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Brief Communication
Analytical solutions for the axisymmetric flow inside a cylindrical
container with a rod along the axis at low Reynolds numbers
Juan Carlos Sturzenegger, Luis G. Sarasu
´a, Arturo C. Martı
´
n
Instituto de Fı
´sica, Igua
´4225, Universidad de la Repu
´blica, 11400 Montevideo, Uruguay
article info
Article history:
Received 1 June 2010
Accepted 2 November 2011
Available online 26 November 2011
Keywords:
Vortex breakdown
Recirculation flow
Control
abstract
The axisymmetric flow inside a cylindrical container with a rod along its symmetry axis
is studied. The flow is produced by the rotation of one of the cylinder end walls, of both
end walls, or of the sidewall. Analytical expressions (for low Reynolds numbers) of the
azimuthal velocity field are presented, thus extending the solutions of the case without
rod. Graphics of constant azimuthal velocity lines are obtained for two configurations
(both end walls in rotation, one end wall in rotation). The calculations are based on
summations with different number of terms. The introduction of Lanczos factors in the
summations is shown to play an important role improving their convergence.
&2011 Elsevier Ltd. All rights reserved.
1. Introduction
The dynamics of viscous and incompressible flows inside cylindrical containers with rotating end or lateral walls has
received considerable attention (Brown and Lo
´pez, 1990;Khalili and Rath, 1994;Lo
´pez, 1990,1995,1996). Analytical
solutions of the flow inside the container are not possible in general but only in some special cases (Khalili and Rath, 1994;
Pao, 1972;Schmeiden, 1928;Schultz-Grunow, 1935). In this contribution we present an example in which an analytical
solution can be calculated.
One of the first studies of the flow produced by a rotating disk inside a cylindrical container was made by Schultz-
Grunow (1935). That work includes an analytical solution of the flow without satisfying the non-slip boundary conditions
on the container sidewall. The obtained solution is rather close to the experimental results when the aspect ratio
d
¼H=R
(where Hand Rare respectively the container height and radius) is very small. It is also worth mentioning that analytical
solutions of the flow in a cylindrical container which verify all the boundary conditions but with the restriction that the
Reynolds number is small have been obtained in previous works by Schmeiden (1928),Pao (1972),Khalili and Rath (1994),
Muite (2004) and Cao et al. (2010).
The dynamics of closed flows inside cylindrical containers is relevant not only from an academic point of view. Indeed,
for certain values of the characteristic parameters, when the swirl is sufficiently large, vortex breakdown (VB) occurs. This
phenomenon is characterized by the appearance of a stagnation point near the axis, followed by regions of reversed axial
flow with a bubble structure. This structural transformation is also accompanied by a sudden change of the core size and
the appearance of disturbances located downstream of the core enlargement. Several experimental studies on this
phenomenon have been performed, one of the most outstanding is that of Escudier (1984). As it can be seen in that article,
depending on the parameters VB produces one, two, or three recirculating bubbles at the symmetry axis.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/jfs
Journal of Fluids and Structures
0889-9746/$ - see front matter &2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfluidstructs.2011.11.002
n
Corresponding author. Tel.: þ598 25258624; fax: þ598 25250580.
E-mail address: marti@fisica.edu.uy (A.C. Martı
´).
Journal of Fluids and Structures 28 (2012) 473–479
Due to the great number of practical implications of VB, the development of mechanisms for controlling its emergence
is of considerable interest. The introduction of a cylindrical rod along the symmetry axis is one of the methods used to
control the flow and the appearance of VB.
There are several experimental articles in which the behavior of VB is studied when adding a cylindrical rod (which can
be straight or slightly conical), see, for instance, Mullin et al. (2000),Husain et al. (2003), and Cabeza et al. (2010). In the
work by Mullin et al. (2000), the effect of adding a small rod along the axis of the cylinder is shown to be robust despite the
qualitative change in the boundary conditions. It is also reported that sloping the inner cylinder has a dramatic effect
enhancing or suppressing the recirculation. The approach to control VB presented by Husain et al. (2003) is based on the
addition of co- or counter-rotation near the axis using a small central rod rotating independently of the bottom or rotating
end wall. It is concluded that such a method may be an effective way to either enhance or suppress VB. More recently,
using a device which does not require the addition of swirl near the axis (and therefore is more feasible in engineering
devices), Cabeza et al. (2010) have shown that the rod may increase or decrease the critical Reynolds number for the
emergence of VB depending on the aspect ratio and the radius of the rod.
Other approaches to the control of VB are based on the introduction of a small rotating disk in the end wall opposite to
the rotating wall (Mununga et al., 2004;Yu et al., 2006,2007), or the use of axial temperature gradients (Herrada and
Shtern, 2003). More recently, Lo Jacono et al. (2008) studied the effect of the length of the rod. In coincidence with other
previous results, they found that for large rods the co-rotation suppresses VB, unlike counter-rotation which enhances VB
region. On the other hand, for short rod lengths, the behavior of the recirculation zone is more complex and as the rod
shortens, the behavior approaches the limit of a flat disk and the recirculation zone bubble vanishes for sufficient counter-
rotation speed.
In this paper we extend previous analytical solutions of the flow inside a cylindrical container to the case in which a
small stationary rod is placed along the axis. Moreover, we study the convergence of the series and the influence of the so-
called Lanczos factors. Firstly, in Section 2 we formulate the problem and present its solution in the classical configuration
without rod. Next, in Section 3 we obtain the analytical solution to the problem with rod. The velocity fields are presented
in Section 4 for some particular cases. Finally, in Section 5, we present a summary and the conclusion.
2. Problem formulation without rod
We consider the flow in a cylinder of radius Rand height Hwith top, bottom and lateral walls either at rest or rotating
with arbitrary angular velocities. The cylinder is filled with fluid of constant kinematic viscosity
n
. In the case of one single
angular velocity
O
, the two dimensionless numbers that characterize this problem are the Reynolds number Re ¼
O
R
2
=
n
and the aspect ratio
d
¼H=R.
In this study, as it is restricted to low Reynolds numbers, we assume that the flow is axisymmetric (Brown and Lo
´pez,
1990;Lo
´pez, 1990). Then, the flow is governed by the axisymmetric Navier–Stokes equations which can be written using a
cylindrical polar coordinate system ðr,
y
,zÞwhose symmetry axis coincides with the axis of the cylinder. The corresponding
velocity field is denoted as v
r
,vy,v
z
, and the pressure as p. We make the governing equations non-dimensional by scaling
the radius of the cylinder and one characteristic angular velocity. Then the top and bottom wall are located at z¼0 and
z¼
d
while the sidewall is located at r¼1. Through all this work we consider non-slip boundary conditions. Moreover, the
introduction of the stream function
c
where v
r
¼ð1=rÞ@
c
=@z, and v
z
¼ð1=rÞ@
c
@rwill prove to be useful.
Broadly speaking, the rotation of the cylinder lid has the following effects. The fluid inside the cylinder in the vicinity of
the moving disk is driven by it towards the periphery, in a spiral movement (with azimuthal and radial components). On
the other hand, there is also an axial and radial recirculation: the fluid moves axially inside the cylinder along the
periphery from the region near the rotating disk into the region near the opposite disk. In this latter region, it moves
radially from the periphery towards the center. It then continues moving along the symmetry axis towards the region near
the rotating disk, completing the cycle. The conservation of angular momentum causes the fluid driven towards the
symmetry axis to increase its angular velocity, creating a vortex along the axis.
An analytical solution for the flow inside the cylindrical container – when VB has not yet taken place – is given by Muite
(2004). Using asymptotic expansions in Re, it can be obtained (Hills, 2001)
vy¼vy
,0
þðReÞvy
,1
þðReÞ
2
vy
,2
þ,ð1Þ
and similar expansions for the stream function. These expressions are replaced in the Navier–Stokes equations and
discriminated for different orders of Re. The zero-order flow equations, which have only azimuthal component, are
obtained. Thus,
c
0
¼0 and
1
d
2
@
2
vy
,0
@z
2
þ@
2
vy
,0
@r
2
þ1
r
@vy
,0
@rvy
,0
r
2
¼0:ð2Þ
We write hereafter zinstead of z=
d
. The zero-order analytical solution is obtained
vy
,0
¼zr þ2X
1
n¼1
sin½n
p
ðz1Þ
n
p
I
1
½
l
n
r
I
1
½
l
n
,ð3Þ
J.C. Sturzenegger et al. / Journal of Fluids and Structures 28 (2012) 473–479474
where I
1
is the first-order modified Bessel function of the first kind and
l
n
¼n
p
=
d
. The flow first-order analytical solution
can be similarly obtained (Muite, 2004).
In those systems with sliding walls, there is a discontinuity in the flow in the periphery of the rotating lid as the sidewall is
at rest. The lid drags the fluid next to it and the sidewall, on the other hand, imposes low velocities in the fluid nearby. Muite
(2004) has also considered how this affects discontinuity in the analytical solutions found. The discontinuous boundary
conditions reduce the convergence rate of series solutions, and to get a physically reasonable solution of the velocity field, the
conditions along the discontinuous edge cannot be modeled as a discontinuous jump.
One solution is to truncate the series representation. But simple truncation does not assure that the result is an accurate
representation of the velocity field. In a trigonometric expansion, a procedure that results in a truncated series expansion
that approximates the required function is the implementation of the so-called Lanczos correction factors (Lanczos, 1956;
Muite, 2004). These factors replace the discontinuous velocity jump by a region of rapid transition that preserves the
properties of the solution away from the discontinuity. The resulting zero-order solutions are
vy
,0
¼zr þ2X
k
n¼1
sin½n
p
ðz1Þ
n
p
I
1
½
l
n
r
I
1
½
l
n
sin
n
p
k
n
p
k
:ð4Þ
In a somewhat similar approach Khalili and Rath (1994) have also considered this problem. Under the assumptions of
axisymmetry and that the radial and axial components of flow velocities are small compared to the azimuthal component
v
r
5vy,v
z
5vy, which meets approximately reality when Rer10, using the Navier–Stokes equations, the azimuthal
equation takes the form
d
2
@
2
v
@r
2
þ1
r
@v
@rv
r
2
þ@
2
v
@z
2
¼0,ð5Þ
where v¼vy=R
o
is the azimuthal dimensionless velocity. This equation is solved by substituting the following kind of
solution
vðr,zÞ¼½ðs1Þzþ1rþf
1
ðzÞf
2
ðrÞ,ð6Þ
and separating variables. The lower lid and the upper lid rotate respectively with angular velocities
o
1
and
o
2
, and the
parameter s¼
o
2
=
o
1
is introduced. Solving the two ordinary differential equations obtained for f
1
ðzÞand f
2
ðrÞthe
analytical solution is obtained
vðr,zÞ¼½ðs1Þzþ1rþX
1
n¼1
A
n
I
1
ð
l
n
rÞsinðn
p
zÞ,ð7Þ
where the coefficients A
n
take the following form
A
n
¼2
n
p
I
1
ð
l
n
Þ½sð1Þ
n
1:ð8Þ
This problem can be generalized to the case in which the lower lid, the upper lid, and the sidewall rotate respectively
with angular velocities
o
1
,
o
2
and
o
3
. The parameters s
1
¼
o
1
=
o
,s
2
¼
o
2
=
o
and s
3
¼
o
3
=
o
are defined, where
o
is some
characteristic angular velocity. In that case, Eqs. (7) and (8) take the following forms
vðr,zÞ¼½ðs
2
s
1
Þzþs
1
rþX
1
n¼1
A
n
I
1
ð
l
n
rÞsinðn
p
zÞ,ð9Þ
A
n
¼2
n
p
I
1
ð
l
n
Þ½ðs
2
s
3
Þð1Þ
n
þðs
3
s
1
Þ:ð10Þ
3. Solution with rod along the axis
The objective of our work is to obtain analytical solutions of the Stokes flow inside the cylindrical container, but with
the addition of a cylindrical rod which coincides with the cylinder symmetry axis. The rod extends from z¼0toz¼1, and
its radius is r¼do1. Under the assumptions of axisymmetry and of radial and axial flow components being small
compared to the azimuthal component, Eq. (5) is still valid but the boundary conditions are different. In this case, the
analytical solution can be written as
vðr,zÞ¼½ðs1Þzþ1rþX
1
n¼1
½A
n
I
1
ð
l
n
rÞþB
n
K
1
ð
l
n
rÞsinðn
p
zÞ,ð11Þ
where also appears K
1
, the first-order modified Bessel function of the second kind. The parameter sis the same one as
in Eq. (7).
J.C. Sturzenegger et al. / Journal of Fluids and Structures 28 (2012) 473–479 475
So as to complete the search of the velocity field solutions, it is necessary to calculate the coefficients A
n
and B
n
using
the boundary conditions. Taking the last equation into account, at the top (z¼1) and bottom (z¼0) walls the boundary
conditions are always satisfied. From the boundary conditions at the rod, r¼d, and the sidewall r¼1(0ozo1) and
projecting these two equations on the function family sinðn
p
zÞthe coefficients A
n
and B
n
can be easily derived
A
n
¼2
n
p
½sð1Þ
n
1K
1
ð
l
n
dÞK
1
ð
l
n
Þd
I
1
ð
l
n
ÞK
1
ð
l
n
dÞI
1
ð
l
n
dÞK
1
ð
l
n
Þ,ð12Þ
B
n
¼2
n
p
½sð1Þ
n
1I
1
ð
l
n
dÞI
1
ð
l
n
Þd
K
1
ð
l
n
ÞI
1
ð
l
n
dÞK
1
ð
l
n
dÞI
1
ð
l
n
Þ:ð13Þ
This problem can be also generalized to the case where the lower lid, the upper lid and the lateral wall are rotating with
different angular velocities. Using the same notation of the previous section, in this case, Eq. (11) takes the following form
vðr,zÞ¼½ðs
2
s
1
Þzþs
1
rþX
1
n¼1
½A
n
I
1
ð
l
n
rÞþB
n
K
1
ð
l
n
rÞsinðn
p
zÞ:ð14Þ
The coefficients A
n
and B
n
can be analogously found
A
n
¼2
n
p
½ðs
2
s
3
Þð1Þ
n
þs
3
s
1
K
1
ð
l
n
dÞK
1
ð
l
n
Þd
I
1
ð
l
n
ÞK
1
ð
l
n
dÞI
1
ð
l
n
dÞK
1
ð
l
n
Þ,ð15Þ
B
n
¼2
n
p
½ðs
2
s
3
Þð1Þ
n
þs
3
s
1
I
1
ð
l
n
dÞI
1
ð
l
n
Þd
K
1
ð
l
n
ÞI
1
ð
l
n
dÞK
1
ð
l
n
dÞI
1
ð
l
n
Þ:ð16Þ
4. Velocity fields
In this section we use Eq. (14) to calculate the azimuthal velocity field inside the cylindrical cavity with two lids and a
rod along its axis corresponding to two configurations. The calculated velocity fields are compared with those of the case
where there is no rod along the axis studied by Khalili and Rath (1994).
The first configuration, shown in Figs. 1 and 2, corresponds to both lids in rotation with the same angular velocity
(s
1
¼s
2
¼1), and still sidewall (s
3
¼0). These conditions are imposed on Eqs. (9) and (14) respectively in both cases (with
and without rod along the axis). As the calculations cannot be performed with infinite terms, the summations must be
truncated at a certain k. It is expected that for some k, the truncated summation will be fairly close to the infinite
summation. After several tests, it was found that when taking 50 terms, smooth lines of azimuthal velocity field are
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
r
z
Fig. 1. Constant azimuthal velocity lines, for s
1
¼1, s
2
¼1, s
3
¼0, and
d
¼1; calculated as a summation of 50 terms, for the cases with rod (thick lines)
and without rod (thin lines). The values of azimuthal velocity on the lines are equally spaced, ranging from 0.1 to 0.8, and d¼0:1.
J.C. Sturzenegger et al. / Journal of Fluids and Structures 28 (2012) 473–479476
obtained, as expected. These are similar to those presented in the paper by Khalili and Rath (1994). The graphics obtained
can be seen in Fig. 1 in which both cases (with and without rods) are superimposed. In this figure we can observe that most
of the lines match, except for the lines with small vnear the symmetry axis, which, in the case with rod, move away from
that axis.
In Fig. 2(a) we can see the velocity fields corresponding to the case with rod, with k¼15 and k¼50 superimposed. The
irregularities in the lines of constant azimuthal velocity near ðr;zÞ¼ð1;0Þand ðr;zÞ¼ð1;1Þclearly increase when k¼15.
The next step is to introduce the so-called Lanczos factors which replace the discontinuous velocity jumps by a region
of rapid transition that preserves the properties of the solution away from the discontinuity (Muite, 2004). In the case
Fig. 2. Case with rod at the axis. Constant azimuthal velocity lines for s
1
¼1, s
2
¼1, s
3
¼0, and
d
¼1; calculated without Lanczos factors for 15 (dashed
lines) and 50 (solid lines) terms of the series expansion (a) and calculated as a summation of 50 terms without Lanczos factors and 15 terms with Lanczos
factors (b). The insets show details in the region around the corner.
J.C. Sturzenegger et al. / Journal of Fluids and Structures 28 (2012) 473–479 477
without rod along the axis the expression reads as
vðr,zÞ¼rþX
k
n¼1
A
n
I
1
ð
l
n
rÞsinðn
p
zÞsin
n
p
k
n
p
k
,ð17Þ
while in the case with rod along the axis as
vðr,zÞ¼rþX
k
n¼1
½A
n
I
1
ð
l
n
rÞþB
n
K
1
ð
l
n
rÞsinðn
p
zÞsin
n
p
k
n
p
k
:ð18Þ
As expected, the introduction of Lanczos factors reduces the irregularities in the lines of constant azimuthal velocity
near ðr;zÞ¼ð1;0Þand ðr;zÞ¼ð1;1Þ, which is noticed specially when graphics with k¼15 with Lanczos factors and k¼50
without Lanczos factors (the latter considered close to the real field) are superimposed (Fig. 2(b)). Thus, it can be seen that
Lanczos factors smooth velocity lines, but these are a little separated from real values in the regions near ðr;zÞ¼ð1;0Þand
ðr;zÞ¼ð1;1Þ. It should be noted that there is also a small discrepancy in the middle region (r0:5, z0:5 for k¼15),
therefore the effect of the Lanczos factors can be favorable or not, depending on the region of the flow.
For the second configuration, with the lower lid in rotation and the top lid fixed, the conditions s
1
¼1, s
2
¼s
3
¼0 and
d
¼1 are imposed on Eqs. (9) and (14). Similarly to the previous case the infinite summations must be truncated at a
certain kobtaining in the case without rod
vðr,zÞ¼½zþ1rþX
k
n¼1
A
n
I
1
ð
l
n
rÞsinðn
p
zÞ,ð19Þ
and in the case with rod along the axis
vðr,zÞ¼½zþ1rþX
k
n¼1
A
n
I
1
ð
l
n
rÞþB
n
K
1
ð
l
n
rÞ
sinðn
p
zÞ:ð20Þ
Following the previous procedure, it was found that when taking 50 terms, smooth lines for the azimuthal velocity field
were obtained. In Fig. 3 the results corresponding to both cases (with and without rods) are superimposed, this shows that
most of the lines match, except for the lines with small vnear the symmetry axis, which in the case with rod move away
from that axis.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
r
z
Fig. 3. Constant azimuthal velocity lines, for s
1
¼1, s
2
¼0, s
3
¼0, and
d
¼1; calculated as a summation of 50 terms. Thin lines: configuration without rod;
thick lines: configuration with rod ðd¼0:1Þ. The values of azimuthal velocity on the lines are equally spaced, ranging from 0.1 to 0.8.
J.C. Sturzenegger et al. / Journal of Fluids and Structures 28 (2012) 473–479478
5. Conclusion
In this paper we have obtained analytical expressions for the Stokes flow of the azimuthal velocity field inside a
cylindrical container with lids, with a cylindrical rod along its axis. We have thus extended the analytical solutions
obtained by Khalili and Rath (1994) and Muite (2004) including the addition of a rod along the cylinder axis.
We have obtained graphics of constant azimuthal velocity lines for two configurations (both lids in rotation, one of the
lids in rotation) based on summations with different number of terms. We found that 50 terms are enough to obtain good
accuracy. When the number of terms is reduced, there appear defects in the lines.
We have also found that the use of Lanczos factors smoothes constant velocity lines reducing defects. Thus lines of good
accuracy throughout the domain of ðr;zÞvalues can be obtained, except in the vicinity of singular points (1; 0) and (1; 1).
With 15 terms and the use of Lanczos factors, we obtained similar results (except near singular points) to those we
obtained with 50 terms without Lanczos factors.
Moreover, when inspecting the graphics where constant velocity lines corresponding to configurations with and without
rods are superimposed, it can be seen that they match in almost all the domain of ðr;zÞvalues, except in the region near the
cylinder symmetry axis (r¼0). Thus we see that the effect of the modification in the boundary conditions is quite large near
the rod. This result, in the limit of low Reynolds numbers, could be related to the experimental observation that for
intermediate Re, the boundary layer generation near the rod can significantly influence the local flow, and particularly, VB and
the necessary Re to obtain it, in spite of the fact that the overall flow is not drastically modified (Cabeza et al., 2010).
Finally, it is worth emphasizing the present approach could be extended to different setups, for example, a rod rotating
(Husain et al., 2003), instead of being at rest, or a small disk rotating near the end wall opposite to the rotating wall
(Mununga et al., 2004). We hope that obtaining analytical expressions for the solutions in those cases will contribute to a
better understanding and control of VB phenomena.
Acknowledgments
We thank the anonymous reviewers for their careful and thoughtful reading. We acknowledge financial support from
PEDECIBA and CSIC (Uruguay).
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