Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T18:53:22.826Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similarity in coupled Brinkman/Navier–Stokes flows

Published online by Cambridge University Press:  24 April 2012

Ilenia Battiato
Ilenia Battiato*
Affiliation:
Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, Göttingen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we derive self-similar solutions of flows through both a porous medium and a pure fluid. Self-similar filtration velocity and hydrodynamic shear profiles are obtained by means of asymptotic analysis in the limit of infinitely small permeability, and for both laminar and turbulent regimes over the porous medium. We show that a spatial length scale, related to the porous layer thickness, naturally emerges from the limiting process and suggests a more formal definition of thick and thin porous media. We finally specialize the analysis to porous media constituted of patterned cylindrical obstacles, which can freely deflect under the aerodynamic shear exerted by the fluid flowing through and over the forest. A self-similar solution for the bending profile of the elastic cylindrical obstacles is obtained as intermediate asymptotics, and applied to carbon nanotube (CNT) forests’ response to aerodynamic stresses. This self-similar solution is successfully used to estimate flexural rigidity of CNTs by linear fit of appropriately rescaled maximum deflection and average velocity measurements.

JFM classification

Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 699 , 25 May 2012 , pp. 94 - 114
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Auriault, J.-L. 2009 On the domain of validity of Brinkman’s equation. Transp. Porous Med. 79, 215–223.CrossRefGoogle Scholar
2. Barenblatt, G. I. & Zel’dovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4, 285–312.CrossRefGoogle Scholar
3. Bars, M. L. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.CrossRefGoogle Scholar
4. Battiato, I., Bandaru, P. R. & Tartakovsky, D. M. 2010 Elastic response of carbon nanotube forests to aerodynamic stresses. Phys. Rev. Lett. 105, 144504.CrossRefGoogle ScholarPubMed
5. Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.CrossRefGoogle Scholar
6. Breugem, W. P., Boersma, B. J. & Uittenbogaard, R. E. 2006 The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 35–72.CrossRefGoogle Scholar
7. Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.CrossRefGoogle Scholar
8. Cieszko, M. & Kubik, J. 1999 Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid. Transp. Porous Med. 34, 319–336.CrossRefGoogle Scholar
9. Deck, C. P., Ni, C., Vecchio, K. S. & Bandaru, P. R. 2009 The response of carbon nanotube ensambles to fluid flow: applications to mechanical property measurement and diagnostics. J. Appl. Phys. 106, 074304.CrossRefGoogle Scholar
10. Falvo, M. R., Clary, G. J., Taylor, R. M., Chi, V., Brooks, F. P., Washburn, S. & Superfine, R. 1997 Bending and buckling of carbon nanotubes under large strain. Nature 389, 582.CrossRefGoogle ScholarPubMed
11. Finnigan, J. J., Shaw, R. H. & Patton, E. G. 2009 Turbulent structure above a vegetation canopy. J. Fluid Mech. 637, 387–424.CrossRefGoogle Scholar
12. Ghisalberti, M. 2009 Obstructed shear flows: similarities across systems and scales. J. Fluid Mech. 641, 51–61.CrossRefGoogle Scholar
13. Ghisalberti, M. & Nepf, H. M. 2002 Mixing layers and coherent structures in vegetated aquatic flows. J. Geophys. Res. 107 (C2), 3011.Google Scholar
14. Ghisalberti, M. & Nepf, H. 2009 Shallow flows over a permeable medium: the hydrodynamics of submerged aquatic canopies. Transp. Porous Med. 78 (3), 385–402.CrossRefGoogle Scholar
15. Ghosh, S., Sood, A. K. & Kumar, N. 2003 Carbon nanotube flow sensors. Science 299 (5609), 1042.CrossRefGoogle ScholarPubMed
16. Goharzadeh, A., Khalili, A. & Jorgensen, B. B. 2005 Transition layer thickness at a fluid-porous interface. Phys. Fluids 17, 057102.CrossRefGoogle Scholar
17. Happel, J. 1959 Viscous flow relative to arrays of cylinders. AIChE J. 5, 174–177.CrossRefGoogle Scholar
18. Huang, J. Y., Chen, S., Wang, Z. Q., Kempa, K., Wang, Y. M., Jo, S. H., Chen, G., Dresselhaus, M. S. & Ren, Z. F. 2006 Superplatic carbon nanotubes. Nature 439, 281.CrossRefGoogle ScholarPubMed
19. Jäger, W. & Mikelić, A. 2000 On the interface a boundary conditions of Beavers, Joseph, and Saffman. SIAM J. Appl. Maths 60 (4), 1111–1127.Google Scholar
20. Joseph, P., Cottin-Bizonne, C., Benoit, J.-M., Ybert, C., Journet, C., Tabeling, P. & Bocquet, L. 2006 Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys. Rev. Lett. 97, 156104.CrossRefGoogle ScholarPubMed
21. Kim, P. & Lieber, C. M. 1999 Nanotube nanotweezers. Science 286 (5447), 2148.CrossRefGoogle ScholarPubMed
22. Kim, S. J. & Choi, C. Y. 1996 Convective heat transfer in porous and overlying fluid layers heated from below. Intl J. Heat Mass Transfer 39, 319–329.CrossRefGoogle Scholar
23. Kruijt, B., Malhi, Y., Lloyd, J., Nobre, A. D., Miranda, A. C., Pereira, M. G. P., Culf, A. & Grace, J. 2000 Turbulence statistics above and within two Amazon rain forest canopies. Boundary-Layer Meteorol. 94, 297–331.CrossRefGoogle Scholar
24. Lu, J. P. 1997 Elastic properties of carbon nanotubes and nanopores. Phys. Rev. Lett. 79 (7), 1297.CrossRefGoogle Scholar
25. Manes, C., Poggi, D. & Ridolfi, L. 2011 Turbulent boundary layers over permeable walls: scaling and near-wall structure. J. Fluid Mech. 687, 141–170.CrossRefGoogle Scholar
26. Manes, C., Pokrajac, D., McEwan, I. & Nikora, V. 2009 Turbulence structure of open channel flows over permeable and impermeable beds: a comparative study. Phys. Fluids 21 (12), 125109.CrossRefGoogle Scholar
27. Meyyappan, M. 2005 Carbon Nanotubes: Science and Applications. CRC Press.Google Scholar
28. Ochoa-Tapia, J. & Whitaker, S. 1995 Momentum transfer at the boundary between a porous medium and a homogenous fluid: I-theoretical development. Intl J. Heat Mass Transfer 38, 2635–2646.CrossRefGoogle Scholar
29. Poggi, D., Porporato, A., Ridolfi, L., Albertson, J. D. & Katul, G. G. 2004 The effect of vegetation density on canopy sub-layer turbulence. Boundary-Layer Meteorol. 111 (3), 565–587.CrossRefGoogle Scholar
30. Poncharal, P., Wang, Z. L., Ugarte, D. & de Heer, W. A. 1999 Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283 (1513), 1513–1516.CrossRefGoogle ScholarPubMed
31. Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
32. Srivastava, A., Srivastava, O. N., Talapatra, S., Vajtai, R. & Ajayan, P. M. 2004 Carbon nanotube filters. Nature Mater. 3, 610.CrossRefGoogle ScholarPubMed
33. Tachie, M. F., James, D. F. & Currie, I. G. 2004 Slow flow through a brush. Phys. Fluids 16 (2), 445–451.CrossRefGoogle Scholar
34. Taylor, G. I. 1971 A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 42 (2), 319–326.CrossRefGoogle Scholar
35. Tien, C. L. & Vafai, K. 1990 Convective and radiative heat transfer in porous media. Adv. Appl. Mech. 27, 225–281.CrossRefGoogle Scholar
36. Treacy, M. M. J., Ebbesen, T. W. & Gibson, J. M. 1996 Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381, 678.CrossRefGoogle Scholar
37. Vafai, K. & Kim, S. J. 1990 Fluid mechanics of the interface region between a porous medium and a fluid layer-an exact solution. Intl J. Heat Transfer Fluid Flow 11 (3), 254–256.CrossRefGoogle Scholar
38. Weinbaum, S., Zhang, X., Han, Y., Vink, H. & Cowin, S. C. 2003 Mechanotransduction and flow across the endothelial glycocalyx. Proc. Natl Acad. Sci. USA 100 (13), 7988–7995.CrossRefGoogle ScholarPubMed
39. Wilson, M. 2009 Superhydrophobic surfaces reduce drag. Phys. Today 62 (10), 16.Google Scholar
40. Zhang, M., Fang, S., Zakhidov, A. A., Lee, S. B., Aliev, A. E., Williams, C. D., Atkinson, K. R. & Baughman, R. H. 2005 Strong, transparent, multifunctional, carbon nanotube sheets. Science 309 (5738), 1215–1219.CrossRefGoogle ScholarPubMed