Propagation of discontinuities in a pipe flow
of suspension of motile microorganisms
(A thread of motile algae for real-time bio-monitoring)
3 image/sec
Petr Denissenko, University of Warwick, 25 June 2008
Microorganism motility. Diffusion, low Re
Stationary
microorganism
Thick
depleted
zone
Thin
depleted
zone
Moving
microorganism
To provide thrust
motion of flagella must be irreversible
For the experiments we used
Chlamydomonas nivalis (phototrophic regime), a biflagellate
Crypthecodinium cohnii (heterotrophic regime), a dynoflagellate
Motility of bacteria and unicellular Algae. Flagellates
salmonella
Bioconvection. Examples
The reason for the bioconvection is that
microorganisms are heavier than water.
Oxytactic bacteria in a Petri dish.
Pattern selection
(from PhD thesis by Martin Bees)
Gyrotactic algae in a flask.
Standing plumes
Bioconvection. Mechanism
**
**
*
Chemotaxis. Cells swim towards O2
*
**
*
*
*
*
*
*
*
* **
** *
**
**
*
Downwelling
pipe flow
*
g
*
*
**
*
*
**
***
**
***
*
Gravitaxis + gyrotaxis:
cells swim upwards
and turned by the flow shear
The reason for the bioconvection
is inhomogeneity in concentration of
microorganisms which are
heavier than surrounding water.
*
*
O2
*
*
*
*
*
*
**
*
**
*
* **
*
**
Upwelling
pipe flow
Kessler, J.
Hydrodynamical focusing of motile algal cells.
Nature 313 (1985)
Patterns formed by C. nivalis
Wall plumes in a shaker
Wall plumes
in upwelling pipe flow
Dendrites above the water surface
Thread in the
downwelling
pipe flow
Microorganism motility. Random walk
Cells advance forward with constant velocity
performing
Biased Random Walk in swimming directions
Thermal noise motion in
flagella etc
Bottom-Heavy cells (gravitaxis),
gyrotaxis, phototaxis
…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=20oC
nodules
train-like disturbance
Air
Flow
Thread of algae
Cell suspension
Light sheet
Laser
PIV field of view
Pipe flow. Experimental setup, observations
P. Denissenko, S. Lukaschuk, Physics Letters A 362, 298-304 (2007)
g
w
r
Axial velocity
Cell concentration
Evolution of nodules. Change of the propagation rate
z
Pipe flow of the suspension. Velocity profile
Flow velocity 400 m/s
Cell forward velocity 70 mm/s
Cell drift velocity 10 m/s
Cell “gyration” radius 0.5 mm
Navier Stokes equation in cylindrical coordinates,
z - independent axisymmetric flow:
P
1    w  
    r   
z
r  r  r  
w  c1 
Poiseuille flow
1 P 2
 r  c2 ln r
4  z
General solution
Singular at r=0
(at the axis)
The model. Pipe flow with the heavy core
Non-dimensional numbers
Microorganism concentration
Vertical velocity
General solution for w
Solution for w,
satisfying boundary
and continuity conditions
r=0
r=b
Non-dimensional pressure gradient
r=1
Discontinuities (as in shock waves and bores)
Continuity Eqn.
+ kinematic condition at r=b
Notation: A = b2 = thread cross-sectional area / p
Cell conservation in the core
A system of PDE in conservative form
Notation: N = An
= cell linear concentration
real l1, l2:
hyperbolic
Rankine-Hugoniot conditions
across the discontinuity
Lax conditions
Discontinuities (as in shock waves and bores)
State 1
Nodule
Discontinuity
D
State 0
Hyperbolic system
A(z,t)
N(z,t)
Train-like
Discontinuity
(bore)
State 1
State 0
Velocity profile in a pipe with algae suspension
Distinct nodules
P. Denissenko, S. Lukaschuk, Physics Letters A 362, 298-304 (2007)
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
An established technique, but
slow (few days) + the pollutant may decay
Video-tracking:
assessing individual motility
complicated hardware (microscope, lighting),
not instantaneous since
needs averaging over many cells,
needs the controlled culture stirring
Nodules on the thread:
assessing motility in bulk
by measuring nodule spacing
and propagation speed
Measurements may be done by a naked eye,
instant response to change in motility
Reliability and repeatability questionable,
needs testing
Electronic noses:
detecting chemicals by
luminescence or change of
the resistance of the substrate
Maintenance problems:
requires cleaning of sensor surfaces,
Sensor calibration etc.