This research explores the transient flow formation in concentric porous annuli filled with porous material having variable porosity. The study focuses on the fluid flow between two horizontal concentric cylinders, assuming the fluid is viscous and incompressible. To solve the governing partial differential equation, the Laplace transform technique is applied, transforming it into an ordinary differential equation in the Laplace domain. Exact solutions are obtained in the Laplace domain, and then inverted to the time-dependent domain using the Riemann-sum approximation method.The accuracy of the Riemann-sum approximation method is validated by comparing the numerical values obtained with those of the exact solution for steady-state flow and transient solution. The results are presented graphically, illustrating the variations of velocity and skin friction with respect to the Darcy number and suction/injection parameter.The study reveals that suction accelerates the flow, while injection retards it. The Darcy number, which represents the permeability of the porous medium, significantly affects the flow characteristics. The skin friction, which is a measure of the shear stress at the surface, is also influenced by the suction/injection parameter.The findings of this study have implications for various engineering applications, such as groundwater flow, oil reservoirs, and chemical engineering processes. Understanding the transient flow formation in porous annuli can help optimize the design and operation of these systems.The use of the Laplace transform technique and Riemann-sum approximation method provides an efficient and accurate solution to the problem. The graphical representation of the results allows for a clear understanding of the flow characteristics and the effects of various parameters. The list of symbols, notation and meaning of parameters can be found in appendix A and B attached in section below.

Avramenko A.A., Trinov A.I, Shevchuk I.V. (2015). An analytical and numerical study on the start-up flow of slightly rarefied gases in a parallel plate channel and a pipe. J. Porous Media 27,231-246. Berman A.S.(1958). Laminar flow in an annulus with porous walls. J Appl Phys 29, 71-75. Deo S. and Srivastava B.G. (2013). Effect of magnetic field on the viscous fluid flow in a channel filled with porous medium of variable permeability. Journal of Applied Mathematics and Computation. Vol. 219 pp. 8959-8964 Ebrahim N.H., El-khatib N., and Awang M. (2013). Numerical solution 0f power-law fluid flow through Eccentric annular geometry American journal of Numerical Analysis 1, 1-7. Jha B.K., Apere C.A. (2013). Time dependent MHD Couette flow in a porous annulus. J Communication in non-linear science and Numerical simulation 188, 1959-1969. Jha B.K., Apere C.A. (2010). Unsteady MHD Couette flow in an annuli: the Riemann-sum approximation approach. J Phys Soc Jpn 79, 124403/1-3/5 Jha B.K., Apere C.A.(2012).Magnetohydrodynamics transient free-convective flow in a vertical annulus with thermal boundary condition of the second kind J heat transfer 134/042502-1. Kandasamy A., Nadiminti S.R. (2015). Entrance Region flow in concentric annuli with rotating inner wall for Herschel-Bulkley fluids. Int. J. Appl comput. Math 1, 235-249 Khalil M.F., Kassab S.Z., Adam I.G., Samaha M.(2008). Laminar flow in concentric annulus with a moving core. IWTC 12, 439-452. Kim, Y.H. (2013). Flow of Newtonian and Non Newtonian fluids in a concentric annulus with a rotating inner cylinder. Korea-Australia Rheolohy, 77-85. Kuznetsov A.V. (1996). Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium. Appl. Sci. Res. 56, 53-67. Mishra S.P. and Roy J.S. (1967). Flow of Elasticoviscous Liquid between Rotating cylinders with suction and injection J Phys. Fluids 10, 2300. Sagir, A.M., M. Abdullahi, and F. Balogun.(2023). An optimal Implicit Block Method for Solutions of the Tumor-Immune Interaction Model of ODEs. Transnational Journal of Mathematical Analysis and Applications. 11(1),45-60. JyotiAcademic Press http://jyotiacademicpress.org Pantokratoras A., Fang T.(2010).Flow of a weekly conducting fluid in a channel filled with a porous medium. J Transport porous Media. D0I 10.1007/s11242-009-9470-6. Rothfus R.R., Monrad C.C., Senacal V.E. (1950). Velocity distribution and fluid friction in smooth concentric annuli. J Industrial and Engineering chemistry 4212, 2511-2520. Srivastava B.G., Deo S.(2013). Effect of magnetic field on the viscous fluid flow in a channel filled with porous medium of variable permeability J Applied Mathematics and Computation 219, 8959-8964. Tzou D.Y. (1997). Macro to Microscale Heat Transfer: The lagging Behaviors. Washington: Taylor and Francis Vadasz P. (1993). Fluid flow through heterogeneous porous media in a rotating square channel. J Transport porous media 12, 43-5. Verma B.K., Datta S. (2012). Flow in an annular channel filled with a porous medium of variable permeability. J Porous Media 15, 891-899. Verma V.K., Singh S.K. (2014). Flow between coaxial rotating cylinders filled by porous medium of variable permeability. Int J Porous media 54,355-359. Verma V.K., Singh S.K. (2015). Magnetohydrodynamics flow in a circular channel filled with a porous medium. J Porous Media 189, 923-928. Yale, I.D., Sa’adu, A., Hamza, M.M. and Bello, Y.(2025). Tranverse Magnetic Field’s Impact on Mixed Convection Flow of an Exothermic Fluid over a porous Material – filled Channel. Journal of Basics and Applied Sciences Research,3(3), 251-260. https://dx.dox.doi.org/10.4314/jobasr.v3i3.27