# American Institute of Mathematical Sciences

June  2011, 15(4): 971-990. doi: 10.3934/dcdsb.2011.15.971

## Perturbation solution of the coupled Stokes-Darcy problem

 1 Laboratoire d'Ingénierie Mathématique, Ecole Polytechnique de Tunisie, Université de Carthage, B.P. 743 - 2078 La Marsa, Tunisia, Tunisia 2 LIMSI, UPR 3251 CNRS, BP 133, Bât. 508,91403 Orsay cedex, France

Received  May 2010 Revised  June 2010 Published  March 2011

Microfiltration of particles is modelled by the motion of particles embedded in a Stokes flow near a porous membrane in which Darcy equations apply. Stokes flow also applies on the other side of the membrane. A pressure gradient is applied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces. This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is much larger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order. The method is applied to the example case of a spherical particle moving normal to a membrane. The solution, limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step the boundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.
Citation: Sondes khabthani, Lassaad Elasmi, François Feuillebois. Perturbation solution of the coupled Stokes-Darcy problem. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 971-990. doi: 10.3934/dcdsb.2011.15.971
##### References:
 [1] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: doi:10.1017/S0022112067001375. [2] G. S. Beavers, E. M. Sparrow and B. A. Masha, Boundary condition at a porous surface which bounds a fluid flow, AIChE. J., 20 (1974), 596-597. doi: doi:10.1002/aic.690200323. [3] J. R. Blake, A note on the image system for a Stokeslet in a no-slip boundary, Proc. Cambridge Philos. Soc., 70 (1971), 303-310. doi: doi:10.1017/S0305004100049902. [4] M. Bonnet, "Boundary Integral Equation Methods for Solids and Fluids," John Willey and Sons LTD, 1995. [5] L. Elasmi, Singularity method for Stokes flow with slip boundary condition, IMA Journal of Applied Math., 73 (2008), 724-739. doi: doi:10.1093/imamat/hxn021. [6] L. Elasmi and F. Feuillebois, Integral equation method for creeping flow around a solid body near a porous slab, Q. J. Mech. Appl. Math., 56 (2003), 163-185. doi: doi:10.1093/qjmam/56.2.163. [7] B. Goyeau, D. Lhuillier, D. Gobin and M. G. Velarde, Momentum transport at a fluid-porous interface, Int. J. Heat Mass Transfer, 46 (2003), 4071-4081. doi: doi:10.1016/S0017-9310(03)00241-2. [8] J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics," Kluwer Academic Publishers, Dordrecht, Boston, London, 1991. [9] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph and Saffman, SIAM J. Appl. Maths, 60 (2000), 1111-1127. doi: doi:10.1137/S003613999833678X. [10] C. Kunert and J. Harting, Roughness induced boundary slip in microchannel flows, Phys. Rev. Letters, 99 (2007), 1-4. doi: doi:10.1103/PhysRevLett.99.176001. [11] N. Lecoq, R. Anthore, B. Cichocki, P. Szymczak and F. Feuillebois, Drag force on a sphere moving towards a corrugated wall, J. Fluid Mech., 513 (2004), 247-264. doi: doi:10.1017/S0022112004009942. [12] A. Niavarani and N. Priezjev, The effective slip length and vortex formation in laminar flow over a rough surface, Phys. Fluids, 21 (2009), 1-10. doi: doi:10.1063/1.3121305. [13] M. E. O'Neill and B. S. Bhatt, Slow motion of a solid sphere in the presence of a naturally permeable surface, Q. J. Mech. Appl. Math., 44 (1991), 91-104. doi: doi:10.1093/qjmam/44.1.91. [14] H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics," Computational Mechanics Publications, 1995. [15] C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow," Cambridge University Press, 1992. [16] C. Pozrikidis, "A Practical Guide to Boundary Element Methods with the Software Library Bemlib," CRC Press, 2002. doi: doi:10.1201/9781420035254. [17] P. G. Saffman, On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50 (1971), 93-101. [18] M. Sahraoui and M. Kaviany, Slip and no-slip velocity boundary conditions at interface of porous, plain media, Int. J. Heat Mass Transfer, 35 (1992), 927-943. doi: doi:10.1016/0017-9310(92)90258-T. [19] P. Schmitz, D. Houi and B. Wandelt, Hydrodynamic aspects of crossflow microfiltration. Analysis of particle deposition at the membrane surface, J. Membrane Sci., 71 (1992), 29-40. doi: doi:10.1016/0376-7388(92)85003-2. [20] A. Sellier, Settling motion of interacting solid particles in the vicinity of a plane solid boundary, Compte-Rendus Acad. Sci., 333 (2005), 413-418. [21] O. I. Vinogradova and G. E. Yakubov, Surface roughness and hydrodynamic boundary conditions, Phys. Rev. E., 73 (2006), 1-4. doi: doi:10.1103/PhysRevE.73.045302.

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##### References:
 [1] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: doi:10.1017/S0022112067001375. [2] G. S. Beavers, E. M. Sparrow and B. A. Masha, Boundary condition at a porous surface which bounds a fluid flow, AIChE. J., 20 (1974), 596-597. doi: doi:10.1002/aic.690200323. [3] J. R. Blake, A note on the image system for a Stokeslet in a no-slip boundary, Proc. Cambridge Philos. Soc., 70 (1971), 303-310. doi: doi:10.1017/S0305004100049902. [4] M. Bonnet, "Boundary Integral Equation Methods for Solids and Fluids," John Willey and Sons LTD, 1995. [5] L. Elasmi, Singularity method for Stokes flow with slip boundary condition, IMA Journal of Applied Math., 73 (2008), 724-739. doi: doi:10.1093/imamat/hxn021. [6] L. Elasmi and F. Feuillebois, Integral equation method for creeping flow around a solid body near a porous slab, Q. J. Mech. Appl. Math., 56 (2003), 163-185. doi: doi:10.1093/qjmam/56.2.163. [7] B. Goyeau, D. Lhuillier, D. Gobin and M. G. Velarde, Momentum transport at a fluid-porous interface, Int. J. Heat Mass Transfer, 46 (2003), 4071-4081. doi: doi:10.1016/S0017-9310(03)00241-2. [8] J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics," Kluwer Academic Publishers, Dordrecht, Boston, London, 1991. [9] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph and Saffman, SIAM J. Appl. Maths, 60 (2000), 1111-1127. doi: doi:10.1137/S003613999833678X. [10] C. Kunert and J. Harting, Roughness induced boundary slip in microchannel flows, Phys. Rev. Letters, 99 (2007), 1-4. doi: doi:10.1103/PhysRevLett.99.176001. [11] N. Lecoq, R. Anthore, B. Cichocki, P. Szymczak and F. Feuillebois, Drag force on a sphere moving towards a corrugated wall, J. Fluid Mech., 513 (2004), 247-264. doi: doi:10.1017/S0022112004009942. [12] A. Niavarani and N. Priezjev, The effective slip length and vortex formation in laminar flow over a rough surface, Phys. Fluids, 21 (2009), 1-10. doi: doi:10.1063/1.3121305. [13] M. E. O'Neill and B. S. Bhatt, Slow motion of a solid sphere in the presence of a naturally permeable surface, Q. J. Mech. Appl. Math., 44 (1991), 91-104. doi: doi:10.1093/qjmam/44.1.91. [14] H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics," Computational Mechanics Publications, 1995. [15] C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow," Cambridge University Press, 1992. [16] C. Pozrikidis, "A Practical Guide to Boundary Element Methods with the Software Library Bemlib," CRC Press, 2002. doi: doi:10.1201/9781420035254. [17] P. G. Saffman, On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50 (1971), 93-101. [18] M. Sahraoui and M. Kaviany, Slip and no-slip velocity boundary conditions at interface of porous, plain media, Int. J. Heat Mass Transfer, 35 (1992), 927-943. doi: doi:10.1016/0017-9310(92)90258-T. [19] P. Schmitz, D. Houi and B. Wandelt, Hydrodynamic aspects of crossflow microfiltration. Analysis of particle deposition at the membrane surface, J. Membrane Sci., 71 (1992), 29-40. doi: doi:10.1016/0376-7388(92)85003-2. [20] A. Sellier, Settling motion of interacting solid particles in the vicinity of a plane solid boundary, Compte-Rendus Acad. Sci., 333 (2005), 413-418. [21] O. I. Vinogradova and G. E. Yakubov, Surface roughness and hydrodynamic boundary conditions, Phys. Rev. E., 73 (2006), 1-4. doi: doi:10.1103/PhysRevE.73.045302.
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