Oscillating flow in a circular pipe with adverse pressure gradient.
The phenomenal growth of energy requirements in recent years has been attracting considerable attention all over the world. This has resulted in a continuous exploration for new ideas avenues in harnessing various conventional energy sources like tidal waves, wind power, geo-thermal energy, etc. It is obvious that in order to utilize the geo-thermal energy to the maximum, one should have a complete and precise knowledge of the amount of perturbations needed to generate convection currents in geo-thermal fluid. Also, the knowledge of the quantity of perturbations essential to initiate convection currents in mineral fluids found in the earth's crust helps one to utilize minimal energy to extract the minerals. There is an extensive literature on the flow through a porous media, which is governed by generalized Darcy's law. Generalized Darcy's law is derived taking into account the viscous and convective acceleration (Brinkman , Yamamoto ). Iwamurd  expressed the equations of flow through a highly porous medium. Raptis et al. -, using the above equations studied the influences of the free convective flow and the mass transfer on the flow through a porous medium. Raptis et al. , also studied the influences on the oscillatory flow through a porous medium. Hiremath and Dharwad  studied the effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature. Baranyi and Shirakashhi  investigated the present a finite- difference solution of the two-dimensional, time dependent incompressible Navier-Stokes equations for laminar flow about fixed oscillating cylinders placed in an otherwise uniform flow. Asghar at el  studied the flow of a non-newtonian fluid induced due to the oscillations of a porous plate. Recently, Chen, XIAO-BO and Guoxiong, Wu  using the singular and highly oscillatory properties of the Green function for ship motions. Recently Aslanov  studied the oscillations of a body with an orbital tethered system. Concerning the studying of the free convection effective perfectely conducting couple stress fluid and state space formulation for magnetohydrodynamic free convection flow with two relaxation times, we may refer to [13 and 14]. Ahmed et al  investigated the laminar boundary layers in oscillatory MHD flow past a porous vertical plate.
In this paper, we discuss the problem of oscillating flow of a viscous incompressible fluid in a circular horizontal pipe with adverse pressure gradient under the assumption of parallel flow to the axis of the pipe. The velocity at the entrance section is assumed constant. A retarded flow is expected due to the adverse pressure gradient. Also a slip velocity may occur in the entrance part at the surface of the pipe but far from the entrance, the motion is laminar. The velocity field through the pipe is obtained and inlet length section [z.sub.L] is calculated after which a parabolic velocity profile occurs.
We consider the flow of two- dimensional incompressible viscous fluid along a circular horizontal pipe. The z'-axis is taken along the axis of the pipe from the entrance section and the r'-axis along the radius of the pipe normal to the z'-axis.
Let u', v' be the velocities along z'-axis and r'-axis respectively. The equations of motion and continuity in cylindrical polar coordinate system are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
[partial derivative]/[partial derivative]r'(r'v') + r' [partial derivative]u'/[partial derivative]z' = 0. (3)
And since v' is of order [delta] in the boundary layer , then from eq.(2)
- 1/[rho]' [partial derivative]p'/[partial derivative]r' = 0 so that p' [equivalent to] p'(z', t')
Introducing the non dimensional quantities defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Where V' is the mean velocity across a section in the pipe in the parabolic velocity profile section, a is the radius of the pipe, U' the velocity at the entrance section, R the Reynold's number in the parabolic velocity profile section and [omega]' the frequency of oscillation.
Substitute in eq. (1) and eq. (3) using eq. (4) we get
[partial derivative]u/[partial derivative]t + v [partial derivative]u/[partial derivative]r + u [partial derivative]u/[partial derivative]z = - [partial derivative]p/[partial derivative]z = - [partial derivative]p/[partial derivative]z + 1/R ([[partial derivative].sup.2]u/[partial derivative][r.sup.2] + 1/r [partial derivative]u/[partial derivative]r (5)
[partial derivative]/[partial derivative]r (vr) + [partial derivative]/[partial derivative]z (ur) = 0 (6)
With the boundary conditions
v = 0, [partial derivative]u/ [partial derivative]r = 0, or u = [u.sub.max] on r = 0
u = v = 0 on r = 1 for z [greater than or equal to] [z.sub.L]
Let us assume the adverse pressure gradient be of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
The equation of continuity (6) is satisfied by a stream function [psi] (r, z, t) such that
u = 1/r [partial derivative][psi]/[partial derivative]r, v = - 1/r [partial derivative][psi]/[partial derivative]z (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[[psi].sub.1](r, z)=1+z/R [f.sub.1]([eta]), [[psi].sub.2](r,z)=1+z/R [f.sub.2]([eta]) (11)
[eta] = [square root of R] r/1 + z, [t.sup.*] = t/[(1 + z).sup.2] (12)
Substituting equations (9), (10), (11) and (12) into eq. (5) and eq.(7) and often one integration we get two ordinary differential equations in [eta] as
[eta][f".sub.1] - [f'.sub.1] + [f.sub.1][f'.sub.1] = 1/2 [[eta].sup.3] (13)
[eta][f".sub.2] + ([f.sub.1] - 1)[f'.sub.2] + ([f.sub.1] - i [omega] [eta])[f.sub.2] = 1/2 [[eta].sup.3] (14)
[f.sub.1](0) = 0, [f'.sub.1](0) = 0 and [f.sub.2](0) = 0, [f'.sub.2](0) = 0 (15)
Solution of the problem
The solution of equation (13) is easily obtained in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
where (c > 1) is given by c = U + 1/U - 1 and (U > 1) is the non-dimensional velocity at the entrance of the pipe where the pressure [rho] [U.sup.2] is constant. The velocity along the axis of the pipe is U/1 + z which decreases as we move along the pipe. The velocities u, v in the entrance part do not vanish at r = 1 but as z [right arrow] [z.sub.L], the inlet length, u, v [right arrow] 0 at the beginning of the parabolic velocity profile section.
Solution of equation (14) for the oscillating part of the velocity is obtained as a power series, using regular singular point method, in the form 
[f.sub.2]([eta]) = [[infinity].summation over (n=0)][a.sub.n][[eta].sup.n+m] (17)
and on substituting in eq.(14) using eq. (15) the coefficients [a.sub.n] are calculated and numerical values for the fluctuating part of the velocity field are plated in figures (5) and (6) for values of [omega]t = [pi]/4 and R = 10000.
The inlet length [z.sub.L]
To determine the inlet length along the pipe from the position z = 0, r = 0, we use the steady part of the velocity field only, since the contribution from the fluctuating part of the velocity field is negligible for large z.
Using the condition
u = f'([[eta].sub.1])/(1 + [z.sub.L])[[eta].sub.1] = 0,
[[eta].sub.1] = [square root of R]/1 + [z.sub.L],
leads to an algebraic equation of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Whose approximate solution gives [z.sub.L] in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
For 1 < C < 2 and R = 10000, 20 < [z.sub.L] < 132, approximately.
Discussion of the results
From figures (1), (2), (3) and (4) for the steady part of the velocity field, the velocity decreases as we move downstream along the pipe. From figures (2) and (3), the backward flow increases as the Reynold's number increases while this backward flow decreases with increasing z along the axis of the pipe. Also the slip velocity at the walls of the pipe decreases with increasing z.
From figures (5) and (6), the fluctuating part of the velocity field increases with increasing the frequency a and increasing U.
It is clear from figure (6) that the fluctuating part of the velocity field decreases with increasing z and is very small for large z.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
Nomenclature u,v the velocities U velocity at the entrance V mean velocity a radius of the pipe R Reynolds number [rho]' density t time p pressure [omega] frequency [z.sub.L] inlet length (r, z) cylindrical coordinates [psi](r, z, t) stream function [eta] = [square root of R]/1 + z r [t.sup.*] = 1/(1+z) c=U + 1/U - 1
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Angail A. Samaan
Department of Mathematics, Woman's University College, Ain Shams University, Cairo-Egypt
E-mail: Angail 123@ Yahoo.com
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|Author:||Samaan, Angail A.|
|Publication:||International Journal of Dynamics of Fluids|
|Date:||Dec 1, 2009|
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