v) the wallshear stress enters directly in the formulation as a dependent variable and (vi) the director theory has been shown to be use-ful for modeling flow in curved tubes. A detailed discussion about the Cosserat theory can be found in Green and Naghdi(1993) and Green et al. (1993).

Using this director theory, we can intend to predict the main properties of a three-dimensional given problem, where thefluid three-dimensional velocity field # ¼ #iei is approximated by1 (see Caulk and Naghdi (1987)):

1 In tindex.

# ¼ v þXk

N¼1

xa1 . . . xaN Wa1 ...aN ; ð1Þ

with

v ¼ v iðz; tÞei; Wa1 ...aN ¼Wia1 ...aN

ðz; tÞei: ð2Þ

This velocity field approximation (1) satisfies both the incompressibility condition and the kinematic boundary conditionexactly. In condition (1), v represents the velocity along the axis of symmetry z at time t; xa1 . . . xaN are the polynomialweighting functions with order k (this number identifies the order of the hierarchical theory and is related to the numberof directors), the vectors Wa1 ...aN are the director velocities which are symmetric with respect to their indices and ei arethe associated unit basis vectors. The selection of such weighting functions represents an important aspect of the formula-tion of our problem. A good choice of these weighting functions can reduce the complexity of the system of partial differ-ential equations in the director formulation of the theory. This choice should be consistent with the hierarchical structureof the basic theory so that the equations for each level of the hierarchy include the equations of all lower orders. The vectorsWa1 ...aN are related to physical features of the fluid, in particular the swirling motion – also called rotational motion. Usingthis approach with nine directors (i.e., k ¼ 3 at condition (1)) and integrating the equations for the conservation of linearmomentum over a circular cross-section of the fluid domain, we obtain unsteady relations between mean pressure gradient,volume flow rate and the swirling scalar function, over a finite section of the tube. Furthermore, we obtain the correspondingunsteady equation for the wall shear stress, which enters directly in the formulation as a dependent variable, and the also apartial differential equation for the swirling scalar function. Some numerical simulations are provided for unsteady flow re-gimes in a constricted rigid tube.

2. Equations of motion

Let xi be the rectangular cartesian coordinates system and x ¼ ðx1; x2; x3Þ where, for convenience, we set x3 ¼ z. We con-sider the isothermal flow of an homogeneous fluid in a (three-dimensional) straight tube of variable circular cross-section X(see Fig. 1). Also, let us consider the surface scalar function /ðz; tÞ, that is related with the straight tube of circular cross-sec-tion by the following relation

/2ðz; tÞ ¼ x21 þ x2

2: ð3Þ

The boundary @X consists in the inlet cross-section C1, the outlet cross-section C2 and the lateral wall of the tube, denoted byCw. Considering the flow of an incompressible viscous fluid without body forces in X, the equations of motion, stating theconservation of linear momentum and mass are given, in X� ð0; TÞ, by

he sequel, latin indices subscript take the values 1;2;3; greek indices subscript 1;2, and the usual summation convention is employed over a repeated

Pe

1

Z

2

1

X1

X2

w

(z,t)

Fig. 1. General fluid domain X with the tangential components of the surface traction vector s1; s2 and pe , where /ðz; tÞ denote the radius of the domainsurface along the axis of symmetry z at time t.

F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116 109

q@#

@tþ # � r#

� �¼ r � T ;

r � # ¼ 0;

T ¼ �pI þ l r#þ ðr#ÞT� �

; tw ¼ T � g;

8>>>><>>>>: ð4Þ

where #ðx; tÞ ¼ #iðx; tÞei is the three-dimensional velocity field, p is the pressure, �pI is the spherical part of the stress due tothe constraint of incompressibility, l is the constant viscosity of the fluid and q is the constant density of the fluid. Eq. (4)1

arises from the conservation of linear momentum and (4)2 from the conservation of mass. In Eq. ð4Þ3, the expression for thetotal stress T defines the constitutive relation for the fluid, tw denotes the stress vector on the surface whose outward unitnormal vector is gðx; tÞ ¼ giðx; tÞei. The components of the outward unit normal vector to the surface /ðz; tÞ are given by

g1 ¼x1

/ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ /2

z

q ; g2 ¼x2

/ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ /2

z

q ; g3 ¼ �/zffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ /2z

q ; ð5Þ

where the subscript variable denotes partial differentiation. Since equation (3) defines a material surface, the velocity field #must satisfy the kinematic condition

ddt

/2ðz; tÞ � x21 � x2

2

� �¼ 0;

i.e.,

//t þ //z#3 � x1#1 � x2#2 ¼ 0; ð6Þ

on the boundary (3), where the material time derivative ddt ð�Þ is given by

ddtð�Þ ¼ @

@tð�Þ þ # � rð�Þ:

Averaged quantities such as volume flow rate and average pressure appear naturally in one-dimensional models. ConsiderSðz; tÞ as a generic axial section of the tube at time t defined by the spatial variable z and bounded by the circle defined in (3)and let Aðz; tÞ be the area of this section Sðz; tÞ. Then, the volume flow rate Q is defined by

Qðz; tÞ ¼Z

Sðz;tÞ#3ðx; tÞda ð7Þ

and the average pressure �p, by

�pðz; tÞ ¼ 1Aðz; tÞ

ZSðz;tÞ

pðx; tÞda: ð8Þ

Using the director theory approach (1) with k ¼ 3, it follows (see Caulk and Naghdi, 1987) that the approximation of thevelocity field #ðx; tÞ ¼ #iðx; tÞei, with nine directors, is given by

#ðx; tÞ ¼ x1ðnþ rðx21 þ x2

2ÞÞ � x2ðxþ wðx21 þ x2

2ÞÞ

e1 þ x1ðxþ wðx21 þ x2

2ÞÞ þ x2ðnþ rðx21 þ x2

2ÞÞ

e2

þ v3 þ cðx21 þ x2

2Þ

e3 ð9Þ

where n;x; c;r;w are scalar functions of the spatial variable z and time t. According to Caulk and Naghdi (1987), the physicalsignificance of these scalar functions in (9) is the following: c is related to transverse shearing motion, x and w are related torotational motion about e3, while n and r are related to transverse elongation.

Now, using the representation of the velocity field given by equation (9), the kinematic condition (6) on the lateral wallreduces to

/t þ v3 þ /2c� �

/z � nþ /2r� �

/ ¼ 0 ð10Þ

110 F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116

and the incompressibility constraint (4)2 becomes

ðv3Þz þ 2nþ x21 þ x2

2

� �cz þ 4rð Þ ¼ 0: ð11Þ

For Eq. (11) to hold at every point in the fluid, the velocity coefficients must satisfy the conditions

ðv3Þz þ 2n ¼ 0; cz þ 4r ¼ 0: ð12Þ

Hence, the boundary condition (6) and the constraint (4)2 are satisfied exactly by the velocity field (9), provided we imposeconditions (10) and (12).

Also, from Caulk and Naghdi (1987) the stress vector tw (see (4)3) on the lateral surface Cw, can be written in terms theoutward unit normal vector g and the tangential components s1; s2; pe (see Fig.1), where s1 is the wall shear stress, given by

tw ¼ s1k� pegþ s2eh; ð13Þ

with k; eh the unit tangent vectors defined by

k ¼ g� eh; eh ¼ ðxa=/Þeabeb; ð14Þ

where e11 ¼ e22 ¼ 0 and e12 ¼ �e21 ¼ 1. Using conditions (5) and (14), the expression for the stress vector (13) on the lateralsurface Cw, can be rewritten as

tw ¼1

/ð1þ /2z Þ

1=2 s1x1/z � pex1 � s2x2ð1þ /2z Þ

1=2� �" #

e1 þ1

/ð1þ /2z Þ

1=2 s1x2/z � pex2 þ s2x1ð1þ /2z Þ

1=2� �" #

e2

þ 1

ð1þ /2z Þ

1=2 s1 þ pe/zð Þ" #

e3: ð15Þ

Instead of satisfying the momentum equation (4)1 pointwise in the fluid, we impose the following integral conditions

ZSðz;tÞ

r � T � q@#

@tþ # � r#

� �� �da ¼ 0; ð16Þ

ZSðz;tÞ

r � T � q@#

@tþ # � r#

� �� �xa1 . . . xaN da ¼ 0; ð17Þ

where N ¼ 1;2;3. Using the divergence theorem and integration by parts, Eqs. (16), (17) for nine directors, can be reduced tothe four vector equations:

@n@zþ f ¼ a; ð18Þ

@ma1 ...aN

@zþ la1 ...aN ¼ ka1 ...aN þ ba1 ...aN ; ð19Þ

where n; ka1 ...aN ; ma1 ...aN are resultant forces defined by

n ¼Z

ST3da; ka ¼

ZS

Tada; ð20Þ

kab ¼Z

STaxb þ Tbxa� �

da; ð21Þ

kabc ¼Z

STaxbxc þ Tbxaxc þ Tcxaxb

� �da ð22Þ

and

ma1 ...aN ¼Z

ST3xa1 . . . xaN da: ð23Þ

The quantities a and ba1 ...aN are inertial terms defined by

a ¼Z

Sq

@#

@tþ # � r#

� �da; ð24Þ

ba1 ...aN ¼Z

Sq

@#

@tþ # � r#

� �xa1 . . . xaN da ð25Þ

and f ; la1 ...aN , which arise due to surface traction on the lateral boundary, are defined by

F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116 111

f ¼Z@S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ /2

z

qtwds; ð26Þ

la1 ...aN ¼Z@S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ /2

z

qtwxa1 . . . xaN ds: ð27Þ

The equation for the mean pressure gradient, wall shear stress and swirling scalar function will be obtained using the result-ing quantities (20)–(27) in Eqs. (18), (19).

3. Flow in rigid tubes

In this section we derive, in the case of a rigid tube, the unsteady relations between volume flow rate, mean pressure andswirling function. So, let us consider flow in a rigid walled tube, i.e.,

/ ¼ /ðzÞ: ð28Þ

On the boundary of the rigid tube, we impose no-slip conditions, requiring that the velocity field (9) vanishes identically onthe surface (3). It follows that

nþ /2r ¼ 0; xþ /2w ¼ 0; v3 þ /2c ¼ 0: ð29Þ

Now, taking into account (28), that Eq. (10) is satisfied identically, the two independent incompressibility conditions (12)reduce to

ðv3Þz þ 2n ¼ 0; /2v3� �

z ¼ 0: ð30Þ

Conditions (7), (9), (29)3 and (30)2 imply that the volume flow rate Q is just a function of time t, given by

QðtÞ ¼ p2

/2ðzÞv3ðz; tÞ: ð31Þ

Replacing the velocity field (9) and the stress vector (15) in Eqs. (20)–(27) with conditions (29), (12)1, (28), (31) we can cal-culate explicitly the forces n; ka1 ...an ;ma1 ...an , the inertial terms a;ba1 ...aN and the surface tractions f ; la1 ...aN that arise from thenine-directors theory. Hence, plugging the solutions of Eqs. (20)–(27) into Eqs. (18), (19) and using (8), we get an equationfor the average pressure

�pzðz; tÞ ¼ �8lp/4

�A QðtÞ � 4q3p/2

�B Q tðtÞ �4q

p2/5�C Q 2ðtÞ þ 3qx2//z

20þ qxxz/

2

20; ð32Þ

where

�A ¼ 1þ 13

/2z þ

116

/4z þ

13

//zz þ14

//2z /zz þ

132

/2/2zz �

196

/3/zzzz; ð33Þ

�B ¼ 1þ 316

/2z þ

116

//zz ð34Þ

and

�C ¼ �/z �3

40//z/zz þ

140

/2/zzz: ð35Þ

Moreover, the equation for the wall shear stress is given by

s1ðz; tÞ ¼4l

p/3ð1þ /2z Þ

A0QðtÞ þ q6p/ð1þ /2

z ÞB0QtðtÞ þ

2q3p2/4ð1þ /2

z ÞC 0Q 2ðtÞ þ q/2

1þ /2z

x2/z

30þxxz/

40

� �; ð36Þ

where

A0 ¼ 1þ 13

/2z þ

116

/4z þ

13

//zz þ38

//2z /zz þ

132

/2/2zz þ

124

/2/z/zzz �1

96/3/zzzz; ð37Þ

B0 ¼ 1� 14

/2z þ

14

//zz ð38Þ

and

C 0 ¼ �/z þ12

/3z �

1940

//z/zz þ3

40/2/zzz: ð39Þ

Finally, we obtain also the following partial differential equation for the swirling scalar function xðz; tÞ:

2 In cRoberts

112 F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116

0 ¼ x 16� 2//zz þ 6/2z þ

12q/zQðtÞ5lp/

� �þxz

6qQðtÞ5lp

� 4//z

� ��xzz/

2 þxtq/2

l: ð40Þ

Now, integrating condition (32) over a finite section of the tube with z1 < z2, we obtain

GðtÞ ¼ 8lAp

� �QðtÞ þ 4qB

3p

� �Q tðtÞ þ

4qCp2

� �Q2ðtÞ � q

20

� �NðtÞ; ð41Þ

where

A ¼ 1L

Z z2

z1

�A

/4 dz; B ¼ 1L

Z z2

z1

�B

/2 dz; C ¼ 1L

Z z2

z1

�C

/5 dz; ð42Þ

N ¼ 1L

Z z2

z1

3x2//z þxxz/2� �

dz ð43Þ

and

GðtÞ ¼�pðz1; tÞ � �pðz2; tÞ

L; L ¼ z2 � z1: ð44Þ

Function GðtÞ is the mean pressure gradient over the interval ½z1; z2� at time t. The constants A;B and C are just determined bythe geometry of the tube over the interval ½z1; z2�, and N is determined by the geometry of the tube over the interval ½z1; z2�and also by the swirling scalar function. The first term on the right hand side of (41) represents the contribution from viscouseffects, the second one arises from unsteady effects, the third term from convective acceleration, and the fourth term fromswirling motion effects.

Remark 1. The case xðz; tÞ ¼ 0 in equations (32), (36), (40) and (41) was studied by Caulk and Naghdi (1987) for a straighttube of variable circular cross-section, and the results were validated on the special case of a straight tube of constantcircular cross-section (see Caulk and Naghdi, 1987). The steady case for swirling motion, with

xðzÞ ¼ -0exp � bz/

� �;

where -0 is a material constant and b represents the swirl decay, given by

b ¼ 12� 6qQ

5p/lþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi64þ 6qQ

5p/l

� �2s8<:

9=;;

was studied and validated by Caulk and Naghdi (1987) for a straight tube of constant circular cross-section /.

Now, let us consider the following dimensionless variables2

z ¼ z/0; / ¼ /

/0; t ¼ x0 t; Q ¼ 2q

p/0lQ ; x ¼ fx; ð45Þ

where /0 is the characteristic radius of the tube, x0 is a characteristic frequency for unsteay flow and f is the Coriolis fre-quency. Substituting the new variables (45) in Eq. (32), we obtain

�pz ¼ �4A

/4Q ðtÞ � 2

3W2

0B

/2Q t ðtÞ �

C

/5Q2 ðtÞ þ R0

203w2//z þ wwz/

2� �

; ð46Þ

where

W0 ¼ /0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqx0=l

p

is the Womersley number, and

R0 ¼q2/4

0

f 2l2

is the Rossby number. A small Rossby number means that the system is strongly affected by Coriolis forces, and a large Ross-by number means that a system is dominated by inertial and centrifugal forces. The Womersley number is the most com-monly used parameter to reflect the pulsatility of the flow. Moreover, in Eq. (46) we have

ases where a steady flow rate is specified, the nondimensional flow rate bQ is identical to the classical Reynolds number used for flow in tubes, seeon and Sequeira (2005).

F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116 113

A ¼ 1þ 13

/2z þ

116

/4z þ

13

//zz þ14

//2z /zz þ

132

/2/2zz �

196

/3/zzzz; ð47Þ

B ¼ 1þ 316

/2z þ

116

//zz ð48Þ

and

C ¼ �/z �3

40//z/zz þ

140

/2/zzz: ð49Þ

Now, integrating condition (46) over a finite section of the tube with z1 < z2, we obtain the following relationship betweenthe mean pressure gradient, tube geometry, volume flow rate, Womersley number, Rossby number and swirling motioneffects

GðtÞ ¼�pðz1; tÞ � �pðz2; tÞ

L¼ 4A1Q ðtÞ þ 2

3W2

0A2Q t ðtÞ þ A3Q 2ðtÞ � R0

20NðtÞ; ð50Þ

where

A1 ¼1

L

Z z2

z1

A

/4

!dz; A2 ¼

1

L

Z z2

z1

B

/2

!dz; A3 ¼

1

L

Z z2

z1

C

/5

!dz ð51Þ

and

N ¼ 1

L

Z z2

z1

3x2//z þ xxz/2

� �dz: ð52Þ

Moreover, the dimensionless form of the wall shear stress, i.e., Eq. (36), is given by

s1ðz; tÞ ¼2D

/3ð1þ /2z Þ

Q ðtÞ þ 112W2

0E

/ð1þ /2z Þ

Q t ðtÞ þ16

F

/4ð1þ /2z Þ

Q 2 ðtÞ þ R0/2

1þ /2z

x2/z

30þ xxz/

40

" #; ð53Þ

where

D ¼ 1þ 13

/2z þ

116

/4z þ

13

//zz þ38

//2z /zz þ

132

/2/2zz þ

124

/2/z/zzz �1

96/3/zzzz; ð54Þ

E ¼ 1� 14

/2z þ

14

//zz ð55Þ

4 2 2 4

2

1

1

2

Fig. 2. Schematics of a constricted tube geometry with lateral wall defined by (58).

Fig. 3. Illustration of the nondimensional swirling scalar function, computed from equation (57) with QðtÞ ¼ sin2 ðtÞ and W20 ¼ 1.

R0 = 0 25

R0 = 7 5

R0 = 1 5

R0 = 15

Fig. 4. Nondimensional unsteady wall shear stress, given by (53), related with different values of the Rossby number.

114 F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116

and

F ¼ �/z þ12

/3z �

1940

//z/zz þ3

40/2/zzz: ð56Þ

Finally, the dimensionless form of Eq. (40), is given by

0 ¼ x 16/� 2/2/zz þ 6//2z þ

65

/zbQ ðtÞ� �

þ xz35

/Q ðtÞ � 4/2/z

� �� xzz/

3 þ xtW20/

3: ð57Þ

2 4 6 8 10t

0.005

0.010

0.015

0.020

0.025

0.030

Fig. 5. Difference in nondimensional unsteady mean pressure gradient, given by (50), between R0 ¼ 0:25 and R0 ¼ 15.

F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116 115

3.1. Flow in a rigid constricted tube

In this subsection we apply the proposed model to investigate flow in a constricted tube, providing details and some pre-liminary numerical results. In particular, we consider flow in a rigid constricted tube (see Fig. 2), where the lateral wall isdefined by

/ðzÞ ¼ 1þ z2; z 2 ½z1; z2� ¼ ½�2;2�; ð58Þ

here z1; z2 are fixed, and the maximum tube constriction occurs at z ¼ 0 and corresponds to a diameter of 1.Taking into account this particular expression for the surface of revolution (58), Eq. (57) can be solved numerically, choos-

ing appropriate initial and boundary conditions. In this preliminary work, the equation was solved by the method of lines,i.e., the space derivatives were discretised by means of finite difference formulas and the resulting system of ordinary dif-ferential equations was solved by a Runge–Kutta method of order 4. The stiffness of this system of ODEs may require specialattention for more general flow conditions.

In Fig. 3 we represent the scalar swirling function in the full length of the tube, for the time interval ½0;10�. The results areconsistent with the reality, with the swirling being higher as the fluid approaches the constriction and lower near theconstriction.

Fig. 4 displays the nondimensional unsteady wall shear stress, given by (53), for increasing values of the Rossby number.We observe that, as the Rossby number increases, there is a break of symmetry in the stress conditions before and after theconstriction. For low Rossby numbers the fluid returns to its initial state, while for larger Rossby numbers the stress functionkeeps increasing after the constriction. Although he simplicity of the model prevents us form drawing any solid physical con-clusions, this may be understood as an installation of turbulence downstream to the constriction.

Finally, we present numerical results regarding unsteady mean pressure gradient, given by Eq. (50). In Fig. 5 we can ob-serve that, due to the imposed periodic volume flow rate, the pressure gradient does not suffer much variation across thisrange of Rossby numbers (½0:25;15�).

4. Conclusions

A nine director theory has been used to derive a one-dimensional model in a straight rigid tube of variable circular cross-section providing a tool for predicting some of the main properties of the three-dimensional viscous fluid model with swirl-ing motion. We have obtained unsteady relations between mean pressure gradient (wall shear stress, respectively), volumeflow rate and swirling scalar function, for a given set of tube geometry, Womersley number and Rossby number. In this pre-liminary work, we presented numerical results regarding unsteady mean pressure gradient and unsteady wall shear stress,for a specific constricted tube with QðtÞ ¼ sin2ðtÞ;W2

0 ¼ 1 and for different values of the Rossby number. Possible extensionsof this work are the application of this hierarchical approach theory to improve the understanding of specific physical phe-nomenas related with swirling motions, and also the coupling of Cosserat models in geometrical multi-scale models.

Acknowledgements

This work has been partially supported by Centro de Investigação em Matemática e Aplicações from Universidade de Évo-ra (CIMA/UE) through Fundação para a Ciência e a Tecnologia (FCT), and by Centro de Matemática Aplicada á Previsão e Dec-isão Económica, Instituto Superior de Economia e Gestão da Universidade Técnica de Lisboa (CEMAPRE/ISEG) through theGrant PEst-OE/EGE/UI0491/2011 of Fundação para a Ciência e Tecnologia (FCT).

116 F. Carapau, J. Janela / International Journal of Engineering Science 72 (2013) 107–116

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