The document summarizes a study of the near-contact-line dynamics of evaporating sessile drops containing bacteria. As the drop evaporates, an internal flow develops that transports bacteria outward and concentrates them near the contact line, where experiments show they form periodic jets. The study develops a theoretical model and numerical simulation of the averaged flow properties and bacteria behavior. Initial 1D simulations show bacteria transitioning from isotropic to aligned with flow near the edge, and bacteria density peaking near the edge matching experiments. Future 2D simulations aim to reproduce the periodic jet pattern observed.

1. Study of the near-contact-line dynamics
of evaporating sessile drop containing bacteria
A Theoretical and Numerical Study
Chiyu Jiang
Cornell University,
Department of Biological and Environmental Engineering
cj336@cornell.edu — 1 (607) 379 4895
Abstract
The near-contact-line dynamics of evaporating sessile drop
containing bacteria is studied theoretically and numerically. As
a result of the pinned contact-line and evaporation, an internal
flow inside the droplet drives the bacteria outward and concen-
trates them near the contact-line. Published experimental works
have revealed a collective behavior of bacteria near the contact-
line. Along the circumference of the drop, bacteria appear in the
form of spatially periodic bacteria jets. In this research, we pro-
pose a theoretical model for the averaged flow properties and
bacteria behavior, and numerically simulate the process using
these governing equation in order to draw a comparison with
the published experimental results.
Introduction
Living Fluids, such as fluids that contain living self-
propelling microorganisms such as bacteria, have unique
rheological properties compared to regular fluids [1]. Pre-
vious experimental and theoretical studies have revealed
that living bacteria which travel in the fluid in a run-and-
tumble manner exert an active stress on the surrounding
fluid [2]. This unique type of locomotion arises from a
mechanism where bacteria are propelled by rotating heli-
cial fliaments called flagellas [3]. As there is no net force or
net torque acting on a bacterim as it swims, the swimming
of the bacterium causes a Stokesian hydrodynamic distur-
bance which behaves like force dipole disturbances in the
farfield [4]. Experiments have shown evidence of orga-
nized bacteria motion correlated on a length scale much
larger than the size of each bacterium [5], and numeri-
cal simulations on hydrodymanically interacting self pro-
pelled particles [6] suggest that these instabilities has been
a result of these force-dipole like disturbances. Another
motivation for this study is the coffee ring effect. The cof-
fee ring effect refers to the phonomenon where passive
particles such as the coffee particles are transported radi-
ally outward of a evaporating drop and deposited along
the edge of the drop to form a dark ring. However living
bacteria are not passive particles like the coffee particles
and instead they apply an active stress to the fluid. Re-
cent experimental studies have revealed that in the case
of evaporating sessile drop containing living E. Coli, the
concentrated bacteria population at the drop edge exert
a collective behavior in the form of periodic bacteria jets
[7]. But there have yet been published theoretical model
to explain this phenomenon. Thus, to come up with a the-
oretical model for this unique transport process, and to
numerically simulate the process is the focus of this re-
search.
Main Objectives
1. Develop a theoretical model for the averaged velocity
field, pressure field bacteria number density and bacte-
ria orientation close to the edge of the drop.
2. Develop a Finite Difference Method code in MATLAB
to solve the coupled governing equation and resolve the
quantities above.
3. Compare the numerical simualtion results with pub-
lished experimental research.
Theory & Numerical Simulation
Governing Equations
Consider the region very close to the drop (y R), where
curvature of the boundary region can be neglected, and a
cartesian coordinate system can be set up. The velocity of
the evaporative flow can be approximated with the equa-
tion below:
uy
e =
R
4(td − t)
2y
R
−λ
= −y−λ (in scaled form) (1)
the parameter λ(θc) reflects the uniformity of evapora-
tion. It can be calculated by:
λ(θc) =
1
2
−
θc
π
≈
1
2
−
(1/9)
π
= 0.4646 (2)
The transport of the bacteria can be modelled using
a classic convection-diffusion process. But due to the
uniqueness of the geometry at the edge of the drop, it is
expressed in a cylindrical-coordinate-like form.
∂n
∂t
+
∂
∂x
(uxn)+
1
y
∂
∂y
(yuyn)−D
∂2
∂x2
+
1
y
∂
∂y
y
∂n
∂y
= 0 (3)
where ux and uy are the velocity in x and y averaged across
the height of the drop. n is the bacteria density, and D is
the nondimensionalized form of the diffusion coefficient.
Balancing the pressure field with the shear stress result-
ing from the flow, as well as the active stress created by
bacteria motion, we get to the equation below:
− p + 2u − β · n < pp > −
I
3
= 0 (4)
where < pp > is the bacteria orientation tensor and β is a
coefficient to scale bacteria concentration.
Taking a lubrication approximation, flow velocity can be
related to the pressure field, and by averaging the velocity
across the drop height, equation 4 can be written explicitly
in the form below:
∂2P
∂x2
+ β
∂2
∂x2
[n(< pxpx > −
1
3
)] + β
∂2
∂x∂y
[n < pxpy >]
+
1
y
∂
∂y
(y
∂P
∂y
) + β
1
y
∂
∂y
[y
∂
∂y
[n < pypx >]]
+ β
1
y
∂
∂y
[y
∂
∂x
[n(< pxpx > −
1
3
)]] =
−3(1 − λ)
(tan θc
2)
y−2−λ (5)
Also from lubrication approximation, the averaged veloc-
ity in x and y can be calculated using the equation below:
ux = −
tan2(θc) y2
3
∂p
∂x
+ β
∂
∂y
n(< pxpx >)
+ β
∂
∂x
n(< pxpx > −
1
3
) (6)
uy = −
tan2(θc) y2
3
∂p
∂y
+ β
∂
∂y
n(< pypy > −
1
3
)
+ β
∂
∂x
n(< pypy >) (7)
The rotation of the swimmers is shear-induced. In mathe-
matical terms, it is governed by the equation below:
∂
∂t
< pipj > +ux
∂
∂x
< pipj > +uy
∂
∂y
< pipj > +
3
2y tan θc
< pipz > uj+ < pjpz > ui −
2
l=1
2 < pipj : pzpl > ul
= −De < pipj > −1/3δij (8)
The fourth moments are approximated using the first
Hinch and Leal closure (HL1)[8], given below:
< pipjpzpl >= δij < pzpl > −
3
m=1
< pzpm >< pmpl >
+ δzl < pipj > − < pipm >< pmpj >
+
3
5
< pipz >< pjpl > + < pipl >< pjpz >
−
1
3
< pipj >< pzpl > (9)
Results and Discussion
Numerical simulation results for 1D simulation have been
achieved. For the present stage, simulations have been
performed only in the y dimension, since evaporative ve-
locity is predominant in y. We also assume that the varia-
tion of pressure would be small in the x dimension, hence
the flow velocity in x can be considered zero, and all spa-
cial derivative values in x are considered to be zero. The
numerical results generally match our expectations. For
the simulation results for bacteria orientation (Fig 1), we
observe that in a region far from the drop edge, the bac-
teria are in an isotropic state, since the component values
for < pxpx >=< pypy >=< pzpz >= 1
3, while close to the
edge, < pypy >→ 1 and < pxpx > and < pzpz > stay at
a very low value, suggesting that being induced by the
shearing flow, bacteria are forced to align in the direction
of the flow, perpendicular to the edge. The simulation re-
sults for the bacteria number density (Fig 2 (a)) matches
experimental results, where bacteria area density peaks at
a region near the edge. Though the experimental data is
affected by noise, it compares well to the simulation re-
sults and the trend of convergence is obvious.
Figure 1: A plot of the values for bacteria orientation tensor compo-
nents versus distances form the edge.
Figure 2: Theoretical versus experimental results of bacteria den-
sity versus distance from the drop edge. (a) Results from theoretical
computation using the 1D model. (b) Experimental results. Image is
reproduced from Kasyap et al.[7]. The image has been processed and
background has been subtracted with a rolling ball radius of 100 pix-
els. The sample region is highlighted and enclosed by the rectangle as
shown.
Conclusions
• The numerical solution for 1D simulation of bacteria
area density agrees with experimental results as well as
intuition.
• The numerical solution for 1D simulation of bacteria ori-
entation suggest that bacteria transit from an isotropic
distribution at farfield to shear induced alignment with
the evaporative velocity.
Forthcoming Research
Simulations will be performed in 2D to resolve the full
problem. We will expect to see periodic ”jets” of bacteria
with comparable wave lengths as those observed in pub-
lished experimental work by Kasyap et.al(2014).
References
[1] Chaouqi Misbah and Christian Wagner. Living fluids.
Comptes Rendus Physique, 14(6):447–450, 2013.
[2] Donald L Koch and Ganesh Subramanian. Collective
hydrodynamics of swimming microorganisms: Living
fluids. Annual Review of Fluid Mechanics, 43:637–659,
2011.
[3] Howard C Berg. E. coli in Motion. Springer Science &
Business Media, 2004.
[4] Eric Lauga and Thomas R Powers. The hydrodynam-
ics of swimming microorganisms. Reports on Progress in
Physics, 72(9):096601, 2009.
[5] Xiao-Lun Wu and Albert Libchaber. Particle diffusion
in a quasi-two-dimensional bacterial bath. Physical Re-
view Letters, 84(13):3017, 2000.
[6] Juan P Hernandez-Ortiz, Christopher G Stoltz, and
Michael D Graham. Transport and collective dynamics
in suspensions of confined swimming particles. Physi-
cal Review Letters, 95(20):204501, 2005.
[7] TV Kasyap, Donald L Koch, and Mingming Wu. Bacte-
rial collective motion near the contact line of an evap-
orating sessile drop. Physics of Fluids (1994-present),
26(11):111703, 2014.
[8] EJ Hinch and LG Leal. Constitutive equations in sus-
pension mechanics. part 2. approximate forms for a
suspension of rigid particles affected by brownian ro-
tations. Journal of Fluid Mechanics, 76(01):187–208, 1976.
Acknowledgements
The author would like to acknowledge Prof. Donald Koch
for his mentor and guidance throughout the research pro-
cess, Dr. Anubhab Roy for his great help in the devel-
opment of theoretical framework and numerical methods,
Dr. T. V. Kasyap for his underlying theoretical work, and
Prof. Mingming Wu and Prof Ashim Datta for their help-
ful conversations regarding the research.