Microfluidics and Nanofluidics

, Volume 15, Issue 1, pp 109–115 | Cite as

Instability in Poiseuille flow in a porous medium with slip boundary conditions

  • B. StraughanEmail author
  • A. J. Harfash
Research Paper


We study a model for Poiseuille flow instability in a porous medium of Brinkman type. In particular, we analyse the effect of slip boundary conditions on the onset of instability. Due to numerous applications in micro-electro-mechanical-systems and other microfluidic devices, such a study is essential. We accurately analyse as to when instability will commence and determine the critical Reynolds number as a function of the slip coefficient.


Poiseuille flow Slip boundary conditions Chebyshev collocation Brinkman porous material Porous metallic foams 



Research supported by a Research Grant of the Leverhulme Trust, “Tippings Points: Mathematics, Metaphors and Meanings”. We should like to thank three anonymous referees for very helpful comments which have led to improvements in the paper. Research supported by a Research Grant of the Iraq Ministry of Higher Education and Scientific Research.


  1. Avila R, Ramos E, Atluri SN (2009) The Chebyshev tau spectral method for the solution of the linear stability equations for Rayleigh–Bénard convection with melting. Comput Model Eng Sci 51:73–92MathSciNetzbMATHGoogle Scholar
  2. Badur J, Karcz M, Lemanski M (2011) On the mass and momentum transport in the Navier–Stokes slip layer. Microfluid Nanofluid 11:439–449CrossRefGoogle Scholar
  3. Bandyopadhyay D, Reddy P.D.S, Sharma A, Joo S.W, Qian S (2012) Electro-magnetic field induced flow and interfacial instabilities in confined stratified liquid layers. Theor Comput Fluid Dyn 26:23–28CrossRefGoogle Scholar
  4. Bassom AP, Blyth MG, Papageorgiou DT (2012) Using surfactants to stabilize two-phase pipe flows of core-annular type. J Fluid Mech 704:333–359MathSciNetzbMATHCrossRefGoogle Scholar
  5. Cercignani C (1988) The Boltzmann equation and its applications. Springer, BerlinGoogle Scholar
  6. Chu WK (2000) Stability of incompressible helium II: a two fluid system. J Phys Condens Matter 12:8065–8069CrossRefGoogle Scholar
  7. Chu AK (2004) Instability of Navier slip flow of liquids. Comptes Rendue Mécanique 332:895–900zbMATHGoogle Scholar
  8. Dongarra JJ, Straughan B, Walker DW (1996) Chebyshev tau—QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl Numer Math 22:399–435MathSciNetzbMATHCrossRefGoogle Scholar
  9. Dragomirescu FI, Gheorghiu CI (2009) Analytical and numerical solutions to an electrohydrodynamic stability problem. Appl Math Comput 59:3718–3727Google Scholar
  10. Duan Z (2012) Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels. Microfluid Nanofluid 12:805–820CrossRefGoogle Scholar
  11. Duan Z, Muzychka YS (2007) Slip flow in non-circular microchannels. Microfluid Nanofluid 3:473–484CrossRefGoogle Scholar
  12. Gheorghiu CI, Dragomirescu FI (2009) Spectral methods in linear stability. Applications to thermal convection with variable gravity field. Appl Numer Math 59:1290–1302MathSciNetzbMATHCrossRefGoogle Scholar
  13. Gheorghiu CI, Rommes J (2012) Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems. Int J Numer Meth Fluids. doi: 10.1002/fld.3669
  14. Hibino K, Ishikawa H, Ishioka K (2012) Effect of a capping inversion on the stability of an Ekman boundary layer. J Meteorol Soc Japan 90:311–319CrossRefGoogle Scholar
  15. Hill AA, Straughan B (2010) Stability of Poiseuille flow in a porous medium. In: Rannacher R, Sequeira A (eds) Advances in mathematical fluid mechanics. Springer, Heidelberg, pp 287–293Google Scholar
  16. Khoshnood A, Jalali MA (2012) Long-lived and unstable modes of Brownian suspensions in microchannels. J Fluid Mech 701:407–418zbMATHCrossRefGoogle Scholar
  17. Lauga E, Brenner MP, Stone H.A (2007) Microfluidics: the no-slip boundary condition. In: Tropea C, Yarin A, Foss JF (eds) Handbook of experimental fluid dynamics. Springer, Berlin, pp 1219–1240Google Scholar
  18. Lauga E, Cossu C (2005) A note on the stability of slip channel flows. Phys Fluids 17:088106CrossRefGoogle Scholar
  19. Lefebvre LP, Banhart J, Dunand DC (2008) Porous metals and metallic foams: current status and recent developments. Adv Eng Mater 10:775–787CrossRefGoogle Scholar
  20. Malik SV, Hooper AP (2007) Three-dimensional disturbances in channel flows. Phys Fluids 19:052102–052102CrossRefGoogle Scholar
  21. Massa L, Jha P (2012) Linear analysis of the Richtmeyer–Meshov instability in shock flame interactions. Phys Fluids 24:056101CrossRefGoogle Scholar
  22. Maxwell JC (1879) On stresses in rarefied gases arising from inequalities of temperature. Philos Trans R Soc Lond 170:231–256zbMATHCrossRefGoogle Scholar
  23. Morini GL, Lorenzini M, Spiga M (2005) A criterion for experimental validation of slip-flow models for incompressible rarefied gases through microchannels. Microfluid Nanofluid 1:190–196CrossRefGoogle Scholar
  24. Navier CLMH (1823) Mémoire sur les lois du mouvement des fluides. Mémoires de l’ Académie Royale des Sciences de l’ Institut de France 6:389–440Google Scholar
  25. Nield DA (2003) The stability of flow in a channel or duct occupied by a porous medium. Int J Heat Mass Transf 46:4351–4354zbMATHCrossRefGoogle Scholar
  26. Priezjev NV (2012) Molecular dynamics simulations of oscillatory Couette flows with slip boundary conditions. Microfluid Nanofluid. doi: 10.1007/s10404-012-1040-5
  27. Rahman MM, Al-Lawatia MA, Eltayeb IA, Al-Salti N (2012) Hydromagnetic slip flow of water based nanofluids past a wedge with convective surface in the presence of heat generation or absorption. Int J Therm Sci 57:172–182CrossRefGoogle Scholar
  28. Shojaeian M, Shojaeian M (2012) Analytical solution of mixed electromagnetic/pressure driven gaseous flows in microchannels. Microfluid Nanofluid 12:553–564CrossRefGoogle Scholar
  29. Spille A, Rauh A, Bühring H (2000) Critical curves of plane Poiseuille flow with slip boundary conditions. Nonlinear Phenom Complex Syst 3:171–173Google Scholar
  30. Stebel J (2012) On shape stability of incompressible fluids subject to Navier’s slip condition. J Math Fluid Mech 14:575–589MathSciNetzbMATHCrossRefGoogle Scholar
  31. Straughan B (1998) Explosive instabilities in mechanics. Springer, HeidelbergGoogle Scholar
  32. Straughan B (2008) Stability, and wave motion in porous media, volume 165 of Appl Math Sci Springer, New YorkGoogle Scholar
  33. Straughan B (2012) Triply resonant penetrative convection. Proc R Soc Lond A 468:3804–3823MathSciNetCrossRefGoogle Scholar
  34. Webber M (2007) Instability of fluid flows, including boundary slip. PhD thesis, Durham UniversityGoogle Scholar
  35. Webber M, Straughan B (2006) Stability of pressure driven flow in a microchannel. Rend Circolo Matem Palermo 29:343–357MathSciNetGoogle Scholar
  36. Yong X, Zhang LT (2012) Slip in nanoscale shear flow: mechanisms of interfacial friction. Microfluid Nanofluid. doi: 10.1007/s10404-012-1048-x
  37. Zhang H, Zhang Z, Ye H (2012) Molecular dynamics-based prediction of boundary slip of fluids in nanochannels. Microfluid Nanofluid 12:107–115CrossRefGoogle Scholar
  38. Zhang WM, Meng G, Wei X (2012) A review on slip models for gas microflows. Microfluid Nanofluid. doi: 10.1007/s10404-012-1012-9

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DurhamDurhamUK

Personalised recommendations