Transport in Porous Media

, Volume 98, Issue 1, pp 147–172 | Cite as

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

  • Jun YaoEmail author
  • Ke Tao
  • Zhaoqin Huang


The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.


Two-phase flow Particulate-fluid suspension  Permeable boundary  Beavers and Joseph slip condition Perturbation method 



This study was supported by the National Basic Research Program of China (”973” Program) (Grant No. 2011CB201004), the National Natural Science Foundation of China (Grant No. 51234007), and the Fundamental Research Funds for the Central Universities (Grant No. 11CX06026A).


  1. Arbogast, T., Lehr, L.H.: Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci. 10, 291–302 (2006)CrossRefGoogle Scholar
  2. Ali, N., Hussain, Q., Hayat, T., Asghar, S.: Slip effects on the peristaltic transport of MHD fluid with variable viscosity. Phys. Lett. A 372, 1477–1489 (2008)CrossRefGoogle Scholar
  3. Aguilar-Madera, C.G., Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A.: One-domain approach for heat transfer between a porous medium and a fluid. Int. J. Heat Mass Transf. 54, 2089–2099 (2011)CrossRefGoogle Scholar
  4. Beavers, G., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)CrossRefGoogle Scholar
  5. Brown, S.R., Stockman, H.W., Reeves, S.J.: Applicability of the Reynolds equation for modeling fluid flow between rough surfaces. Geophys. Res. Lett. 22, 2537–2540 (1995)CrossRefGoogle Scholar
  6. Baber, K.: Modeling the transfer of therapeutic agents from the vascular space to the tissue compartment (a continuum approach). Universitat Stuttgart, Master thesis (2009)Google Scholar
  7. Charm, S.E., Kurland, G.S.: Blood Flow and Microcirculation. Wiley, New York (1974)Google Scholar
  8. Chandesris, M., Jamet, D.: Boundary conditions at a planar fluid-porous interface for a Poiseuille flow. Int. J. Heat Mass Transf. 49, 2137–2150 (2006)CrossRefGoogle Scholar
  9. Drew, D.: Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291 (1983)CrossRefGoogle Scholar
  10. Furman, A.: Modeling coupled surface-subsurface flow processes: a review. Vadose Zone J. 7, 741–756 (2008)CrossRefGoogle Scholar
  11. Goyeau, B., Lhuillier, D., Gobin, D.: Momentum transport at a fluid-porous interface. Int. J. Heat Mass Transf. 46, 4071–4081 (2003)CrossRefGoogle Scholar
  12. Hayat, T., Hussain, Q., Ali, N.: Influence of partial slip on the peristaltic flow in a porous medium. Phys. Lett. A 387, 3399–3409 (2008)Google Scholar
  13. Huang, C., Shy, S., Chien, C., Lee, C.: Parametric study of anodic microstructures to cell performance of planar solid oxide fuel cell using measured porous transport properties. J. Power Sources 195, 2260–2265 (2009)CrossRefGoogle Scholar
  14. Huang, Z., Yao, J., Li, Y.: Permeability analysis of fractured vuggy porous media based on homogenization theory. Sci. China Tech. Sci. 53, 839–847 (2010)CrossRefGoogle Scholar
  15. Ishii, M.: Thermo-Fluid Dynamic Theory of Two-phase flow. Volume 22 of Direction des etudes et recherches d’électricité de France. Eyrolles, Paris (1975)Google Scholar
  16. Iliev, O., Laptev, V.: On numerical simulation of flow through oil filters. Comput. Vis. Sci. 6, 139–146 (2004)Google Scholar
  17. Jamet, D., Chandesris, M.: Boundary conditions at a fluid-porous interface: an a priori estimation of the stress jump coefficients. Int. J. Heat Mass Transf. 50, 3422–3436 (2007)CrossRefGoogle Scholar
  18. Jamet, D., Chandesris, M., Goyeau, B.: On the equivalence of the discontinuous one- and two-domain approaches for modeling of transport phenomena at a fluid-porous interface. Transp. Porous Media 78, 403–418 (2009)CrossRefGoogle Scholar
  19. Jiménez-Lozano, J., Sen, M., Corona, E.: Analysis of peristaltic two-phase flow with application to ureteral biomechanics. Acta Mech. 219, 91–109 (2011)CrossRefGoogle Scholar
  20. Kleinstreuer, C.: Two-Phase Flow: Theory and Applications. Taylor and Francis, London (2003)Google Scholar
  21. Larson, R.E., Higdon, J.J.L.: Microscopic flow near the surface of two-dimensional porous-media: 1. Axial-flow. J. Fluid Mech. 166, 449–472 (1986)CrossRefGoogle Scholar
  22. Larson, R.E., Higdon, J.J.L.: Microscopic flow near the surface of two-dimensional porous-media: 2. Transverse flow. J. Fluid Mech. 178, 119–136 (1987)CrossRefGoogle Scholar
  23. Le Bars, M., Worster, M.G.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173 (2006)CrossRefGoogle Scholar
  24. Mekheimer, K.S., El Shehawey, E.F., Elaw, A.M.: Peristaltic motion of a particle-fluid suspension in a planar channel. Int. J. Theor. Phys. 37, 2895–2920 (1998)CrossRefGoogle Scholar
  25. Mariamma, N.K., Majhi, S.N.: Flow of a Newtonian fluid in blood vessel with permeable wall–a theoretical model. Comput. Math. Appl. 40, 1419–1432 (2000)CrossRefGoogle Scholar
  26. Mishra, M., Rao, A.R.: Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract. J Biomech. 38, 779–789 (2005)CrossRefGoogle Scholar
  27. Morales-Zárate, E., Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A.: Diffusion and reaction in three-phase systems:Average transport equations and jump boundary conditions. Chem. Eng. J. 138, 307–332 (2008)CrossRefGoogle Scholar
  28. Neale, G., Nader, W.: Practical significance of Brinkman’s extension of Darcy’s law : coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Eng. 52, 415–478 (1974)CrossRefGoogle Scholar
  29. Nield, D.A.: The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow 12, 269–272 (1991)CrossRefGoogle Scholar
  30. Niceno, B., Nobile, E.: Numerical analysis of fluid flow and heat transfer in periodic wavy channels. Int. J. Heat Fluid Flow 22, 156–167 (2001)CrossRefGoogle Scholar
  31. Nield, D.A.: The Beavers-Joseph boundary condition and related matters: a historical and critical note. Transp. Porous Media 78, 537–540 (2009)CrossRefGoogle Scholar
  32. Ochoa-Tapia, J.A., Whitaker, S.: Momentum-transfer at the boundary between a porous medium and a homogeneous fluid: 1. Theoretical development. Int. J. Heat Mass Transf. 38, 2635–2646 (1995a)CrossRefGoogle Scholar
  33. Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid. II. Comparison with experiment. Int. J. Heat Mass Transf. 38, 2635–2656 (1995b)CrossRefGoogle Scholar
  34. RaviKumar, Y.V.K., KrishnaKumari, S.V.H.N., Raman Murthy, M.V., Sreenadh, S.: Unsteady peristaltic pumping in a finite length tube with permeable wall. Trans. ASME J. Fluids Eng. 32, 1012011–1012014 (2010)Google Scholar
  35. Sudicky, E.A., Frind, E.O.: Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Water Resources Res. 18, 1634–1642 (1982)CrossRefGoogle Scholar
  36. Sahraoui, M., Kaviany, M.: Slip and no-slip velocity boundary conditions at interface of porous, plain media. Int. J. Heat Mass Transf. 35, 927–943 (1992)CrossRefGoogle Scholar
  37. Srivastava, V.P., Srivastava, L.M.: Influence of wall elasticity and Poiseuille flow on peristaltic induced flow of a particle-fluid mixture. Int. J. Eng. Sci. 35, 1359–1386 (1997)CrossRefGoogle Scholar
  38. Shavit, U.: Special issue on “transport phenomena at the interface between fluid and porous domains” a preface. Transp. Porous Media 78, 327–330 (2009)CrossRefGoogle Scholar
  39. Tam, C.K.W.: The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537–546 (1969)CrossRefGoogle Scholar
  40. Vafai, K., Kim, S.: Fluid mechanics of the interface region between a porous medium and a fluid layer-an exact solution. Int. J. Heat Fluid Flow 11, 254–256 (1990)CrossRefGoogle Scholar
  41. Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A.: Diffusive mass transfer between a microporous medium and an homogeneous fluid: Jump boundary conditions. Chem. Eng. Sci. 61, 1692–1704 (2006)CrossRefGoogle Scholar
  42. Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A.: Jump momentum boundary condition at a fluid-porous dividing surface: derivation of the closure problem. Chem. Eng. Sci. 62, 4025–4039 (2007)Google Scholar
  43. Valdés-Parada, F.J., Alvarez-Ramirez, J., Goyeau, B., Ochoa-Tapia, J.A.: Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation. Transp. Porous Media 78, 439–457 (2009a)CrossRefGoogle Scholar
  44. Valdés-Parada, F.J., Alvarez-Ramirez, J., Goyeau, B., Ochoa-Tapia, J.A.: Jump condition for diffusive and convective mass transfer between a porous medium and a fluid involving adsorption and chemical reaction. Transp. Porous Media 78, 459–476 (2009b)CrossRefGoogle Scholar
  45. Whitaker, S.: The Method of Volume Averaging. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  46. Wood, B.D., Quintard, M., Whitaker, S.: Jump condition at non-uniform boundaries: The catalytic surface. Chem. Eng. Sci. 55, 5231–5245 (2000)CrossRefGoogle Scholar
  47. Weerakone, W.M.S.B., Wong, R.C.K., Mehrotra, A.K.: Single-phase (brine) and two-phase (DNAPL-brine) flows in induced fractures. Transp. Porous Media 89, 75–95 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Reservoir EngineeringSchool of Petroleum Engineering, China University of Petroleum (Huadong)QingdaoChina

Personalised recommendations