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Transport in Porous Media

, Volume 116, Issue 2, pp 533–566 | Cite as

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

  • M. S. Faltas
  • E. I. SaadEmail author
  • Shreen El-Sapa
Article

Abstract

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.

Keywords

Corrugated microannulus Slip–Brinkman flow Flow rate and pressure gradient 

References

  1. Barrat, J., Bocquet, L.: Large slip effect at a nonwetting fluid-solid interface. Phys. Rev. Lett. 82, 4671–4674 (1999)CrossRefGoogle Scholar
  2. Bergles, A.E.: Some perspectives on enhanced heat-transfer-2nd generation heat-transfer technology. J. Heat Transf. 110, 1082–1096 (1988)CrossRefGoogle Scholar
  3. Brinkman, H.C.: On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A1, 81–86 (1949)CrossRefGoogle Scholar
  4. Choi, C., Westin, J.A., Breuer, K.S.: Apparent slip flows in hydrophilic and hydrophobic microchannels. Phys. Fluids 15, 2897–2902 (2003)CrossRefGoogle Scholar
  5. Chow, J.C.F., Soda, K.: Laminar flow and blood oxygenation in channels with boundary irregularities. J. Appl. Mech 40, 843–850 (1973)CrossRefGoogle Scholar
  6. Chu, Z.K.-H.: Slip flow in an annulus with corrugated walls. J. Phys. D Appl. Phys. 33, 627–631 (2000)CrossRefGoogle Scholar
  7. Chu, W.K.-H., Fang, J.: Slip flow between longitudinally corrugated cylinders. Mech. Res. Commun. 27, 353–358 (2000)CrossRefGoogle Scholar
  8. Churaev, N., Sobolev, V., Somov, A.: Slippage of liquids over lyophobic solid surfaces. J. Colloid Interface Sci. 97, 574–581 (1984)CrossRefGoogle Scholar
  9. Davis, M.H.: Collisions of small cloud droplets: gas kinetic effects. J. Atmos. Sci. 29, 911–915 (1972)CrossRefGoogle Scholar
  10. Duan, Z., Muzychka, Y.S.: Slip flow in non-circular microchannels. Microfluid. Nanofluid. 3, 473–484 (2007)CrossRefGoogle Scholar
  11. Duan, Z., Muzychka, Y.S.: Effects of corrugated roughness on developed laminar flow in microtubes. J. Fluids Eng. 130, 031102 (2008)CrossRefGoogle Scholar
  12. Duan, Z., Muzychka, Y.S.: Effects of axial corrugated roughness on low Reynolds number slip flow and continuum flow in microtubes. J. Heat Transf. 132, 041001 (2010)CrossRefGoogle Scholar
  13. Durlofsky, L., Brady, J.F.: Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 3329–3340 (1987)CrossRefGoogle Scholar
  14. Givler, R.C., Altobelli, S.A.: A determination of the effective viscosity for the Brinkman–Forchheimer flow model. J. Fluid Mech. 258, 355–370 (1994)CrossRefGoogle Scholar
  15. Haddad, O.M., Al-Nimr, M.A., Sari, M.S.: Forced convection gaseous slip flow in circular porous micro-channels. Transp. Porous Media 70, 167–179 (2007)CrossRefGoogle Scholar
  16. Karniadakis, G.E., Beskok, A.: Microflows: Fundamentals and Simulation. Springer, New York (2002)Google Scholar
  17. Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, New York (1991)CrossRefGoogle Scholar
  18. Kennard, E.H.: Kinetic Theory of Gases. McGraw-Hill, New York (1938)Google Scholar
  19. Koplik, J., Levine, H., Zee, A.: Viscosity renormalization in the Brinkman equation. Phys. Fluids 26, 2864–2870 (1983)CrossRefGoogle Scholar
  20. Kunert, C., Harting, J.: Simulation of fluid flow in hydrophobic rough microchannels. Int. J. Comp. Fluid Dyn. 22, 475–480 (2008)CrossRefGoogle Scholar
  21. Lauga, E., Brenner, M.P., Stone, H.A.: Microfluidics: the no-slip boundary condition. In: Foss, J., Tropea, C., Yarin, A. (eds.) Handbook of Experimental Fluid Dynamics, pp. 1219–1240. Springer, Berlin (2007)Google Scholar
  22. Li, W.L., Lin, J.W., Lee, S.C., Chen, M.D.: Effects of roughness on rarefied gas flow in long microtubes. J. Micromech. Microeng. 12, 149–156 (2002)CrossRefGoogle Scholar
  23. Liu, S., Masliyah, J.H.: Non-linear flows in porous media. J. Non-Newton. Fluid Mech. 86, 229–252 (1999)CrossRefGoogle Scholar
  24. Liu, H., Patil, P.R., Narusawa, U.: On Darcy–Brinkman equation: viscous flow between two parallel plates packed with regular square arrays of cylinders. Entropy 9, 118–131 (2007)CrossRefGoogle Scholar
  25. Lundgren, T.S.: Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51, 273–299 (1972)CrossRefGoogle Scholar
  26. Neira, M.A., Payatakes, A.C.: Collocation solution of creeping Newtonian flow through sinusoidal tubes. J. AIChE 25, 725–730 (1979)CrossRefGoogle Scholar
  27. Ng, C.-O., Wang, C.Y.: Darcy–Brinkman flow through a corrugated channel. Transp. Porous Med 85, 605–618 (2010)CrossRefGoogle Scholar
  28. Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Google Scholar
  29. Phan-Thien, N.: On Stokes flow between parallel plates with stationary random roughness. ZAMM 60, 675–679 (1980)CrossRefGoogle Scholar
  30. Phan-Thien, N.: On Stokes flow of a Newtonian fluid through a pipe with stationary random surface roughness. Phys. Fluids 24, 579–582 (1981a)Google Scholar
  31. Phan-Thien, N.: On the effects of the Reynolds and Stokes surface roughnesses in a two-dimensional slider bearing. Proc. R. Soc. Lond. A 377, 349–362 (1981b)Google Scholar
  32. Phan-Thien, N.: On Stokes flows in channels and pipes with parallel stationary random surface roughness. ZAMM 61, 193–199 (1981c)Google Scholar
  33. Pit, R., Hervet, H., Leger, L.: Direct experimental evidence of slip in hexadecane: solid interfaces. Phys. Rev. Lett. 85, 980–983 (2000)CrossRefGoogle Scholar
  34. Pozrikidis, C.: Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495–514 (1987)CrossRefGoogle Scholar
  35. Talbot, L., Cheng, R.K., Schefer, R.W., Willis, D.R.: Thermophoresis of particles in heated boundary layer. J. Fluid Mech. 101, 737–758 (1980)CrossRefGoogle Scholar
  36. Tretheway, D.C., Meinhart, C.D.: Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluids 14, L9–L12 (2002)CrossRefGoogle Scholar
  37. Tretheway, D.C., Meinharta, C.D.: A generating mechanism for apparent fluid slip in hydrophobic microchannels. Phys. Fluids 16, 1509–1515 (2004)CrossRefGoogle Scholar
  38. Wang, C.-Y.: Parallel flow between corrugated plates. J. Eng. Mech. 102, 1088–1090 (1976)Google Scholar
  39. Wang, C.-Y.: On Stokes flow between corrugated plates. J. Appl. Mech. 46, 462–464 (1979)CrossRefGoogle Scholar
  40. Wang, C.-Y.: Stokes flow through a channel with three-dimensional bumpy walls. Phys. Fluids 16, 2136–2139 (2004)CrossRefGoogle Scholar
  41. Wang, H., Wang, Y.: Flow in micro channels with rough walls: flow pattern and pressure drop. J. Micromech. Microeng. 17, 586–596 (2007)CrossRefGoogle Scholar
  42. Wang, K., Tavakkoli, F., Wang, S., Vafai, K.: Forced convection gaseous slip flow in a porous circular microtube: an exact solution. Int. J. Therm. Sci. 97, 152–162 (2015)CrossRefGoogle Scholar
  43. Wang, K., Vafai, K., Wang, D.: Analytical characterization of gaseous slip flow and heat transport through a parallel-plate microchannel with a centered porous substrate. Int. J. Numer. Methods Heat Fluid Flow 26, 854–878 (2016)CrossRefGoogle Scholar
  44. Yaglom, A.M.: An Introduction to the Theory of Stationary Random Functions. Courier Dover Publications, New York (2004)Google Scholar
  45. Yu, L.H., Wang, C.Y.: Darcy–Brinkman flow through a bumpy channel. Transp. Porous Med. 97, 281–294 (2013)CrossRefGoogle Scholar
  46. Zhou, H., Khayat, R.E., Martinuzzi, R.J., Straatman, A.G.: On the validity of the perturbation approach for the flow inside weakly modulated channels. Int. J. Numer. Methods Fluids 39, 1139–1159 (2002)CrossRefGoogle Scholar
  47. Zhu, Y., Granik, S.: Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys. Rev. Lett. 87, 096105 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceAlexandria UniversityAlexandriaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceDamanhour UniversityDamanhourEgypt

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