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Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque
1. | Department of Mathematics, Penn State University, McAllister Bldg., University Park, PA 16802, United States |
2. | Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, United States |
3. | Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802 |
4. | Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, United States |
References:
[1] |
G. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech., 41 (1970), 545-570.
doi: 10.1017/S0022112070000745. |
[2] |
G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order $c^2$, J. Fluid Mech., 56 (1972), 401-427.
doi: 10.1017/S0022112072002435. |
[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.
doi: 10.1137/S0036141003424708. |
[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.
doi: 10.1007/s00205-008-0152-2. |
[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.
doi: 10.1017/S0022112007004922. |
[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.
doi: 10.1016/0021-9797(72)90111-7. |
[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.
doi: 10.1007/s11538-009-9442-6. |
[10] |
B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres, preprint, arXiv: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.
doi: 10.1088/1478-3975/5/4/046003. |
[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.
doi: 10.1103/PhysRevE.80.041922. |
[13] |
Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, Rheology of active-particle suspensions, Phys. Rev. Lett., 92 (2004), 118101.
doi: 10.1103/PhysRevLett.92.118101. |
[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.
doi: 10.1103/PhysRevLett.95.204501. |
[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.
doi: 10.1017/S002211207200271X. |
[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.
doi: 10.1017/S0022112007007835. |
[17] |
K. Ito, Stochastic integral, Proc. Imperial Acad., Tokyo, 20 (1944), 519-24.
doi: 10.3792/pia/1195572786. |
[18] |
K. Ito, Stochastic differential equations on a differentiable manifold 2, mem. Coll. Sci. Kyôto 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.
doi: 10.1098/rspa.1922.0078. |
[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.
doi: 10.1115/1.2427083. |
[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.
doi: 10.1017/S0022112071000788. |
[23] |
T. Levy and E. Sanchez-Palencia, Suspension of solid particles in a newtonian fluid, J. Non-Newt. Fluid Mech., 13 (1983), 63-78.
doi: 10.1016/0377-0257(83)85022-8. |
[24] |
R. M. Macnab, The bacterial flagellum: Reversible rotary propellor and type iii export apparatus, J. Bacteriology, 181 (1999), 7149-7153. |
[25] |
H. P. McKean, "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.
doi: 10.1017/S0022112084001105. |
[27] |
T. J. Pedley and J. O. Kessler, A new continuum model for suspensions of gyrotactic micro-organisms, J. Fluid Mech., 212 (1990), 155-182.
doi: 10.1017/S0022112090001914. |
[28] |
H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,'' Springer-Verlag, New York, 1989.
doi: 10.1115/1.2897281. |
[29] |
D. Saintillan, The dilute rheology of swimming suspensions: A simple kinetic model, Experimental Mechanics, 50, 2010.
doi: 10.1007/s11340-009-9267-0. |
[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. |
[31] |
A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria, Phys. Rev. Lett., 103 (2009), 148101.
doi: 10.1103/PhysRevLett.103.148101. |
[32] |
A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.
doi: 10.1103/PhysRevLett.98.158102. |
[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.
doi: 10.1103/PhysRevE.80.031903. |
[34] |
G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming, J. Fluid Mech., 632 (2009), 359-400.
doi: 10.1017/S002211200900706X. |
[35] |
L. Turner, W. S. Ryu and H. C. Berg, Real-time imaging of fluorescent flagellar filaments, J. Bacteriol., 182 (2000), 2793-801.
doi: 10.1128/JB.182.10.2793-2801.2000. |
[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
doi: 10.1073/pnas.0406724102. |
[37] |
M. Wu, J. W. Roberts, S. Kim, D. L. Koch and M. P. Delisa, Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique, Applied and Environmental Microbiology, 72 (2006), 4987-94.
doi: 10.1128/AEM.00158-06. |
[38] |
X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath, Phys. Rev. Lett., 84 (2000), 3017-20.
doi: 10.1103/PhysRevLett.84.3017. |
show all references
References:
[1] |
G. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech., 41 (1970), 545-570.
doi: 10.1017/S0022112070000745. |
[2] |
G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order $c^2$, J. Fluid Mech., 56 (1972), 401-427.
doi: 10.1017/S0022112072002435. |
[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.
doi: 10.1137/S0036141003424708. |
[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.
doi: 10.1007/s00205-008-0152-2. |
[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.
doi: 10.1017/S0022112007004922. |
[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.
doi: 10.1016/0021-9797(72)90111-7. |
[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.
doi: 10.1007/s11538-009-9442-6. |
[10] |
B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres, preprint, arXiv: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.
doi: 10.1088/1478-3975/5/4/046003. |
[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.
doi: 10.1103/PhysRevE.80.041922. |
[13] |
Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, Rheology of active-particle suspensions, Phys. Rev. Lett., 92 (2004), 118101.
doi: 10.1103/PhysRevLett.92.118101. |
[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.
doi: 10.1103/PhysRevLett.95.204501. |
[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.
doi: 10.1017/S002211207200271X. |
[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.
doi: 10.1017/S0022112007007835. |
[17] |
K. Ito, Stochastic integral, Proc. Imperial Acad., Tokyo, 20 (1944), 519-24.
doi: 10.3792/pia/1195572786. |
[18] |
K. Ito, Stochastic differential equations on a differentiable manifold 2, mem. Coll. Sci. Kyôto 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.
doi: 10.1098/rspa.1922.0078. |
[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.
doi: 10.1115/1.2427083. |
[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.
doi: 10.1017/S0022112071000788. |
[23] |
T. Levy and E. Sanchez-Palencia, Suspension of solid particles in a newtonian fluid, J. Non-Newt. Fluid Mech., 13 (1983), 63-78.
doi: 10.1016/0377-0257(83)85022-8. |
[24] |
R. M. Macnab, The bacterial flagellum: Reversible rotary propellor and type iii export apparatus, J. Bacteriology, 181 (1999), 7149-7153. |
[25] |
H. P. McKean, "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.
doi: 10.1017/S0022112084001105. |
[27] |
T. J. Pedley and J. O. Kessler, A new continuum model for suspensions of gyrotactic micro-organisms, J. Fluid Mech., 212 (1990), 155-182.
doi: 10.1017/S0022112090001914. |
[28] |
H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,'' Springer-Verlag, New York, 1989.
doi: 10.1115/1.2897281. |
[29] |
D. Saintillan, The dilute rheology of swimming suspensions: A simple kinetic model, Experimental Mechanics, 50, 2010.
doi: 10.1007/s11340-009-9267-0. |
[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. |
[31] |
A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria, Phys. Rev. Lett., 103 (2009), 148101.
doi: 10.1103/PhysRevLett.103.148101. |
[32] |
A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.
doi: 10.1103/PhysRevLett.98.158102. |
[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.
doi: 10.1103/PhysRevE.80.031903. |
[34] |
G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming, J. Fluid Mech., 632 (2009), 359-400.
doi: 10.1017/S002211200900706X. |
[35] |
L. Turner, W. S. Ryu and H. C. Berg, Real-time imaging of fluorescent flagellar filaments, J. Bacteriol., 182 (2000), 2793-801.
doi: 10.1128/JB.182.10.2793-2801.2000. |
[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
doi: 10.1073/pnas.0406724102. |
[37] |
M. Wu, J. W. Roberts, S. Kim, D. L. Koch and M. P. Delisa, Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique, Applied and Environmental Microbiology, 72 (2006), 4987-94.
doi: 10.1128/AEM.00158-06. |
[38] |
X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath, Phys. Rev. Lett., 84 (2000), 3017-20.
doi: 10.1103/PhysRevLett.84.3017. |
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