Non-linear effects in flow in porous duct.

1 Introduction

The problems of fluid flow through porous duct have arouse the interest of Engineers and Mathematicians, the problems have been studied for their possible applications in cases of membrane filtration, transpiration cooling, gaseous diffusions and drinking water treatment as well as biomedical engineering. Such flows are very sensitive to the Reynold number.

Berman was the first researcher who studied the problem of steady flow of an incompressible viscous fluid through a porous channel with rectangular cross section, when the Reynold number is low and the perturbation solution assuming normal wall velocity to be equal was obtained .

Sellars , extended the problem studied by Berman by using very high Reynold numbers.

Also wall suctions were recognize to stabilize the boundary layer and critical Reynold number for natural transition 46130 was obtained . The stabilization effects of wall suction is due to the change of mean velocity profiles.

In the review of Joslin , it is also noticed that the uniform wall suction is not only a tool for laminar flow control but can also be used to damped out already existing turbulence.

The effects of Hall current on the steady Hartman flow subjected to a uniform suction and injection at the boundary plates has been studied .

Other reviews of flow in porous duct tend to focus only on one specific aspect of the subject at a time such as membrane filteration , the description of boundary conditions  and the existence of exact solutions .

In this paper, we consider the steady two-dimensional laminar flow of an incompressible viscous fluid between two parallel porous plates with equal suction and assume that the wall velocity is non uniform.

2 Formulation of the problem

The steady laminar flow of an incompressible viscous fluid between two parallel porous plates with an equal suction at walls and non uniform cross flow velocity is considered. The well known governing equations of the flow are:

Continuity equation

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0. (1)

Momentum equations (without body force)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Let us consider channel flow between uniformly parallel plates with equal suction. Assuming that we are far downstream of the entrance, the boundary conditions can be defined as

y = h, u = 0, v = [v.sub.w], (4)

y = -h, u = 0, v = -[v.sub.w]. (5)

Let [bar.u](0) denote the average axial velocity at an initial section (x = 0). Then it is clear from a gross mass balance that [bar.u](x) will differ from [bar.u](0) by the amount [v.sub.w]/h x. This observation led Berman (1953) to formulate the following relation for the stream in the channel .

[psi](x, y) = (h[bar.u](0) - [v.sub.w]x) f([y.sup.*]). (6)

Where [y.sup.*] = y/h, [psi](x, y) is a stream function, [bar.u](0) is initial average axial velocity and f is dimensionless function to be determined. The velocity components follow immediately from the definition of [psi]:

u(x, [y.sup.*]) = [partial derivative][psi]/[partial derivative]y = ([bar.u](0) - [v.sub.w]x/h)f'([y.sup.*]) = [bar.u](x)f'([y.sup.*]), (7)

v(x, [y.sup.*]) = - [partial derivative][psi]/[partial derivative]x = [v.sub.w]f([y.sup.*]) = v(y). (8)

The stream function must now be made to satisfy the momentum equations (2) and (3) for steady flow (2) and (3) will now become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Using (7) and (8) in (9) and (10), the momentum equations reduces to,

-1/[rho] [partial derivative]p/[partial derivative]x = ([bar.u](0) - [v.sub.w]X/h)([v.sub.w]/h(ff" - [f'.sup.2]) - v/[h.sup.2] f"\), (11)

-1/[rho]h [partial derivative]p/[partial derivative][y.sup.*] = [v.sup.2.sub.w]/h ff' - v[u.sub.w]/[h.sup.2] f". (12)

Now differentiating (12) w.r.t x, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Differentiating (11) w.r.t [y.sup.*], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

From (13), (14) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Let the suction Reynold number be Re = h[v.sub.w]/v and substitute into above expression, we get

f"" + Re(f'f" - ff'") = 0. (16)

(16) has no known analytic-closed form solution, but it can be integrated once i.e integrate (16) w.r.t [y.sup.*], we get

f'" + Re([f'.sup.2] - ff") = K = const. (17)

The boundary conditions on f([y.sup.*]) 0f (4) and (5) can now be written as,

f(1) = 1, f(-1) = -1, f (1) = 0, f (-1) = 0. (18)

Hence, the solution of the equations of motion and continuity is given by non-linear fourth order differential equation (16) subject to the boundary condition (18).

3 Results

3.1 Approximate analytic solution (perturbation)

The non-linear ordinary differential equation (16) subject to condition (18) must in general be integrated numerically. However for special case when "Re" is small, approximate analytic results can be obtained by the use of a regular perturbation approach. Note that perturbation method has been used because the equations (16 and 18) are non-linear by using that technique, we get a linear approximated version of the true equations. The solution of f([y.sup.*]) may be expanded in power of Re 

f([y.sup.*]) = [[infinity].summation over (n=0)] [Re.sup.n][f.sub.n]([y.sup.*]) (19)

where [f.sub.n]([y.sup.*]) satisfies the symmetric boundary conditions

[f.sub.0](0) = [f'.sub.0](1) = [f".sub.0](0) = 0, [f.sub.0](1) = 1 (20)

and

[f.sub.n](0) = [f'.sub.n](1) = [f".sub.n](0) = 0, [f.sub.n](1) = 1. (21)

Here [f.sub.n] are independent of Re. Substituting (19) in (16), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equating coefficients of Re, we get

[f"".sub.o] = 0, (22)

[f"".sub.1] + [f'.sub.o][f".sub.o] - [f.sub.o][f"'.sub.o] = 0, (23)

[f"".sub.2] + [f'.sub.o][f".sub.1] + [f'.sub.1][f".sub.o] - [f.sub.o][f".sub.1] - [f.sub.1]f["'.sub.o] = 0. (24)

The solution of (22) is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A, B, C and D are constants.

Applying the boundary condition (20) to the above equation, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

The solutions of Eq (23) and (24) subject to the boundary condition (21), are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Hence, the first order perturbation solution for f ([y.sup.*]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

The second order perturbation of solution for f([y.sup.*]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Hence, the first order expression for the velocity components are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

For pressure distribution, from Eq. (11) we get [h.sup.2]/[rho]v [partial derivative]p/[partial derivative]x = [[bar.u](0) - [v.sub.w]x/h][f"'([y.sup.*]) + Re([f'.sup.2]([y.sup.*]) - f([y.sup.*])f"([y.sup.*]))], and since f"'([y.sup.*]) + Re [f'.sup.2]([y.sup.*]) - f([y.sup.*]) f" ([y.sup.*])) = K, from (17), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Now, from Eq. (12), we have

[partial derivative]p/[partial derivative][y.sup.*] = [mu][v.sub.w]/h f"([y.sup.*]) - [rho][v.sup.2]f([y.sup.*])f'([y.sup.*]). (33)

Since dp = [partial derivative]p/[partial derivative]x dx + [partial derivative]p/[partial derivative][y.sup.*] d[y.sup.*], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Integrating (34), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

The pressure drop in the major flow direction is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

3.2 Numerical solution

The approximate results of the previous section are not reliable when the Reynold number is not small. To obtain the detail information on the nature of the flow for different values of Reynold number (i.e. Re = 0, 10, 20, 30), a numerical solution to the governing equations is necessary. The RungeKutta program App.C is used to solve Eq. (17) numerically. One initial condition and constant (K) are unknown; i.e. starting at [y.sup.*] = 1, then f"(1) and K were guessed and the solution double-iterated until f(-1) = -1 and f (-1) = 0. The most complete sets of profiles are shown in the figs. 1 and 2.

4 Discussion

The velocity profiles have been drawn for different values of Reynold number (i.e. Re = 0,10,20,30). The shapes change smoothly with Reynold number and show no odd or unstable behaviour. Suction tends to draw the profiles toward the wall. From fig. (1), it is observed that for Re > 0 in the region 0 [less than or equal to] [y.sup.*] [less than or equal to] 1, f ([y.sup.*]) decreases with the increase of Reynold number Re. Also from fig. (2), it is observed that, for Re > 0, then f (y*) decreases with an increase of the Reynold number in the range of 0 [less than or equal to] [y.sup.*] [less than or equal to] 1.

5 Conclusion

In this paper, a class of solutions of laminar flow through porous duct has been presented. Numerical approach is necessary for arbitrary values of Re. Also, when a cross flow velocity along the boundary is not uniform, a numerical technique is necessary to solve Eq. (2) and (3). Also, from the results obtained in this article, we can now conclude that, the non-linear effects of a flow of the porous duct is due to non uniform cross flow velocity and non vanishing terms of convective acceleration of momentum equations. The perturbation solution obtained for this problem reduces to the results of Berman .

Acknowledgement

The authors wish to thank Dr. Chifu E. Ndikilar for his time, useful discussions and help in preparing the numerical solution to this problem.

Nomenclature

A,B,C,D: Constants K: Arbitrary Constant

f: Dimensionless function representing lateral velocity profile

h: Height of the channel

P: Pressure

x: Axial distance

y: Lateral distance

[v.sub.w]: Lateral wall velocity

u(x,y): Axial velocity component

v(x,y): Lateral velocity component

[y.sup.*] = y/h: Dimensionless lateral distance

Re = [v.sub.w]h/v: Wall Reynold number

Greek Symbols

[mu]: Shear viscosity

v: Kinematic viscosity

[rho]: Fluid density

[psi](x, y): Stream function.

Submitted on: August 23, 2013/Accepted on: September 03, 2013

References

[1.] Berman A.S. Laminar flow in channels with porous walls. Journal of Applied Physics 1953, v. 24, 1232-1235.

[2.] Sellars J.R. Laminar flow in channels with porous walls at high suction Reynold number. Journal of Applied Physics 1955, v. 26, 489.

[3.] Drazin P.G. and Reid W.H. Hydrodynamic stability, Cambridge University Press, (1981).

[4.] Joslin R.D. Aircraft laminar flow control. Annual Review of Fluid Mechanics 1998, v. 30, 1-29.

[5.] Abu-El-Dhat E. Hartman flow with uniform suction and injection at the boundary plate. M.Sc. Thesis, Helwan University, Egypt, (1993).

[6.] Sahraoui M. and Kaviany M. Slip and no-slip velocity boundary-conditions at interface of porous, plain media. International Journal of Heat and Mass Transfer, 1992, v. 35(4), 927-943.

[7.] Wang C.Y. Exact solutions of the steady-state Navier-Stokes equations. Annual Review of Fluid Mechanics, 1991, v. 23, 159-178.

[8.] Belfort G. Fluid-mechanics in membrane filtration--recent developments. Journal fo Membrane Science, 1989, v. 40, 123-147.

[9.] Frank White. Viscous fluid flow, 2nd edition, McGraw-Hill company New York, (1991).

[10.] Ganesh S. and Krishnambal S. Magnetohydrodynamics flow of viscous fluid between two parallel plates. Journal of Applied Science 2006, v. 6(11), 2420-2425.

Hafeez Y. Hafeez*, Jean Bio-Chabi Orou ([dagger]) and Omololu A. Ojo ([double dagger])

* Physics Department, Federal University Dutse, Nigeria. E-mail: hafeezyusufhafeez@yahoo.com

([dagger]) University Abomey-Calavi, Republic of Benin. E-mail: Jchabi@yahoo.fr

([double dagger]) African University of Science and Technology, Abuja--Nigeria. E-mail: prayerz@yahoo.com
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