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for all
m
0. We see that the different modes decouple, as far as linear stability
is concerned. As long as
b
0
M
is small enough, the resting state is stable against
fluctuations. However, when
b
0
M
exceeds a critical value, the resting state is unstable,
and the droplet spontaneously starts to move. It is straightforward to see that for
k
R
2
, this happens first for the lowest mode at
m
<
3
D
i
/
=
1 which corresponds to
a surface velocity term of the form
u
θ
∼
κ
sin
θ
.
6.3.2 Hydrodynamic Flow Fields
As discussed earlier, a swimmer is characterised by its hydrodynamic flow field.
Knowledge of this flow field is crucial for understanding swimmer interactions. To
elucidate the flow field around the droplet swimmers and for the sake of clarity, we
briefly present here the theoretical model and calculations of the flow field for a
squirmer.
Typical models of squirmers [
24
] assume a 'spherical' particle that is driven by
a purely tangential velocity (distortion) on its surface. The velocity at a point
r
s
on
the surface of a sphere of radius
a
is given by
B
n
ˆ
e
P
n
ˆ
2
v
s
r
s
,
ˆ
e
=
a
· ˆ
2
e
r
s
r
s
a
− ˆ
·
r
s
/
e
(6.5)
n
(
n
+
1
)
a
n
=
1
where the
B
n
are constants,
P
n
(
x
)
is the derivative of the
n
th Legendre polynomial,
ˆ
and
e
is a unit direction vector associated with the sphere. From Eq.
6.5
, the polar
component of the surface velocity can be written as
u
θ
=
θ
+
B
2
/
θ
B
1
sin
2sin2
,
θ
=
(
ˆ
·
r
s
/
)
where
is the polar angle. The relative strengths of
B
1
and
B
2
can be tuned to change the characteristic of the surface velocity as shown in Fig.
6.4
.
For
arccos
e
a
β
≡
B
2
/
B
1
<
0, the propulsion acts from the posterior half of the sphere, while
for
0, it acts from the anterior. These conditions are qualitatively similar to
biological swimmers, so called pushers and pullers respectively. We assign the term
neutral squirmer, corresponding to the case of
β>
0.
The flowfield
v
around the droplet is then calculated by solving the Stokes equation
for axisymmetric incompressible flows with
β
=
∇·
v
=
0 with the boundary conditions
v
0. As we have seen in the discussion above
of the linear stability analysis, we expect a surface velocity of the form
u
θ
(
r
=
a
,θ)
=
u
and
v
r
(
r
=
a
,θ)
=
θ
θ
≡
κ
sin
(θ)
corresponding to that of
0 for the droplet swimmers. The solution of the flow
field for such boundary conditions, shown in full in the appendix (reproduced from
[
29
]), yields a velocity
v
given by
β
=
v
1
cos
a
3
r
3
v
r
(
r
,θ)
=−
−
(θ),
