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Adv. Studies Theor. Phys., Vol. 2, 2008, no. 10, 473 - 478

The Multiple Solutions of Laminar Flow in a

Uniformly Porous Channel with Suction/Injection

Botong Li1

, Liancun Zheng1

, Xinxin Zhang2, Lianxi Ma

3

1Department of Mathematics and Mechanics

University of Science and Technology Beijing, Beijing 100083, China

[email protected], [email protected]

2Mechanical Engineering School

University of Science and Technology Beijing, Beijing 100083, China

3Department of Physics, Blinn College, Bryan, TX 77805, USA

[email protected]

Abstract: This paper presents a numerical investigation for laminar flow in a

uniformly porous channel with suction/injection at both moving walls. The

characteristics for the existence of multiple solutions to the problem are numerically

established for values of Reynolds number and the velocity coefficient.

Mathematics Subject Classification: Numerical hydromechanics

Keywords: Porous channel, moving walls, shooting method, multiple solutions

1 IntroductionFollowing the pioneering work of Berman [1], the problems of the steady,

incompressible, laminar flow in channels or circular pipes with uniformly porous

walls with suction/injection at both walls have attracted considerable attention duringthe last few decades. The main reason for it is probably that fluid flow is produced


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474 Botong Li, Liancun Zheng, Xinxin Zhang,Lianxi Ma

industrially in increasing quantities and is therefore just likely to be pumped in a plant.

The great majority of theoretical investigations in this field described the fluid flow in

the vicinity of the surface with the aid of similarity solutions [2-6].The purpose of this paper is to present a numerical investigation for this

problem. A special emphasis is given to the formulation of boundary layer

equations, which may provide the multiple similarity solutions.

2 Formulation of the problemConsider the steady, incompressible, laminar flow along a two-dimensional

channel with porous walls through which fluid is injected/extracted with uniform

speed wv . Let the channel width be2h and introducing the dimensionless variable:

y

h = (1)

the Navier-Stokes equations and the continuity equation are written as:

2 2

2 2 2

1 1u v u P u uu

x h x x h

+ = + +

(2)

2 2

2 2 2

1 1v v v P v vu

x h h x h

+ = + +

(3)

The boundary conditions are:

( , 1) 0u x = , ( , 1) wv x v = , ( ,0) 0u

x

=

, ( ,0) 0v x = (4)

The problem is reduced to the following equation:

2[ ]f R f ff k + = (5)

where ka constant and the suction Reynolds number of the flow is taken as:

( )wv h

R

(6)

The boundary conditions (4) become:

(0) 0f =

(0) 0f =

(1) 1f =

(1) 0f = (7)


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Multiple solutions of laminar flow 475

Watson investigated analytically a similar flow of fluid which is driven by uniform

steady suction through the porous and accelerating walls of the channel. The problem

is also reduced to equation (5) with the following boundary conditions:

(0) 0f =

(0) 0f =

0(1) 1f k= +

0(1)f k = (8)

Where 0k is the velocity coefficient of accelerating walls ( 0 ( 1, 0]k ).

3 Numerical InvestigationIn order to obtain the numerical solution, we transfer the problems (5) and (8) to a

system of four first-order equations as follows

f u = v w = v w = w Rfw Ruv = (9)

The corresponding boundary conditions are:

(0) 0, (0) 0f v= = 0 0(1) 1 , (1)f k u k= + = (10)

We introduce the parameters of tand s such that

(0)u t= , (0)w s= (11)

The problem now is to find the parameters ,t s . We denote the solutions of (9)-(10)

as ),,( 21 ttf , ),,( 21 ttu , ),,( 21 ttv and ),,( 21 ttw . Thus the following equations

are solved by using the Newtonian technique.

01),,1(),( 02121 == kttftt 0),,1(),( 02121 == kttutt (12)

4 Numerical Results

The Watson problems is discussed by dividing the value of the suction Reynolds

number into three sections for different velocity coefficient and some interesting

numerical results are presented in figures 1-4.

Figures 1-2 show the distribution characteristic of values of (1)f with

R ( 0R > or 0


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For 00 =k , Only a single solution is observed for each value of [0,12.165)R

Triple solutions are found for each value of (12.165, )R + ; Only a single solution

is observed for 0R < .

For 0 0.5k = , only a single solution for each value of [0,25)R Triple

solutions for each value of (25, )R + ; Only a single solution for 0R < .

For 0 0.75k = , only a single solution for each [0, 48)R Triple solutions for

each value of (48, )R + ; Only a single solution for 0R < .

Figures 3-4 show the unique velocity profiles and the unique shear profiles for

10R = with different values of velocity parameters 0k .

5 ConclusionsThe multiple solutions are investigated for laminar uniform flow in a channel with

special suction or injection at both porous walls for certain parameters and the

transfer characteristics are discussed in detail.

Acknowledgement: The work is supported by the National Natural Science

Foundations of China (No.50476083).

References

[1]A.S.Berman, Laminar flow in channels with porous walls, J.Appl. Phys, 24(1953),1232-1235.

[2] A. MG, C Lu and S P H, Asymptotic behavior of solutions of a similarity

equation for laminar flows in channels

with porous walls, SIAM. J. Applied Mathematics, 49(1992), 139-162.

[3]C.L, On existence of multiple solutions of a boundary value problem from pipe

flow, Q. A. M., 2(1994), 361.


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Multiple solutions of laminar flow 477

[4]C.L, On the asymptotic solution of laminar channel flow with large suction, J. M.

A, 28 (1997), 1113-1134.

[5] E.B.B.Watson, W.H.H.Banks, M.B.Zaturska, et. al., On transition to chaos in

two-dimensional channel flow symmetrically driven by accelerating walls,

J.Fluid Mech., 212(1990), 451-485.

[6]S. P. Hastings, C. Lu and A. D. Macgillivray, A boundary value problem with

multiple solutions from the theory of laminar flow, SIAM. J. Math. Anal,

23(1992), 201-208.


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0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

k0 = - 0. 75

k0

= - 0 . 5

k0

= 0

-f''(1)

R -800 -700 -600 -500 -400 -300 -200 -100 01

2

3

4

5

6

7

k0 = - 0 . 7 5

k0

= - 0 . 5

k0

= 0

R

-f''(1)

Fig.1 Multiple values of (1)f with R ( 0R > ) Fig.2 Unique value of (1)f with R ( 0

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