International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
Numerical Modeling of Couette Flow
Problem With Suspended Particulate Matter
Pramodini Samal 1 , T.C.Panda
1
2
2
Orissa Engineering College, Bhubaneswar-751007 (Odisha), India.
Department of Mathematics, Berhampur University, Berhampur-760007 (Odisha), India.
Abstract: The solution of the governing equations of
incompressible Couette flow with suspended particulate
matter is obtained by using finite difference technique
with non uniform grid. Starting from the assumed
initial conditions the velocity and temperature
distribution are computed in steps of time .It is
observed that the velocity and temperature are
changing rapidly near upper plate as to be expected. It
has been observed that the final steady state profiles are
linear for clear fluid where as it is not linear in case of
fluid with particulate matter. The effect of particle
loading is to decrease the velocity and temperature of
the carrier fluid.
Keywords: Suspended Particulate Matter, Particle
Loading, Volume Fraction, Diffusion, Viscocity,
Nomenclature:
C , C → Specific heats of fluid and solid particles
Ec → Eckert numbe
F → dust parameter
h → distance between plates
k → thermal conductivity
Pr →prandtl number
Re → fluid phase Reynolds number
t → time
t ∗ →dimensionless time
( T,T ,) → Temperature of fluid and particle phase
T → Temperature of the plate at η = 0
T → Temperature of the plate at η = 1
(u∗ ,u∗ ) → non-dimensional velocity components of
fluid and particle phase respectively
(u, v) → fluid phase velocities
(u , v )→ particle phase velocities
(x, y) → space co-ordinates
D → diffusion parameter
U → velocity of the upper plate
α → Particle loading
ε = → diffusion parameter
ISSN: 2231-5381
(ν, ν ) → kinetic coefficients of viscosity of fluid and
particle phase respectively
(ρ, ρ ) → density of fluid and particle phase
respectively
ρ∗ → dimensionless density of the particle phase
γ → ratio between C and C
φ → volume fraction of the suspended particles
I.
INTRODUCTION
The study of transport phenomena i.e transfer of
mass, momentum and energy is increasingly
recognized as a unified description of fundamental
importance. The mean velocity does not only
describe the transport of substances but also by the
presence of random chaotic fluctuations in the
velocity field. For calculating adjective transport of
substances, data on the vector field of mean current
velocity for the ocean region are needed with
variation of time. The prediction of the mechanical
transport in the ocean is a difficult task. Parallel flow
through a straight channel provides a good
understanding in connection with flow in estuaries.
With a view to study the transport through estuaries,
we have considered a simple case of flow situation
between two infinite parallel plates.
Malashetty M.S et al [4] have
studied two phase MHD flow and heat transfer in an
inclined channel in which one phase is electrically
conducting and it has been found that the velocity
and temperature can be increased or decreased with
suitable values of the ratios of viscosities, thermal
conductivities, the heights and the angle of
inclination.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
Seth et a;[12] have studied combined free
and forced convention flow of a viscous
incompressible electrically conducting fluid in a
rotating channel and obtained an analytical solution
in closed form and derived the effect of grashof
number and rate of heat transfer at both the plates.
XUJ.L et al[14] have investigated
experimentally an adiabatic concurrent vertical two
phase flow of air and water in vertical rectangular
channels and compare a model which has been
extended to predict the flow regime transition from
bubbly flow to slug flow, slug flow to churn flow
using the bubble rising velocity and the increased
frictional co-efficient for rectangular channels.
ZangLian et al [15] have developed single
and multichannel experimental structures using
plasma-etched silicon with Pyrex glass cover, which
allow uniform heating and spatially resolved
thermometry and provide optical access for
visualization of being regimes and found pressure
drop and wall temperature distribution data, which
are consistent with predictions accounting for solid
conduction and homogeneous two-phase convection.
Lain.Santiogo et al [3] have studied detailed
measurements in a horizontal channel flow laden
with solid particles with different size and loading
ratio using phase-Doppler anemometry. They
validated the numerical calculations based on the
Euler/Lagrange approach. The conservation equation
includes appropriate source term resulting from the
dispersed phase. For modeling the particle phase in
the Lagrangian frame all relevant effects are
accounted for such as transverse lift forces, wall
collisions with roughness and interparticle collision.
Muste.M et.al [6 ] have studied the
comparison of two phase flow with the traditional
mixed flow essentially as flow of a single fluid. They
confirmed that suspended particles may affect a
turbulent flow throughout its depth. Suspended
particles modify flow turbulence, the main effects
quantified being decreases in the bulk water velocity
and in the Von Karman constant, while the flows
friction velocity remains approximately constant.
Rohimzadeh etal [8] have studied the large
eddy simulation method (LES) which was used for
simulating the particle laden turbulent flow. It is
ISSN: 2231-5381
concluded in the model that the presence of particles
has negligible effect on the carrier flow.
WEI,Zhangying et al [13] have investigated
the particle flow of water mixed with sand escaped
from filtering in the labyrinth channel.CFD analysis
has been performed on liquid solid two phase flow in
labyrinth channel emitters so CFD analysis can be
used in optimal design of labyrinth-channel emitters.
There have been several investigations of
Couette flow.H.Schlichting [11] has given the
solutions to those problems for an incompressible
Newtonian fluid. Data and Mishra [2] have studied
unsteady Couette flow and heat transfer of a dusty
fluid filling the gap between two infinite parallel
plates kept at arbitrary temperature and found the
solution to be valid for any time.Panda , Mishra &
Panda [8] have studied the unsteady Couette flow and
heat transfer of a dusty fluid with the inclusion of
volume fraction and Brownian diffusion of SPM in
the mathematicalformulation.They have obtained the
solution by using Cranck-Nicholson finite implicit
scheme.
In the present study the finite volume
fraction, Brownian diffusion of SPM are considered
to show the effect on unsteady two phase coquette
flow with heat and mass transfer.
Schematic of Couette Flow
II.
MATHEMATICAL MODELING
Consider the two phase flow between two parallel
plates separated by a vertical distance L. The upper
plate is moving at the velocity U and the lower plate
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International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
is kept stationary. The flow field between the two
plates is driven exclusively by the shear stress
exerted on the fluid by the moving upper plate
resulting in a velocity profile across the flow.
Let the lower plate be coincident with the plane
y=0 and upper plate be placed at the plane y=L , the
lower plate is held at a constant temperature
and
upper plate at
.
The basic equations for the present investigation
are based on the balance laws of mass, linear
momentum and energy for both phases which are
viewed as interacting continua. In the present work
the hydrodynamic interaction of the particles and the
fluid is restricted to a mutual drug force (based on
Stroke’s linear drag theory). Whose value depends on
the particle concentration and the relative velocity
between the phases. Since the particle phase
Reynolds number is assumed to be small, other
interaction forces such as the virtual mass force, the
shear lift force (Saffiman[ 10 ]) and the spin lift
force(Rubinow and Keller [ 9 ]) are negligible
compared to the Stoke’s drag force. The Thermal
interaction between the phases is limited to energy
transfer or heat flux due to temperature difference
between the particles and the fluid. This depends on
the ratio of the velocity relaxation time to the
temperature relaxation time
(marble [5]).No
radioactive heat transfer from one particle to another
and from mutual particle interaction is assumed to
exist since the fluid acts as a buffer which prevents
direct contacts between the surfaces of different
particles. Models of this type have been given
previously by many investigators such as Marble [ 5]
and Chamakha [1 ]
Here the model for Couette flow stretches from +∞
to−∞ in x-direction. Science there is no beginning or
end of this flow, the flow field variables must be
independent of x.
In the present problem the equation of
continuity for fluid phase is given by
(
)
If we expand V in Taylor Series about the point y=0,
we get
( ) = (0) +
+
2
(ℎ) = (0) +
+… … … ..
= ℎ, we get
Evaluating at the upper plate at
ℎ
+… … … ..
2
ℎ+
Since
(0) = 0, (ℎ) = 0,
0………..
=
=0
= 0
Implies
ℎ
.
This is a physical characteristic of Coutte flow,
namely that there is no vertical component of
velocity anywhere. This leads us to assume ≈ 0 in
accordance with the physical characteristic of
Coutteflow. From particle phase continuity equation,
we get
+
≈ 0,gives,
Thus
(1 − )
=0
= constant everywhere
= (1 − )
+
(1 − )
−
−
−
−
−
(1)
(2)
= (1 − )
+ (1 − )
+
(3)
=0
=−
( )
=0
= 0
Or
=
( ) = 0.
i.e
= 0 As at the lower plate = 0
+
ISSN: 2231-5381
( )
−
+
(4)
=0
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International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
Introducing the non-dimensional variables
∗
=
∗
,
∗
,
=
,
∗
,
=
∗
,
=
∗
,
=
= −2
,
=
,
−
=
,
=
,
=
(
)
=
,
1
1−
+
,
(5)
∗
=
And after dropping stars the equations (5.1) to (5.4)
reduces to
∗
,
∗
– 1.5 , =
+ 0.5
=
, =
=
=
=
Where
= −2
=
−
−
+
−
(7)
,
=
+
−
−
=−
∗
− 1.5 ,
(
−
)
−
−
∗∗
−
Pr
+
−
= 0,
− 1−
4 1
31 −
−
>0
= 1
= 0as
impermeable.
=
(
>0
the
plates
=
METHOD OF SOLUTION
By the method of Finite difference Technique using
Non Uniform grid, equations (6) to (9) are reduced to
the following form
+
∗
,
∗∗
+
+
+
∗
+
∗∗
=
,
+
∗
+
∗∗
+
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,
=
=
=
1
∗
∗∗
(12)
(13)
(14)
∆ +
2 1
31 −
1
1
1−
(
)
−
= −1.5 −
,
∆
+
− 1−
3
Pr
2
(
)
,
=
−
+1
2
3
Pr
2
−
+1
+
+1
−2
IV.
(11)
+
−
)
are
(10)
III.
−
+1
≤0
= 0
= 1,
= 0,
< 1
+ 0.5
−
+
= −1.5 −
∗∗
= −2
= 0 0 ≤
= 1,
∆
The boundary conditions for the present problem are
= 0,
−2
∗∗
,
=
(9)
=
+
=
+ 0.5
=
(8)
−
−2
(6)
∗∗
=
−
∆
−
=
2
+ 0.5
−
∆
RESULT ANALYSIS
Some results for the velocity and
temperature profile at various stages in the time
marching process are shown in fig 1,2, 3, 4.The
initial condition in the figure is labeled as 0.∆ .The
velocity and temperature profile after 2 times steps is
labeled 2.∆ . Clearly the velocity and temperature are
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International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
changing most rapidly near the upper plate as
expected .Other profiles are shown after 12.∆ ,
36.∆ , 60.∆ , 240.∆ respectively. The driving
influence of the shear stress exerted by the upper
plate is gradually communicated to the rest of the
fluid, resulting in a final, steady state profile after 240
time steps after u. This steady state profile is linear,
as expected and perfectly agreed with exact,
analytical solution .It is also observed that the steady
state for u with SPM (∝= 0.1) is reached after 240
time steps. Thus it is concluded that the presence of
particles have no effects for the carrier fluid velocity
to reach it’s steady state (Fig-1). From Fig-2 , it can
be concluded that the steady state achieved after 36
time steps (∝= 0) and also when presence of
particles (∝= 0.1) Fig-2
Fig-3 depicts the particle phase velocity
for different time steps. The steady state profile for
is reached after 240 time steps and the steady state
profile is linear. But the steady state profile for
is
reached after 36 time steps (Fig-4) but it is not linear.
1.20E+00
1.00E+00
T(0)
8.00E-01
T(2)
6.00E-01
T(36)
4.00E-01
T(120)
2.00E-01
T(240)
0.00E+00
-2.00E-01
0
0.5
Fig-2 Variation of U with Y
1
1.5
(α = 0.1)
1.20E+00
1.00E+00
UP(0)
8.00E-01
UP(2)
6.00E-01
UP(36)
4.00E-01
UP(120)
2.00E-01
The above calculations we have carried out
with E=1. Since there is no difference in the behavior
of the solution for larger value of E . However for
large E, number of steps for attending steady state
may require more marching steps to attain steady
state.
Variation of U with Y (α = 0) i.e. Clear fluid
0.00E+00
-2.00E-01
1.00E+00
8.00E-01
8.00E-01
U(0)
6.00E-01
6.00E-01
U(2)
4.00E-01
4.00E-01
U(36)
2.00E-01
2.00E-01
U(12
0)
0.00E+00
0.00E+00
0.5
1
0.5
1
1.5
1.20E+00
1.00E+00
0
0
Fig-3 Variation of T with Y (α = 0.1)
1.20E+00
-2.00E-01
UP(240)
TP(2)
TP(36)
-2.00E-01
1.5
TP(0)
TP(120)
TP(240)
0
0.5
1
1.5
Fig-4
Fig-1 Variation of Temperature with Y (α = 0.1)
ISSN: 2231-5381
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International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 2 – Aug 2014
[9]
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