Propagation of discontinuities in a pipe flow of suspension of motile microorganisms (A thread of motile algae for real-time bio-monitoring) Petr Denissenko, University of Warwick, 25 June image/sec

Microorganism motility. Diffusion, low Re For the experiments we used Chlamydomonas nivalis (phototrophic regime), a biflagellate Crypthecodinium cohnii (heterotrophic regime), a dynoflagellate Thick depleted zone Stationary microorganism Moving microorganism Thin depleted zone To provide thrust motion of flagella must be irreversible

Motility of bacteria and unicellular Algae. Flagellates salmonella

Bioconvection. Examples Oxytactic bacteria in a Petri dish. Pattern selection (from PhD thesis by Martin Bees) Gyrotactic algae in a flask. Standing plumes The reason for the bioconvection is that microorganisms are heavier than water.

Bioconvection. Mechanism O2O2 * * * * * ** * * * * * * Chemotaxis. Cells swim towards O 2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Kessler, J. Hydrodynamical focusing of motile algal cells. Nature 313 (1985) * * * * * * * * * * * * * * * * * * * * Downwelling pipe flow Upwelling pipe flow The reason for the bioconvection is inhomogeneity in concentration of microorganisms which are heavier than surrounding water. Gravitaxis + gyrotaxis: cells swim upwards and turned by the flow shear g

Patterns formed by C. nivalis Wall plumes in a shaker Wall plumes in upwelling pipe flow Thread in the downwelling pipe flow Dendrites above the water surface

Microorganism motility. Random walk Cells advance forward with constant velocity performing Biased Random Walk in swimming directions Bottom-Heavy cells (gravitaxis), gyrotaxis, phototaxis Thermal noise motion in flagella etc …another mechanism of a taxis is Run-and-Tumble, but it is unaffected by the flow shear.

Bioconvection. Modelling Continuum models: Diffusion of admixture (cells) + convection where diffusion tensor is derived from solutions of Fokker-Planck equation for the cell velocity distribution Based on the Biased Random walk model. Linear, weakly non-linear, DNS. Pedley & Kessler (1990), Bees & Hill (1997), Metcalfe & Pedley (2001), Ghorai & Hill (2002). A problem: cell velocity distribution varies in space e.g. faster cells go further up (Vladimirov et al., 2004). Separate simulation of the flow and cell motility: DNS for the viscous flow with variable density, which is defined by the cell concentration at each step. Motility of each cell is simulated separately at each step. Hopkins, Fauci (2002). A problem: hard to learn how the flow depends on parameters.

T=20 o C Air Laser Light sheet Cell suspension PIV field of view Thread of algae Flow nodules train-like disturbance Pipe flow. Experimental setup, observations g P. Denissenko, S. Lukaschuk, Physics Letters A 362, (2007)

Evolution of nodules. Change of the propagation rate Cell concentration Axial velocity z

Pipe flow of the suspension. Velocity profile Navier Stokes equation in cylindrical coordinates, z - independent axisymmetric flow: Flow velocity 400  m/s Cell forward velocity 70 mm/s Cell drift velocity 10  m/s Cell “gyration” radius 0.5 mm Poiseuille flow Singular at r=0 (at the axis) General solution

The model. Pipe flow with the heavy core Microorganism concentration Vertical velocity General solution for w Solution for w, satisfying boundary and continuity conditions r = 0 r = b r = 1 Non-dimensional pressure gradient Non-dimensional numbers

Discontinuities (as in shock waves and bores) A system of PDE in conservative form Rankine-Hugoniot conditions across the discontinuity Lax conditions Continuity Eqn. + kinematic condition at r=b Notation: A = b 2 = thread cross-sectional area /  Cell conservation in the core Notation: N = An = cell linear concentration real    : hyperbolic

Discontinuities (as in shock waves and bores) D Discontinuity State 0 State 1 State 0 State 1 Discontinuity (bore) Nodule Train-like Hyperbolic system A ( z, t ) N ( z, t )

Velocity profile in a pipe with algae suspension P. Denissenko, S. Lukaschuk, Physics Letters A 362, (2007) Distinct nodules

A thread of motile algae for real-time bio-monitoring 3 image/sec

Real-time Biomonitoring tool. Is it competitive? A standard tool: measuring the culture growth rate Video-tracking: assessing individual motility Nodules on the thread: assessing motility in bulk by measuring nodule spacing and propagation speed Electronic noses: detecting chemicals by luminescence or change of the resistance of the substrate An established technique, but slow (few days) + the pollutant may decay complicated hardware (microscope, lighting), not instantaneous since needs averaging over many cells, needs the controlled culture stirring Measurements may be done by a naked eye, instant response to change in motility Reliability and repeatability questionable, needs testing Maintenance problems: requires cleaning of sensor surfaces, Sensor calibration etc.