Fluid Flow in an Asymmetric Channel

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AUSTRALIAN JOURNAL OF ENGINEERING AND TECHNOLOGY RESEARCH (AJETR)

Vol.1, Issue 1 - 2016

43

FLUID FLOW IN AN ASYMMETRIC CHANNEL OF VARIABLE CROSS-SECTION WITH SLIP CONDITION

AT THE WALL

SYED WASEEM RAJA

Department of Mathematics, MANU University,Gachibowli, Hyderabad, India

M. V. RAMANAMURTHY

Department of Mathematics, Osmania University, Hyderabad, India

P. MUTHU

Department of Mathematics, NIT, Warangal, India

MOHAMMED ABDUL RAHIM

Department of General Studies, RCYCI, Yanbu Industrial College, Yanbu, K.S.A

ABSTRACT

This study deals with the effects of slip and phase difference in a steady flow of an incompressible

asymmetric rigid channel with permeable walls. It is assumed that the effect of fluid absorption

through permeable walls is accounted by prescribing flux as a function of axial distance. The

perturbation method is applied to linearize the non-linear governing equations by assuming the ratio

of inlet width to wavelength to be small. Effects of the above parameters on the velocity profile,

mean pressure drop and wall shear stress are studied in detail and explained graphically.

Keywords: Permeable channel, slip parameter, asymmetry.

1 INTRODUCTION

The study of the flow of viscous fluid in an asymmetric channel of varying cross section with

permeable walls is much interested in recent years in view of its numerous applications in many

physiological and engineering problems. Fluid flow in renal tubules was studied by many authors.

Mathematical modeling of the flow in proximal renal tubule was first studied by Macey [16] where

he considered the flow of an incompressible viscous fluid through a circular tube with a linear rate of

reabsorption at the wall. Bulk flow in the proximal tubule decays exponentially with the axial

distance was calculated by Kelman [5]. Then, Macey [17] used this condition to solve the equations of

motion to find the average pressure drop. Marshall et.al [7] and Palatt et al [12] studied the physicalconditions existing at the rigid permeable wall instead of prescribing the flux /radial velocity at the

wall.

In all the above studies the researcher considered the renal tubule to be symmetry. But in general,

renal tubules may not be symmetric throughout their length. A hydrodynamical aspect of an

incompressible viscous fluid in a circular tube of varying cross-section with reabsorption at the wall is

studied by Radhakrishnamacharya et al [14]. Flow in rigid tubes of slowly varying cross-section with

absorbing wall is studied by Peeyush Chandra and Krishna Prasad [13]. Fluid flow through a

diverging/converging tube with variable wall permeability was studied by Chaturani and Ranganatha

[2].


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The concept of slowly varying flow is given by Manton [6] where he obtained an asymptotic series

solution for the low Reynolds number flow through an axisymmetric tube, where radius varies slowly

in the axial direction.

The effects of slope parameter and reabsorption coefficient on the flow of fluid in a symmetric

channel with varying cross section with no-slip velocity at the walls are studied by Muthu andTesfahun [9].

In all the above studies the researchers have taken the boundary condition at the wall to be a no-slip

condition, whereas the no-slip condition is one of the aspects on which the mechanics of the viscous

liquids is built. However, there are many situations where this assumption does not hold [15].

Elshahed [8] illustrated the significance of the effect of slip at the wall. Also, the slip would be most

useful for certain problems in chemical engineering and other applications ([15],[3],[4],[18],[19]).

Fluid flow through the non-uniform channel with permeable wall and slip effect in symmetry channel

is studied by [11]. Further, Muthu and Tesfahun [10] discussed the flow through in renal tubule by

considering the asymmetric channel of varying cross-section, whereas Waseem et al [20] study the

effect of slip on fluid flow in a channel of the slowly varying cross section.

Thus, in this paper, an attempt is made to understand the flow through renal tubule of asymmetry

channel of varying cross-section and a slip velocity at the walls of the channel.

2 MATHEMATICAL FORMULATION

Here we consider an incompressible fluid flow through the asymmetric channel with a slowly varying

cross-section. The boundaries of the channel wall are taken by Muthu et al [10]as

1 1 1

2 2 1

2( ) cos ..... upper wall

2( ) cos ..... lower wall

x x d a

x x d b

, where 0 x (1)

Where1

d and2

d are the half width of the channel from the x-axis to1( ) x and 2 ( ) x

respectively at the inlet (at x=0),1a and

1b are amplitudes and is the wavelength further

1a ,

1b ,

1d ,

2d , satisfies the condition

22 2

1 1 1 1 1 22 cos( )a b a b d d (2)

Figure 1. Geometry of the channel


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We shall consider the motion of the fluid to be laminar and steady and the channel to be long

enough to neglect the initial and end effects. The equations of continuity and momentum are given

by

0u v

x y

(3)

2 2

2 2

1u u p u uu v

x y x x y

(4)

2 2

2 2

1v v p v vu v

x y y x y

(5)

Where u and v are the velocity components along the x and y axes respectively, p is the

pressure, is the density of the fluid and

is kinematic viscosity.

In order to complete the formulation of the problem, the boundary conditions are taken as follows.

(a) The tangential velocity at the wall is not zero. That is,

1 1

1

γdη dηat η ( )

d β d

u vu v y x

x y x y (6)

2 2

2

γdη dηat η ( )

d β d

u vu v y x

x y x y (7)

Where β is slip parameter and γ is the specific permeability of the porous medium.

(b)

The reabsorption has been accounted for by considering the bulk flow as a decreasing

function of x . That is, the flux across a cross-section is given by

1

2

( )

( )

( ) ( , ) ( )

x

o

x

Q x u x y dy Q F x

, (8)

Where ( ) 1 F x when 0 and decreases with x, 0 is the reabsorption coefficient and is a

constant, and oQ is the flux across the cross-section at x=0.

The boundary conditions (6) and (7) are well known Beavers and Joseph[1] condition when applied to

tangential velocity.

We introduce the stream function such that

,u v y x

(9)

And the non-dimensional quantities as

2

1 21 2, , , , , ,

o o

x y d x y p pd d d Q Q


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Where1 2

d d d .

By introducing the above non-dimensional variables the equations (3)—(5) can be written as (the

primes are dropped)

22 2 2 2 2 2

2 2 2

2 2 2 2 2 2R

x y y x y x x x y y

(10)

Whered

and oQ

R

.

Further, the boundary conditions (6—8) becomes

2 2

2sin 2 ξ sin 2 A x A x

y x y x y

at 1 1 1η ( ) cos 2 y x x (11)

2 2

2sin 2 ξ sin 2

B x B x

y x y x yat 2 2 2η ( ) cos 2 y x x (12)

1( )

2 F x at 1 1 1η ( ) cos 2 y x x (13)

1( )

2 F x at 2 2 2η ( ) cos 2 y x x (14)

Where 12

a A , 1

2

b B , 1

1

a

d , 1

2

b

d , 1

1β

d

d , 2

2β

d

d ,

γξ

dβ

The parameter R is the Reynolds number and is the wavenumber (the ratio of inlet width to the

wavelength).1

and 2 are amplitude ratios (the ratios of amplitudes 1a and 1b to the inlet width

respectively) and1

β and2β are ratios of distance from the x-axis to the upper wall and lower wall to

the inlet width respectively. In this problem, we consider exponentially decaying bulk flow [6] that is,

in equation (8), F is taken as

( ) x F x e (15)

3 METHOD OF SOLUTION

It is observed that the flow is quite complex because of nonlinearity of governing equation and theboundary conditions (10)-(14). Thus to solve equation (10) for velocity components, in the present

analysis, we assume the wave number 1 (long wavelength approximation). We shall seek a

solution for stream function ψ , x y in the form of a power series in terms of , as

0 1ψ , ψ , ψ , .. x y x y x y (16)

Substituting equation (16) in equations (10)-(14), and equating the coefficients of like powers of ,

we get the following sets of equations for 0 1ψ , , ψ , ,... x y x y


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Zeroth Order System

4

0

4 0

y

(17)

The corresponding boundary conditions are:2

0 0

2

ψ ψξ

y yat

1( ) y x and 2 ( ) y x (18)

1

1ψ ( ) η ( )

2

x

o F x e at y x (19)

2

1ψ ( ) η ( )

2

x

o F x e at y x (20)

First Order System

3 34

0 0 0 01

4 2 3

ψ ψ ψ ψψR

y y y x x y

(21)

The corresponding boundary conditions are:

22

0 01 1

2

ψ ψψ ψsin 2 ξ sin 2 A x A x

y x y y x

at

1( ) y x (22)

22

0 01 1

2

ψ ψψ ψsin 2 ξ sin 2 B x B x

y x y y x

at2

( ) y x (23)

0ψ 0 at

1η ( ) y x and

2η ( ) y x (24)

Similar expressions can be written for higher orders of . However, since we are looking for an

approximate analytical solution for the problem, we consider up to the order of1

equations.

The solution of equation (17) along with the corresponding boundary conditions (18-20) as

3 2

0 1 2 3 4

1ψ = ( ) ( ) ( ) ( )

2 A x y A x y A x y A x (25)

Following the similar procedure as in equation (25) the solution of equation (21) along with boundaryconditions (22-24) is

7 6 5 4

1 5 6 7 8

1 1 1 1ψ = ( ) ( ) ( ) ( )

840 360 120 24 R A x y A x y A x y A x y

3 2

9 10 11 12

1 1( ) ( ) ( ) ( )

6 2 A x y A x y A x y A x (26)

By substituting the value of0

ψ and1

ψ in equation (16), we get

3 2

1 2 3 4

1ψ= ( ) ( ) ( ) ( )

2 A x y A x y A x y A x

7 6 5 4

5 6 7 8

1 1 1 1δ ( ) ( ) ( ) ( )

840 360 120 24

R A x y A x y A x y A x y


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3 2

9 10 11 12

1 1( ) ( ) ( ) ( )

6 2

A x y A x y A x y A x (27)

Now, the nondimensional pressure , p x y can be obtained by using equations (27), (9) and (4),

and it is given as

2

2

1,

u u u u p x y dx R u dx v dx

x y x y (28)

The mean pressure is given as

1

2

η ( )

1 2 η ( )

1( ) ( , )

η ( ) η ( )

x

x

p x p x y dy x x

(29)

Further, the mean pressure drop between 0 x and0

x x is

0 0( ) (0) ( ) p x p p x (30)

The wall shear stress ( ) w x is defined as

2

2

1

( )

1

yy xx xy

w

dy dy

dx dx x

dy

dx

at1

η ( ) y x and1

η ( ) y x (31)

Where 2μ , 2μ , and μ

xx yy xy

u v u v

x y y x

Using the non-dimensional quantity 1 1

2

0μQ

w w

d

and 2 2

2

0μQ

w w

d

, the wall shear stress 1 w and

2

w

(after dropping the prime) can be written as

1

2

2 2 21 1

2

2 1

η η2 1

η1

w

d d v u u v

y x dx y x dx

d

dx

(32)

2

2

2 2 22 2

2

2 2

η η2 1

η1

w

d d v u u v

y x dx y x dxd

dx

(33)

It may be noted that in equation (28), the integrals are difficult to evaluate analytically to get closed

form expression for , p x y . Therefore, they are calculated by numerical integration.

4 RESULTS AND DISCUSSION

The purpose of the present discussion is to analyze the behavior of a steady incompressible fluid flow

in an asymmetric channel of slowly varying cross-section with absorbing wall by considering a slip

velocity at the walls.


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It may be noted that characterized the phase difference which varies in the range 0 π . Here

0 represents symmetric channel. π represents the asymmetric channel with waves are in

phase. and ξ represents reabsorption coefficient and slip at the channel wall, respectively. It is

observed that in the absence of slip i.e., ξ 0 , our results are in tune with those of Muthu and

Tesfahun[10].

We discuss the effect of these parameters on the transverse velocity ( , )v x y , mean pressure drop

( ) p, and wall shear stress ( )

w. The following parameters are fixed as A=-0.0628, B=0.0628,

1 2 1 2β 0.5, β 0.5, ε 0.1, ε 0.1, δ 0.1 in our numerical calculation. For low Reynolds

number flow, we have taken 1.0 R . To see the effect of ξ we have taken ξ=0, 0.15 and 0.4.

The Transverse velocity v:

The transverse velocity ( , )v x y which is obtained from equations (9) and (27) Here we have

discussed the effects of the phase difference ( ), in the presence of non-zero slip coefficient ( ξ ) on

the transverse velocity by taking the behavior at a different cross-section of the channel. We have

taken 0.1,0.5,0.9 x and π

0, ,2

.

Figure 2(a) displays the effect of ( ) on v at x =0.1 and ξ =0.0. It may be observed that as ( )

increases from 0 to , the magnitude of v decreases. It may be remarked that the reabsorption value

at the wall is fixed at x =0.1 and when ( ) increases, the cross-sectional area is reduced. This results

in lesser v values. Now, if ξ =0.15 similar effect is observed as above. When ( ) varies from 0 to

mixed trends is observed in velocity.

If ξ =0.4 the velocity decreases when varies from 0 to . But comparing with no slip ( ξ =0.0) case

velocity increases in quantity. This may be due to the effect of the slip (see figures 2(b), 2(c)).

Figure 3(a) displays the effect of on v at x =0.5 with ξ =0.0. It may be noted that as increases

the magnitude of v has mixed trends, due to the variation of the cross-section of the channel at x =0.5.

If ξ =0.15 and ξ =0.4, similar mixed trends is observed on v , due to the effect of slip (see figures 3(b),

3(c)).Figures 4(a)-4(c) display the effect of on v when x =0.9 for ξ =0.0, 0.15 and 0.4. It is observed

that as increases the magnitude of v has a mixed trend.

Mean Pressure drop

The value of the mean pressure drop (29) over the length of the channel is calculated from different

values of and ξ . Figure 5(a) represents the effect of when ξ =0.0. It is observed that as the

width of channel contracts, the mean pressure drop increases. Particularly, at the entrance of thechannel, the mean pressure drop for the asymmetrical channel is more than the symmetrical channel.


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It can be understood from figure 5(a) for 0, ,2

. However, due to contraction in the middle

of the channel, the reverse is true at the end of the channel.

When ξ =0.15, a similar trend as mentioned above is observed, with a quality difference (see fig.5

(b)).

As ξ =0.4, the trend is reversed, this shows the effect of slip combined with asymmetry nature of the

channel (fig. 5( c) ).

Magnitude of wall shear stress

The effects of and ξ on the magnitude of the wall shear stress (1 2

| | and | | w w ) are presented in

figures 6 and 7 respectively.

It may be noted from figures 6(a) to 6(b), and 7(a) to 7(b) that the upper wall and lower on shear

stress (in magnitude) increases as the channel changed from symmetry to asymmetry and no-slip to

slip conditions, except in the middle of the channel where there is more contraction.

This indicates that as the width of the channel decreases due to asymmetry nature of walls,

(1 2

| | and | | w w

) increases. But when ξ =0.4, the nature of the curve is oscillatory [See fig. 6(c) and 7

(c )].

5 CONCLUSIONS

The main contribution of this study is to see the effect of the phase difference in the presence of slip

at the walls on the flow of incompressible fluid in an asymmetric channel of the slowly varying cross-

section. The mathematical problem is solved using a regular perturbation method assuming the ratio

of inlet width to wavelength is small. We observe the following observation in the present study.

(i)

As phase difference increases the magnitude of velocity decreases.

(ii)

As the channel changes from symmetric to asymmetric the mean pressure drop increases.

(iii) The wall shear stress increases as the channel changes from symmetry to asymmetry and

no slip to slip.

APPENDIX – IMPORTANT FORMULAS AND GRAPHS

1 3 3 2 2 2 2

2 1 1 2 2 1 2 1

4( )

η -η +3η η -3η η -12ξ η +12ξ η

xe

A x , -α1 2

2 3 3 2 2 2 2

2 1 1 2 2 1 2 1

-6 η +2ξ+η e( )=

η -η +3η η -3η η -12ξ η +12ξ η

x

A x

2 -α1 1 2 23 3 3 2 2 2 2

2 1 1 2 2 1 2 1

12 ξη +2ξ +η η +ξη e( )=

η -η +3η η -3η η -12ξ η +12ξ η

x

A x

-α 3 3 2 2

2 1 1 2 2 1 2 1 1 24 3 3 2 2 2 2

2 1 1 2 2 1 2 1

-e -η -η 3η η 3η η 12ξη +12ξη +12ξη η( )=η -η +3η η -3η η -12ξ η +12ξ η

x

A x


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15 1( )=12 ( )

dA A x A x

dx, 16 2( )=12 ( )

dA A x A x

dx

32 17 2 3 1( )=4 ( ) 6 ( ) 6 ( )

dAdA dA A x A x A x A x

dx dx dx,

2 48 3 1( )=2 ( ) 6 ( )

dA dA A x A x A x

dx dx

9 1 2 1322

1 2 1 2

1( ) 7560 2ξ η η sin 2π +

420 η η 12ξ η η

A x B A x

1 2 147560 2ξ η η sin 2π A x

1 2 2 1

2 2 4 4 2

1 2 1 15 1 23024 η -η ξ 10 η +η η +η ξ 5η 5η

6

A R A A

2 1 12 2 3 3 1

1 2 2 3

5

2ξ η η η η η η6

dA

A dx

2 1 1

2 2 3 317 32 11 2 2

10ξ 5ξ2ξ η η η η η η

9 6

A dAdA A

dx dx

1

2 21 4

1 2 2 5 16

5ξ2ξ η +η η η

2

A dA A A

dx

24 4 3 3 2 2 2 2

1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 5

ξ 1η η η η η η η η η -η η +η 2η +2η +η η

36 216

A

1

22 2 2 2

1 2 2 1 2 2 1 1 2 7

5 1η +η η +η ξ η -η 3η +3η +4η η

72 144

A

22 22 1 1 2 1 2 1 2 85

4ξ η +η +η η η +η η -η72

A

2 31 2 413 2 2 2 2 21 2 1 1 1

( )=η η ξ η η ξ η ξ3 3 2 3 3

dAdA dA dA A x

dx dx dx dx

2 31 2 414 1 1 1 1 11 2 1 1 1

( )=η η ξ η η ξ η ξ3 3 2 3 3

dAdA dA dA A x

dx dx dx dx

4 4 2 2 3 3 5 515 1 2 1 2 1 2 1 2 1 21

( )= η η +η η +η η η η ξ η η2

A x

2 2 2 2

16 1 2 2 1 1 2 1 2 1 2

24 3 2 2 3 4 2 2

1 1 2 2 1 2 1 2 1 2

1

( )= η η η η η η η +η η η ξ72

5 8 9 8 η η η η η η η η η η

1008 5 5 5

A x

2 2 3 217 1 2 1 2 1 2 2 31 1

( )= η η η η ξ η - η ( ) 3 ( )2 2

A x A x A x

2 210 1 2 1 1 182 2 2

1 2 1 2 1 2

1 1 1 1 1( ) 15120 η η ξ η η sin 2π

420 3 6 6η η η 2η η η 12ξ

A x B A x

21 1 2 2 2 191 1 1

15120 η η η η η ξ sin 2π6 6 3

AA x


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11 2 21 1 27 2 3 281 5

15120 η η ξ5 3

dA R A A A A A A

dx

28 21 2 2

3

η η2ξ ξ

3 2 2 3

A A dA A

dx

3 2 3 4

1 1 2 1 2 2 1 2

1 1 1 1ξ η η η η η ξ η η

6 3 3 6

dA A

dx

7 63

1 28 5 1 5 6 2 5 11 1 1 1 1ξ η ξ+ η η3 3780 2520 2160 1890

dA A A A A A Adx

5 4 3 2 2 3

20 1 22 1 23 1 24 2 1 25 1 2 26 2

1 1 1 1 1 1η η η η η η η η

5 5 5 315 315 2520

A A A A A A

3 2 2 31 2 418 2 2 2 2 21

( ) η 3η ξ η 2η ξ η ξ3

dAdA dA dA A x

dx dx dx dx

3 2 2 31 2 419 1 2 1 1 11

( ) η 3η ξ η 2η ξ η ξ3

dAdA dA dA A x

dx dx dx dx

220 5 6 2 6 7 5 21

( ) 24ξ 7 η 7ξ 7 3 η

1512

A x A A A A A

6 6 5

5 4 4 2 3 3 22 2 1 2 121 1 1 2 1 1 2 1 2

η η η η η( ) 2η 6ξη η η ξ+η η ξ+η η ξ

6 6 6 3 A x

325 2 5 6 7 6 7 8

22 2 2

η ξ 7ξ ξ 5( ) η η

504 63 216 108 216 72 432

A A A A A A A A x

4

3 25 223 5 6 2 6 7 2 7 8 2 8

η 1 1 1 5( ) 24ξ 7 η 49ξ 21 η 126ξ 35 η ξ

504 1512 1512 1512 72

A A x A A A A A A A

4

3 25 224 5 6 2 6 7 2 7 8 2 8

η 1 1 1 175( ) 24ξ 7 η 49ξ 21 η 126ξ 105 η ξ

8 24 24 24 8

A A x A A A A A A A

43 2

5 225 5 6 2 6 7 2 7 8 2 8

η 1 1 1 175( ) 24ξ 7 η 49ξ 14 η 126ξ 35 η ξ6 24 24 24 8

A

A x A A A A A A A

4

3 25 226 5 6 2 6 7 2 7 8 2 8

2 η 1 1 1( ) 6ξ 7 η 14ξ 14 η 42ξ 35 η 35ξ

3 6 6 6

A A x A A A A A A A

4

4 3 2 2 4 3 5227 2 1 2 1 1 2 1 1 2 1

5η 5 5 5 1 5( ) η η η η ξ η η ξ 10η 30η ξ η η

18 9 3 3 18 18 A x

4 3 2 3 2 428 2 2 1 2 1 1 1 2 11 1 1 1

( ) η η η η η ξ 2η 6η ξ η η6 3 6 6

A x

11 2 2 21 2 1 1 2 21 12

( )840 η η η 2η η η ξ

A x

1 2 10 310080 sin 2π 5040 sin 2π BB B x AB B x

31 21 2 12 7 1 810 5

1008 η η ξ ξ ξ7 3

dAdA dA R B B A B

dx dx dx

7 2 2 641 9 5 2 1 2 2 1

1 1 1 1 135 ξ ξ η η ξ η ξη η

126 126 84 1008 504

dA A B A

dx

2 2 5 2 2 2 4

2 2 2 1 2 2 2 1

1 1 9 1 1 9η ξ η ξη η η ξ η ξη η

84 2 2 84 2 2

3 2 2 3 4 2 2 2

2 2 2 1 2 2 2 1

1 1 9 1 1 9η ξ η ξη η η ξ η ξη η

84 2 2 84 12 2


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5 2 2 6

2 2 2 1 2 2

1 2 13 1 2η ξ η ξη η η ξ- η ξ

84 3 6 84 3

6 5 2 2 46 2 1 2 1 2 2 2 11 1 1 1 1

ξ η η ξ 2η ξ η η 4ξη ξ η η72 72 36 36 2

A

2 2 2 3 3 22 2 2 1 2 2 11 1 1

η 4ξη ξ η η ξη 4η ξ η36 2 36

2 2 4 5

2 2 2 1 2 2 11 7 4 8

1 1 1 1 52η ξ- η +ξ η η η ξ ξη

36 2 36 2 12 B A B A

2 31 2 41 2 2 2 2 2

1 2 1 1 1 1( ) η η ξ η η ξ η ξ

3 3 2 3 3 3

dAdA dA dA B x

dx dx dx dx

3 2 2

2 1 2 1 2 1 2 2

1 1 1 1( ) η η ξ η η ξ η η η ξ

4 4 2 2

B x

2 31 2 43 1 2 1 1 1

1 2 1 1 1 1( ) η η ξ η η ξ η ξ3 3 2 3 3 3

dAdA dA dA B x

dx dx dx dx

4 3 2 2 2 2 2

4 1 2 1 2 2 2 1 2 2

2 2 2 3

1 2 2 2 2 2

1 1 5 1 1( ) η η ξ η ξ ξη η η η 3ξη η ξ

6 6 3 6 6

5 1 1η η ξη η ξ ξη η ξ

3 6 6

B x

6 5 2 4 3 3 4 2 5 65 2 1 2 2 1 2 1 1 2 2 2 1 2 2 1 21 1

( ) η ξ η η ξ η η 3η η ξ-3η η ξ-3 η ξ η η η η ξ η η ξ2 6

B x

5 4 2 3 2 3 4 56 2 1 2 2 1 2 1 2 1 2 2 2 1 25 5 5 1 1 5 5

( ) ξ η η η ξ η η 5η η ξ-5 η ξ η η η η ξ η η ξ3 3 3 2 6 3 3

B x

4 3 2 2 3 47 2 1 2 2 1 2 2 1 2 2 1 2 2

2

3 1 2 1 2 2

1 1( ) η ξ η η ξ η η 3 η ξ η η η η ξ η ξη

2 6

1 2 19 η ξ η η η ξ η

6 3 6

B x A

A

4 3 2 2 3 48 2 1 2 2 1 2 2 1 2 2 1 21 1

( ) η ξ η η ξ η η 3 η ξ η η η η ξ η ξη2 6

B x

3 2 2 39 2 1 2 2 1 2 2 1 2( ) η ξ η η ξ η η η ξ η η η ξ B x

2 2

10 2 1 2 2 1 2 2

1 1

( ) η ξ η 2 η ξ η η 2 η ξ η4 4

B x

5 2 2 4 2 2 311 2 1 2 2 1 2 2 1 21 11 1 1 1 7 1

( ) η ξ η ξη η ξ η ξη ξ η η η36 72 144 12 12 2 2

B x

2 2 2 2 3 42 1 2 2 2 2 2 1 2 21 1 7 1 1 1 1

η η η ξη ξ 2η ξ η ξ η η η ξ ξη12 12 2 12 6 12 3

4 3 2 2 3 412 5 1 6 2 2 1 2 2 1 2 2 1 2 2 1 2 35 1 1

( ) η ξ η η ξ η η 3 η ξ η η η -η ξ η ξη3 2 6

B x B A B A A

2 5 6 7 81 2 2 2 2 2

1 2 1 1 1 1( ) η η ξ η η ξ η ξ

3 3 2 3 3 3

dA dA dA dAC x

dx dx dx dx


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2 5 6 7 82 1 1 1 1 1

1 2 1 1 1 1( ) η η ξ η η ξ η ξ

3 3 2 3 3 3

dA dA dA dAC x

dx dx dx dx

4 3 2

3 5 2 5 6 2 6 7 2 7 8 2 8

2 7 7 7 35( ) η ξ η ξ - η 7ξ η 35ξ

5 6 3 3 6

C x A A A A A A A A

3 2 2 3

4 2 1 2 1 2 1 2 2 6 2 1 7

1 1 1 1( ) ξ+ η η ξη η η ξ η η ξη 9 η ξ+ η

2 2 6 6

C x A A

5 4 2 3 2 3 4 5

5 2 1 2 1 2 1 1 2 2 2 1 2

1 1( ) ξ+ η η ξη η ξη η ξη η η ξ+ η η ξη

2 2

C x

3 2 2 3

6 2 1 2 1 2 2 1 2

1 1( ) ξ+ η η ξη η η ξ- η η ξη

2 2

C x

4 3 2

7 5 2 5 6 2 2 6 7 7 8 2 8

1 1 1 5( ) η ξ - η η 2ξ 5ξ η 20ξ

7 4 2 4

C x A A A A A A A A

5 4 2 3 2 3 4 58 2 1 2 1 2 1 1 2 2 2 1 2 5

4 3 2 2 3 4

2 1 1 2 1 2 2 2 1 2 6

3 2 2 3

2 1 1 2 2 2 1 2

1 1( ) ξ+ η η ξη η ξη η ξη η η ξ- η η ξη2 2

5 5 5 5 5 1 5ξ η η ξη η ξη η η η ξ η ξη

3 6 3 3 3 2 3

5 1 1ξ+ η η ξη η η η ξ η ξη

3 2 2

C x A

A

7

A


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Figures 2(a)- 2(c) Distribution of Transverse velocity v with y at x=0.1, =1.0


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Figures 5(a)- 5(c) Distribution of Mean Pressure Drop with x


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Figures 6(a)- 6(c) Distribution of Magnitude | w1| of with x


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Figures 7(a)-7(c) Distribution of Magnitude |

w2| of with x


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REFERENCES

1.

Beavers, G.S., Joseph, D.D., (1967). Boundary conditions at a naturally permeable wall. J.

Fluid Mechanics 30, 197 –207.

2. Chaturani, P., Ranganatha, T. R., (1991). Flow of Newtonian fluid in non-uniform tubes with

variable wall permeability with application to flow in renal tubules. ActaMechanica , 11 –26.

3.

Chu, Z., (2000). Slip flow in an annulus with corrugated walls. J. Phys. D ,33, 627 –631.

4.

Joseph, D. D., Ocando, D., (2002). Slip velocity and lift. J. Fluid Mech. 454, 263 –286.

5. Kelman, R. B., (1962). A theoretical note on exponential flow in the proximal part of the

mammalian nephron. Bull. Of Mathematical Biophysics , 303 –317.

6.

Manton, M. J., (1971). Low Reynolds number flows in slowly varying axi-symmetric tubes. J.

Fluid Mech., 49, 451 –459.

7. Marshall, E.A., Trowbridge, E. A., (1974). Flow of a Newtonian fluid through a permeable

tube: The application to the proximal renal tubule. Bull. Of Mathematical Biology, 457 –476.

8. Moustafa, E., (2004). Blood flow in capillary under starling hypothesis. Appl. Math. Comput.,

149, 431 –439.

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Muthu, P., Tesfahun, B., (2010). Mathematical model of flow in renal tubules.Int. J.

Appl.Math.Mech., 6, 94 –107.

10. Muthu, P., Tesfahun, B., (2011). Fluid flow in an asymmetric channel.Tamkang Journal of

mathematics, 42 149-162.

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Muthu, P., Tesfahun, B., (2012). Flow through non-uniform channel with permeable wall and

slip effect. Special Topics & Reviews in Porous Media— An International Journal, 3, 321 –328.

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Paul, J., Henry Sackin, P., Roger, I.,Tanner (1974). A hydrodynamical model of a permeable

tubule. J. Theor.Biol., 287 –303.

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Peeyush Chandra., Krishna Prasad., J. S. V. R., (1992). Low Reynolds number flow In tubes ofvarying cross-section with absorbing walls. Jour. Math. Phy.Science.26(1), 19-36.

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Radhakrisnamacharya.,G, Peeyush Chandra, Kaimal, M. R .,(1981). A Hydrodynamical study

of the flow in renal tubules. Bull. Of Mathematical Biology ,151 –163.

15. Rao, I. J., Rajagopal, T.,(1999). The effect of the slip boundary condition on the flow of fluids

in channel. Acta Mech., 135, 113 –126.

16. Robert, I., Macey., (1963). Pressure flow patterns in a cylinder with reabsorbing walls. Bull.

of Mathematical Biophysics., 1-9.

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Robert, I., Macey .,(1965). Hydrodynamics in renal tubules.Bull. of Mathematical Biophysics,

117 –124.


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18.

Vasudeviah, M., Balamurugan ,K.,(1999). Stokes slip flow in corrugated pipe. Int. J. Eng.Sci.,

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Wang, C. Y., (2009). Low Reynolds number slip flow in a curved rectangular duct, J, Applied

Mech. 69, 189-196.

20. Waseem Raja, Syed., Ramana Murthy, M.V., Muthu, P., Abdul Rahim, Mohammed.,(2014).

Effect of slip velocity on fluid flow in a channel of varying cross-section. Acta Ciencia Indica,

XL M(3),347-369.

Fluid Flow in an Asymmetric Channel

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