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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 2008
3 image/sec
Microorganism motility. Diffusion, low Re
For the experiments we usedChlamydomonas nivalis (phototrophic regime), a biflagellateCrypthecodinium cohnii (heterotrophic regime), a dynoflagellate
Thickdepleted
zone
Stationary microorganism
Moving microorganism
Thindepleted
zone
To provide thrustmotion 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 thatmicroorganisms are heavier than water.
Bioconvection. Mechanism
O2
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Chemotaxis. Cells swim towards O2
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Kessler, J. Hydrodynamical focusing of motile algal cells. Nature 313 (1985)
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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 plumesin upwelling pipe flow
Thread in the downwelling pipe flow
Dendrites above the water surface
Microorganism motility. Random walk
Cells advance forward with constant velocity performingBiased Random Walk in swimming directions
Bottom-Heavy cells (gravitaxis),gyrotaxis, phototaxis
Thermal noise motion in flagella etc
…another mechanism of a taxisis Run-and-Tumble, but it isunaffected 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 distributionBased 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 spacee.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=
20o C
Air
Lase
r
Ligh
t sh
eet
Cell suspension
PIV
fie
ld o
f vi
ew
Thr
ead
of a
lgae
Flo
w
nodu
les
tra
in-li
ke d
istu
rban
ce
Pipe flow. Experimental setup, observations
g
r
w
P. Denissenko, S. Lukaschuk, Physics Letters A 362, 298-304 (2007)
Evolution of nodules. Change of the propagation rateC
ell c
once
ntra
tion
Axi
al v
eloc
ity
z
Pipe flow of the suspension. Velocity profile
rcrz
Pcw
r
wr
rrz
P
ln4
1
1
22
1
Navier Stokes equation in cylindrical coordinates,
z - independent axisymmetric flow:Flow velocity 400 m/sCell forward velocity 70 mm/sCell drift velocity 10 m/sCell “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 = b2 = 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 0State 1
Discontinuity (bore)
Nodule
Train-like
Hyperbolic systemA ( z , t )N ( z , t )
Velocity profile in a pipe with algae suspension
P. Denissenko, S. Lukaschuk, Physics Letters A 362, 298-304 (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 bulkby measuring nodule spacing and propagation speed
Electronic noses:detecting chemicals by luminescence or change of the resistance of the substrate
An established technique, butslow (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 motilityReliability and repeatability questionable,needs testing
Maintenance problems: requires cleaning of sensor surfaces, Sensor calibration etc.