Effective viscosity of bacterial suspensions: A three-dimensional pde model with stochastic torque

Communications on Pure and Applied Analysis

Volumn 11, Issue 1, 2012, Pages 19-48

  Haines, Brian M ^a   Aranson, Igor S ^b   Berlyand, Leonid ^a   Karpeev, Dmitry A ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b ARGONNE NATIONAL LABORATORY   (United States)

Author keywords

Bacterial suspensions; Effective viscosity; Homogenization

Indexed keywords

EID: 82255195734     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2012.11.19     Document Type: Conference Paper

References (38)

1. G. Batchelor, The stress system in a suspension offorce-free particles, J. Fluid Mech., 41 (1970), 545-570.
2. G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order c , J. Fluid Mech., 56 (1972), 401-427.
3. L. Berlyand, L. Borcea and A. Panchenko, Network approximation for effective viscosity of concentrated suspensions with complex geometry, SIAM J. Math. Anal., 36 (2005), 1580-1628. (Pubitemid 41343489)
4. L. Berlyand, Y. Gorb and A. Novikov, Fictitious fluid approach and anomalous blow-up of the dissipation rate in a 2d model of concentrated suspensions, Arch. Rat. Mech. Anal., 193 (2009), 585-622, 2008.
5. L. Berlyand and A. Panchenko, Strong and weak blow up of the viscous dissipation rates for concentrated suspensions, J. Fluid Mech., 578 (2007), 1-34. (Pubitemid 47307555)
6. H. Brenner and D. W. Condiff, Transport mechanics in systems of orientable particles: Iii. arbitrary particles, J. Colloid and Interface Sci., 41 (1972), 228-274.
7. A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications, New York, 1956.
8. G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I," Springer, New York, 1994.
9. V. T. Gyrya, I. S. Aranson, L. V. Berlyand, and D. A. Karpeev, A model of hydrodynamic interaction between swimming bacteria, Bull. Math. Biol., 72 (2009), 148-183.
10. B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension ofspheres, preprint, ar Xiv:1104.1102.
11. B. M. Haines, I. S. Aranson, L. Berlyand and D. A. Karpeev, Effective viscosity of dilute bacterial suspensions: a two-dimensional model, Phys. Biol., 5 (2008), 046003.
12. B. M. Haines, A. Sokolov, I. S. Aranson, L. Berlyand and D. A. Karpeev, Three-dimensional model for the effective viscosity of bacterial suspensions, Phys. Rev. E, 80 (2009), 041922.
13. Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, Rheology of active-particle suspen- sions, Phys. Rev. Lett., 92 (2004), 118101.
14. J. P. Hernandez-Ortiz, C. G. Stoltz, and M. D. Graham, Transport and collective dynamics in suspensions of confined swimming particles, Phys. Rev. Lett., 95 (2005), 204501.
15. E. J. Hinch and L. G. Leal, The effect of brownian motion on the rheological properties of a suspension of non-spherical particles, J. Fluid Mech., 52 (1972), 683-712.
16. T. Ishikawa and T. J. Pedley, The rheology of a semi-dilute suspension of swimming model micro-organisms, J. Fluid Mech., 588 (2007), 399-435.
17. K. Ito, Stochastic integral, Proc. Imperial Acad., Tokyo, 20 (1944), 519-24.
18. K. Ito, Stochastic differential equations on a differentiable manifold 2, mem. Coll. Sci. Kŷoto Univ., 28 (1953), 82-85.
19. G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, R. Soc. London Ser. A, 102 (1922), 161-79.
20. M. J. Kim and K. S. Breuer, Use of bacterial carpets to enhance mixing in microfluidic systems, Jour. Fluids Engin., 129 (2007), 319-25.
21. S. Kim and S. Karrila, "Microhydrodynamics," Dover Publications, New York, 1991.
22. L. G. Leal and E. J. Hinch, The effect of weak brownian rotations on particles in shear flow, J. Fluid Mech., 46 (1971), 685-703.
23. T. Levy and E. Sanchez-Palencia, Suspension of solid particles in a newtonian fluid, J. Non- Newt. Fluid Mech., 13 (1983), 63-78. (Pubitemid 13603735)
24. R. M. Macnab, The bacterial flagellum: Reversible rotary propellor and type iii export apparatus, J. Bacteriology, 181 (1999), 7149-7153.
25. H. P. Mc Kean, "Stochastic Integrals," Academic Press, New York and London, 1969.
26. K. C. Nunan and J. B. Keller, Effective viscosity of a periodic suspension, J. Fluid Mech., 142 (1984), 269-287.
27. T. J. Pedley and J. O. Kessler, A new continuum model for suspensions ofgyrotactic micro- organisms, J. Fluid Mech., 212 (1990), 155-182.
28. H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications," Springer- Verlag, New York, 1989.
29. D. Saintillan, The dilute rheology ofswimming suspensions: A simple kinetic model, Experimental Mechanics, 50, 2010.
30. J. Shioi, S. Matsuura and Y. Imae, Quantitative measurements of proton motive force and motility in bacillus subtilis, J. Bacteriol., 144 (1980), 891-897. (Pubitemid 11219105)
31. A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria, Phys. Rev. Lett., 103 (2009), 148101.
32. A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Concentration dependence of the collective dynamics ofswimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.
33. A. Sokolov, R. E. Goldstein, F. I. Feldchtein and I. S. Aranson, Enhanced mixing and spatial instability in concentrated bacterial suspensions, Phys. Rev. E, 80 (2009), 031903.
34. G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming, J. Fluid Mech., 632 (2009), 359-400.
35. L. Turner, W. S. Ryu and H. C. Berg, Real-time imaging offluorescentflagellarfilaments, J. Bacteriol., 182 (2000), 2793-801.
36. I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, PNAS, 102 (2005), 2277-82 (Pubitemid 40279366)
37. M. Wu, J. W. Roberts, S. Kim, D. L. Koch and M. P. Delisa, Collective bacterial dynam- ics revealed using a three-dimensional population-scale defocused particle tracking technique, Applied and Environmental Microbiology, 72 (2006), 4987-94. (Pubitemid 44078830)
38. X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath, Phys. Rev. Lett., 84 (2000), 3017-20. (Pubitemid 40598951)