Vol. 1, No. 1 | Jan – June 2017 SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 13 Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection Kaleemullah Bhatti Department of Mathematics and Social Science, Sukkur IBA Sukkur, Pakistan kaleem.msmts@iba-suk.edu.pk Abdul Majeed Siddiqui Department of Mathematics, Pennsylvania State University, York Campus York, PA 17403, U.S.A. ams5@psu.edu Zarqa Bano Department of Mathematics and Social Science, Sukkur IBA Sukkur, Pakistan zarqa.bano@iba-suk.edu.pk Abstract This work is concerned with the flow of a Newtonian fluid through a pipe of constant circular cross section which is uniformly porous. The governing unsteady equations are solved analytically after converting them to a third order ordinary differential equation using similarity transformation method. Expressions for axial and radial velocity components and pressure distribution are obtained. The characteristics of complex axial velocity and complex radial velocity for different values of parameters are analysed. The effects of small suction and small injection are delineated through graphs. Results reveal that suction or injection has significant influence on the flow. Keywords: Unsteady Stokes Flow, Porous pipe, circular cross section, Periodic Suction and Injection, No Slip. 1. Introduction Stokes flows or creeping flows are gaining importance because of their wide range of applications. Such type of flows occurs in lubrication process, in swimming of microorganism or sperm, in the flow of lava, in paints and viscous polymers etc. There are many real world flow situations that correspond to the laminar flow through a channel with porous walls. One of the applications is in the modeling of process such as transpiration cooling, in which walls of the channel are protected from the heat by injection of cooler fluids; another application is to model the fluid flow occurring during separation of isotopes of uranium-235 and uranium-238 by gaseous diffusion in order to produce fuel for nuclear reactors. Another situation in which flow through porous channels occur is in the model to control the boundary layer flow of air on the wings of aircrafts by injection or suction of fluids into or out of the wings. Some of such situations require one or both the walls of the channel to be porous. mailto:kaleem.msmts@iba-suk.edu.pk mailto:ams5@psu.edu mailto:zarqa.bano@iba-suk.edu.pk K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 14 Berman [1] (1953) was the first mathematician who encountered the problem of laminar flow through a porous channel of rectangular cross section. There was uniform injection through the lower plate and uniform suction through the upper plate. Narasimhan [6] (1961) extended the problem studied by Berman while considering flow of a non- Newtonian fluid through a uniform circular pipe. Khaled [5] (2004) studied the effect of slip conditions on Stokes and Couette flows due to oscillating wall. Ganesh [2] (2007) studied unsteady Stokes flow through a channel consisting of parallel porous plates when there is periodic suction through the lower plate and periodic injection through the upper plate. Zaman [4] (2013) considered the magneto-hydrodynamic fluid flow between parallel porous plates with uniform suction and injection. After that Kirubhashankar [3] (2014) studied the problem of unsteady magneto-hydrodynamic fluid flow through parallel porous plates when one plate was moving with constant velocity. In this paper the flow of a Newtonian fluid through a uniformly porous pipe of constant circular cross is considered. The effects of small suction and small injection are delineated through graphs. Results reveal that suction or injection has significant influence on the flow. 2. Problem Formulation We consider a circular pipe of uniform cross section and of infinite length and a Newtonian fluid passes through the pipe. The following flow assumptions are considered:  Flow is axisymmetric with negligible body forces.  Flow is unsteady.  Stokes flow is presumed and therefore convective forces may be neglected in governing equations due to very small Reynolds Number.  The walls of the cylindrical pipe are uniformly porous.  The suction or injection occurs periodically through walls with velocity 0 iwt v e . Fig. 1: Uniformly Porous Pipe with Circular Cross Section We use cylindrical polar coordinates ( , , )r x with x -axis as axial axis of pipe and r as radial distance from the axis. Due to axial symmetry  coordinate disappears throughout. Defining a dimensionless parameter = 𝑟 𝑎 , (0 ≤ 𝜉 ≤ 1) , where a is radius of the pipe and we are going to introduce this parameter in flow equations and boundary conditions as follows: x -Component: 2 2 2 2 2 2 1 1 1 , (1) u p u u u t x x a a                        r -Component: 2 2 2 1 1 1 ( ) , (2) v p v v t a x a                            Continuity Equation:     1 0, (3)u v x a          where u and v are the velocity components in x and r direction in flow field at any K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 15 time t. The boundary conditions of the flow at any instant t are: 0 0 ( ,1) 0, (4) 0, (5) ( , 0) 0, (6) . (( 7,1) ) iwt u x u v x v x v e             We chose velocity vector U and pressure P in the form:  ˆ ˆ( , ) ( , ) , (8)i tu x i v x j e   U ( , ) . (9) i t P p x e   Using equation (8) and equation (9) the governing equations (1) to (3) reduce to: x -Component: 2 2 2 2 2 2 1 1 1 , (10) p iwu x u u u x a a                      r -Component: 2 2 2 1 1 1 ( ) , (11) p iwv a v v x a                          Continuity Equation:     1 0, (12)u v x a          The boundary conditions reduce to: ( ,1) 0, (13)u x  0 0, (14) u          ( , 0) 0, (15)v x  0 ( ,1) . (16)v x v 3. Solution to the problem Introducing the stream function  as follows, so that the continuity equation (12) is identically satisfied: 2 1 ( , ) , (17)u x a        1 ( , ) . (18)v x a x        Following the assumption of Berman [1] and Narasimhan [6], writing the stream function  as: ( , ) ( ) ( ), (19)x g x h   so that the velocities (17) and (18) are given as: 2 1 ( , ) ( ) ( ), (20)u x g x h a     1 ( , ) ( ) ( ). (21)v x g x h a      The boundary conditions together with the knowledge of inlet conditions to the pipe give rise to an expression for ( )g x . 2 0 0 1 ( ) , (22) (1) 2 a u g x av x h        Where 0 u is the average axial velocity of fluid at the entrance of pipe and is given as follows: 1 0 0 2 (0, ) . (23)u u d    Assuming that: K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 16 ( ) ( ) , (24) (1) h h       0 ( ) 1 ( ) ( ) . (25) (1) h t t dt h          Using equations (24) and (25) we get the velocity components as follows: 0 0( , ) ( ), (26) 2 u xv u x a           and 0 ( , ) ( ), (27)v x v   and the stream function reduces to: 2 0 0 ( , ) ( ).(28) 2 a u x av x            In above equations the function ( )  is still to be determined. It is also worth to mention that radial velocity becomes function of  only. Now using equations (26) and (27) in equations of motion (10) and (11) gives: 0 0 2 1 2 ( ) ( ) ( ) , (29) u xvp x a iw a                              and 0 0 2 2 1 ( ) ( ) ( ) ( ) .(30) p iwv a v a                         From onwards we would use the notations  and  instead of ( )  and ( ) in order to avoid repetition. Differentiate equation (29) with respect to  to get: 2 0 0 2 2 1 2 , (31) up x v x a iw a                                  and differentiate equation (30) with respect to x to get: 2 1 0. (32) x         Equating the results of equation (31) equation (31) we get: 0 0 2 2 0, (33) 2 u x v iw a a                           since this is to be satisfied for all x , therefore: 2 2 0. (34)iw a                   Assuming 2 iw   equation (34) becomes:  2 2 2 35= 0,a                  The boundary conditions on functions  and  can be obtained using equations (26) and (27) and already prescribed boundary conditions (13) to (16) and are given below: (1) = 0, (0) = 0, (0) = 0, (1) = 1.(36)    K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 17 The third order linear ODE (35) together with the associated boundary (36) constitute an exact solution to equations of motion and continuity as formulated. Equation (35) is well known Modified Bessel’s equation with order " "a . The general solution of equation (35) is easily obtained in terms of Modified Bessel functions as below: 1 2 0 3 0 ( ) = ( ) ( ), (37)c c I a c K a     where ( )I x and ( )K x are Modified Bessel functions of first and second kind respectively and are defined as follows: 2 =0 1 ( ) = , (38) ! ( 1) 2 m m x I x m m               ( ) ( ) ( ) = . (39) 2 sin( ) I x I x K x        Since 1 , 0 ( )I a and 0 ( )K a are three linearly independent solutions, therefore we can drop 0 ( )K a , having less contribution in solution. Also 0 ( )K a diverges at = 0 with singularity of logarithmic type. With this assumption equation (37) reduces to: 1 2 0 ( ) = ( ). (40)c c I a   Using substitution (25) and boundary conditions (36) we can easily find the values of arbitrary constants as given below: 0 1 0 1 2 ( ) = , (41) ( ) 2 ( ) a I a c a I a I a      2 0 1 2 = . (42) ( ) 2 ( ) a c a I a I a       Substituting these constants in general solution (40) we get: 0 0 0 1 2 ( ( ) ( )) ( ) = , (43) ( ) 2 ( ) a I a I a a I a I a           and using equation (25) 0 1 0 1 ( ) 2 ( ) ( ) = . (44) ( ) 2 ( ) a I a I a a I a I a           Substitute ( )  and ( ) in the components of velocity defined in equations (26) and (27) to get: 0 0( , ) = ( ) 2 u xv u x a          0 0 0 0 0 1 2 ( ( ) ( )) ( , ) = , (45) 2 ( ) 2 ( ) u xv a I a I a u x a a I a I a              and 0 ( , ) = ( )v x v  0 1 0 0 1 ( ) 2 ( ) ( , ) = . (46) ( ) 2 ( ) a I a I a v x v a I a I a          Using equation (46) in equation (8), unsteady components of velocity are: K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 18 0 0 0 0 0 1 ( , , ) = 2 2 ( ( ) ( )) , (47) ( ) 2 ( ) iwt u xv u x t a a I a I a e a I a I a                and 0 1 0 0 1 ( ) 2 ( ) ( , , ) = .(48) ( ) 2 ( ) iwta I a I a v x t v e a I a I a          Equations (47) and (48) now fully define the axial and radial velocity components respectively.Further introducing the following non-dimensional parameters: 0= , (49) RE au N  0= , (50) av R  equations (47) and (48) can be written in non-dimensional form as follows: 0 0 0 0 1 ( , , ) 1 = 2 2 ( ( ) ( )) , (51) ( ) 2 ( ) RE iwt u x t x R u a N a I a I a e a I a I a                and 0 1 0 0 1 ( ) 2 ( )( , , ) = .(52) ( ) 2 ( ) iwta I a I av x t e v a I a I a         The velocity field is now fully defined by equation (51) for axial component and by equation (52) for radial component. 4. Pressure Distribution The pressure distribution can be obtained by extracting the pressure gradients from (29) and (30) and by integrating with respect to x and  respectively. Thus: 0 = ( , ) (0, ), (53) x p dx p x p x       0 = ( , ) ( , 0). (54) p d p x p x         It follows from (53) and (54) that: 0 0 =0 = ( , ) (0, 0).(55) x x p p dx d p x p x                Using (29) and (30) we get:                     2 2 0 0 2 20 2 2 20 2 2 2 2 0 2 2 ( , ) = (0, 0) d d d d d d 1 1 d d 2 d d 1 1 d d , (56) 2 d p x p v a v a v x a a u x a                                                                                                    (56) and hence from equation (9) ( , , ) = ( , ) , (57) i t P x t p x e    where p(0,0) is pressure at the entrance of the channel. One can easily complete the calculations for pressure distribution by making the substitutions for ( )  and K. Bhatti et al. Unsteady Incompressible Stokes Flow through Porous Pipe of Uniform Circular Cross Section with Periodic Suction and Injection (pp. 13 - 22) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 19 ( ) from equations (43) and (44) in the equation (56) and (57). 5. Results and Discussion In this chapter the expressions for axial and radial velocity components for the two dimensional unsteady stokes flow through a pipe of uniform cross section and radius are obtained in equations (51) and (52). The case when 0