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.

show all references

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.

[1]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks and Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167

[2]

Martin Mayer, Cheikh Birahim Ndiaye. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 5037-5062. doi: 10.3934/dcds.2022085

[3]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407

[4]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations and Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61

[5]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

[6]

Joyce R. McLaughlin and Arturo Portnoy. Perturbation expansions for eigenvalues and eigenvectors for a rectangular membrane subject to a restorative force. Electronic Research Announcements, 1997, 3: 72-77.

[7]

Ken Abe. Some uniqueness result of the Stokes flow in a half space in a space of bounded functions. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 887-900. doi: 10.3934/dcdss.2014.7.887

[8]

Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525

[9]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure and Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[10]

Marc Briane, Loïc Hervé. Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3431-3463. doi: 10.3934/dcds.2022021

[11]

Anita T. Layton, J. Thomas Beale. A partially implicit hybrid method for computing interface motion in Stokes flow. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1139-1153. doi: 10.3934/dcdsb.2012.17.1139

[12]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[13]

Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks and Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765

[14]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[15]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[16]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[17]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[18]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[19]

María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks and Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004

[20]

Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (4)

[Back to Top]