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 Total Downloads : 128
 Authors : Ramesh Chandra Samal, Dr. Trilochan Biswal
 Paper ID : IJERTV4IS020409
 Volume & Issue : Volume 04, Issue 02 (February 2015)
 Published (First Online): 20022015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Fluctuating Flow of a Second Order Fluid between Two CoAxial Circular Pipes
Mr. Ramesh Chandra Samal
Department of Mathematics, Ajay Binay Institute of Technology,
Cuttack 753014 Orissa, India.
Dr. Trilochan Biswal, Department of Mathemaics VIVTECH,
Bhubaneswar, Khurda Orissa, India
Abstract – In this paper an analytical solution to the flow of a second order fluid is presented expressing the pressure gradient in the form of Fourier series. The effect of the amplitude coefficient of the meanvelocity for different values of frequency of excitation is shown in different graphs.
2000 Mathematics Subject Classification: 76A05
Key words Second order fluid, annular pipe, analytical solution.

INTRODUCTION
In everyday life, we encounter many different kinds of fluids. The study of flows of Newtonian and non
Newtonian fluids through pipes and tubes has became important not only because of their technological
annular region between two coaxial circular pipes and got the solution using Fourier series.

BASIC EQUATIONS
We work through the cylindrical polar coordinates
r, , z. zaxis coincides with the common axis of the
circular pipes. The radius of the outer pipe is a and inner pipe is b. Let 0,0, wbe the unsteady rectilinear flow
between the pipes. All physical quantities are independent of because of axial symmetry. The equation of the continuity reduces to
w
importances but also in view of the interesting mathematical features presented by the equations
z 0
————– (1)
governing the flow. Such studies have a considerable practical relevance because of their applications in petro chemical industries, manufacturing of foods and paper and many other similar activities. Uchida (1) studied the
pulsating flow Newtonian fluid due to the pressure gradient
Thus w is independent of z and we can write w = w (r, t).
The stress components for the problem under discussion are given below
w 2
in the direction of the flow. Rajgopal etal.(2) Pontrelli (3) made important theoretical studies on these fluids, Rath and Jena (4) studied the flow of a viscous fluid generated in response to fluctuations in the axial velocity of the outer
cylinder Biswal etal.(5) studied the above problem incase
rr p
p
22 3 r
w 2
of viscoelastic liquid. Lui Ciqun and Huang Jungi (6)
zz p 3 r
studied the axial flow of second order fluid and analyzed the flow characters of these fluids. Hayat etal.(7) studied the Fluctuating flow of a third grade fluid on porous plate
r
z 0
w
2 w
in a rotating medium.Kaloni(8) analyzed the Fluctuating flow of an elastic viscous fluid past a porous flat
rz
1 r
2 tr
plate.Hayat etal(9),Fetecau(10) studied the above problem
w
—— (2)
on a porous plate.Ozer etal(11) studied the flow of a second grade fluid through a cylindrical permeable tube.Hayat etal(12) considered the MHD flow of the above fluid in a
1 2 t r
The equations of motion becomes
2
porous channel.Similar type of flows were investigated by
1 p 22
w
1 w
0 – (3)
Wang etal.(13),Tadhg etal.(14), Tiwary etal.(15), Hayat etal.(16) made analytical studies on transient rotating flow of a second grade fluid . Hayat etal. (17) studied the peristaltic flow of a second order fluid in the presence of an induced magnetic field.Jamil etal. (18), Hayat et al.(19)
r
1 1 p
r
r r
0 ————— (4)
r r
studied the flow of a second grade fluid in different
w 1 p
2 w
1 w
mediums and got very interesting results.In this paper, we
2
—— (5)
will study the fluctuating flow of a second order fluid in the
t z
t r
r r
Where, = is the density
w0 r RP
wn
remt
————— (10)
= coefficient of viscoelasticity = 2
3
= coefficient of crossviscocity =
And,
n1
w r w r iw
r ————— (11)
And
= coefficient of kinematicviscocity = 1
n cn sn
On the basis of the equation (1), (3) & (5) we may assume
the function (11) satisfies the differential equation
1
f (t)
————— (6)
z
w'' r 1 w' r in w
r

an
Equation (5) becomes
n r n in n
in
w f t
2w
1 w
——— (7)
—————– (12)
t
t r 2
r r
Boundary conditions are The boundary conditions (8) reduced to
r a,
wr,t 0
w a 0
r b,
wr,t 0
— (8)
n
n 0,1,2 ————— (13)
wn b
0


SOLUTION OF THE PROBLEM
The pressuregradient (7) can be expressed in the form of a Fourier series as
The solution of (12) subject to the boundary conditions

is
1 p
a0 2
2 a0 a 2 b 2 log a log r
z
f t a0
acn cos n t asn sin n t
n1
in t
w r, t a r
4
a
4log a log b
3
a0 Rp ane
———– (9)

R n
1 J
kri
2
n1
p
n1
in
0
Where, a0 = steady pressure gradient
k kai 12 k kbi 12
a a ia , R = real part of the expression
0 0
n cn
sn P
and
acn
& asn
are constants which represents the
k kri 12 J
kai 32 J
kbi 32
amplitudes of the elemental vibrations of a pulsating
0 0
0
pressure gradient superposed on pressuregradient.
a0 ,where
a0 is steady
J kai 32 k
kbi 12 J
kbi 32 k
kai 12 eint
0
2
0
0 0
We assume the period of excitation as .
————— (14)
In view of the periodic pressure distribution, we can assume the solution for the velocity field as,
In the above
J 0 and
K 0 are Bessel functions of zeroth
order of first and second kind respectively where
wr,t w0 r wcn rcos nt wsn rsin nt n
n1
k in
2
————— (15)
in
and with mn in ————— (16)


reduces to
wr, t a0 a 2 r
4
2 a0 a 2 b 2 log a log r
4log a log b
In the above expressions the suffix n denote the quantity in the nth mode of excitation, which is dropped out in the case of the flow under a signal pulse. With the help of the above nondimensioal quantities, the velocity field can be written as
R
an 1 J
im
Rk
m a k
m b
2 2
2 2
p in
n1
0 n 0 n 0 n
wr, t
a0 a 1 4
a0 a 1
4
n
log

k0
mn
RJ 0
imn
a J 0
im
n b
a 2
acn
n
1 psin n t Q cos n t
J im
ak
m b J
im
bk
m ae
it
n1 k 2
0 n 0 n
0 n 0 n

asn
p 1cos n t Q sin n t ——— (19)
————— (17)
To simplify the above equation, we introduce the following nondimensional parameters,
k 2
n
where
P At1 Bt2 Af1 Bf2
k a
Frequency parameter
Q Bt At Bf Af
1 2 1 2
1
and
————— (20)
tan
n Nondimensional viscoelastic
f1 1t1 f 2 1t2
1
parameter
A f1 1
f1
f
t1 1
f 2 t2
1
kn k
n Frequency parameter in the nth

f
1 f
f1 1t2
f 2 1t1
mode of excitation.
2 2 t 1f t 1f
1 2 2 1
kn
f 1t f 1t f t 1 t t 12
1 1 2 2 1 1 2 2
1
2
2
1
1
2
2
1
mn a rn isn
f 1t f 1t t 1t t 1f 2
n
————— (21)
Where
rn
cos n cos 4 2
B f
1 f
f1
1t1 f 2
1t2
1
2 2 f
t1 1
f 2 t2
1
s cos sin n
— (18)
f 1t
f
1t
n n
4 2

f
1 f
1 2
2 1
1
1 t 1f t 1f
1 2 2 1
J0 imn r f1 if 2
f1 1t1 f 2
1t2 f1 t1 1 f 2 t2
12
f1 1t2
f 1t t 1f t 1f 2
2 1 1 2 2 1
J0 imn a f1 1 if 2 1
2 1 1 2 2 1
————— (22)
J0 imnb f1 if 2
f 1t f 2 1t2
1
1
A t1 1 t1
k0 mn r t1 it2
f1 t1 1 f 2 t2 1
k m a t 1 it 1
f1 1t2 f 2 1t1
0 n 1 2
t2 1 t2 t 1f
t
1f
1 2 2 1
k0 mnb t1 it2
f 1t f
1
1
1t f t 1 f t 12
2
2
1
1
2
2
1
1
f 1t f
1t t 1f t 1f 2
r b 1 2
2 1 1 2 2 1
where
a
and
a
————— (23)
B t
1 t
f1 1t1 f 2 1t2
1
2 2 f
t1 1
f 2 t2
1
21 2
21 2
log
t 1 t f1
1 1
1t2
f 2 1t1
* a
t1 1f 2 t2 1f1
w
, t 1
cn 1 sin n t Q cos n t
2
c s
a k 2
0
n
f1 1 t1




f2
1 t2

f1
t1
1 f2
t2 1
8 a
1cos n t
f 1t f 1t t 1f t 1 f
2
n1 8 sn
1 2 2 1 1 2 2 2
a k 2 Q sin n t
————— (24)
The mean velocity over one period across the crosssection is denoted by w and is defined by
n1 0 n
————— (27)
w 2 dt 1 b wr,t 2 rdr
The expression for the nondimensional pressure gradient is as follows
2 0 a 2 b2 a
*
1 p
e z
2a
1 pw 2

p
z
a0 2
2 2
2
a 2 b2 2
2
8a 2 b2 a b a b
log a log b
64
acn
asn
R S 1 a
cos n t a
sin n t
a0 2
2
a 2 b2
e
n1 0
n1 0
8 a b
a a 2
log a log b
Where
2aw
Re Reynolds number
0
8 s
————— (25)
the starred quantities denote the corresponding non dimensional expressions.
where s
1
2 1 2
cos


SECTIONAL MEAN VELOCITY
The mean pressure gradient G over one period is given by,
The expression for the instantaneous mass flow across a section of tubes is derived from the sectional
meanvelocitie. But the sectionalmean velocity "wMv " is
G 2
2
0
f t dt a0
——— (26)
given as
The mean velocity in the pulsating motion under the
w 1 a wr,t 2 rdr
influence of a periodic pressure gradient (9) is identified with that in the steadystate flow under the same value of pressure gradient as that in the pulsating flow and is not
affected by the presence of the viscoelastic parameter .
The nondimensional expression for the velocity now reduces to
Mv a 2 b2 b
a 2
0 a 2 b 2
a a 2 b 2 2 log a
0
2 16
8 log a log b
0
a 2 b 2
a a 2 b 2 2
16log a log b
a a 2 b 2 a 2 log a b 2 log b
S C
0
8log a log b
n
a 1
n
1 2
cos n t
2 2 2

8 sn

acn
a b a S
C
sin n t
a k 2
T D
n n
n 1 0
n n
n sin n t
n1
2 2
1 2
a 2
————— (30)
2 Tn
a
Dn
a 2
cos n
t
a 2 b 2
We define the amplitude coefficient and phase lag in the
nth mode of the sectional meanvelocity from the wave of the pressuregradient by the following expressions

cn
n
2 S n
Cn
2 cos n t
respectively.
1
n1
2
S C
2 2
n
a T D sin n t ——— (28)
1 n
2 n n
A 8
1 2
— (31)
MV n
K 2 S
2
Where
n Tn
Dn
a rdr
b
Sn Cn
a 2
2
and
1 2
n
2 1
Tn Dn
a Qrdr a T
D
tan
MV
(1 )2 S D —
b 2 n n
———— (32)
n n
S At 1 Bt / 1 Af 1 Bf 1
with help of the equations (31) and (32) we get the non
n 2 1 2
1 dimensional form of sectional meanvelocity as follows
C At
Bt / Af
Bf
W n 1 A
acn sinn t
a
a
0
n
MV n
– (33)
n 2 1 2 1
MV
n1
MV asn
1
2
cos n t
0
MV n
Tn Bt2
1 At / 1 Bf
1 Af1
1


RESISTANCE COEFFICIENTS
The shearing stress on the wall is given by,
/
S dW
——— (34)
Dn Bt2 At1 Bf2 Af1
F 1 dR
————— (29)
The sectional mean velocity in dimensionless form is given by
where 1 is the coefficient of viscosity. We denote the
*
nondimensional frictional force at the outer wall by S Fa n
1 2 log 1 2
W n
S n
and that at the inner wall by corresponding expressions as
Fb and get their
*
W MV n
MV
W
1
s 8
acn
log
1
*
S n
Fa
S Fa n 1
2
a
k
n1 0 n
W 2
2
S C
1 1 2 / 2 log
1 n n sin n t
1 2
16 S
T D
n n

cos n t
Re
acn
2
S n cos n t Tn sin n t
1 2
n1 a0 kn
a
and

sn T
cos n t S
sin n t — (35)
n
0 n
n
n1 a k 2
1 2 / 2 log
* S n
S
and,
S Fb n
Fb
1 W 2
* 16
n
acn sinn t
2 S Fb
a
Re A 0
SFb n
SF n a
1 2 / 2 log
n1
b sn cosn t
SF n
16
S
C cos n t
a0
————— (40)
b
Re acn n
n1 a0 kn
2 D
sin n t
n
Mean rate of work done is given by

asn D
cos n t C
sin n t
W a 2 b2 W
p
a k 2 n
n (36)
e MV
z ———– (41)
n1 n n
n
n
We define the amplitude coefficient and phase lag in the nth mode of the resistance coefficient behind the wave of the imposed pressure gradient by the expressions
The total meanrate of work done is
0
2
2
A n
SFa
1
n
K 2 S
S 2 T 2
We
We dt ————– (42)
A n
SFb
1
n
K 2 S
C 2 D 2
———– (37)
But
n
n
a
n tan
1 Sn
W n
W
cn sin n t
a0
MV n
SFa
T
MV 1 AMV n
n n1
asn cosn t
1 Cn
a
MV n
b
SFn
tan
Dn
———– (38)
0
————— (43)
Here the suffixes a and b denote the corresponding values on the outer and inner wall
and the pressure gradient is given by
respectively.
p 81W
acn
asn
With these substitutions the equations (35) and
(36) respectively reduce to
1 1 2 / 2 log

z
1
a 2 a
n1 0
cos n t sin n t
a
n1 0
————— (44)
S
W a 2 b2 W
p
* 16
n
acn sinn t
e
—
MV n
z
S Fa
Re
a
ASF 0
SFa n
a n a
W 2
n1
sn cos n t
SF n
8 1 2 1
a0
a S
———————(39)
S
a 1 S c
8 cn 1 n n sin n t
n
n1 K 2
a0
1 2
Tn Dn cos n t
1 2
Sn cn
S
a 1
1 2
1 cos n t
2
8 sn
k
2
2 2
a
cn
n1 an
n Tn Dn
8 1W 1
a
2
sin n t
S 4
Tn Dn 0
1
n1
1 2 K 2
asn
n
2
asn
acn cos n t
sin n t —— (45)
a
1 a
a
0
n1 0 n1 0
On simplification, we get
Sn Cn
———— (47)
The total mean rate of change of kinetic energy across the
cross section is
1
W 2 1 a
1 2
sin n t 2
W 8
1 2 S 8
cn
e 1 S
K 2 a T D
Wk 2
Wk dt 0
——— (48)
n 1 n 0 n
n cos n t 0
1 2
where Wk is the rate of increase of kinetic energy of the fluid in a unit length pipe which is given by
2
S
C
1 a w
1 n
n cos n t
2
Wk 2 b t
2RdR
———— (49)
8 1
asn
1
K a
2
n1 n
0 T
n
D
n sin n t
The total rate of change of dissipation of energy due to internal friction is given by
1 2
RZ RZ
Wi
S C
Where Wi stands for the total rate of change of dissipation
of energy. The total meanrate of change of dissipation of
1 n
n sin n t
energy due to internal friction is given by
a 1 a
1 2
2 a


cn cos n tS 8 cn
n1 a
n1 K 2 a
T D
Wi
0
dt b 2RdR
0 n 0 n
n cos n t
2
1 2
S C
a
2
1 n
n cos n t
cn
1 asn
1 2
8 W 2 1 2 T D a
8 2
1 S 4 n
n 0
n1 K n a0
Tn Dn S
1 2 K 2 2
2 sin n t
n1
n asn
1
a
S C
0
1 n
n sin n t
——— (50)
a 1 a
1 2
Work done = The total mean rate r

sn sin n tS 8 cn
0 n
n1 a
n1 K 2 a
0 T D
n n cos n t
Thus we get the meanrate of work done = The total mean rate of change of dissipation of energy and this
1 2
fact leads to the same conclusions as in Uchida(1) that the
S C
1 n n cos n t
pressure gradient does work equal to the energy loss due to
dissipation of energy after a full cycle of the motion. Also
1 a
1 2
– (46)
the kinetic energy changes instantaneously but there is not
8 sn
n 0 n n sin n t
n1 K 2 a T D
1 2
loss in it after a complete cycle. Thus we see the energy loss is caused by the dissipation and is increased by the existence of the components in the fluctuating motion.
and
We
2
e
W dt
2 0
We define the coefficient of excess work as the extra energy dissipated due to the pulsation of amplitude
a

a
2
.
2
which is equal to a0 cn sn
Then we have in the nth mode of vibration, the coefficient of excess work is given by
n
C.E.W
4Tn Dn
n 1 2 K 2


DISCUSSION OF THE RESULTS
In this paper we have studied the flow of a second order fluid in the annual region between two coaxial circular pipes. The pressure gradient expressed in the form
of Fourier series, The following conclusions are made.
Fig1 and Fig2 shows the effect of the amplitude
coefficient of the mean velocity
AMV for different values
of K which is the frequency of excitation for
0.2 and

we see that until K=1.6.
AMV
does not rise above the value zero
For small values of i.e. 0.2 and 0.4,
AMV
records larger values in the case of a Newtonian fluid
i.e 0, the maximum value occurring for values of K
between 3 and 4, with
600
the mean velocity
amplitude coefficient has negligible values whatever be the values of . This is also seen in Table1 and Table 2.
In the Fig.3 and Fig. 4 we see that for low frequency there is not much difference in the amplitude
coefficient AMV when there is change in the values of
though as a rule the Newtonian value 00 are smaller than the corresponding values of the nonNewtonian case
00 . It is to be noted that unlike the for going
intermediate frequency case,
AMV have their largestvalues
with extremely slow pulsation and drop to almost to zero value when K has a value slightly greater than 1. It records
a slight rise for larger value of K. the value of
AMV for
slow pulsation in the case of low frequency is higher, the larger the radii ratio.
The effect of k on MV , ASFa , ASFb and C.E.W is shown in Table 1 &Table 2 for 0.2 and 0.4.
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