This paper presents pressure variation in fluid flow over a porous media. In the model, we considered water as an incompressible fluid: the flow as nonsteady and uniform. We derived an equation for the nonuniform bottom topography (flow depth) and substituted into the governing equation for shallow water flow with nonuniform bottom topography. We made use of Darcy’s law to construct equation for Darcy flux, which in turn related pressure gradient to the flow velocity, the porosity, and the permeability of the porous media. From the governing equation of shallow water flow with nonuniform bottom topography, we solved for the flow velocity using Homotopy Perturbation Method (HPM). We incorporated the flow velocity into the equation for the pressure gradient and solved for the pressure variation in the channel. We analyzed and found out that, the higher the permeability the lower the pressure within the flow and the lower the permeability the higher the pressure, because there is going to be a pressure build-up under this condition. We also found that the higher the flow height (H) the higher the pressure.
Cite this paper
Mbah, G. C. E. and Oshilim, E. O. (2020). Pressure Variation in a Fluid Flow over Non-Uniform, Porous Bottom Topography. Open Access Library Journal, 7, e6033. doi: http://dx.doi.org/10.4236/oalib.1106033.
Craik, A.D. (2004) The Origins of Water Wave Theory. Annual Review of Fluid Mechanics, 36, 1-28. https://doi.org/10.1146/annurev.fluid.36.050802.122118
Mbah, G.C.E. and Udogu, C.I. (2015) Open Channel Flow Over a Permeable River Bed. Open Access Library Journal, 2, 1-7. https://doi.org/10.4236/oalib.1101475
Okeke, E.O. (1999) On the Linearised Shallow Water Waves Over a Sloping Bottom. Nuovo Cimento della Societa Italiana di fisica C, No. 6, 72. https://doi.org/10.1007/BF02511373
Mbah, G.C.E. and Ezeorah, J.N. (2007) On the Wave Equations of Shallow Water with Non-Uniform Bottom Topography. Journal of Nigerian Association of Mathematical Physics, 12, 143-150.
Mbah, G.C.E. (2008) Shallow Water Flow Over a Varying Non-Uniform Bottom Region. Proceedings of the First International Seminar on Theoretical Physics & National Development, 1, 238-254.
El, G.A., Grimshaw, R.H.J. and Smyth, N.F. (2009) Transcritical Shallow Water Flow Past Topography: Finite-Amplitude Theory. Journal of Fluid Mechanics, 640, 187-214. https://doi.org/10.1017/S0022112009991315
Mbah, G.C.E. and Isienyi, S.U. (2018) Shallow Water Flow with Non-Uniform Bottom Topography. Journal of Nigerian Society of Mathematical Biology, 1, 67-93.
Abbas, M.N. (2011) Modeling of Porosity Equation for Water Flow through Packed Bed of Monosize Spherical Packing. Journal of Engineering and Development, 15, 205-226.
Bear, J. (2018) Modeling Phenomena of Flow and Transport in Porous Media. Springer International Publishing AG, Switzerland. https://doi.org/10.1007/978-3-319-72826-1
Peregrine, D.H. (1972) Equations for Water Waves and the Approximation behind them. Journal of Fluid Mechanics, 59, 95-121. https://doi.org/10.1016/B978-0-12-493250-0.50007-2