Analytical Solution to Uniform Flow over a Porous Plane with Downward Suction
Publication: Journal of Engineering Mechanics
Volume 142, Issue 9
Abstract
Studies on open channel flows generally focus on the flow profiles in the longitudinal direction. When generating the analytical solutions, the simplified governing equations are usually employed by neglecting the vertical velocity component, which is much less in quantity than the horizontal one. However, the vertical velocity is actually not negligible, especially at the permeable bottom, as well as the horizontal velocity. In this study, the authors investigate a two-dimensional flow field composed of a fluid (upper) layer and a homogeneous porous medium (lower) layer with downward suction. In the upper layer, Navier-Stokes equations are employed to describe the flow, whereas the porous medium flow theory is addressed in the lower layer. Setting the stream function for the velocity components associated with corresponding boundary conditions, the authors successfully obtain the analytical solutions by the six-order power series method (PSM) and the differential transform method (DTM) respectively, and then acquire the velocity profiles in both layers. Comparing these solutions with previous research, the authors find that the present approach can simplify the algorithm process and the results are in very good agreement.
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Acknowledgments
This study was financially supported by the Ministry of Science and Technology of Taiwan under Grant No.: MOST 103-2313-B-005-007-MY3.
References
Arikoglu, A., and Ozkol, I. (2006). “Solution of differential–difference equations by using differential transform method.” Appl. Math. Comput., 181(1), 153–162.
Battiato, I. (2012). “Self-similarity in coupled Brinkman/Navier-Stokes flows.” J. Fluid Mech., 699(May), 94–114.
Beavers, G. S., and Joseph, D. D. (1967). “Boundary conditions at a naturally permeable wall.” J. Fluid Mech., 30(1), 197–207.
Berman, A. S. (1953). “Laminar flow in channels with porous walls.” J. Appl. Phys., 24(9), 1232–1235.
Biot, M. A. (1956a). “Theory of propagation elastic waves in a fluid saturated porous solid. I: Low-frequency range.” J. Acoust. Soc. Am., 28(2), 168–178.
Biot, M. A. (1956b). “Theory of propagation elastic waves in a fluid saturated porous solid. II: High-frequency range.” J. Acoust. Soc. Am., 28(2), 179–191.
Cieszko, M., and Kubik, J. (1999). “Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid.” Transp. Porous Media, 34(1–3), 319–336.
Deng, C., and Martinez, D. M. (2005). “Viscous flow in a channel partially filled with a porous medium and with wall suction.” Chem. Eng. Sci., 60(2), 329–336.
Hsieh, P. C., and Chen, Y. C. (2013). “Surface water flow over a pervious pavement.” Int. J. Numer. Anal. Meth. Geomech., 37(9), 1095–1105.
Hsieh, P. C., and Yang, J. S. (2013). “A semi-analytical solution of surface flow and subsurface flow on a slopeland under a uniform rainfall excess.” Int. J. Numer. Anal. Meth. Geomech., 37(8), 893–903.
Hsieh, P. C., Yang, J. S., and Chen, Y. C. (2012). “An integrated solution of the overland and subsurface flow along a sloping soil layer.” J. Hydrol., 460–461, 136–142.
Jager, W., and Mikelic, A. (2000). “On the interface a boundary conditions of Beavers, Joseph, and Saffman.” SIAM J. Appl. Math., 60(4), 1111–1127.
Kazezyılmaz-Alhan, C. M. (2012). “An improved solution for diffusion waves to overland flow.” Appl. Math. Model., 36(9), 4165–4172.
Kazezyılmaz-Alhan, C. M., and Medina, M. A. (2007). “Kinematic and diffusion waves: Analytical and numerical solutions to overland and channel flow.” J. Hydraul. Eng., 217–228.
Maple 6 [Computer software]. Waterloo Maple, Waterloo, Canada.
Ochoa-Tapia, J., and Whitaker, S. (1995). “Momentum transfer at the boundary between a porous medium and a homogenous fluid: I-theoretical development.” Int. J. Heat Mass Transfer, 38(14), 2635–2646.
Song, C. H., and Huang, L. H. (2000). “Laminar poroelastic media flow.” J. Eng. Mech., 358–366.
Stehfest, H. (1970). “Numerical inversion of Laplace transforms.” Commun. ACM, 13(1), 47–49.
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Published In
Journal of Engineering Mechanics
Volume 142 • Issue 9 • September 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Sep 13, 2015
Accepted: Mar 23, 2016
Published online: May 11, 2016
Published in print: Sep 1, 2016
Discussion open until: Oct 11, 2016
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