SECONDARY FLOW OF TWO IMMISCIBLE LIQUIDS IN A ROTATING

International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN: 2319-3972 Vol.1, Issue 2 Nov 2012 37-49 © IASET

SECONDARY FLOW OF TWO IMMISCIBLE LIQUIDS IN A ROTATING ANNULAR PIPE OF CIRCULAR CROSS SECTION 1

T. POORNA KANTHA, 2 A.S.N. MURTI & 3V.V. RAMANA RAO

1

Assistant Professor, Department of Mathematics, GITAM Institute of Technology, GITAM University, Visakhapatnam, India

2

Professor, Department of Mathematics, GITAM Institute of Technology, GITAM University, Visakhapatnam, India 3

Professor(Retd), Department of Applied Mathematics, Andhra University, Visakhapatnam, India

ABSTRACT Ramana Rao and Narayana (1981) studied the flow of two incompressible immiscible liquids occupying equal heights between two parallel plates in a rotating system under the action of constant pressure gradient. They also studied the associated thermal distribution, assuming equal and different plate temperatures. This branch of fluid mechanics has developed rapidly in recent times as an obvious consequence of interest in geophysical flow problems, earth’s atmosphere, oceans and core and of stars and galaxies. We considered the flow of two incompressible immiscible fluids occupying equal heights in rotating circular pipe. Immiscible fluids we mean, superposed fluids of different densities and viscosities.

The rotating pipe that we consider here has the following physical meaning. If we introduce a pipe in a rotating flow, for example, rotating flow due to earth’s rotation, the pipe also rotates.

KEYWORDS: Pressure gradient, Flux INTRODUCTION This sets up the primary and secondary flows. Ramana Rao and Narayana (1981) suggested that olive oil and water can be taken as two immiscible liquids to test their theoretical conclusions for setting up an experiment.The uniqueness for two immiscible fluids in one-dimensional porous medium was studied by Baiocchi, Evans, Lawrence C, frank, Leonid, Friedman, Anver (1980). Patrudu (2001) studied the laminar flow of two incompressible immiscible liquids under a constant pressure gradient through a channel of circular cross section in a rotating straight pipe, rotating with a uniform angular velocity about an axis perpendicular to the channel. This problem was later extended by Sivarama Prasad (2006) for the hydro magnetic case.

The flow of two incompressible, immiscible liquids through a straight pipe in the annular region bounded by two concentric circles of radii a and a,  < 1, under two constant pressure gradients, occupying equal heights is considered.

BASIC EQUATIONS AND THEIR NON DIMENSIONAL FORM We consider the steady laminar flow of two incompressible liquids under the action of constant pressure gradient through a channel of arbitrary cross section rotating with a uniform angular velocity about an axis perpendicular to the channel. The equations of motion in steady state flow, relative to a set of rectangular Cartesian coordinates z1) rotating with a constant angular velocity ’ with respect to an inertial system are

1

1

r  r (x1, y1,

38

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

 u ' v' u '1 v'12    2 ρ1 ' w1 sin θ  ρ1 u1 1  1    r ' r ' θ ' r '  =-

u'  ' 2 v    1 1  '2 u'1  21  2 1  r ' r ' r '   

 u ' v' u '1 u1v'12  2 ρ'1 ' w1 cos θ  ρ1  u1 1  1  r'  r ' r ' θ ' =-

(1.1)

   

v' 1  ' 2 u    1 1  ' 2 u'1  12  2 1  r ' r ' r ' r '   

(1.2)

 w'1 v'1 w'1   2  '1 ' u1 sin   v'1 cos  u1    r ' r '   = 

 '   1 1  ' 2 w1 z '

u ' u '1 1 v'1   0 r ' r ' r θ where ' = P' -

' 2 



2

r

2

(1.3)

(1.4)

1 ρ1  ' 2 ( r ' 2 sin 2 θ  z ' 2 ) 2

(1.5)

2



1  1   r ' r r 2 θ 2

(1.6)

For fully developed laminar flow, the form of ’ is restricted to ' = -cz' + F (r',)

(1.7)

where ‘c’ is a constant and may be termed as the gradient of ’ along the axis of the pipe.

The above equations are the equations of motion of a viscous incompressible liquid characterized by viscosity v1 and density ρ1 occupying the space between r = a  and r' = a in the circular pipe for the upper liquid. Here (u’1, v’1, w’1) are the components of the velocity in the direction of (r’,, z’) where ‘’ is the angle between the radius and the axis of rotation and z1 is measured from the axis of the pipe.

The corresponding equations for the lower liquid occupying the space between r’ = 0 to r’ = a  of the circular pipe, are 2  u ' 2 v' 2 u ' 2 v' 2     2 ρ2 ' w2 sin θ  ρ2 u ' 2      r ' r '  θ ' r '  

=-

u'  ' 2 v     2 2  ' 2 u ' 2  22  2 2  r ' r ' r '   

(1.8)

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

 v' 2 v' 2 v' 2 u 2 v ' 2  2 ρ' 2 ' w2 cos θ  ρ2  u 2    r ' r ' θ ' r'  =-

39

   

v' 1  ' 2 u     2 2  ' 2 u' 2  22  2 2  r '  r ' r '   

(1.9)

 w' 2 v' 2 w' 2  2 ρ' 2 ' (u 2 sin θ  v' 2 cos θ  ρ2  u 2   r ' r ' θ   = 

 '   2 2  ' 2 w 2 z '

u ' 2 u ' 2 1 v' 2   0 r ' r' r θ where ’ = P’ -

1 ρ 2  ' 2 ( r ' 2 sin 2 θ  z ' 2 ) 2

(1.10)

(1.11)

(1.12)

For the upper liquid, we introduce the stream function '1 such that r' u’1 =

 '1 

v’1 = =

 '1 r '

(1.13)

where ’1 is a function of r’ and  only. Eliminating ’ from (1.1) and (1.2) we get

1  2  ' ( D  w1 )   ( '1 ,  ' 2  '1 )  v1  ' 4 1 r

(1.14)

Using eq. (1.13) in eq. (1.3), we get

1  2 ' ( D  1 )   ( '1 , w'1 )  v1  ' 2 w1 r where D* = cos And

 sin θ   r ' r ' θ

(1.15)

(1.16)

 (X, Y) stands for the Jacobian of X and Y with respect to r’ and  respectively. i.e. J ( X , Y ) 

 (X , Y )  (r ' ,  )

For the lower liquid following similar analysis, we get

(1.17)

40

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

1  2  ' ( D  w2 )   ( ' 2 ,  ' 2  ' 2 )  v 2  ' 4  2 r

(1.18)

1 c 2 ' ( D   2 )   ( ' 2 , w'2 )  v 2  ' 2 w2 r 2

(1.19)

In terms of the non dimensional variables,

w'1 =

R'1 =

c a 2 W1 , 4 ρ1 v1 c a3 4 ρ1 v 2 1

1 

c a 3 1 , 4 1 v1

2 ' a 2 T1  v1

,

(1.20)

w'2 =

c a 2 W2 , 4 ρ2 v 2

2 

c a 3 2 , 4  2 v2

R'2 =

c a3 , 4 1 v 2 2

T2 

2 ' a 2 v2

where ‘c’ is the pressure gradient, R1, R2 stand for the Reynolds numbers of the upper and lower liquids respectively, T1 and T2 are the Taylor numbers for these liquids respectively, we get for Upper Liquid

T1 ( D  1 ) 

R  ( '1 , w'1 )  4   2 w1 r

(1.21)

R  ( '1 ,  21 )   41 r

(1.22)

R2  ( ' 2 , w' 2 )  4   2 w2 r

(1.23)

R2  ( ' 2 ,  2 2 )   4 2 r

(1.24)

 T1 ( D  W1 )  Lower Liquid

T2 ( D   2 ) 

 T2 ( D  W2 ) 

where D* = cos

 sin θ  r r θ

(1.25)

and when k = 1 the pressure gradient is same for both the liquids. For different values of k, the pressure gradient is varied.

Boundary Conditions To find the solutions for W1, 1, W2, 2 we use the following boundary conditions given in non dimensional form,

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

at r = 1, W1 = 0 and

at r =  , W2 = 0 and

1 = 1 = 0 r

2 = 2 = 0 r

41

(1.26)

(1.27)

at r = b, (line between  and 1 i.e.  < b < 1)



1 1 1  2 =  r r r r



 1  2 = , W1 = W2 r r

W1 W2 μ L where L  2 r r μ1

(1.28)

Method of Solution For the upper liquid, we assume 1 = T111 +

2

T1 12 + ……

W1 = W10 + T1W11 +

2

T1 W12 +…..

(1.29) (1.30)

For the lower liquid, we assume 2 = T221 + T22 22 + ……

(1.31)

W2 = W20 + T2W21 + T22 W22 +…..

(1.32)

Solutions Substituting equations (1.29) and (1.30) in equation (1.21) & equations (1.31) and (1.32) in equation (1.23) and to the zeroth power in both T1 and T2, we get 2W10 = - 4

(1.33)

2W20 = - 4 K

(1.34)

which are to be solved with the appropriate boundary conditions to this approximation, namely W10 = 0 at r = 1 W10 = W20 at r = ε

W10 W 20 L at r  b r r and W20 = 0 at r = 

(1.35)

On solving eqs. (1.33 ) and ( 1.34 ), we get W10 = C1 + C2 logr – r2

(1.36)

W20 = C3 + C4 logr – kr2

(1.37)

42

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

where

C4 =

1  b 2 (2 log b  1 )  kb 2 (2L log b  1)  k 2 (1  L) log b  log 

C3 = k  2 – C4 log  C2 = LC4 – 2kLb2 + 2b2 C1 = 1 To the first order approximation in T1 and T2 , using the expressions (1.22 ), ( 1.24 ), ( 1.36 ) and ( 1.37 ) for w10 and w20 we get for the upper liquid 4 11 = -D * W10

(1.38)

and for the lower liquid 4 21 = -D * W20

(1.39)

 sin    r r 

where D * = cos

D * W10 = C2

Cos θ  2r cos θ r

(1.40)

D * W20 = C4

Cosθ  2kr cos θ r

(1.41)

These are to be solved subject to the conditions 11 = 0 and

11 =0 r

r = ,

W21 = 0, 21 = 0 and

 21 =0 r

r = b,

-

at

r = 1, W11 = 0 ,

at

at



1 11 1  21 =r r r r

11  21 = r r

W11 = W21,

W11 W 21 =L r r

(1.42)

and writing 11 = g11(r) cos 21 = g21(r) cos in eqs. (1.38), (1.39) we get

(1.43)

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

43

g11 =

A  1  A1 r  A2 r log r  3  A4 r 3  2r 5  12c 2 r 3 log r   192  r 

(1.44)

g21 =

A  1  A5 r  A6 r log r  7  A8 r 3  2kr 5  12c 4 r 3 log r   192  r 

(1.45)

A4 =

P3 P4  P1 P6 P2 P4  P1 P5

A3 =

P3 P5  P2 P6 P2 P5  P2 P5

A2 = 12C2 – 8 + 2A3 – 2A4 A1 = -2 – A3 – A4 P1 = 2b logb +

1 b b

P2 = b3 – b –2b log b P3 = 12C2 b3 log b – 2b5 + 2b – 12 C2 b logb + 8b log b P4 = 1 + 2 lob b -

1 b2

P5 = -3 –2 log b + 3b2 P6 = 36C2 b2 logb + 12C2b2 – 10b4 + 10 – 12C2-12C2 logb + 8 log b A5 =

p 9 p11  p 8 p12 p 7 p11  p 8 p10

A6 =

p 9 p10  p 7 p12 p 8 p10  p 7 p11

A 7  b[12C4 b 3 logb  2kb5  A 5 b  A 6 blogb  A8 b 3 ] {12C 4 4 log   36C 4 b 4 logb  12C 4 b 4  10kb 4  2k 6  A 5 (2  b 2 )  A 6 (b 2  b 2 logb A8 

 2 logb)} 4 3b 4

p 7  2 2 b 4  2 4 b 2

p 8  2 b 4 4 b 2  2 2 b 4 log  2 4 b 2 logb p 9  48C4 4 b 4 log  48C4 4 b 4 logb  12k 6 b 4  12k 4 b 6

44

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

p10  - 2 6 6 2 b 4  4 4 b 2 p 11  6  2 6 log  3 2 b 4  6 2 b 4 log  4 4 b 2  4 4 b 2 log b p 12  144 C 4 4 b 4 log( b )  12C 4 4 b 4  3 6k 6 b 4  40 k 4 b 4  4C 4 10  12C 4 8 To the first order of approximation in T1 and T2 from equations (1.21) and (1.23), we get 2w11 =

R1 (11 , w 10 ) r

(1.46)

2w21 =

R2  ( 21 , w 20 ) . r

(1.47)

Using the expressions for 11, 21, w10, w20 from equations (1.43), (1.40) and (1.41) in eqs. (1.46), (1.47) we get 2w11 =

R1 C 2  2r 2 g 11 ( r ) sin  r2

2w21 =

R2 C 4  2 kr 2 g 21 ( r ) sin  2 r











(1.48)



(1.49)

The boundary conditions on w11 and w21 are at

r = 1, w11 = 0

at

r = b, w11 = w21,

at

r =  , w21 = 0,

and  =

w11 w21 = L r r (1.50)

T2 .

T1

Solving equations (1.48) and (1.49), we get w11 = h11(r) sin w21= h21(r) sin where h11(r)=A 9r+

(1.51)

A 10 R1 + {A11rlogr + A12r3logr + A13r 5logr + r 96

A14r (logr)2 +A15

log r + A16r3 + A17 r5 + A18 r7} r

(1.52)

45

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

A 20 R 2 + {A21rlogr + A22r3logr + A23r5logr + r 96

h21(r) = A19 r +

A24r (logr)2 +A25

log r + A26r3 + A27 r5 + A28 r7} r

(1.53)

and A9 = –p13 – A10 A10 =

p14   p13 2  A20 2

A19 =

p 21p 23  p 20 p 24 p19 p 23  p 20 p 22

A20 =

p 21p 22  p19 p 24 p 20 p 22  p19 p 23 p13 =

p14 =

p15 =

R1 {A16 + A17 + A18} 96 R2 96

 5log + A24  (log )2 + A25 log  

+ A26  3 + A27  5 + A28  7}

R1 log b + A b3 + A b5 + A b7} {A11blogb + A12b3logb + A13b5logb + A14b(logb)2 + A15 16 17 18 96 b

p16 = 

p17 =

{A21  log  + A22  3log  +A23

R2 log b + A b3 + A b5 + A b7} {A21blogb + A22b3logb + A23 b5logb + A24 b(logb)2 + A25 26 27 28 96 b

R1 1 - log b ) {A11 + A11logb + A12b2 + 3A12b2logb +A13(b4 + 5b4logb) + A14((logb)2 + 2logb) +A15( 96 b2 b2

+3A16 b2 + 5A17b4 + 7A18b6}

p18 = L

R2 1 {A21 + A21logb + A22b2 + 3A22b2logb +A23(b4 + 5b4logb) + A24((logb)2 + 2logb) + A25( 96 b2

log b ) + 3A b2 + 5A b4 + 7A b6} 26 27 28 b2 p19 = -b2  2 p20 = 1- b2-  2 p21 = (p16 – p15 )b  2 – p14  + p14b2  + p13  2

46

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

p22 = - Lb2  2 p23 = -1 – b2 - L  2 p24 = (p18 – p17 )b2  2 + p14b2  + p14  - p13  2

A11 =

A14 =

A 1C 2 A 3 A 2 C 2   4 2 8

A12 = ,

C  A 2 3C 22 A13 = 2  8 4 , 2 ,

A 2C2  A 3C 2  C2 A 4 2A 4 C 2  4A 1  3A 2  18C 22 A15 = A16= A17=  8 , 4 32 6 24 , ,

A18=

2C 4 A 5  4kA 7  C 4 A 6 kC 4 C A  6C 42  kA 6 1 A21= A22= A23= A24= 4 6 24 , 8 8 2 , 8 , ,

A25=

 C4A 7  kA 8  4kC4 2C 4 A 8  4kA 5  18C 42  3kA 6  k2 A26= A27= , A28= 4 32 24 24 , ,

DISCUSSIONS OF THE STREAMLINES IN THE CENTRAL PLANE In the central plane perpendicular to the axis of rotation θ =

 3 or , it can be seen that from the equations 2 2

(1.43), v' = 0 in either case. So a particle of liquid once in this plane does not leave it in the subsequent motion. The motion in the two halves of the pipe is therefore quite distinct from each other.

The differential equation of the streamlines in the central plane of the pipe is

dy du

1 1



dz 1 dw 1

(θ 

π 2

or 3

π ) 2

(1.54)

dz 1 

w1 dr 1 u1

dz 1 

where u 1  

1  1 r 1 

w2 where 21  g 21 (r ) cos  1  1  1 r 

(1.55)

(1.56)

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

rw

T 2 dz 

 g







2

dr

21 - sin

r(w







rw 2 dr g 21 sin θ

 T 2 w 21 )dr sin 

20

g

(A

(A

5

r

2

(1.57)

21

r(w 1 192r

47

5

r

2

20

 T2w

 A 6 r 2 logr  A 8 r

21

4

)dr

 2kr

6

192r 2 (w 20  T 2 w 21 )dr  A 6 r 2 logr  A 8 r 4  2kr 6  12c

 12c

4

4

r 4 cos  )

r 4 cos  )

To a sufficient approximation these streamlines for the lower liquid are given by

T2 ( z 1  z 10 ) 

192 [l1 r 3  l 2 r 5  l 3  l 4 ] A7

where 3k 2 3c 4 2 log  c 4 9  5kA7  5kA5 2 5kA5 c 4 log   A5 c 4  kA6 2 25 A7

l1



l2



l3



c 4 r 3 log r 3



A 5 c 4 r 3 logr 5A 7

l4

(1.58)

and zo' is a constant of integration which is different for different stream lines and equal to the distance of the point r=0 on the curve from the axis of rotation.

Values of

T1 ( z 1  z 10 ) for  = 0.5 b = 0.75 r

Values of

T2 ( z 1  z 10 ) for  = 0.5 b = 0.75

T1 ( z 1  z 10 )

0.75

0.19313

0.8

1.11275

0.85

2.64434

0.90

5.00022

0.95

8.43429

1.0

0

48

T. Poorna Kantha, A.S.N. Murti & V.V. Ramana Rao

r

T2 ( z 1  z 10 )

0.1

0.01633

0.2

0.10848

0.3

0.26647

0.4

0.52794

0.5

0.87765

To a sufficient approximation these streamlines for the upper liquid are given by

T1 ( z 1  z 10 )



 c2 3 192  r 3 5 r log r  l 6 r 3 log r    l5 r  A3  3 3 

(1.59)

where

l5



l6



  A1  A3     5 A3    A1 c 2  A2  5 A2 

  

Table shows the same stream line in the plane of symmetry for ε =0 and ε =1. We note that no stream line in the central plane ever reach the edge of the pipe. As the angular velocity Ω' is increased, the distance which must be covered by the central stream line to be within a given distance from the edge gets smaller., this results holds good for all values of ε between 0.5 and 1.

For a fixed value of T2, the effect of decreasing ε from unity to 0.5, is to increase the distance that the liquid particle in the central plane travel in going from points near edge of the pipe i.e., r=0 to points r= ε. The similar conclusions can be drawn for the upper liquid also.

STREAM LINES OF THE SECONDARY FLOW To the first order of approximation in T1 and T2, the equation for the projection of the stream lines on the cross section of the channel for the upper liquid is given by 192000 g 11 cosθ = k.

Figure gives the projection of the stream lines on the cross section of the channel when  = 0.5 for some values of k.

49

Secondary Flow of two Immiscible Liquids in a Rotating Annular Pipe of Circular Cross Section

CONCLUSIONS The stream line pattern in the central plane of the pipe has been discussed and the pattern of general stream lines for both upper and lower liquids have been obtained and shown graphically also. For different values of k, we get a set of closed curves which are symmetrical about the axis of rotation θ= 0. For an equal and opposite value of k, the curves in the lower half are obtained by the reflection of the curves on the diameter of the cross section perpendicular to the axis of 

rotation. The upper half of the cross section is divided into two regions by a circle of radius r= r , which corresponds to a stream line when k= 0. The other branches of the curve with k= 0 are the peripheries of the channel r=ε, r= 1 and the parts of the axis θ=



  between two peripheries. In the region I (ε ≤ r ≤ r ), the stream lines are obtained as k assumes 2 1

negative values between 0 and minimum value k (corresponding to the degenerate stream line for r= r min ). In the region II 

11

(r ≤ r ≤ 1), the stream lines are obtained as k assumes positive values between 0 and maximum value k ( corresponding to the degenerate stream lines for r= r max ). At these degenerate points, both the radial and transverse components of velocity in the cross section vanish. Thus the stream lines of motion through the points (r min , 0), (r max , 0), (r min ,  ), (r max

,  ) are straight lines and the motion in opposite directions about two pairs of straight lines. It is found for ε= 0.5, r min 1



11

= 0.79, k = -0.02404, r = 0.87, r max = 0.96, k = 0.00003. The figure gives the projection of the stream lines on the cross section of the channel when ε= 0.5 for some values of k. Similarly for the lower liquid it is found that for ε= 0.5, r min = 1



11

0.0.64, k = -0.02404, r = 0.523, r max = 0.74, k = 0.00003.

REFERENCES 1.

Patrudu, M. V. N (2001) Ph.D Thesis submitted to Andhra University, Waltair.

2.

Vidyanidhi, V., and Ramana Rao, V.V. (1968) J. Phys. Soc. Japan, 25, P. 1694.

3.

Vidyanidhi, V., and Ramana Rao, V.V. (1969) J. Phys. Soc. Japan. 27, P. 1027.

4.

Vidyanidhi, V., Bala Prasad, V., and Ramana Rao, V.V.(1977) J. Phys. Soc. Japan, 39, P. 107.

5.

Vidyanidhi, V (1969) J. Maths. Phys. Sci. 3, P. 193.