propagation of discontinuities in a pipe flow of suspension of motile microorganisms (a thread of...

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.