numerical study for flow past an impermeable sphere - IJAMM

NUMERICAL STUDY FOR FLOW PAST AN IMPERMEABLE SPHERE
D. V. Jayalakshmamma 1 , M. Sankar 2 , P. A. Dinesh 3 , and D. V. Chandrashekhar 4
1 Department of Mathematics, Vemana Institute of Technology, Bangalore –34, Karnataka,
India.
Email: jaya.dvj@gmail.com
2 Department of Mathematics, East Point College of Engineering and Technology, Bangalore-
49, Karnataka, India.
Email: manisankarir@yahoo.com
3 Department of Mathematics, M.S.Ramaiah Institute of Technology, Bangalore –54,
Karnataka, India.
Email: dineshdpa@hotmail.com
4 Department of Mathematics, Vivekananda Institute of Technology, Bangalore –74,
Karnataka, India.
Email: dvchandrur@yahoo.com
Received 14 March 2012; accepted 7 September 2012
ABSTRACT
A numerical study of Brinkman flow is considered, for a steady, incompressible, viscous fluid
past an impermeable sphere embedded in a sparsely packed porous media, by assuming
uniform shear away from the sphere. Similarity transformation is employed to convert the
governing partial differential equation to linear ordinary differential equation. The resulting
ordinary differential equation is solved numerically by using shooting technique, which uses
Runge-Kutta algorithm and Newton-Raphson correction for the guessed initial values. The
effect of non-dimensional parameters, on the velocity and shear stress is investigated and
results are shown graphically.
Keywords: Brinkman flow, uniform-shear, shooting technique, Runge-Kutta,
Newton-Raphson
1 INTRODUCTION
The study of uniform flow or uniform shear flow past a porous body in stokes flow is of
practical interest in many industrial problems, particularly in chemical industry and hydrology
to understand the control (i.e., increases and decreases) of drag and torque on the body. Early
studies on these problems were mainly concerned with the use of Darcy model since the
dimensionless particle diameter (i.e., the ratio of particle diameter to the characteristic length
of the problem) is usually vanishingly small. Flows in rocks, soil, sand and other media
involved in hydrology and geothermal study fall in this category.
In the literature, we find several studies on the flow past a sphere in porous media using
Darcy’s or Brinkman equation under different boundary conditions. Joseph and Tao
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D. V. Jayalakshmamma et al.
(Joseph and Tao1964) used Darcy model to study the flow past a porous sphere. They have
used no-slip condition, under the assumption that the permeability is small relative to the
square of the sphere radius. Wolfersdorf (Wolfersdorf1989) has studied Stokes flow past a
sphere embedded in a permeable media.
Many practical problems involve porous materials such as metals and fibrous media of high
porosity where dimensionless particle diameter is small but not vanishingly small. In such
situation, Darcy’s law is shown to be inadequate. Non-Darcy equation which incorporates
both boundary and inertia effects is considered. In the study of flow past a porous body in
stokes flow the inertia effects are negligible compared to boundary effects. These boundary
effects are considered using Brinkman model (Brinkman 1947).
The problem of stokes flow past porous bodies using Brinkman model in the interior region
has been studied by Higdon and Kojima (Higdon and Kojima 1981), and derived some
asymptotic results for small and large permeability’s. Qin and Kaloni (Qin and Kaloni1988)
have developed a general solution of the Brinkman equation based on Cartesian form. Pop
and Cheng (Pop and Cheng 1992) have used the Brinkman model to study stokes flow past a
circular cylinder embedded in a porous medium. Padmavathi et al. (Padmavathi et al.1993)
have studied Stokes flow past a porous sphere using Brinkman model. They have found the
velocity and pressure field for general non-axisymmetric stokes flow, both inside and outside
a porous spherical shell. Barman (Barman 1996) has studied the flow of a Newtonian fluid
past an impervious sphere embedded in a porous medium, using Brinkman model. He
obtained exact solutions by specifying constant velocity away from the sphere. He found that
the viscous sublayer increases with the increase of the permeability of the porous medium.
Pop and Ingham (Pop and Ingham 1996) presented a closed form, exact solution for the
forced flow past a sphere which is embedded in a porous medium using the Brinkman model.
Srinivasacharya and Murthy (Srinivasacharya and Murthy 2002) investigate flow past an
axisymmetric body embedded in a saturated porous medium using Brinkman’s extension.
Rudraiah et al (Rudraiah et al 2003) studied non-linear convection in porous media. They
have shown that in many practical applications of flow in porous media, non-Darcy equation
is more realistic to describe the flow instead of Darcy equation. Srinivasacharya and Shiferaw
(Srinivasacharya and Shiferaw 2008) analyzed the steady flow of an incompressible and
electrically conducting micropolar fluid flow between two concentric porous cylinders. They
solved the governing non-linear ordinary differential equations numerically using
qusilinearization technique. Shivashankar et al. (Shivashankar et al. 2010) presented a
numerical study on natural convection to study the effect of magnetic field in a vertical double
cylindrical annular cavity filled with porous media. They solved the governing equation by
alternate direction implicit method and stream function by successive line over relaxation
method. The Drag on a fluid sphere embedded in a porous medium was obtained by
Deo et al. (Deo et al. 2010), using Brinkman model.
Recently, Eldesoky (Eldesoky 2012) presented the influence of slip condition on peristaltic
flow in an axisymmetric cylindrical tube. Darcy’s law was used to describe the flow field and
closed form of analytic solution was obtained by perturbation analysis. Sheth and Patel
(Sheth and Patel 2012) analyzed the similarity solution of the motion of two immiscible fluids
in homogeneous porous media. Bala Anki Reddy et al. (Bala Anki Reddy et al. 2012)
analyzed the boundary layer flow and heat transfer over a wavy surface embedded in a
quiescent electrically conducting fluid. The similarity transformation method was used to
transform governing partial differential equation into a system of ordinary differential
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Numerical Study For Flow Past An Impermeable Sphere
71
equation which are solved numerically Runge-Kutta fourth order method with shooting
technique.
Many practical problems, like removing impurities in the integrated circuits used in
computers, lubrication process in porous bearings, cooling of machineries etc. involves the
flow past a sphere / cylinder with uniform shear away from it, which change significantly the
streamline patterns, drag and torque rather than uniform velocity. Therefore, in this problem
we considered the study of steady, an incompressible fluid flow past an impermeable sphere
embedded in a constant and high porosity porous medium based on Brinkman model
assuming uniform shear far from the sphere. The similarity solution method is used to
transform partial differential equation (PDE) of order four to get the ordinary differential
equation (ODE) of order four with variable co-efficient. The resultant ODE is solved
numerically using shooting technique method. The method transforms the boundary value
problem in ODE of order four into the initial value problem with system of four equations of
first order. The tangential and normal components of velocity and shear stress are computed
numerically for different non-dimensional parameters.
2 MATHEMATICAL FORMULATION
We consider the study of steady, axi-symmetric flow of a viscous and incompressible fluid
past an impermeable sphere of radius a with uniform velocity u
, embedded in a sparsely
packed porous medium. Here we considered the Brinkman model specifying uniform shear
away from the sphere. The governing equations for the problem under the consideration are:
q 0
(1)
2
p
q
k
q , (2)
where q qu,
v,
w
is the velocity of the fluid, is the viscosity of the fluid, k is the
permeability of the porous medium is the effective or Brinkman viscosity and p is the
hydrostatic pressure.
We use a spherical coordinate system r ,,
with the origin at the center of the sphere and
the axis 0, along the direction of the uniform velocity u
as shown in Figure. 1. Due to
the symmetry of the problem, we have 0 .
The variables involved in the governing equations are non-dimensionalised using the
transformation:
r u v ap
r ; u ; v ; p . (3)
a u u u
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D. V. Jayalakshmamma et al.
Figure 1: Physical Configuration
Using these non- dimensional variables from equation (3) in governing equations (1) and (2)
for spherical polar co-ordinate system and omitting astrics, we get:
r
2 r
r
u
v
sin
0
sin
, (4)
2 2
2
p u 2 u
1 u cot u
2u
2 v
2v
cot
u , (5)
2
2 2 2
2 2
2
r
r
r r
r
r
r r
r
2 2
2
2
1 p v 2 v
1 v cot v
2 u
vcos
ec
v , (6)
2
2 2 2
2
2
r
r
r r
r
r
r
r
where is the porous parameter and is the viscosity ratio.
As the motion is axi-symmetric and for spherical polar co ordinates we can introduce the
r , as:
stream function
1
u
;
2
r sin
1
v . (7)
r sin r
Using equation (7) in the equations (5) and (6), the pressure term is eliminated by cross
r,
as:
differentiation and reduces to fourth order partial differential equation in
2
4 2
E E 0 , (8)
where E
2
2
r
2
sin
1
is the Laplacian operator.
2
r
sin
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Numerical Study For Flow Past An Impermeable Sphere
73
To solve this physical system, the boundary conditions considered are no-slip condition at the
surface of the solid sphere,
0, at r 1. (9)
r
For far away from the sphere, the flow has uniform shear from Batchelor (1993)
(taking n 2 in the equation 6.8.5),
3
r 2
r,
sin cos
, as r .
3
(10)
The boundary condition (10) i.e. far away from the surface of the sphere enables us to
consider the solution of equation (8) by similarity solution method as:
2
, f rsin
cos
r . (11)
Substituting equation (11) in equation (8), fourth order partial differential equation in r
,
reduces to an ordinary differential equation of order four in f r
i.e.
f iv
12
r
24
3
r
6
2
r
2
2
r f r
f r
f r
f r 0
, (12)
and the corresponding boundary condition from equations (9) and (10) reduces to:
r f r
0
f at r 1, (13)
3
r
f r
as r .
3
(14)
The shearing stress at any point on the surface of the sphere is
r
r
v
r r
1 u
r
. (15)
The dimensionless shear stress at any point on the surface of the sphere is given by
u
v u
r
. (16)
r
r r
1
r
a
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D. V. Jayalakshmamma et al.
3 NUMERICAL PROCEDURE
The equation (12) with the boundary conditions of equations (13) and (14) has been solved by
using most effective Newton’s Raphson shooting method along with Runge-kutta fourth order
integration method. Here we transform the higher order ordinary differential equation (12)
into system of simultaneous equations of first order. Further, they are transformed into initial
value problem by applying shooting technique. The resultant initial value problem is then
solved by employing Runge-Kutta fourth order technique.
Let,
d f1 f f 1,
f2,
dr
d f
d r
2
f3
d f3 , f4
,
dr
d f
dr
2
12 24
f3
f2
f3
f
2 3
2 1 , (17)
r r r
4
6
subjected to the following initial conditions:
0, f 1
0, f 1
,
1
f . (18)
1
1
2
3
f4
In a shooting method, the guessed initial conditions, and are assumed and equation (17)
is then integrated numerically as an initial valued problem to a given terminal point. The
accuracy of the assumed initial conditions is checked by comparing the calculated values of
the depended variable at the terminal point with its given value at that point. If any difference
exists, improved values of the assumed initial conditions must be obtained and process is
repeated. The computations are carried out using C-programming language.
4 RESULTS AND DISCUSSION
In this paper we studied, 2- dimensional steady, viscous, incompressible fluid flow past an
impermeable sphere, embedded in a sparsely packed porous medium. Brinkman model is
used to describe the fluid flow specifying uniform shear far away from the surface of the solid
sphere. The basic governing equations are reduced to fourth order partial differential equation
in terms of stream function r
,
. Using similarity solution method, the obtained partial
differential equation is transferred to fourth order ordinary differential equation in f .
The fourth order ordinary differential equation in f r , is solved numerically using Newton’s
Raphson shooting method along with Runge-kutta fourth order integration method. In this
method, the fourth order boundary value problem is converted to initial value problem of
solving first order simultaneous equations. The missed initial values are assumed initially in
such a way that, these assumed values are verified with calculated value of the dependent
variable value at the terminal point. The computations have been carried out for tangential and
normal components of velocities, shear stress for various values of non-dimensional
parameters namely porous parameter and viscosity ratio embedded in the problem.
r
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Numerical Study For Flow Past An Impermeable Sphere
75
The tangential component of velocity is computed for different values of porous parameter
and viscosity ratio as a function of r at 4 and the results are depicted in Figure2
(a)-(b). We observe that, the tangential velocity increases at small distance from the wall with
an increase in porous parameter for a fixed value of viscosity ratio , and then decreases to
its asymptotic value as the fluid moves far away from the sphere, given in Figure 2 (a).Similar
effect is noticed for increasing value of viscosity ratio for fixed value of porous parameter ,
which is shown in Figure 2 (b). It is also noticed that the thickness of the viscous sublayer
increases with decrease in porous parameter . However, viscous sublayer decreases for
increasing the values of viscosity ratio .
Figure 2 (a): Variation of tangential velocity for fixed value of and different values of
Figure 2(b): Variation of tangential velocity for fixed value of and different values of
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D. V. Jayalakshmamma et al.
The variation of normal component velocity is also studied as a function of r and, .For
4 we considered the variation of normal velocity and shown in Figure 3 (a)-(b). We
found that the normal component of velocity increases with decrease in viscosity ratio for
fixed value of porous parameter is shown in Figure 3 (a). On the other hand, for a fixed
viscosity ratio , it declines with increase in porous parameter , which is given in
Figure 3 (b). This shows that, increase in the value of porous parameter dampens the effect of
viscosity ratio on the normal component of the velocity.
Figure 3 (a) Variation of normal velocity for fixed value of and different values of
Figure 3(b): Variation of normal velocity for fixed value of and different values of
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Numerical Study For Flow Past An Impermeable Sphere
77
Figure 4(a): Variation of shear stress for fixed value of and different values of
Figure 4(b): Variation of shear stress for fixed value of and different values of
Also, shearing stress which is non-linear in contrast to linear relationship that exits for
uniform flow past a sphere. The non- dimensional shearing stress is computed from equation
(16) as function for various values of porous parameter and viscosity ratio. In both cases, it
remains periodic in nature, attains its maximum value for and vanishes at the
4
end points, shown in Figures 4 (a)-(b).
3 ACKNOWLEDGEMENT
The authors are grateful to the research centers of East Point College of Engineering and
Technology, M S Ramaiah Institute of Technology, Vemana Institute of Technology, and
Vivekananda Institute of Technology, Bangalore, India for their support and encouragement
to carry out our research work.
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D. V. Jayalakshmamma et al.
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