Fluctuating Flow of a Second Order Fluid between Two Co-Axial Circular Pipes

DOI : 10.17577/IJERTV4IS020409

Text Only Version

Fluctuating Flow of a Second Order Fluid between Two Co-Axial 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 mean-velocity 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.

1. 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.

2. BASIC EQUATIONS

We work through the cylindrical polar coordinates

r, , z. z-axis 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 visco-elastic 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 visco-elasticity = 2

3

= coefficient of cross-viscocity =

And,

n1

w r w r iw

r ————— (11)

And

= coefficient of kinematic-viscocity = 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

3. SOLUTION OF THE PROBLEM

The pressure-gradient (7) can be expressed in the form of a Fourier series as

The solution of (12) subject to the boundary conditions

1. 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

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

a0 ,where

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)

2. 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 non-dimensioal 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 non-dimensional 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 Non-dimensional 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 cross-section 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 non-dimensional 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

4. 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

mean-velocitie. But the sectional-mean 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 steady-state flow under the same value of pressure gradient as that in the pulsating flow and is not

affected by the presence of the visco-elastic parameter .

The non-dimensional 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 mean-velocity from the wave of the pressure-gradient 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 mean-velocity 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

5. 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

*

non-dimensional 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 mean-rate 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 mean-rate 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 mean-rate 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 n-th mode of vibration, the coefficient of excess work is given by

n

C.E.W

4Tn Dn

n 1 2 K 2

6. 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.

Fig-1 and Fig-2 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

1. 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 Table-1 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 non-Newtonian case

00 . It is to be noted that unlike the for going

intermediate frequency case,

AMV have their largest-values

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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