# American Institute of Mathematical Sciences

January  2012, 11(1): 19-46. doi: 10.3934/cpaa.2012.11.19

## 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

Received  December 2009 Revised  March 2010 Published  September 2011

We present a PDE model for dilute suspensions of swimming bacteria in a three-dimensional Stokesian fluid. This model is used to calculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background flow. A bacterium is modeled as an impenetrable prolate spheroid with self-propulsion provided by a point force, which appears in the model as an inhomogeneous delta function in the PDE. The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation). Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background flows, such as a pure straining or planar shear flow, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experiments on suspensions of Bacillus subtilis.
Citation: B. M. Haines, Igor S. Aranson, Leonid Berlyand, Dmitry A. Karpeev. Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque. Communications on Pure & Applied Analysis, 2012, 11 (1) : 19-46. doi: 10.3934/cpaa.2012.11.19
##### References:
 [1] G. Batchelor, The stress system in a suspension of force-free particles,, J. Fluid Mech., 41 (1970), 545. doi: 10.1017/S0022112070000745. Google Scholar [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. doi: 10.1017/S0022112072002435. Google Scholar [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. doi: 10.1137/S0036141003424708. Google Scholar [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. doi: 10.1007/s00205-008-0152-2. Google Scholar [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. doi: 10.1017/S0022112007004922. Google Scholar [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. doi: 10.1016/0021-9797(72)90111-7. Google Scholar [7] A. Einstein, "Investigations on the theory of the Brownian movement,'', Dover Publications, (1956). Google Scholar [8] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I,'', Springer, (1994). Google Scholar [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. doi: 10.1007/s11538-009-9442-6. Google Scholar [10] B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres,, preprint, (). Google Scholar [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). doi: 10.1088/1478-3975/5/4/046003. Google Scholar [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). doi: 10.1103/PhysRevE.80.041922. Google Scholar [13] Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, Rheology of active-particle suspensions,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.118101. Google Scholar [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). doi: 10.1103/PhysRevLett.95.204501. Google Scholar [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. doi: 10.1017/S002211207200271X. Google Scholar [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. doi: 10.1017/S0022112007007835. Google Scholar [17] K. Ito, Stochastic integral,, Proc. Imperial Acad., 20 (1944), 519. doi: 10.3792/pia/1195572786. Google Scholar [18] K. Ito, Stochastic differential equations on a differentiable manifold 2,, mem. Coll. Sci. Ky\^oto Univ., 28 (1953), 82. Google Scholar [19] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid,, R. Soc. London Ser. A, 102 (1922), 161. doi: 10.1098/rspa.1922.0078. Google Scholar [20] M. J. Kim and K. S. Breuer, Use of bacterial carpets to enhance mixing in microfluidic systems,, Jour. Fluids Engin., 129 (2007), 319. doi: 10.1115/1.2427083. Google Scholar [21] S. Kim and S. Karrila, "Microhydrodynamics,'', Dover Publications, (1991). Google Scholar [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. doi: 10.1017/S0022112071000788. Google Scholar [23] T. Levy and E. Sanchez-Palencia, Suspension of solid particles in a newtonian fluid,, J. Non-Newt. Fluid Mech., 13 (1983), 63. doi: 10.1016/0377-0257(83)85022-8. Google Scholar [24] R. M. Macnab, The bacterial flagellum: Reversible rotary propellor and type iii export apparatus,, J. Bacteriology, 181 (1999), 7149. Google Scholar [25] H. P. McKean, "Stochastic Integrals,'', Academic Press, (1969). Google Scholar [26] K. C. Nunan and J. B. Keller, Effective viscosity of a periodic suspension,, J. Fluid Mech., 142 (1984), 269. doi: 10.1017/S0022112084001105. Google Scholar [27] T. J. Pedley and J. O. Kessler, A new continuum model for suspensions of gyrotactic micro-organisms,, J. Fluid Mech., 212 (1990), 155. doi: 10.1017/S0022112090001914. Google Scholar [28] H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,'', Springer-Verlag, (1989). doi: 10.1115/1.2897281. Google Scholar [29] D. Saintillan, The dilute rheology of swimming suspensions: A simple kinetic model,, Experimental Mechanics, 50 (2010). doi: 10.1007/s11340-009-9267-0. Google Scholar [30] J. Shioi, S. Matsuura and Y. Imae, Quantitative measurements of proton motive force and motility in bacillus subtilis,, J. Bacteriol., 144 (1980), 891. Google Scholar [31] A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.148101. Google Scholar [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). doi: 10.1103/PhysRevLett.98.158102. Google Scholar [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). doi: 10.1103/PhysRevE.80.031903. Google Scholar [34] G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming,, J. Fluid Mech., 632 (2009), 359. doi: 10.1017/S002211200900706X. Google Scholar [35] L. Turner, W. S. Ryu and H. C. Berg, Real-time imaging of fluorescent flagellar filaments,, J. Bacteriol., 182 (2000), 2793. doi: 10.1128/JB.182.10.2793-2801.2000. Google Scholar [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. doi: 10.1073/pnas.0406724102. Google Scholar [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. doi: 10.1128/AEM.00158-06. Google Scholar [38] X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath,, Phys. Rev. Lett., 84 (2000), 3017. doi: 10.1103/PhysRevLett.84.3017. Google Scholar

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##### References:
 [1] G. Batchelor, The stress system in a suspension of force-free particles,, J. Fluid Mech., 41 (1970), 545. doi: 10.1017/S0022112070000745. Google Scholar [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. doi: 10.1017/S0022112072002435. Google Scholar [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. doi: 10.1137/S0036141003424708. Google Scholar [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. doi: 10.1007/s00205-008-0152-2. Google Scholar [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. doi: 10.1017/S0022112007004922. Google Scholar [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. doi: 10.1016/0021-9797(72)90111-7. Google Scholar [7] A. Einstein, "Investigations on the theory of the Brownian movement,'', Dover Publications, (1956). Google Scholar [8] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I,'', Springer, (1994). Google Scholar [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. doi: 10.1007/s11538-009-9442-6. Google Scholar [10] B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres,, preprint, (). Google Scholar [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). doi: 10.1088/1478-3975/5/4/046003. Google Scholar [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). doi: 10.1103/PhysRevE.80.041922. Google Scholar [13] Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, Rheology of active-particle suspensions,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.118101. Google Scholar [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). doi: 10.1103/PhysRevLett.95.204501. Google Scholar [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. doi: 10.1017/S002211207200271X. Google Scholar [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. doi: 10.1017/S0022112007007835. Google Scholar [17] K. Ito, Stochastic integral,, Proc. Imperial Acad., 20 (1944), 519. doi: 10.3792/pia/1195572786. Google Scholar [18] K. Ito, Stochastic differential equations on a differentiable manifold 2,, mem. Coll. Sci. Ky\^oto Univ., 28 (1953), 82. Google Scholar [19] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid,, R. Soc. London Ser. A, 102 (1922), 161. doi: 10.1098/rspa.1922.0078. Google Scholar [20] M. J. Kim and K. S. Breuer, Use of bacterial carpets to enhance mixing in microfluidic systems,, Jour. Fluids Engin., 129 (2007), 319. doi: 10.1115/1.2427083. Google Scholar [21] S. Kim and S. Karrila, "Microhydrodynamics,'', Dover Publications, (1991). Google Scholar [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. doi: 10.1017/S0022112071000788. Google Scholar [23] T. Levy and E. Sanchez-Palencia, Suspension of solid particles in a newtonian fluid,, J. Non-Newt. Fluid Mech., 13 (1983), 63. doi: 10.1016/0377-0257(83)85022-8. Google Scholar [24] R. M. Macnab, The bacterial flagellum: Reversible rotary propellor and type iii export apparatus,, J. Bacteriology, 181 (1999), 7149. Google Scholar [25] H. P. McKean, "Stochastic Integrals,'', Academic Press, (1969). Google Scholar [26] K. C. Nunan and J. B. Keller, Effective viscosity of a periodic suspension,, J. Fluid Mech., 142 (1984), 269. doi: 10.1017/S0022112084001105. Google Scholar [27] T. J. Pedley and J. O. Kessler, A new continuum model for suspensions of gyrotactic micro-organisms,, J. Fluid Mech., 212 (1990), 155. doi: 10.1017/S0022112090001914. Google Scholar [28] H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,'', Springer-Verlag, (1989). doi: 10.1115/1.2897281. Google Scholar [29] D. Saintillan, The dilute rheology of swimming suspensions: A simple kinetic model,, Experimental Mechanics, 50 (2010). doi: 10.1007/s11340-009-9267-0. Google Scholar [30] J. Shioi, S. Matsuura and Y. Imae, Quantitative measurements of proton motive force and motility in bacillus subtilis,, J. Bacteriol., 144 (1980), 891. Google Scholar [31] A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.148101. Google Scholar [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). doi: 10.1103/PhysRevLett.98.158102. Google Scholar [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). doi: 10.1103/PhysRevE.80.031903. Google Scholar [34] G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming,, J. Fluid Mech., 632 (2009), 359. doi: 10.1017/S002211200900706X. Google Scholar [35] L. Turner, W. S. Ryu and H. C. Berg, Real-time imaging of fluorescent flagellar filaments,, J. Bacteriol., 182 (2000), 2793. doi: 10.1128/JB.182.10.2793-2801.2000. Google Scholar [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. doi: 10.1073/pnas.0406724102. Google Scholar [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. doi: 10.1128/AEM.00158-06. Google Scholar [38] X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath,, Phys. Rev. Lett., 84 (2000), 3017. doi: 10.1103/PhysRevLett.84.3017. Google Scholar
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