1 Osborne Reynolds Research Student Award: The

1
Osborne Reynolds Research Student Award:
The Hydrodynamics of Swimming Microorganisms
Douglas Brumley
University of Cambridge
Department of Applied Mathematics and Theoretical Physics
Supervisor: Prof. Timothy J. Pedley, F.R.S.
I.
INTRODUCTION
The study of microorganisms can be traced back to the Dutch scientist, Antonie Philips van Leeuwenhoek.
His pioneering work with microscopes in the 17th century allowed him to see bacteria, protozoa and other singlecelled organisms for the first time. Microorganisms occupy all areas of the biosphere and form crucial components
of all ecosystems. Bacteria and other types of microorganisms are the earliest forms of life on Earth. They have
lived and evolved for billions of years, and have proved to include the most resilient lifeforms on the planet, outliving
the dinosaurs and other physiologically more advanced organisms. They are used in an extraordinary number of
applications, including agriculture, the preparation of food, water treatment, medicine, energy production, warfare,
science and industry. The work presented here can be divided into several distinct categories. We begin in
Section II by outlining the results of some recent experiments. In particular, we have been able to accurately
analyse the time-dependent flow fields around the colonial alga Volvox carteri and characterise the metachronal
wave propagating on its surface. In Section III, we use results from the time-averaged flow fields, in conjunction
with the spherical envelope method for ciliary propulsion present by Blake [1], to calculate the forces and torques
acting on two interacting squirmers in the lubrication limit. This interaction becomes significant when the spacing
between organisms is sufficiently small. We follow elements of the work by Ishikawa et al. [2], finding small
corrections to some of the terms. We also perform the lubrication analysis to higher order, allowing us to calculate
the first non-vanishing contributions to the normal force. Importantly, the lubrication theory permits the analysis
of suspensions with a high volume fraction of organisms, through pairwise addition of interactions. We use this
theory in Section IV to assess the stability of a monolayer of close-packed spherical squirmers. We find that
the monolayer is always unstable without the introduction of an additional mechanism, such as a repulsive force
between adjacent squirmers, or a gravitational torque arising from their bottom-heaviness. In some circumstances,
we are able to calculate critical conditions, beyond which perturbations to the squirmers’ configuration decay.
II.
EXPERIMENTAL RESULTS
Several different types of swimming microorganisms feature frequently in theoretical and experimental studies, because they are readily cultured and possess geometries which are amenable to mathematical modelling.
The family Volvocaceae consists of a number of genera of motile green flagellates. These motile colonies are
perfectly spherical and comprise large numbers of biflagellated cells held together by some extracellular matrix.
For the species Volvox carteri, a typical colony consists of 1-5 × 104 cells. These cells embedded on the surface
of the colony beat their flagella in a coordinated fashion, producing a net fluid motion. Interestingly, the daughter
colonies situated within the main colony are, on average, displaced from the geometric centre. This gives rise
to a “bottom-heaviness”, which acts as a self-righting mechanism, much like the keel of a yacht. The radius of a
Volvox colony may be anywhere between 100 µm and 500 µm depending upon the position within the 48 hour
lifecycle. In order to characterise the fluid motion around a colony of Volvox carteri, we use high-speed imaging
and particle image velocimetry (PIV). This enables us to accurately calculate the time-dependent flow field around
such colonies and extract the relevant parameters involved in locomotion. Figure 1(c) shows a photograph of a
Volvox colony, viewed with a microscope. The surrounding fluid is seeded with passive tracer particles, the motion
of which is recorded at 1000 frames per second.
We define θ to be the angle from the anterior end of the colony, as shown in Fig. 1. The orientation of
the colony is depicted in Fig. 1 with an arrow, and is the direction in which the free colony would swim. The
fluid velocity field may be written as u(r, θ, t) = ur (r, θ, t)êr + uθ (r, θ, t)êθ , where ur (r, θ, t) and uθ (r, θ, t) are the
radial and tangential components respectively. Figures 1(a)-1(b) show the time-dependent results from the PIV,
evaluated at the radius r = a corresponding to the tips of the flagella. The diagonal bands present in Fig. 1(a)
represent a metachronal wave propagating from the anterior end (θ = 0) to the posterior end (θ = π) of the colony,
the speed and wavelength of which can be accurately calculated using a two-dimensional Fourier analysis.
Rt
The time-averaged flow field uav (r, θ) = 1/t0 0 0 u(r, θ, t)dt (for sufficiently large t0 ) can be readily calculated, and the results are presented in Figs. 1(d) - 1(f). We adopt the squirmer model of Blake [1] to describe the
2
(a)
(b)
s -1
μm/s
400
400
400
120
300
5
100
200
200
θ
θ
100
μm
0
0
60
4
θ
100
80
μm
100
μm
6
300
300
200
3
0
2
−100
−100
−100
−200
1
40
−200
−200
−300
0
−300
20
−300
0
−400
−200
−400
−1
−400
−200 −100
0
100
200
μm
(c)
(d)
−200 −100
0
100
200
−100
0
μm
μm
(e)
(f)
100
200
FIG. 1: Figures showing the (a) radial and (b) tangential components of the velocity as a function of t and θ. Results shown are
for a fixed radius r = a, corresponding to the tips of the flagella. Also shown is (c) a photograph of a Volvox colony which is held
by a glass micropipette. For particle image velocimetry (PIV), we use yellow green beads (505/515nm) with diameter 0.5µm
and volume fraction 2 × 10−4 . The remaining figures show the time-averaged (d) velocity field, (e) velocity magnitude and (f)
vorticity. PIV analysis is conducted using frames n and n + 10, corresponding to a time interval of 1/100 sec. The subsequent
results are averaged over 1300 frames.
boundary condition at the surface of the colony:
X
ur r=a =
An (t)Pn (cos θ),
X
uθ r=a = sin θ
Bn (t)Wn (cos θ),
n
(1)
n
2
where Pn is the nth Legendre polynomial and Wn is defined as Wn (cos θ) = n(n+1)
Pn0 (cos θ). We will consider a
time-independent squirmer with zero radial velocity on the sphere surface (An (t) = 0 and Bn (t) = Bn ∀ n). In
these circumstances, the model boundary conditions presented in Eq. (1) are in good agreement with the timeaveraged flow found experimentally, facilitating the use of this “squirmer” model in the subsequent sections. The
parameters B1 and B2 determine the swimming speed and stresslet strength respectively. The ratio β = B2 /B1 is
a quantity that measures the forwards/backwards asymmetry of the squirmer, and is a parameter which features
prominently throughout our study.
III.
LUBRICATION THEORY
We consider the problem of two closely-separated spherical squirmers of radius a and αa, with a minimum
separation given by a for 1. The Reynolds number is considered to be sufficiently small so as to permit use
of the Stokes equations. By linearity of these equations, the problem involving two squirming spheres in a fluid
that is at rest infinitely far away can be broken down into two distinct problems. The first has the squirming-sphere
boundary condition on sphere 1 and zero velocity boundary condition on sphere 2. The second problem has zero
velocity on sphere 1 and the squirming-sphere boundary condition on sphere 2. The orientation vector of the
squirmer, e, is the unit vector along the axis of symmetry. Without loss of generality, the frame is chosen such that
the orientation of the squirmer lies in the x-z plane. ie e · ey = 0. We solve the Stokes equations to find the fluid
velocity in the gap between the spheres. This is subsequently used, in conjunction with the associated pressure
distribution, to calculate the forces and torques acting on sphere 1 due to its squirming action. The forces and
torques acting on sphere 2 can also be found and are of similar form. We find that these quantities scale as ∼ log ,
indicating a singular form in the lubrication limit 1.
The pair of squirmers also experience forces and torques arising due to their linear and angular velocities,
the precise form of which is outlined by Kim and Karrila [3]. We introduce an additional torque acting due to
the presence of bottom-heaviness, as well as a repulsive force between the squirmers which acts at very short
3
length-scales (compared to the separation width). The precise forms of these are given by
4
i
Tgrav
= − πa3 ρh ei × g,
3
Frep = α1 α2
exp(−α2 ) r
,
1 − exp(−α2 ) r
(2)
where ρ is the density, g is the acceleration due to gravity, and h is the distance between the centre of mass
and geometric centre of the squirmer. The ratio of the gravitational torque to the viscous torque is given by
Gbh = 2πρgah
µB1 .
IV.
STABILITY OF A UNIFORM MONOLAYER OF STEADY SPHERICAL SQUIRMERS
The lubrication analysis outlined above corresponds to two interacting spherical squirmers in Stokes flow.
The results can be used for a suspension with a high volume fraction of organisms, through pairwise addition of
interactions. In such circumstances, the total force and torque acting on any given squirmer will be the sum of
the contributions arising from interactions with any of its neighbours that are sufficiently close for lubrication forces
to be significant. In this fashion, a global matrix-vector equation can be constructed, which encapsulates the net
force and torque on every squirmer in a given configuration.
We investigate the stability of a uniform monolayer of steady spherical squirmers, in an infinite fluid. This
monolayer is composed of a d × d diamond of squirmers, subject to periodic boundary conditions. The equilibrium orientation of each squirmer is upright and lies in the plane of the monolayer, and the spacing between all
squirmers is assumed to be uniform. Moreover, the position of each squirmer is restricted to lie in the plane of
the monolayer. We investigate the stability of this monolayer, from both analytical and numerical standpoints. In
particular, we identify the stability of the system subject to small translational and rotational perturbations. The two
parameters which we vary are β, which essentially measures the stresslet strength, and Gbh , which quantifies the
extent of bottom-heaviness. Without the inclusion of the inter-particle repulsive force presented in Eq. (2), we find
that the monolayer is always unstable to small perturbations. The presence of this repulsive force can establish
translational stability, though in addition, a sufficiently large value of Gbh is required to ensure the squirmers are
stable to orientational perturbations. This is in very good agreement with simulations conducted by Ishikawa et
al. in which a full boundary element method is employed to describe the dynamics of a suspension of squirmers.
We see that the stability properties of the monolayer can be accounted for through lubrication forces and torques
alone. For small perturbations, Figure. 2 shows the evolution in time of the orientations of all squirmers in the
monolayer, for various values of Gbh . Importantly, the repulsive force is included, which gives rise to translational
stability. We see that there exists a critical value of Gbh , above which the orientational perturbations decay. It turns
out that both the repulsive force and the presence of bottom-heaviness are required to stabilise this monolayer.
0.06
0.01
0.01
0.008
0.009
0.008
0.006
0.004
Gbh= 40
0
0
−0.002
−0.02
−0.004
−0.04
10
20 30
40
50
t
60 70
80
90 100
0.006
0.005
Gbh= 40
0.004
0.003
0.002
−0.008
−0.06
0
Gbh= 50
−0.006
Gbh= 35
0.007
Gbh= 45
0.002
ζ
ζ
0.02
std(ζ)
0.04
Gbh= 35
initially appears
to be stable
Gbh= 45
Gbh= 50
0.001
−0.01
0
10
20
30
40
50
t
60 70
80 90 100
0
10
20
30 40
50 60
70
80
90 100
t
FIG. 2: Figures showing the orientation, ζ, of every squirmer in an 8 × 8 diamond, as a function of time. Results have been
computed for Gbh = 35, 40, 45 and 50 over the interval t ∈ [0, 100]. Also shown is the standard deviation of ζ as a function of
time. We have used 0 = 2 × 10−3 , α1 = 1, α2 = 103 and β = 1.
We have also extended this study to investigate the dynamics of a monolayer of steady spherical squirmers
situated between two plane parallel walls, with the squirmers permitted to move in any of the three spatial dimensions. The corresponding results are qualitatively very similar, with both the repulsive force and bottom-heaviness
required to guarantee translational and orientational stability.
[1] Blake, J. R. (1971) Journal of Fluid Mechanics 46, 199–208.
[2] Ishikawa, T., Simmonds, M. P., and Pedley, T. J. (2006) Journal of Fluid Mechanics 568, 119–160.
[3] Kim, S. and Karrila, S. J. (2005) Microhydrodynamics - Principles and Selected Applications, Dover Publications, .