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International Journal of Systems Science and Applied Mathematics
2020; 5(3): 32-35
http://www.sciencepublishinggroup.com/j/ijssam
doi: 10.11648/j.ijssam.20200503.12
ISSN: 2575-5838 (Print); ISSN: 2575-5803 (Online)
Axisymmetric Flow Through a Cylinder with Porous
Medium
Muhammad Umar Farooq
*
, Abdul Rehman, Naveed Sheikh, Manzoor Iqbal
Department of Mathematics, University of Balochistan, Quetta, Pakistan
Email address:
*
Corresponding author
To cite this article:
Muhammad Umar Farooq, Abdul Rehman, Naveed Sheikh, Saleem Iqbal. Axisymmetric Flow Through a Cylinder with Porous Medium.
International Journal of Systems Science and Applied Mathematics. Vol. 5, No. 3, 2020, pp. 32-35. doi: 10.11648/j.ijssam.20200503.12
Received: October 2, 2020; Accepted: October 20, 2020; Published: October 27, 2020
Abstract:
The present work is a struggle to establish a mathematical appearance of the conduct of axisymmetric fluid flow
in a moving cylinder confined in a porous medium. The fluid is assumed to be flowing through the annular region formed
between two concentric smooth cylinders for the case when the outer cylinder is kept fixed while the inner cylinder is assumed
to be moving with a constant velocity along the axial direction and is also assumed to be rotating with a constant angular
velocity with reference to the centre line along the axial axis. Firstly, the conducting equations of motion are obtained in the
form of a system of coupled non-linear partial differential equations with corresponding boundary conditions. The system is
then transformed into a new set of coupled non-linear ordinary differential equations using a set of suitable similarity
transformation. The problem is then solved using the fourth order numerical technique, the Runge-Kutta-Shooting method. The
concluding results are derived for non- dimensional coupled differential equations. In the end the results are graphically
presented and the behaviour of porosity parameter over the fluid flow is examined. The observed results indicated that with
increasing values of the Reynolds’s numbers the non-dimensional linear and axial velocities also increases.
Keywords:
Moving Cylinder, Porous Medium, Runge-Kutta 4
th
Order Shooting Method
1. Introduction
From the last quarter of the previous century, the study
of lubricants protecting the moving cylinder in the
automobiles as well as in industrial machines had become
a research target. Considering the required portion of
medicine for a particular disease in a syrup, the solid
particles are quite effectively used. Although the petro
chemical liquid is area of interest of scientists for a long
period to get new substances but the nature of different
solid particles present in the fluids changes the results in
targeted areas. The stable state viscous glide as well as
thermal conductivity of a fluid with density variation
connected to temperature change, concludes the result of
the exact solution of the Navier–Stokes velocity and
energy equation extracted in the case, here temperature of
disc or its associated wall heat is treated as fixed. Putra et
al. [1] studied the behaviour of a nanofluid depending
upon the ingredients composition, correlation and size of
the particle. Kuznetsov & Nield [2] explained the thermal
irregularity. Khan and Pop [3] studied the boundary layer
flow on an extended covering. Nadeem and Rehman [4]
discussed the stagnation-flow of a nanofluid over an axial
cylinder. From infinite circular cylinder heat transfer and
fluid glide around the surface was studied by Khan et al.
[5]. An analytic approach was performed for heat transfer
and fluid flow along elliptical cylinders by Khan et al. [6].
Covering Yawed circular cylinders by real fluid flow was
presented in mathematical relation considering crosswise
and piecewise velocity by Chiu and Lien hard [7]. Around
a round cylinder, similarities of flow and heat transfer
along drag coefficient idea were floated by Ma and Duan
[8]. Few other related works are cited in [9-11].
A stagnation point is a point in a flow field where the local
velocity of the fluid is zero. Stagnation points exist at the
surface of objects in the flow field [12-15]. The purpose of
the present effort is to study the behavior of fluid flowing
through the region formed between two concentric cylinders
[16-19] through some porous medium.
International Journal of Systems Science and Applied Mathematics 2020; 5(3): 32-35 33
2. Formulation of the Problem
The aim of present paper is to study the problem of fluid
flow through the annular region formed between two
concentric cylinders, where the inner cylinder with the radius
R is assumed to be rotating with a constant angular velocity
and is also translating with a constant velocity along the
axial axis, the outer cylinder with radius
is assumed
fixed while fluid is injected from the top of the outer cylinder
towards the inner cylinder with a constant velocity along
radial axis. The obtained governing set of coupled, nonlinear
partial differential equations for conservation of mass and
momentum is of the form [20-22]
(1)
!
"
(2)
!
"
(3)
!
"
(4)
Where (u, v, w) are the velocity components along
#$# %-axes and & is the density.
The associated boundary conditions are
#% '# % # % (5)
# % '
#%
#% (6)
Define the transformation [23-24]
()*
+*
#' ,-, (7)
./
0
-1 2-, (8)
-
3
#1
3
, (9)
With the help of the above transformations, Equation (1) is
identically satisfied and take the following form.
-
4
5
)
4*
5
.
4
6
)
4*
6
7 /
4
6
)
4*
6
4)
4*
4
)
4*
89
4
)
4*
, (10)
-
4
:
4*
4:
4*
7 /
4:
4*
2
4)
4*
892 , (11)
;-
4
<
4*
;
4<
4*
<
*
7 ;/
4<
4*
=)<
*
89, , (12)
Where 89
3
>!
?
is the porosity parameter, 7
(3
=
is
the cross-flow Reynolds number and
@
The boundary conditions in non-dimensional form are
/A #
4)
4*
#/B#
4)C
4*
, (13)
2A A#2 #,A A# , , (14)
$A A#$ #
DA A#D , (15)
Introduce the following functions
E
F/
F- #E
=
FE
F- #E
G
FE
=
F- #H F2
F- IJFK F,
F-
Then the set of equations (10-12) and the boundary
conditions (13) take the following form
-
4L
6
4*
.
4L
4*
7 /
4L
4*
4)
4*
M
4L
N
4*
89
4
)
4*
, (16)
-
4O
4*
4:
4*
7 /
4:
4*
4)
4*
2 89 , (17)
;-
4P
4*
;
4<
4*
<
*
7 ;/
4<
4*
=)<
*
89 (18)
The boundary condition in non-dimensional form are
/A #E
A #/B# E
# (19)
2A #2 #,A A# , @ (20)
3. Results and Discussion
The presented research paper is an effort to provide a
numerical pattern that observes the flow along the surface of
a rotating as well as vertically moving cylinder in a porous
medium. The numerical results are calculated for
nondimensional system of first order differential equations
by using the 4
th
order Runge-Kutta shooting method [25-28].
In Figure 1 it is observed that the velocity profile increases.
Figure 1 shows the behaviour of velocity profile for different
values of Reynolds Number Re. Figure 2 shows the
behaviour of velocity gradient for different values of the
Reynolds Number Re. From figure a dual behaviour is
observed. In Figure 3 the non-dimensional velocity 2
decreased for increasing values of the porosity parameter89.
Figure 3 shows the behaviour of velocity profile 2 for
different values of the porosity parameter89. Figure 4 shows
the behaviour of velocity profile h for different values of the
porosity parameter89. From the figure it is observed that
with increase in the non-dimensional velocity profile h
increases for increasing values of the porosity parameter89.
Figure 1. Behaviour of non-dimensional velocity profile f for different values
of the Reynolds number7.
34 Muhammad Umar Farooq et al.: Axisymmetric Flow Through a Cylinder with Porous Medium
Figure 2. Behaviour of non-dimensional velocity gradient /Q for different
values of the Reynolds number7.
Figure 3. Behaviour of non-dimensional velocity 2for different values of the
porosity parameter89.
Figure 4. Behaviour of non-dimensional velocity ,for different values of the
porosity parameter89.
4. Conclusion
Main conclusions from the above analysis are:
With increase in the Reynolds numbers the non-
dimensional velocity profile f increases.
With increase in the porosity parameter the non-
dimensional velocity profile g decreases.
With increase in the Reynolds numbers the non-
dimensional velocity profile h increases.
References
[1] N. Putra, W. Roetzel, S. K. Das, Natural convection of nano-
fluids. Heat and mass transfer, 39 (8-9), (2003), 775-784.
[2] D. A. Nield, A. V. Kuznetsov, Thermal instability in a porous
medium layer saturated by a nanofluid. International Journal
of Heat and Mass Transfer, 52 (25-26), (2009), 5796-5801.
[3] W. A. Khan, I. Pop, Boundary layer flow past a wedge moving
in a nanofluid. Mathematical Problems in Engineering,
(2013).
[4] Nadeem, S., & Rehman, A. (2013). Axisymmetric stagnation
flow of a nanofluid in a moving cylinder. Computational
mathematics and modeling, 24 (2), 293-306.
[5] Khan, W. A., J. R. Culham, and M. M. Yovanovich. "Fluid
flow around and heat transfer from an infinite circular
cylinder." (2005), 785-790.
[6] W. A. Khan, J. R. Culham, M. M. Yovanovich, Fluid flow
around and heat transfer from elliptical cylinders: analytical
approach. Journal of thermophysics and heat transfer, 19 (2),
(2005), 178-185.
[7] W. S. Chiu, J. H. Lienhard, On real fluid flow over yawed
circular cylinders (1967).
[8] H. Ma, Z. Duan, Similarities of Flow and Heat Transfer
around a Circular Cylinder. Symmetry, 12 (4), (2020), 658.
[9] S. Nadeem, Abdul Rehman, Mohamed Ali, The boundary
layer flow and heat transfer of a nanofluid over a vertical
slender cylinder, J. NanoEngineering and NanoSystems
(2012) 1-9.
[10] S. Nadeem, Abdul Rehman, Changhoon Lee, Jinho Lee,
Boundary layer flow of second grade fluid in a cylinder with
heat transfer, Mathematical Problems in Engineering, Volume
2012, Article ID 640289.
[11] S. Nadeem, Abdul Rehman, K. Vajravelu, Jinho Lee,
Changhoon Lee, Axisymmetric stagnation flow of a
micropolar nanofluid in a moving cylinder, Mathematical
Problems in Engineering, Volume 2012, Article ID 378259.
[12] Abdul Rehman, S. Nadeem, Mixed convection heat transfer in
micropolar nanofluid over a vertical slender cylinder, Chin.
Phy. Lett. 29 (12) (2012) 124701-5.
[13] S. Nadeem, Abdul Rehman, Axisymmetric stagnation flow of
a nanofluid in a moving cylinder, Comp. Math. Mod. 24 (2)
(2013) 293-306.
[14] Abdul Rehman, S. Nadeem, M. Y. Malik, Stagnation flow of
couple stress nanofluid over an exponentially stretching sheet
through a porous medium, J. Power Tech. 93 (2) (2013) 122-
132.
[15] Abdul Rehman, S. Nadeem, M. Y. Malik, Boundary layer
stagnation-point flow of a third grade fluid over an
exponentially stretching sheet, Braz. J. Che. Eng. 30 (3)
(2013) 611-618.
International Journal of Systems Science and Applied Mathematics 2020; 5(3): 32-35 35
[16] Abdul Rehman, S. Nadeem, Heat transfer analysis of the
boundary layer flow over a vertical exponentially stretching
cylinder, Global J. Sci. Fron. Res. 13 (11) (2013) 73-85.
[17] M. Y. Malik, M. Naseer, S. Nadeem, Abdul Rehman, The
boundary layer flow of Casson nanofluid over a vertical
exponentially stretching cylinder, Appl. NanoSci. DOI:
10.1007/s13204-012-0267-0.
[18] Abdul Rehman, S. Nadeem, S. Iqbal, M. Y. Malik, M. Naseer,
Nanoparticle effect over the boundary layer flow over an
exponentially stretching cylinder, J. NanoEngineering and
NanoSystems (2014) 1-6.
[19] M. Y. Malik, M. Naseer, S. Nadeem, Abdul Rehman, The
boundary layer flow of hyperbolic tangent fluid over a vertical
exponentially stretching cylinder, Alexandria Eng. J., 53
(2014) 747-750.
[20] Abdul Rehman, R. Bazai, S. Achakzai, S. Iqbal, M. Naseer,
Boundary Layer Flow and Heat Transfer of Micropolar Fluid
over a Vertical Exponentially Stretched Cylinder, App Comp
Math, 4 (6) (2015) 424-430.
[21] Abdul Rehman, G. Farooq, I. Ahmed, M. Naseer, M. Zulfiqar,
Boundary Layer Stagnation-Point Flow of Second Grade Fluid
over an Exponentially Stretching Sheet, American J App Math
Stat, 3 (6) (2015) 211-219.
[22] Abdul Rehmana, S. Achakzai, S. Nadeem, S. Iqbal, Stagnation
point flow of Eyring Powell fluid in a vertical cylinder with
heat transfer, Journal of Power Technologies 96 (1) (2016)
57–62.
[23] Abdul Rehman, Saleem Iqbal, Syed Mohsin Raza,
Axisymmetric Stagnation Flow of a Micropolar Fluid in a
Moving Cylinder: An Analytical Solution, Fluid Mechanics, 2
(1) (2016) 1-7.
[24] Naheeda Iftikhar, Abdul Rehman, Peristaltic flow of an Eyring
Prandtl fluid in a diverging tube with heat and mass transfer,
International Journal of Heat and Mass Transfer 111 (2017)
667–676.
[25] Abdul Rehman, Naveed Sheikh, Boundary Layer Stagnation-
Point Flow of Micropolar Fluid over an Exponentially
Stretching Sheet, International Journal of Fluid Mechanics &
Thermal Sciences, 2017; 3 (3): 25-31.
[26] Haroon Rasheed, Abdul Rehman, Naveed Sheikh, Saleem
Iqbal, MHD Boundary Layer Flow of Nanofluid over a
Continuously Moving Stretching Surface, Applied and
Computational Mathematics, 2017; 6 (6): 265-270.
[27] Najeeb Alam Khan, Umair Bin Saeed, Faqiha Sultan, Saif
Ullah, and Abdul Rehman, Study of velocity and temperature
distributions in boundary layer flow of fourth grade fluid over
an exponential stretching sheet, AIP ADVANCES 8, 025011
(2018).
[28] Naheeda Iftikhar, Abdul Rehman, Hina Sadaf, Muhammad
Najam Khan, Impact of wall properties on the peristaltic flow
of Cu-water nano fluid in a non-uniform inclined tube,
International Journal of Heat and Mass Transfer 125 (2018)
772–779.
