A study of an incompressible two-dimensional flow i n a channel with one porous wall is presented in this research. The governing continuity and momentum equations together with the associated boundary conditions are first reduced to a set of self simil ar non- linear coupled ordinary differential equations usin g similarity transformations. Then we solved the ordi nary differential equation by DTM and the numerical meth od.

he objective of the present study was to apply DTM to obtain an explicit analytic solution of laminar, isothermal, incompressible viscous flow in a rectan gular domain bounded by two moving porous walls, which enable the fluid to enter or exit. Figure II shows the effects of changing the Reynolds number while maintaining the values of Non- dimensional wall dilation rate. The result shows th at as the Reynolds number increases, the normal component of velocity decrease. In Figure II a proper comparison is also made between the numerical solution.Runge Kutta method and RVIM. A great agreement between analytical solutions and numerical ones are illustrated. In Figure III , the effects of Non-dimensional wall dilation rate with constant Reynolds number on radi al velocity can be illustrated. For every level of injection or suction, in the cas e of expanding wall, increasing a leads to higher radial velocity near the center and the lower radial veloc ity near the wall. The reason is that the flow toward the ce nter becomes greater to make up for the space caused by the expansion of the wall and as a result, the radial v elocity also becomes greater near the center.

he flow of Newtonian and non-Newtonian fluids in a porous surface channel has attracted the interest o f many investigators in view of its applications in engine ering practice. One of these applications is to treat the internal motion of the gases in solid rocket motors as the superposition of a steady average flow and a conglomeration of unsteady fields 1 . The average flow, also commonly known as the mean flow, represents the bulk motion of the gases in the rocket and can be approximated by the steady flow in a porous pipe. Most scientific probl ems such as two-dimensional viscous flow between slowly expanding or contracting walls with weak permeabili ty and other fluid mechanic problems are inherently nonlinear. Except a limited number of these problem s, most of them do not have analytical solution. There fore, these nonlinear equations should be solved using ot her methods. Stebel 2 conducted a study on shape stability of incompressible fluids subject to Navier slip, focus ing on he equations of motions for incompressible fluids that slip at the wall. It was noted that the issue of bo undary conditions in fluid mechanics has been studied for over two centuries by many distinguished scientists but still it is subject to discussion in the mathematical commun ity. Makinde and Osalusi 3 investigated the steady flow in a channel with slip at the permeable boundaries. They reported that an increase in the positive value of flow Reynolds number(Re) represents an increase in the f luid suction while an increase in the negative value of Re represents an increase in the fluid injection. They also noticed that wall skin friction increases with suct ion and decreases with injection and that, both slip parame ter and magnetic field have great influence on wall skin fr iction. A similar study was done by Makinde 4 on extending the utility of perturbation series in problems of lamin ar flow in a porous pipe and diverging channel, by consider ing a steady ax symmetric flow of a viscous incompressibl e fluid driven. The flow of an incompressible viscous fluid between a uniformly porous upper plate and a lower impermeabl e plate that is subjected to a Navier slip is modeled and analyzed in this study using analytical approaches.Consider the laminar, isothermal and incompressible flow in a cylindrical domain bounded by permeable surfaces with one end closed at the head well while the other remains open. A schematic diagram of the prob lem is shown in Fig. I . The walls expand radially at a time- dependent rate . Furthermore, the origin is assumed to be the center of the classic squeeze fil m problem. This enables us to assume flow symmetry ab out . Under these assumptions, the transport equation for the unsteady flow is given as follows: Using some modification and special variable 13 , and the we have: The resulting equation.5 is the classic Berman's formula 14 , with a = 0 (channel with stationary walls). After the flow field is found, the normal pressure gradient can be obtained by substituting the veloci ty components into Equations.1-3.

Culick FEC (2006) Unsteady motions in combustion chambers for propulsion systems. Agardograph, Advi sory Group for Aerospace Research and Development. 2. Stebel J On shape stability of incompressible fl uids subject to Navier?s slip 23(2010)35-57 3. Makinde OD & Osalusi E MHD steady flow in a cha nnel with slip at the permeable boundaries. Applied Mathematics Department, University of Limpopo, Sout h Africa (2005). Makinde OD Laminar flow in a channel of varying width with permeable boundaries Romanian Journal of Physics 40 (1995) 403-417. 5. Yogesh MJ, Denn MM Planar contraction flow with a slip boundary condition, Newyork, NY 10031, USA (2003) 6. Ganji DD, Azimi M Application of Max Min Approach and Amplitude Frequency Formulation to Nonlinear Oscillation Systems U.P.B. Scientific Bulletin 74( 2012)131 -140. 7. Ganji DD, Azimi M, Mostofi M Energy Balance Meth od and Amplitude Frequency Formulation Based of Strongl y Nonlinear Oscillators Indian Journal of Pure & Applied Physics 50 (2012) 670-675. 8. Ganji DD & Azimi M Application of DTM on MHD Jeffery Hamel Problem with Nanoparticle U.P.B. Scientific Bulletin , Series D. 75 (2013)223-230. 9. Shakeri F, Ganji DD & Azimi M Application of HPM - Pade Technique to Jeffery Hamel Problem International Azimi M, Azimi A, Mirzaei M Investigation of th e unsteady graphene oxide nanofluid flow between two moving plates Journal of Computational and Theoritical Nanoscience 11 (2014) 1-5. 11. Gorji-Bandpy M , Azimi M & Mostofi M Analytical Methods to a Generalized Duffing Oscillator Australian Journal of Basic and Applied Science 5 (2011) 788-796 12. Karimian S & Azimi M Periodic Solution for Vibr ation of Euler-Bernoulli Beams Subjected to Axial Load Using DTM and H A Scientific Bulletin Series D 2 (2014) 69-76 13. Majdalani J, Zhou C & Dawson CA Two-dimensional viscous flow between slowly expanding or contractin g walls with weak permeability Journal of Biomech 35 (2002) 1399-1403. 14. Berman AS Laminar flow in channels with porous w alls Journal of Applied Physics 24 (1953) 1232-1235.