Models of bacteria swimming in a nematic liquid crystal
Authors:
Mochong Duan and Noel J. Walkington
Journal:
Quart. Appl. Math. 79 (2021), 695-715
MSC (2020):
Primary 92-10, 37M25, 35Q92
DOI:
https://doi.org/10.1090/qam/1598
Published electronically:
July 9, 2021
MathSciNet review:
4328144
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Abstract: Models of dilute systems of bacteria swimming in a nematic liquid crystal are developed and analyzed. The motion and orientation of the bacteria are simulated using ordinary differential equations coupled with the partial differential equations modeling the nematic liquid crystal (Ericksen Leslie equations). The analysis and numerical simulations of this system are shown to predict interesting phenomena observed experimentally.
References
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- J. Walton, G. McKay, M. Grinfeld, and N. J. Mottram, Pressure-driven changes to spontaneous flow in active nematic liquid crystals, The European Physical Journal E, 43 (2020), pp. 1–14.
- C. W. Wolgemuth, Collective swimming and the dynamics of bacterial turbulence, Biophysical Journal, 95 (2008), pp. 1564–1574.
- S. Zhou, Recent progresses in lyotropic chromonic liquid crystal research: elasticity, viscosity, defect structures, and living liquid crystals, Liquid Crystals Today, 27 (2018), pp. 91–108.
- S. Zhou, K. Neupane, Y. A. Nastishin, A. Baldwin, S. Shiyanovskii, O. Lavrentovich, and S. Sprunt, Elasticity, viscosity, and orientational fluctuations of a lyotropic chromonic nematic liquid crystal disodium cromoglycate, Soft matter, 10 34 (2014), pp. 6571–81.
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References
- François Alouges, Antonio DeSimone, and Aline Lefebvre, Optimal strokes for low Reynolds number swimmers: an example, J. Nonlinear Sci. 18 (2008), no. 3, 277–302. MR 2411380, DOI 10.1007/s00332-007-9013-7
- G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
- Xiuqing Chen and Jian-Guo Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Differential Equations 254 (2013), no. 7, 2764–2802. MR 3017030, DOI 10.1016/j.jde.2013.01.005
- Antonio DeSimone, François Alouges, and Aline Lefebvre, Biological fluid dynamics, non-linear partial differential equations, Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, pp. 26–31. MR 3220660, DOI 10.1007/978-1-4614-1806-1_3
- Antonio DeSimone, François Alouges, and Aline Lefebvre, Biological fluid dynamics, non-linear partial differential equations, Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, pp. 26–31. MR 3220660, DOI 10.1007/978-1-4614-1806-1_3
- J. L. Ericksen, Nilpotent energies in liquid crystal theory, Arch. Rational Mech. Anal. 10 (1962), 189–196. MR 169513, DOI 10.1007/BF00281186
- F. C. Frank, On the theory of liquid crystals, Discuss. Faraday Soc., 25 (1958), pp. 19–28.
- M. Genkin, A. Sokolov, O. Lavrentovich, and I. Aranson, Topological defects in a living nematic ensnare swimming bacteria, Physical Review X, 7 (2017).
- M. M. Genkin, A. Sokolov, and I. S. Aranson, Spontaneous topological charging of tactoids in a living nematic, New Journal of Physics, 20 (2018), p. 043027.
- G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. Royal Society London, 102 (1922), pp. 161–179.
- R. Koizumi, T. Turiv, M. M. Genkin, R. J. Lastowski, H. Yu, I. Chaganava, Q.-H. Wei, I. S. Aranson, and O. D. Lavrentovich, Control of microswimmers by spiral nematic vortices: Transition from individual to collective motion and contraction, expansion, and stable circulation of bacterial swirls, Phys. Rev. Research, 2 (2020), p. 033060.
- F. Leslie, Theory of flow phenomenum in liquid crystals, in The Theory of Liquid Crystals, W. Brown, ed., vol. 4, New York, 1979, Academic Press., pp. 1–81.
- M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys., 85 (2013), pp. 1143–1189.
- G. Marrucci and F. Greco, Flow behavior of liquid crystalline polymers, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds., vol. LXXXVI, John Wiley & Sons, Inc., 1993, pp. 331–404.
- A. Panchenko, D. F. Hinz, and E. Fried, Spatial averaging of a dissipative particle dynamics model for active suspensions, Physics of Fluids, 30 (2018).
- C. Peng, T. Turiv, Y. Guo, Q.-H. Wei, and O. Lavrentovich, Command of active matter by topological defects and patterns, Science 354 (2016), pp. 882–885.
- Florian Rathgeber, David A. Ham, Lawrence Mitchell, Michael Lange, Fabio Luporini, Andrew T. T. McRae, Gheorghe-Teodor Bercea, Graham R. Markall, and Paul H. J. Kelly, Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Software 43 (2017), no. 3, Art. 24, 27. MR 3615280, DOI 10.1145/2998441
- D. Saintillan and M. J. Shelley, Active suspensions and their nonlinear models: Living fluids / Fluides vivants, Comptes Rendus Physique, 14 (2013), pp. 497–517.
- André M. Sonnet and Epifanio G. Virga, Dissipative ordered fluids, Springer, New York, 2012. Theories for liquid crystals. MR 2893657, DOI 10.1007/978-0-387-87815-7
- J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical $xy$ model: How birds fly together, Physical Review Letters, 75 (1995), pp. 4326–4329.
- John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI 10.1103/PhysRevE.58.4828
- John Toner, Yuhai Tu, and Sriram Ramaswamy, Hydrodynamics and phases of flocks, Ann. Physics 318 (2005), no. 1, 170–244. MR 2148645, DOI 10.1016/j.aop.2005.04.011
- Noel J. Walkington, Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 3, 523–540. MR 2804649, DOI 10.1051/m2an/2010065
- J. Walton, G. McKay, M. Grinfeld, and N. J. Mottram, Pressure-driven changes to spontaneous flow in active nematic liquid crystals, The European Physical Journal E, 43 (2020), pp. 1–14.
- C. W. Wolgemuth, Collective swimming and the dynamics of bacterial turbulence, Biophysical Journal, 95 (2008), pp. 1564–1574.
- S. Zhou, Recent progresses in lyotropic chromonic liquid crystal research: elasticity, viscosity, defect structures, and living liquid crystals, Liquid Crystals Today, 27 (2018), pp. 91–108.
- S. Zhou, K. Neupane, Y. A. Nastishin, A. Baldwin, S. Shiyanovskii, O. Lavrentovich, and S. Sprunt, Elasticity, viscosity, and orientational fluctuations of a lyotropic chromonic nematic liquid crystal disodium cromoglycate, Soft matter, 10 34 (2014), pp. 6571–81.
- S. Zhou, O. Tovkach, D. Golovaty, A. Sokolov, I. S. Aranson, and O. D. Lavrentovich, Dynamic states of swimming bacteria in a nematic liquid crystal cell with homeotropic alignment, New Journal of Physics, 19 (2017), p. 055006.
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Additional Information
Mochong Duan
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
ORCID:
0000-0002-0496-1189
Email:
mduan@andrew.cmu.edu
Noel J. Walkington
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID:
180170
Email:
noelw@andrew.cmu.edu
Keywords:
Swimming,
bacteria,
liquid crystal
Received by editor(s):
March 2, 2021
Received by editor(s) in revised form:
April 27, 2021
Published electronically:
July 9, 2021
Additional Notes:
The second author was supported in part by National Science Foundation Grants DREF-1729478 and DMS–2012259. This work was also supported by the NSF through the Center for Nonlinear Analysis
Article copyright:
© Copyright 2021
Brown University
