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velocity of the tracer particle shows two clear peaks, but the details of the formation
of the loop lie in the component velocities in the X and the Y directions respectively.
The velocity components in the X and Y directions are shown in the bottom panel
of Fig.
7.6
. It can be clearly seen from the X-component of the tracer velocity (which
in this case is the directional lateral to the swimmer direction) that there are two
peaks in the velocity, each corresponding to the movement of the tracer away from
and then towards the swimmer, respectively. Similarly, the peak seen in the velocity
of the tracer in Y-direction (which is the direction of the swimmer) corresponds
to when the tracer reaches the mid-point along the swimmer body, where it moves
exactly opposite to the swimmer, whereas the valleys correspond to the front and the
back of the swimmer. It is easy to see that for a tracer that goes around the surface
of a sphere these are the direct consequence of a tangential velocity on a sphere of
the form
u
. As we discussed in the previous chapter, this is the flow that
is to be expected at the droplet surface due to the Marangoni stresses that propel it.
It must be noted that the tracer velocity in the rest frame of the swimmer will also
have the same form except that they will be shifted by an amount corresponding to
the swimmer velocity.
As the distance from the squirmer increases, we still see loopy tracer paths, but
they are not closed (Fig.
7.5
)—however, this is simply because of the finite time effect.
Asymptotically, all tracer paths must close upon themselves. In fact, it was recently
shown analytically by Dunkel et al. [
23
] that for self-motile force free swimmers,
asymptotically, tracer paths converge to a closed loop. While loopy patterns were
seen previously in experiments with eukaryotic swimmers [
21
], the formation of
closed trajectories was not resolved. The results presented in this section demonstrate
experimentally, for the first time, the dynamics of closed loop formation. Further,
it can be seen from the X and Y velocities of the tracer particles that the details
of the shape and orientations of the tracer loops are a signature of the flow field
due to the swimmer and could throw more light on the flow field of the droplet
squirmers, complementing the PIV measurments presented in the previous chapter.
These investigations are currently ongoing.
Walls, as opposed to fellow squirmers or passive tracers, are fixed entities. There-
fore, squimer-wall interactions are of a different kind as we will see in the section to
follow.
θ
∼
κ
sin
θ
7.3.3 Swimmers at Walls
It has been observed and studied in many experiments that microscopic swimmers
are attracted to surfaces (walls) [
24
]. Bacteria and sperm, both flagellated swimmers
are known to aggregate at surfaces where they perform circular motions which can
lead to large scale organization [
24
,
25
]. By invoking the method of images,
1
it has
1
The method of images is is a mathematical tool for solving differential equations in which the
domain of the sought function is extended by the addition of its mirror image with respect to
