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Concentration fluctuations in a bacterial suspension

Tuesday 25th June 2013 - 11:00 to 11:45
INI Seminar Room 1
Recent analyses and simulations have identified an instability of a quiescent bacterial suspension above a threshold concentration, (nL3)crit = (5/C)(L/U\tau), where n is the bacterium number density, L and U the bacterium length and swimming speed, t the mean interval between tumbles, and C a measure of the intrinsic force-dipole. This instability is thought to underlie the large-scale coherent motions observed in experiments. There, however, remains a discrepancy between theory and simulations. While the former predicts a spatially homogeneous instability with coupled orientation and velocity fluctuations, simulations have observed large-scale concentration fluctuations. Even in the stable regime, solutions of the linearized equations reveal significant concentration fluctuations.

We will formulate an analytical solution that illustrates the linearized evolution of the velocity, orientation and concentration fields in a bacterial suspension starting from an arbitrary initial condition. The analysis relies on a remarkable correspondence between orientation fluctuations in a bacterial suspension and vorticity fluctuations in an inviscid fluid. The governing operators in both cases possess singular continuous spectra in addition to discrete modes. The dynamics of the singular orientation modes leads to transient growth of concentration fluctuations in the manner that the singular vorticity modes lead to kinetic energy growth in high-Reynolds-number shearing flows. We will discuss the velocity, orientation and stress correlations, emerging from an uncorrelated Poisson field, both below and above the critical concentration.

We also analyze the role of tumbling as a source of fluctuations. Regarding a tumble as a ‘linear collision’ governed by Poisson statistics allows one to write down the orientation-space noise, and this in turn leads to the analog of the fluctuating hydrodynamic equations for a bacterial suspension.

Co-author: Donald Koch (Chemical and bio-molecular engineering, Cornell University, NY, USA.)

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons