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Generalized Navier-Stokes equations for active suspensions

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Abstract

We discuss a minimal generalization of the incompressible Navier-Stokes equations to describe the complex steady-state dynamics of solvent flow in an active suspension. To account phenomenologically for the presence of an active component driving the ambient fluid flow, we postulate a generic nonlocal extension of the stress-tensor, conceptually similar to those recently introduced in granular flows. Stability and spectral properties of the resulting hydrodynamic model are studied both analytically and numerically for the two-dimensional (2D) case with periodic boundary conditions. Future generalizations of this theory could be useful for quantifying the shear properties of active suspensions.

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References

  1. M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R.A. Simha, Rev. Mod. Phys. 85, 1143 (2013)

    Article  ADS  Google Scholar 

  2. L.H. Cisneros, R. Cortez, C. Dombrowski, R.E. Goldstein, J.O. Kessler, Exp. Fluids 43, 737 (2007)

    Article  Google Scholar 

  3. H.P. Zhang, A. Be’er, E.L. Florin, H.L. Swinney, Proc. Natl. Acad. Sci. USA 107, 13626 (2010)

    Article  ADS  Google Scholar 

  4. V. Schaller, C. Weber, C. Semmrich, E. Frey, A.R. Bausch, Nature 467, 73 (2010)

    Article  ADS  Google Scholar 

  5. A. Sokolov, I.S. Aranson, Phys. Rev. Lett. 109, 248109 (2012)

    Article  ADS  Google Scholar 

  6. T. Sanchez, D.T.N. Chen, S.J. DeCamp, M. Heymann, Z. Dogic, Nature 491, 431 (2012)

    Article  ADS  Google Scholar 

  7. J. Dunkel, S. Heidenreich, K. Drescher, H.H. Wensink, M. Bär, R.E. Goldstein, Phys. Rev. Lett. 110, 228102 (2013)

    Article  ADS  Google Scholar 

  8. C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein, J.O. Kessler, Phys. Rev. Lett. 93(9), 098103 (2004)

    Article  ADS  Google Scholar 

  9. C.W. Wolgemuth, Biophys. J. 95, 1564 (2008)

    Article  ADS  Google Scholar 

  10. A. Baskaran, M.C. Marchetti, Proc. Natl. Acad. Sci. 106, 15567 (2009)

    Article  ADS  Google Scholar 

  11. D. Saintillan, M. Shelley, J. R. Soc. Interface 9, 571 (2011)

    Article  Google Scholar 

  12. H.H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R.E. Goldstein, H. Löwen, J.M. Yeomans, Proc. Natl. Acad. Sci. USA 109, 14308 (2012)

    Article  ADS  Google Scholar 

  13. F.G. Woodhouse, R.E. Goldstein, Phys. Rev. Lett. 109, 168105 (2012)

    Article  ADS  Google Scholar 

  14. T. Brotto, J.B. Caussin, E. Lauga, D. Bartolo, Phys. Rev. Lett. 110, 038101 (2013)

    Article  ADS  Google Scholar 

  15. L. Giomi, M.J. Bowick, X. Ma, M.C. Marchetti, Phys. Rev. Lett. 110, 228101 (2013)

    Article  ADS  Google Scholar 

  16. A. Zöttl, H. Stark, Phys. Rev. Lett. 112, 118101 (2014)

    Article  ADS  Google Scholar 

  17. X.L. Wu, A. Libchaber, Phys. Rev. Lett. 84, 3017 (2000)

    Article  ADS  Google Scholar 

  18. K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, R.E. Goldstein, Phys. Rev. Lett. 103, 198103 (2009)

    Article  ADS  Google Scholar 

  19. J. Dunkel, V.B. Putz, I.M. Zaid, J.M. Yeomans, Soft Matter 6, 4268 (2010)

    Article  ADS  Google Scholar 

  20. I.M. Zaid, J. Dunkel, J.M. Yeomans, J. R. Soc. Interface 8, 1314 (2011)

    Article  Google Scholar 

  21. A. Sokolov, I.S. Aranson, Phys. Rev. Lett. 103(14), 148101 (2009)

    Article  ADS  Google Scholar 

  22. S. Rafai, L. Jibuti, P. Peyla, Phys. Rev. Lett. 104, 098102 (2010)

    Article  ADS  Google Scholar 

  23. S.D. Ryan, B.M. Haines, L. Beryland, F. Ziebert, I.S. Aranson, Phys. Rev. E 83, 050904(R) (2011)

    Article  ADS  Google Scholar 

  24. G. Foffano, J.S. Lintuvuori, A.N.M. nd K. Stratford, M.E. Cates, D. Marenduzzo, Eur. Phys. J. E 35, 98 (2012)

    Article  Google Scholar 

  25. H. Wioland, F.G. Woodhouse, J. Dunkel, J.O. Kessler, R.E. Goldstein, Phys. Rev. Lett. 110, 268102 (2013)

    Article  ADS  Google Scholar 

  26. S.P. Thampi, R. Golestanian, J.M. Yeomans, Phys. Rev. Lett. 111, 118101 (2013)

    Article  ADS  Google Scholar 

  27. E. Lushi, H. Wioland, R.E. Goldstein, Proc. Natl. Acad. Sci. USA 111, 9733 (2014)

    Article  ADS  Google Scholar 

  28. J. Dunkel, S. Heidenreich, M. Bär, R.E. Goldstein, New J. Phys. 15, 045016 (2013)

    Article  ADS  Google Scholar 

  29. L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Pergamon, London, 1959)

  30. I.S. Aranson, L.S. Tsimring, Rev. Mod. Phys. 78, 641 (2006)

    Article  ADS  Google Scholar 

  31. J. Swift, P.C. Hohenberg, Phys. Rev. A 15, 319 (1977)

    Article  ADS  Google Scholar 

  32. J. Toner, Y. Tu, Phys. Rev. Lett. 75, 4326 (1995)

    Article  ADS  Google Scholar 

  33. J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  34. J. Toner, Y. Tu, S. Ramaswamy, Ann. Phys. 318, 170 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. S. Ramaswamy, Annu. Rev. Cond. Mat. Phys. 1, 323 (2010)

    Article  ADS  Google Scholar 

  36. R. Großmann, P. Romanczuk, M. Bär, L. Schimansky-Geier, Phys. Rev. Lett. 113, 258104 (2014)

    Article  ADS  Google Scholar 

  37. R.A. Simha, S. Ramaswamy, Phys. Rev. Lett. 89, 058101 (2002)

    Article  ADS  Google Scholar 

  38. D.L. Henann, K. Kamrin, Proc. Nat. Acad. Sci. 110, 6730 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004)

    Article  ADS  Google Scholar 

  40. C.M. Pooley, G.P. Alexander, J.M. Yeomans, Phys. Rev. Lett. 99, 228103 (2007)

    Article  ADS  Google Scholar 

  41. K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E. Goldstein, Proc. Natl. Acad. Sci. USA 108, 10940 (2011)

    Article  Google Scholar 

  42. J.S. Guasto, K.A. Johnson, J.P. Gollub, Phys. Rev. Lett. 105, 168102 (2010)

    Article  ADS  Google Scholar 

  43. E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  44. H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, 2nd edn. (Springer, Berlin, 1996)

  45. U. Frisch, Turbulence (Cambridge University Press, Cambridge, England, 2004)

  46. R. Abraham, J. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, Vol. 75 of Applied Mathematical Sciences (Springer, 1989)

  47. C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1987)

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E17, Cambridge, MA, 02139-4307, USA

    J. Słomka & J. Dunkel

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  1. J. Słomka
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  2. J. Dunkel
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Słomka, J., Dunkel, J. Generalized Navier-Stokes equations for active suspensions. Eur. Phys. J. Spec. Top. 224, 1349–1358 (2015). https://doi.org/10.1140/epjst/e2015-02463-2

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  • DOI: https://doi.org/10.1140/epjst/e2015-02463-2

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